Archive of events
Mathematical and Computational Finance Lab
Past STEM Fellowship Internship Events
Lunch at the Lab past events
FALL 2024
November 19, 2024
Speaker: Luca Lalor (Department of Mathematics & Statistics, University of Calgary, Calgary, AB, Canada)
TOPIC: Reinforcement Learning in Non-Markov Market-Making
Abstract:
We develop a deep reinforcement learning (RL) framework for an optimal market-making (MM) trading problem, specifically focusing on price processes with semi-Markov and Hawkes Jump-Diffusion dynamics. We begin by discussing the basics of RL and the deep RL framework used, where we deployed the state-of-the-art Soft Actor-Critic (SAC) algorithm for the deep learning part. The SAC algorithm is an off-policy entropy maximization algorithm more suitable for tackling complex, high-dimensional problems with continuous state and action spaces like in optimal market-making (MM). We introduce the optimal MM problem considered, where we detail all the deterministic and stochastic processes that go into setting up an environment for simulating this strategy. Here we also give an in-depth overview of the jump-diffusion pricing dynamics used, our method for dealing with adverse selection within the limit order book, and we highlight the working parts of our optimization problem. Next, we discuss training and testing results, where we give visuals of how important deterministic and stochastic processes such as the bid/ask, trade executions, inventory, and the reward function evolved. We include a discussion on the limitations of these results, which are important points to note for most diffusion models in this setting.
November 5, 2024
Speaker: Yang Yang (Department of Mathematics & Statistics, University of Calgary, Calgary, AB, Canada)
TOPIC: Stochastic Path-Dependent Volatility Models for Price-Storage Dynamics in Natural Gas Markets and Discrete-Time Swing Option Pricing
Abstract:
This work is devoted to the price-storage dynamics in natural gas markets. A novel stochastic path-dependent volatility model is introduced with path-dependence in both price volatility and storage increments. Model calibrations are conducted for both the price and storage dynamics. Further, we discuss the pricing problem of discrete-time swing options using the dynamic programming principle, and a deep learning-based method is proposed for numerical approximations. A numerical algorithm is provided, followed by a convergence analysis result for the deep-learning approach.
October 22, 2024
Speaker: Yu Li (Department of Mathematics & Statistics, University of Calgary, Calgary, AB, Canada)
TOPIC: A Weighted Multilevel Monte Carlo Method
Abstract:
The Multilevel Monte Carlo (MLMC) method, introduced by Giles, has been widely applied across various fields. In this presentation, we introduce a Weighted Multilevel Monte Carlo (WMLMC) method, combining it with the multiple control variate approach, and derive the recursive formulas for both the MLMC method and the WMLMC method. To demonstrate its performance, we apply the WMLMC to option valuation problems and showcase two applications: one with the PDE method and another using Richardson Extrapolation, both of which yield significant cost savings. We further extend WMLMC to a multi-dimensional version inspired by the Multi-Index Monte Carlo method, applying it to 2-dimensional examples that demonstrate substantial computational efficiency.
October 8, 2024
Speaker: Joshua McGillivray (Department of Mathematics & Statistics, University of Calgary)
TOPIC: The Subordination of a Hawkes Process Into a Standard Brownian Motion and Its Application to Energy Markets
Abstract:
We define a new model using a Hawkes process as a subordinator in a standard Brownian motion. We demonstrate that this Hawkes subordinated Brownian motion can be fit to 2018 and 2019 natural gas and crude oil front-month futures log returns. We also compare the simulations of a square of a Hawkes subordinated Brownian motion with its Ito formula. We simulate both processes and compare their distributions, trajectories, and percent errors across multiple runs. We derive the generator relating to this Hawkes subordinated Brownian motion, calculate several moments, and conjecture its distribution. We also provide explicit solutions to the second moments of the Hawkes process and its intensity as well as the cross moment between the Hawkes process and its intensity in the case of an exponential kernel.
September 24, 2024
Speaker: Sudeesha Navodh Arachchige (Department of Mathematics & Statistics, University of Calgary)
TOPIC: Stochastic Modeling and Simulation of Wind Power, Electricity Load and Gas Prices for Texas Energy Market
Abstract:
The simulation of wind power, electricity load and gas prices will allow commodity traders to see the future movement of prices in a more probabilistic manner. The ability to observe possible paths for each process enables traders to get valuable insights for placing their trades on electricity prices. Since these processes have a high seasonality factor, the seasonality component was modelled using a Truncated Fourier series and the random component using stochastic differential equations. It is evident from the literature that all the above processes are mean-reverting processes, thus three mean-reverting OU processes were considered as the model for each process. The industry experts believe there is a correlation between wind power, electricity load and gas prices. For example, when wind power is higher, and the electricity load is lower gas prices are relatively low. Thus, to account for this correlation structure in the simulation, we utilized a Vine Copula and integrated it into the simulation. The study is conducted for the Texas energy market and uses daily time scales for the simulations.
Summer 2024
July 2, 2024
Speaker: Yevhen Havrylenko (Department of Mathematical Sciences, University of Copenhagen, Denmark)
TOPIC: Value-at-Risk constrained portfolios in incomplete markets: a dynamic programming approach to Heston's model
Abstract:
This talk is based on Escobar-Anel, M., Havrylenko, Y., and Zagst, R. (2023) and is centered on dynamic portfolio optimization with terminal-wealth constraints. We start with a brief overview of the seminal paper H. Kraft and M. Steffensen (2013), in which the authors generalize the dynamic programming approach to optimal-investment problems with terminal-wealth constraints in a complete Black-Scholes market. Building on that, we extend the dynamic programming approach to constrained portfolio optimization problems in an incomplete financial market, where the stock price follows the Heston stochastic volatility model. We demonstrate that the value function in the constrained problem can be represented as an expected modified utility of a vega-neutral financial derivative on the optimal wealth in the unconstrained problem. Furthermore, we show how the optimal wealth and the optimal investment strategy in the constrained problem are linked to the optimal wealth and the optimal investment strategy in the unconstrained problem. We demonstrate the details using the example of a power-utility maximizing investor with a Value-at-Risk constraint on terminal wealth. At the end of the talk, we illustrate the results of our numerical studies and highlight the potential of our methodology for solving other utility-maximization problems with terminal-wealth constraints in incomplete markets.
1.Kraft, H., and Steffensen, M. (2013). A dynamic programming approach to constrained portfolios. European Journal of Operational Research, 453–461. https://doi.org/10.1016/j.ejor.2013.02.039
2.Escobar-Anel, M., Havrylenko, Y., and Zagst, R. (2023). “Value-at-Risk constrained portfolios in incomplete markets: a dynamic programming approach to Heston's model”. Working paper. https://doi.org/10.48550/arXiv.2208.14152
Winter 2024
March 12, 2024
Speaker: Chinonso Nwankwo (Department of Mathematics & Statistics, University of Calgary)
TOPIC: Ensemble Deep Learning Method and Free Boundary American Style Stochastic Volatility Models
Abstract:
We present deep learning method based on ensemble configurations for solving free boundary American style stochastic volatility models. To this end, we formulate our solution as a free boundary problem which enables the computation of the early exercise boundary surface simultaneously with the value function and Greeks. For precise computation of the free boundary plane, we first fix the free boundary using the Landau transformation and normalize the value function and the time domain. We then develop a novel ensemble auxiliary operator (EANO) involving a suite of configurations based on the ensemble neural network output (EDNN). The early exercise boundary surface, value function, and Greeks are predicted from the EDNN, EANO, and the derivatives of EANO after training. The performance of our neural network configuration is verified and validated with some existing methods. It provides an alternative approach for solving free boundary American style stochastic volatility models.
February 13, 2024
Speaker: Mark Reesor (Department of Mathematics & Statistics, Wilfrid Laurier University)
TOPIC: Incorporating Climate Risk into Portfolio Credit Risk Models via Distortion
(please note that the presentation/slides will be posted when all the results are published).
Abstract:
Regulatory requirements are evolving towards mandating financial institutions to estimate and report their climate-related financial risks. Climate risks classify into two broad categories --- physical and transition --- and, being medium- to long-term in nature, they are important risk factors for credit portfolios. Threshold models for portfolio credit risk specify account level models with both systematic and idiosyncratic effects. These aggregate to generate the portfolio loss distribution from which risk metrics are calculated. Augmenting the systematic factor with climate factors is one method to incorporate climate risk into existing models. Distortion provides a method for re-weighting a probability distribution. The amount of deformation depends on the choice of distortion function and its parameter. Here, we propose distortion as a way of incorporating climate risk into existing credit risk models. Some properties of the distorted credit risk models are derived and explored. The connection between distortion functions and constrained relative entropy optimisation provides insight into distortion function structure and parameter values. The relation between our proposed models and existing climate-adjusted portfolio credit risk models will be discussed. This is joint work with Arie Zeldenrijk, Mark Drmac, and Walid Mnif.
Fall 2023
December 5, 2023
Speaker: Chinonso Nwankwo (Department of Mathematics & Statistics, University of Calgary)
TOPIC: Fast and Enhanced Shallow Neural Network for Solving Free Boundary Pricing Models
(please note that the presentation/slides will be posted when all the results are published).
Abstract:
We present an enhanced, fast, and very shallow learning framework for solving free boundary options pricing problems with auxiliary neural operators (ANOs). To this end, we first rigorously explore the efficacy of some featured activation functions (FAFs) for solving these models with auxiliary neural operators (ANOs). We observe that some of the existing and well performing activation functions can be enhanced and adapted further to specific model characteristics and conditions. By regulating some of the existing activation functions and accustoming new ones using both linear and nonlinear neural network outputs, we then obtain a fast and enhanced shallow learning configuration for solving both single and dual free boundary models. Results further confirm that our learning method can admit one hidden layer and very small hidden nodes for the approximation of free boundary pricing models.
October 24, 2023
Speaker: Sudeesha Nawodh Arachchige (Department of Mathematics & Statistics, University of Calgary)
TOPIC: Stochastic Modelling of Wind Derivatives in Energy Markets for Alberta
(please note that the presentation/slides will be posted when all the results are published).
Abstract:
Wind energy is becoming a major source of renewable energy in the world. Due to the unpredictable nature of wind, financial contracts known as wind derivatives are introduced in energy markets to protect power generators from financial risks. Wind power generators face two risks, one due to changes in wind intensity and energy production, and the second due to changes in electricity retail prices. To hedge these risks simultaneously, the quanto option is an ideal financial tool. In this talk, we model the logarithm of spot prices of electricity with a Variance Gamma (VG) and Normal Inverse Gaussian (NIG) process while the wind speed and wind power production is with an Ornetein-Uhlenbeck (OU) process. Since the risk from changing wind power production and spot prices is highly correlated, we must model this correlation as well. This is reproduced by replacing the small jumps of the Lévy process with a Brownian component and correlating it with wind power and speed OU processes. Then we will study the income of the wind energy company from a stochastics point of view and finally, we will price the European put-type quanto option for the wind energy producer. We will compare the quanto option prices obtained from the VG process and NIG process.
October 10, 2023
Speaker: Joshua McGillivray (Department of Mathematics & Statistics, University of Calgary)
TOPIC: Swaps in Energy Commodities Markets
Abstract:
We discuss and value variance, volatility, covariance, and correlation swaps in the Vasicek, Schwartz one-factor, and Heston models in continuous time. The data used is primarily 2019 natural gas and crude oil futures closing prices due to the liquidity and size of the options market in the commodity energy sector. We derive approximations for covariance and correlation swap fair strikes in the Heston model following the continuous time regime, using the discrete regime for reference. We check the accuracy of our approximation using simulated error distributions of the calibrated parameters from the CIR component of the Heston model. We present the effect of varied parameters on the value of the fair strikes for covariance and correlation swaps. Finally, we evaluate the fair strikes of covariance and correlation swaps using three different approximations, yielding values and error bounds of dramatically varying sizes.
September 26, 2023
Speaker: Luca Lalor (Department of Mathematics & Statistics, University of Calgary & Futures First Intern (Montreal, QC))
TOPIC: High Frequency Market Making with Short-Term Alpha and Adverse Selection
Abstract:
An important type of financial market participant is the market maker (MM). The MM aims to facilitate trades, profit from the spread (best ask-best bid), optimize their trade executions, and adapt to changing market conditions. One can build a stochastic optimal control problem whereby the MMs aim is to maximize terminal wealth by trading in and out of positions using limit orders. The financial marketplace is populated by traders that come to the market for different reasons and with varying degrees of information. This leads to the MM being exposed to adverse selection risk. For example, when trading with informed market participants, the MM is exposed to having a sell limit order filled right before prices go up, or a buy limit order before prices go down. Here, we will incorporate this adverse selection effect by specifying that an assets mid-price has a drift with a short-term alpha component. This drift is affected by the arrival of market orders. After introducing the problem and showing how to obtain a solution to the stochastic optimal control problem, some back test results on a short-term oil futures contract will be shown. Here a simulation of the strategy is performed which will depict how trading, alpha and inventory evolved over a 5-minute time interval. There are some limitations to these results which we hope to address in future research and the talk will end with some initial ideas and recent academic work in the area.
Winter 2023
April 18, 2023
Speaker: Syeda Fareeha Ali (Department of Mathematics & Statistics, University of Calgary)
TOPIC: Locational Spread Option with Stochastic Correlation
Abstract:
Contrary to the common assumption, the correlation between financial derivatives may not be constant across time. This thesis analyses the role of stochastic correlation in modeling locational spread options for natural gas. We first derive a model with the Ornstein-Uhlenbeck process between two spread assets with constant correlation and then a combination of the Ornstein-Uhlenbeck and Jacobi process is used to model a stochastic correlation. The Margrabe formula is employed to evaluate option prices with constant correlation, the solution for which is used to compare with Monte Carlo simulations for stochasticity. In the last Implied correlation is calculated to conduct facts about stochasticity.
April 4, 2023
Speaker: Luca Lalor (Department of Mathematics & Statistics, University of Calgary)
TOPIC: A Numerical Solutions to an Algorithmic and HFT Problems with a Jump-Diffusion Price Processes
Abstract:
The main subject of this talk is to introduce an algorithmic and High-Frequency Trading model where the price process is of the jump-diffusion type. This talk begins with a brief introduction on how to apply Stochastic Optimal Control theory to algorithmic trading problems. A price process in the Jump-Diffusion setting is then introduced along with its infinitesimal generator, which encompasses one of the major modelling adjustments in this research. Previous research modelled the jumps through a diffusion approximation, while here the jumps are modelled directly. Preliminary results, using an Implicit-Explicit Finite Difference Scheme, for an Optimal Acquisition algorithmic trading problem will be presented. Here the jump part of the Jump-Diffusion price process will be a function of a Poisson process. This talk will end with a discussion on proposed modifications to the discussed algorithmic trading problem, so that the future models will account for the non-Markovian property seen in LOB data.
March 21, 2023
Speaker: Joshua McGillivray (Department of Mathematics & Statistics, University of Calgary)
TOPIC: The Hawkes Processes in Commodity Energy Markets
Abstract:
It is not uncommon for asset prices to be at the whims of market perception. The market functions to efficiently assign prices to the value of the goods being sold but market forces can distort this value based on the available supply and demand of the asset. Thus, if the market perception of an asset radically changes, the price may as well. We will investigate this phenomenon through the lens of the Hawkes process and attempt to model crude oil and natural gas futures prices by interpreting the driver of price spikes as a self-excitation phenomenon of a stochastic process. The analysis that followed led to a surprising and counter-intuitive result.
March 7, 2023
Speaker: Sudeesha Nawodh (Department of Mathematics & Statistics, University of Calgary)
TOPIC: Stochastic Modelling of Wind Derivatives in Energy Markets for Alberta
Abstract:
Wind energy is becoming a major source of renewable energy in the world. Due to the unpredictable nature of wind, financial contracts known as wind derivatives are introduce in energy markets to protect power generators from financial risks face due to this. Wind power generators face two risks, one due to change in wind intensity and energy production, and the second due to change in electricity retail prices. To hedge these risks simultaneously, quanto option is an ideal financial tool. In this project, we have modeled the logarithm of spot prices of electricity with Normal Inverse Gaussian (NIG) process while the wind speed and wind power is production is with an Ornetein-Uhlenbeck (OU) process. Since the risk from changing wind power production and spot prices is highly correlated, we must model this correlation as well. This is reproduced by replacing the small jumps of NIG process by a Brownian component and correlate it with wind power and speed OU processes. The project is conduct for the Alberta region and we will discuss only up to developing the correlation and income formula’s for a wind energy company. As for future work, we will price the European put-type quanto option for Alberta region.
February 28, 2023
Speaker: Chinonso Nwankwo (Department of Mathematics & Statistics, University of Calgary)
TOPIC: Deep Learning and American Options via Free Boundary Framework
Abstract:
We propose a deep learning method for solving the American options model with a free boundary feature. To extract the free boundary known as the early exercise boundary from our proposed method, we introduce the Landau transformation. For efficient implementation of our proposed method, we further construct a dual solution framework consisting of a novel auxiliary function and free boundary equations. The auxiliary function is formulated to include the feed forward deep neural network (DNN) output and further mimic the far boundary behaviour, smooth pasting condition, and remaining boundary conditions due to the second-order space derivative and first-order time derivative. Because the early exercise boundary and its derivative are not a priori known, the boundary values mimicked by the auxiliary function are in approximate form. Concurrently, we then establish equations that approximate the early exercise boundary and its derivative directly from the DNN output based on some linear relationships at the left boundary. Furthermore, the option Greeks are obtained from the derivatives of this auxiliary function. We test our implementation with several examples and compare them with the existing numerical methods. All indicators show that our proposed deep learning method presents an efficient and alternative way of pricing options with early exercise features.
February 7, 2023
Speaker: Anatoliy Swishchuk (Department of Mathematics & Statistics, University of Calgary)
TOPIC: Introduction to Telegraph Processes/Equations and Their Applications in Finance
Abstract:
In this talk, I’ll introduce classical/symmetric telegraph process (a.k.a Kac process (1951)), and show its connection with wave, diffusion, Schroedinger equtions and Feynman-Kac formula. Asymmetric telegraph process will be also introduced. Then, I’ll present two models for stock prices based on symmetric and asymmetric telegraph processes, and option pricing formulas for them. Margrabe’s spread options valuations for two stocks modelled by two different telegraph processes will be presented as well. Those models and formulas are supported by numerical examples and graphs. (This talk is based on our joint research collaboration with Prof. A. Pogorui (Ukraine) and Prof. R. Rodriguez-Dagnino (Mexico)).
January 24, 2023
Speaker: Ana Karen Roldan Contreras (Department of Mathematics & Statistics, University of Calgary)
Abstract:
In this talk, I will review solutions for optimal control problems such as optimal acquisition, optimal liquidation, and market making, which are considered primary purposes in trading activity. Diffusion processes model the price. I will compare the optimal solution using the Semi-Markov process versus the general compound Hawkes process in the limit order market.
On the one hand, the counting process corresponding to the Semi-Markov process changes states following a Markov chain but takes a random amount of time between changes. For in predicting the future, not only would we want to know the present state but also the length of time that has been spent in that state. On the other hand, the Hawkes process N(t) is a counting process with a self-exciting property, a clustering effect, and long-run memory properties. These two processes have their particularities, and both results are consistent with what has been observed in the empirical data on limit order books.
The models used in this study describe the controlled inventory process Q(t) of the agent, the midprice with the corresponding SDE, and the cash process X(t) of the agent in order to evaluate and maximize the agent’s value function H(t, x, S, q), by solving the Hamilton–Jacobi–Bellman equation and designing a unique strategy to trade assets with prices that behaves as suggested. The final results are the optimal inventory Q(t)*, the optimal speed of trading ν(t)*, and the optimal depth δ∗,±(t, q). I will compare the optimal solutions expressed in terms of parameters describing the arrival rates and the midprice and find a more general solution that describes the observed price process in HFT markets more authentically. In addition, the comparison will be explicit with real-world data from LOBster data.
FALL 2022
December 6, 2022
Speaker: Kirill Golubnichiy (Department of Mathematics & Statistics, University of Calgary)
Abstract:
In the previous paper (Inverse Problems, 32, 015010, 2016), a new heuristic mathematical model was proposed for accurate forecasting of prices of stock options for 1-2 trading days ahead of the present one. This new technique uses the Black-Scholes equation supplied by new intervals for the underlying stock and new initial and boundary conditions for option prices. The Black-Scholes equation was solved in the positive direction of the time variable, This ill-posed initial boundary value problem was solved by the so-called Quasi-Reversibility Method (QRM). This approach with an added trading strategy was tested on the market data for 368 stock options and good forecasting results were demonstrated. We use the geometric Brownian motion to provide an explanation of that effectivity using computationally simulated data for European call options. We also provide a convergence analysis for QRM. The key tool of that analysis is a Carleman estimate. To enhance these results, the Neural Network Machine Learning is applied on the second stage. Real market data are used. Results of Quasi-Reversibility Method and Machine Learning method are compared in terms of accuracy, precision and recall.
November 15, 2022
Speaker: Zhouzhou Gu (Department of Mathematics & Statistics, University of Calgary)
TOPIC: Affine GARCH option pricing models, stochastic interest rates, and diffusion limits
Abstract:
We proposes a derivative pricing framework when the asset returns and the short-term rate process are modelled with affine GARCH models driven by correlated Gaussian innovations. The risk neutral dynamics are derived based on a co variance dependent pricing kernel and semi-closed form solutions are derived for European style options and bond prices. We further derive the weak diffusion limits of the underlying processes under both physical and risk-neutral measure and we investigate the consistency between the proposed pricing kernel with the well-known Girsanov's principle in continuous time. A variety of numerical exercises are provided to analyze the validity of our pricing formulae, the sensitivity of the option prices relative to the pricing kernel parameters, and the convergence of option prices to those based on the limiting diffusions. Finally, we illustrate an empirical analysis based on a joint estimation using historical asset returns and short-term rates, and cross sections of options and bond prices.
October 25, 2022
Speaker: Tim J. Boonen (University of Amsterdam, Netherlands)
TOPIC: (No-)Betting Pareto-Optima under Rank-Dependent Utility
Abstract:
In this talk, I discuss a pure-exchange economy with no aggregate uncertainty, and I characterize in closed-form and in full generality Pareto-optimal risk-sharing allocations between two agents who maximize rank-dependent utilities (RDU). I then derive a necessary and sufficient condition for Pareto-optima to be no-betting allocations (i.e., deterministic allocations - or full insurance allocations). This condition depends only on the probability weighting functions of the two agents, and not on their (concave) utility functions. Hence with RDU preferences, it is the difference in probabilistic risk attitudes given common beliefs, rather than heterogeneity or ambiguity in beliefs, that is a driver of a bet. As by-product of our analysis, I answer the question of when sunspots matter in this economy.
October 4, 2022
Speaker: Tylar Jia (Department of Mathematics & Statistics, University of Calgary)
Abstract:
This talk will present a methodology to incorporate large-scale atmospheric information into short-term wind speed forecast in Alberta using two publicly accessible datasets. The first dataset is used for atmospheric clustering by applying the k-means algorithm and the hidden Markov model on atmospheric variables related to wind speed. The second dataset is used to test the proposed time series regime-switching models and mixture models that integrate the clustering results to predict 6-hour ahead wind speed at 23 weather stations in Alberta, Canada. The predictive performance is compared for atmospheric clustering methods and forecasting models.
September 20, 2022
Speaker: YuYu Chen (University of Waterloo, ON)
TOPIC: An unexpected stochastic dominance: Pareto distributions, catastrophes, and risk exchange
Abstract:
We show the perhaps surprising inequality that the weighted average of iid ultra-heavy-tailed (i.e., infinite mean) Pareto losses is larger than a standalone loss in the sense of first-order stochastic dominance. This result is further generalized to allow for random total number and weights of Pareto losses and for the losses to be triggered by catastrophic events. We discuss several implications of these results via an equilibrium analysis in a risk exchange market. First, diversification of ultra-heavy-tailed Pareto losses increases portfolio risk, and thus a diversification penalty exists. Second, agents with ultra-heavy-tailed Pareto losses will not share risks in a market equilibrium. Third, transferring losses from agents bearing Pareto losses to external parties without any losses may arrive at an equilibrium which benefits every party involved. The empirical studies show that our new inequality can be observed empirically for real datasets that fit well with ultra-heavy tails.
SPRING 2022
June 14, 2022
Speaker: Anatoliy Pogoruy (Zhytomyr Ivan Franko University, Zhytomyr, Ukraine)
TOPIC: Random Motions Based on Telegraph Processes and Their Applications
Abstract: In this talk, I’ll present the main results about the telegraph processes [1], their generalizations and some applications. I’ll review one-dimensional random motion in the generalized Erlang environment [2] and mention some results on other non-Markovian random walks [3], [4] and fading evolutions [5]. I will also mention multidimensional random motion with uniformly distributed changes of direction [5], and a system of interactive particles with Markovian switching [5]-[6]. In conclusion, I consider application of the telegraph process as an alternative model to the diffusion process in the Black-Scholes formula [6].
References:
1. Goldstein S. On diffusion by discontinuous movements and on the telegraph equation Quart. J. Math. Mech. – 1951. Vol. 4. – P. 129–156.
2. Pogorui A., Rodriguez-Dagnino R. M. One-Dimensional Semi-Markov Evolution with General Erlang Sojourn Times. Random Operators and Stochastic Equations, Vol.13, No.4, pp.399-405 (2005).
3. Franceschetti M. When a random walk of fixed length can lead uniformly anywhere inside a hypersphere, J. Theor. Probab. – 2007.– Vol. 20. –Р. 813–823.
4. Le Caer G.: A Pearson random walk with steps of uniform orientation and Dirichlet distributed lengths. J. Stat. Phys. – 2010. – Vol. 140. – P. 728–751.
5. Pogorui A., Swishchuk A., Rodriguez-Dagnino R. M. Random Motions in Markov and Semi-Markov Environments 1 Homogeneous Random Motions and their Applications, Wiley, 2021
6. Pogorui A., Swishchuk A., Rodriguez-Dagnino R. M. Random Motions in Markov and Semi-Markov Random Environments 2: High-dimensional Random Motions and Financial Applications, Wiley, 2021
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Prof. Pogoruy’s Short Bio: Anatoliy Pogoruy is a Professor at Zhytomyr Ivan Franko University, Zhytomyr, Ukraine. He’s got his BSc and MSc from Kyiv State University, and Ph.D. and D.Sc. from the Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine. His areas of research are semi-Markov processes and evolutions, reliability of systems, transformed telegraph processes and their applications, including finance. A. Pogoruy published around 55 scientific papers, co-authored two-volume book “Random Motions in Markov and Semi-Markov Random Environments” (vol. 1: Homogeneous Random Motions and their Applications, vol. 2: High-dimensional Random Motions and Financial
Applications), John Wiley & Sons, 2021 (joint with R.M. Rodriguez-Dagnino and A. Swishchuk), and also textbook “Concepts and Problems for Mathematical Competitors”, Dover Publications Inc., 2020 (joint with A. Sarana and R.M. Rodriguez-Dagnino). A. Pogoruy has many awards including the diploma of the Ministry of Education of Ukraine for achievements in education and science, the Medal of Zhytomyr University, and the Medal of the Ministry of Science and Education of Ukraine “Excellence in Education”.
WINTER 2022
April 12, 2022
Speaker: Molla Hasib Uddin
TOPIC: Numerical Approximations of Coupled Forward-Backward SPDEs
Abstract: The forward-backward stochastic partial differential equations (FBSPDEs) arise naturally in many applications of probability theory and stochastic processes. Here we use a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled FBSPDEs with homogeneous Dirichlet boundary conditions where coefficients may be nonlinear and nonlocal. The existence and uniqueness of the strong solution of such FBSPDEs are derived under certain monotonicity conditions. The finite element method in the spatial domain leads to approximations of FBSPDEs by finite-dimensional forward-backward stochastic differential equations (FBSDEs) which are numerically computed by using some deep learning-based schemes. We also address the wellposedness of such FBSDEs as well as the convergence rate for spatial discretization. Numerical examples indicate the very good efficiency of our method.
April 5, 2022
Speaker: Luca Lalor
TOPIC: Cloud Deployment of Algorithmic Trading Strategies
Abstract: This presentation will be the last in a four-part lecture series dedicated to introducing Python for Algorithmic and High Frequency Trading. There will be two main parts to this presentation. The first part of this presentation will introduce how one can work with real time data for which sockets are in general the tool of choice. Here the basics of socket programming will be introduced. Data arrives in real time and usually in massive amounts, making real-time processing of the data and the real-time decision making based on the streaming data a necessity. I will proceed to give examples on 1) how to run a simple tick data server, 2) how one can connect a simple tick data server, 3) generating signals in real time, and 4) visualizing streaming data with Plotly and Real Time Streams. The second part of this presentation will give live examples on how one can deploy algorithmic trading strategies via Python in an actual account (through the Oanda-broker, with a demo account being used for presentation purposes). A brief introduction on how to set this up will be given along with some examples which will include 1) Retrieving historical data, backtesting and factoring in leverage and margin, 2) Working with Streaming Data, Placing Orders, and Implementing Strategies, 3) Retrieving Account Information. All these examples will be performed on live data in real time.
March 29, 2022
Speaker: Sarath Kumar Jayaraman
TOPIC: Long memory in option pricing: A fractional discrete-time framework
Abstract: This article studies the impact of long memory on asset return modelling and option pricing. We propose a general discrete-time pricing framework based on affine multi-component volatility models that admit ARCH(∞) representations, which not only nests a plethora of option pricing models from the literature, but also allows for the introduction of novel fractionally integrated processes for option valuation purposes. Using an infinite sum characterization of the unconditional cumulant generating function of the log-asset price, we derive semi-explicit expressions for European option prices under a volatility-dependent stochastic discount factor. We carry out an extensive empirical analysis which includes returns-only as well as return and option joint estimations of a variety of short- and long-memory models for the S&P 500 index. Our results indicate that the inclusion of long memory into return modelling substantially improves the option pricing performance. Using a set of out-of-sample option pricing errors, we show that a single-component long-memory model outperforms a richer-parametrized two component model with short-memory dynamics. Authors: Maciej Augustyniak, Alexandru Badescu, Jean-Francois Begin, Sarath Kumar Jayaraman.
March 22, 2022
Speaker: Ye Wang
TOPIC: Optimal Reinsurance Under Vajda Condition and Range-Value-At-Risk
Abstract: In this project we study an optimal reinsurance problem where the insurer’s risk-adjusted liability gets minimized. To better reflect the spirit of reinsurance, we impose exogenously Vajda condition on indemnity functions which requires the reinsurer to pay an increasing proportion of loss. To consider both robustness and tail risk, the insurer is assumed to apply Range-Value-at-Risk (RVaR) to evaluate its risk. Under the expected value premium principle, we derive the closed-form solution to our problem, which includes the results in Chi and Weng (2013) as special cases. Some comparative studies and sensitivity analysis are also carried out through numerical examples.
March 15, 2022
Speaker: Luca Lalor
TOPIC: Predicting Market Movements with Machine Learning
Abstract: This presentation will be the third of a four-part lecture series dedicated to introducing Python for Algorithmic and High Frequency Trading. There will be 3 parts to this presentation. The first part will discuss how linear regression can be used for market movements predictions. There will be a brief review on linear regression, the basic idea on how it can be used in price prediction, its use in predicting a financial securities future price and/or direction and will conclude with how one can use a vectorized backtesting approach to streamline this process. The second part will discuss how one can use machine learning for market movement prediction. It will begin with a simple linear regression example using the Python Scikit-learn package, followed by examples with simple classification and logistic regression algorithms. The third part of this presentation will focus on how deep learning methods can be used for future market movement predictions. In this section, I will discuss the basics of how Neural Networks work, revisit the simple classification problem, and then how a dense neural network (DNN) can be used for making predictions in financial markets. I will also briefly discuss how one can add different types of features to try and improve a DNN model. All models discussed in this presentation will coincide with live examples, through the Python Jupyter Notebook application.
March 1, 2022
Speaker: Zuming Sun
TOPIC: Polynomial Process and Polynomial Regression Model for French Electricity Prices
Abstract: Empirical experience reveals that electricity prices can be affected by the dynamics of residual demand, power generation capacity of each commodity and spot prices of each generation technology. In this talk, I’ll present a model involving a polynomial map of polynomial processes, a so called PMPP model, for electricity spot prices. Each reasonable underlying factor in PMPP model is modeled by a suitable polynomial process, such as the Ornstein-Uhlenbeck process or geometric Brownian motion. The polynomial map is determined by polynomial regression, revealing the relationship between the underlying factors and spot prices. The PMPP model provides the advantage of cheap and convenient computation for forward prices because of the great property of polynomial processes that conditional expectations of polynomial function of the future state of polynomial process conditional on current state are given by polynomials of the current state. My work mainly focuses on the French electricity market. The optimal PMPP model is selected by statistical criterion, meanwhile maximum likelihood estimation and Kalman filter are applied for model calibration. The primary results indicate that our PMPP model can capture the electricity prices well, when compared with the classic structural model, and could be useful for risk management purposes.
February 15, 2022
Speaker: Luca Lalor
TOPIC: Mastering Vectorization to Backtest Trading Strategies
Abstract: This presentation will be the second of a four-part lecture series dedicated to introducing Python for Algorithmic and High Frequency Trading. It will begin with an introduction to vectorization in Python using the packages NumPy and Pandas. Then I will discuss some of the most common return and risk performance metrics for a trading strategy and how they can be easily calculated in Python. Next the presentation will go on to discuss three simple examples of how one can backtest certain trading strategies using Python. These strategies will be based on some of the most common technical analysis (i.e., price analysis) trading strategies studied by traders and investors. The first example will be for a Simple Moving Average crossover trading strategy, the second for a momentum trading strategy and the third for a mean reversion trading strategy. At the end, I will briefly touch on data snooping and overfitting, and why it is very important to take this into account when Backtesting a trading strategy. Most parts of this presentation will coincide with examples in action, through the Python Jupyter Notebook application.
February 1, 2022
Speaker: Myles Sjogren
TOPIC: General Compound Hawkes Processes for Mid-Price Prediction
Abstract: High frequency financial data is burdened by a level of randomness that is unavoidable and obfuscates the task of modelling. This idea is reflected in the intraday evolution of limit orders book data for many financial assets and suggests several justifications for the use of stochastic models. For instance, the arbitrary distribution of inter arrival times and the subsequent dependence structure between consecutive book events. This has led to the development of many stochastic models for the dynamics of limit order books. In this paper we look to examine the adaptability of one family of such models, the General Compound Hawkes Process (GCHP) models, to new data and new tasks. We further focus on the prediction problem for the mid-price within a limit order book and the practical applications of these stochastic models, which is the main contribution of this paper. To this end we examine the use of the GCHP for predicting the direction and volatility of futures and stock data and discuss possible extensions of the model to help improve its predictive capabilities.
January 18, 2022
Speaker: Luca Lalor
TOPIC: Introduction to Python for Finance and Algorithmic Trading I
Abstract: This lecture will be the first in a four-part lecture series dedicated to introducing Python for Algorithmic and High Frequency Trading. This lecture will first introduce the book Python for Algorithmic Trading: From Idea to Cloud Deployment by Yves Hilpisch and give an overview of the book’s contents and structure. Then, it will proceed to give three examples of Python Applications in Finance using the Python application Jupyter Notebook. The first example will introduce the use of the NumPy library and how it can be used to solve the Euler Discretization Geometric Brownian Motion model. The second example will introduce the pandas library and the DataFrame class to show how one can work with real data. The third example will show how Python can be used to efficiently solve many of the classic PDEs in finance. The lecture will then go on to discuss why Python is a great choice in the field of Algorithmic and High Frequency trading.
FALL 2021
December 7, 2021
Speaker: Ana Karen Roldan Contreras
TOPIC: Stochastic Optimal Control Problems in Limit Order Books for General Compound Hawkes Process
Abstract: In this talk, I will review solutions for optimal control problems as optimal acquisition, optimal liquidation, and market making, which are considered as main purposes in the trading activity. Using the price approximated by diffusion process for the general compound Hawkes process (GCHP, with results from Swishchuk and Huffman, 2018) in the limit order market. In these models, the agent is to maximize her own utility or value function by solving the HJB equation, designing a unique strategy to trade assets with prices that behaves as suggested. In addition, applications with real data will be reviewed.
November 23, 2021
Speaker: Joshua McGillivray
TOPIC: Swaps in Energy Markets
Abstract: Derivatives markets as big as ever, and their importance is only growing as more are being developed. In this presentation, we are delving into how forward contracts, variance, volatility, correlation, and covariance swaps are created and valued. The proliferation of these contracts into commodities markets is dramatically increasing the proportion of investor portfolios which rely on commodities. The largest sector within the commodity market is easily the energy market, specifically oil and gas. Swaps are very similar to forward contracts however the underlying asset determining their payoff is not a stock price, instead it is a property of the stock price such as the variance of a stock or the correlation between stocks. The valuation of these contracts can range from tricky to completely intractable. The addition of convenience yields into these equations only complicates them further. Thus, the application of swaps into energy markets is both a valuable and difficult task which may have many real-world effects.
November 16, 2021
Speaker: Qi Guo
TOPIC: Machine Learning for Stock Selection: Framework and Results
Abstract: Stock selection is one of the fundamental parts in the asset management (investment) problem. An increasing number of publications claim varying degrees of success in applying ML to asset pricing problems. However, deploying machine learning (ML) to stock selection (return & risk forecasting) is still an ambitious project. In this work, we presented a framework of ML for stock selection and tested several classical ML models with past 30-year SP&500 stock data such as SVM, GDBT, and Lasso. We also conducted back-testing for an industrial model which is called Multi-Branch Boosted Tree (MBBT). Some deep learning methods were also considered such as Neural Network and Learning to Rank (LTR). Most of this project were done during my internship at TD asset management with Dr. Tianyu Tan. Partial results and codes were developed at the 11th Montreal Industrial Problem-Solving Workshop (IPSW).
October 26, 2021
Speaker: Yang Yang
TOPIC: Stochastic Path-Dependent Models and Analysis
Abstract: This talk is devoted to research on stochastic path-dependent models. The whole content will be threefold, covering the modeling issues, including model setups, discretization, simulations, and result analysis; pricing problems, including derivation of pricing partial differential equations, market data fitting; optimization, focusing on the well-posedness in solutions of a stochastic path-dependent optimal control problem with an infinite dimensional setup. The purpose of this talk is to explore, analyze and demonstrate the obstacles and broad applications if path-dependence is introduced to general stochastic differential equation systems.
October 12, 2021
Speaker: Yu Li
TOPIC: Generalized Multi-level Monte Carlo Method
Abstract: The Multilevel Monte Carlo (MLMC) method has been applied successfully in a wide range of settings over the last decade or so since its first introduction by Giles. When using only two levels, the method can be viewed as a kind of control-variate approach to reduce variance by Kebaier. In this article, we extend this approach to any number of levels, and under mild assumptions, we can derive meaningful values for the coefficients. To illustrate this method, we apply it to Geometric Brownian Motion (GBM), Inhomogeneous Geometric Brownian Motion (IGBM), and Cox-Ingersoll-Ross (CIR) processes using Euler and Milstein discretization, and we can show that the improvement it makes by numerical results when either the first two level approximations are poor related or they are high related and converges quickly.
September 28, 2021
Speaker: Ehsan Fooladamoli
TOPIC: Merton problem for the risk model based on general compound Hawkes process
Abstract: The goal of insurance companies, like any other financial institutions, is to maximize their wealth. In doing so, there are different parameters they have to consider, such as premium rate, number of claim arrivals, size of claim arrival, etc. Moreover, they can invest their money in risk-free and risky asset to earn some income from those resources as well. In this talk, we will give an overview of classic risk models, and then move on to finding an optimal control for the risk model based on general compound Hawkes process. Some simulation results and their interpretations will be discussed as well.
WINTER 2021
March 23, 2021
Speaker: Anatoliy Swishchuk
TOPIC: Merton Investment Problem for Hawkes Risk Model in Insurance
Abstract: Merton optimal investment and consumption stochastic problem is one of the most studied classical problem in finance (see [1,2]). In this talk, we will show how to solve Merton optimal investment stochastic control problem for Hawkes risk model R(t) in insurance. Namely, we will show how to find an optimal investment for the risk model R(t) based on general compound Hawkes process (GCHP) (see [3,4]), when an investor decides to invest some capital A(t) in risky assets (e.g., stocks) and the rest, (R(t)-A(t)) in risk-free assets (e.g., bonds or bank account). The talk consists of three parts: 1) Introduction and review of Merton investment problem in finance; 2) Review of the most popular risk models in insurance, including the one based on GCHP; 3) Merton investment problem for Hawkes risk model in insurance and its solution.
References.
- Merton, R. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. The Review of Econ. Stat., 247-257.
- Merton, R. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. of Economic Theory. 3, 373-413.
- Swishchuk, A. (2018): Risk model based on general compound Hawkes process. Wilmott, v. 2018, Issue 94.
- Swishchuk, A., Zagst, R. and Zeller, G. (2020): Hawkes processes in insurance: Risk model, application to empirical data and optimal investment. Insurance: Mathematics and Economics, https://doi.org/10.1016/j.insmatheco.2020.12.005.
March 9, 2021
Speaker: Ana Karen Roldan Contreras
TOPIC: Book’s review -‘Deep Learning’ by Goodfellow I., Bengio J. and Courville A., The MIT Press, 2016. Chapter 12: Applications
Abstract: In this chapter, we describe how to use deep learning to solve applications in computer vision, speech recognition, natural language processing, and other areas of commercial interest. The chapter begins with discussion the large-scale neural networks implementations required for most serious AI applications. Then, we review several specific application areas that deep learning has been used to solve.
Feb. 23, 2021
Speaker: Ehsan Fooladamoli
TOPIC: Paper’s review -‘Deep Learning: An Introduction for Applied Mathematicians’ by Catherine F. Higham and Desmond J. Higham, SIAM Review, vol. 61, No. 4, pp. 860-891.
Abstract: Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computational mathematics, notably from calculus, approximation theory, optimization, and linear algebra. This article provides a very brief introduction to the basic ideas that underlie deep learning from an applied mathematics perspective. The paper focuses on three fundamental questions: What is a deep neural network? How is a network trained? What is the stochastic gradient method? The ideas are illustrated with a short MATLAB code that sets up and trains a network.
Feb. 2, 2021
Speaker: Devin Kwok
TOPIC: Review of ‘Deep Learning’ book by I. Goodfellow, Y. Bengio and A. Courville, The MIT Press, 2016. Chapter 20: Deep Generative Models
Abstract: In this chapter, we present several of the specific kinds of generative models that can be built and trained using the techniques presented in chapters 16–19. All of these models represent probability distributions over multiple variables in some way. Some allow the probability distribution function to be evaluated explicitly. Others do not allow the evaluation of the probability distribution function, but support operations that implicitly require knowledge of it, such as drawing samples from the distribution. Some of these models are structured probabilistic models described in terms of graphs and factors, using the language of graphical models presented in chapter 16. Others cannot easily be described in terms of factors, but represent probability distributions nonetheless.
January 19, 2021
Speaker: Weiliang Lu
TOPIC: Review of ‘Deep Learning’ book by I. Goodfellow, Y. Bengio and A. Courville, The MIT Press, 2016. Chapter 19: Approximate Inference
Abstract: Many probabilistic models are difficult to train because it is difficult to perform inference in them. In the context of deep learning, we usually have a set of visible variables v and a set of latent variables h. The challenge of inference usually refers to the difficult problem of computing p(h|v) or taking expectations with respect to it. Such operations are often necessary for tasks like maximum likelihood learning. Exact inference requires an exponential amount of time in these models. Even some models with only a single layer, such as sparse coding, have this problem. In this Chapter 19, we introduce several of the techniques for confronting these intractable inference problems.
FALL 2020
December 1, 2020
Speaker: Joshua McGillivray
TOPIC: Review of ‘Deep Learning’ book by I. Goodfellow, Y. Bengio and A. Courville, The MIT Press, 2016: Chapter 17: Monte Carlo Methods and Chapter 18: Confronting the Partition Function (sec. 18.1 (The Log-Likelihood Gradient)+18.2 (Stochastic Maximum Likelihood and Contrastive Divergence))
Abstract: Many problems in applied mathematics are infeasible to solve algebraically, thus we must rely on numerical methods to approximate a solution. Some numerical methods incorporate randomness to account for the unpredictability of the world around us. Monte Carlo methods are a group of randomized algorithms which rely on randomness to create a solution within a certain error. Monte Carlo methods are ubiquitous in machine learning and are all but a necessity to solve fundamental problems underlying the machine learning process. This presentation presents an overview of Monte Carlo methods within the context of deep learning as well as discusses some problems that arise due to their use. The log-likelihood gradient (sec. 18.1) and stochastic maximum likelihood and contrastive divergence (sec. 18.2) from Chapter 18 will be also discussed.
November 17, 2020
Speaker: Myles Sjogren
TOPIC: Review of ‘Deep Learning’ book by I. Goodfellow, Y. Bengio and A. Courville, The MIT Press, 2016. Chapter 16: Structured Probabilistic Models for Deep Learning
Abstract: Deep learning draws upon many modeling formalisms that researchers can use to guide their design efforts and describe their algorithms. One of these formalisms is the idea of structured probabilistic models. A structured probabilistic model is a way of describing a probability distribution, using a graph to describe which random variables in the probability distribution interact with each other directly. This chapter provides basic background on some of the most central ideas of graphical models, and focuses on the concepts that have proven most useful to the deep learning research community. The use of graphs to describe probability distributions is common practice when dealing with complex scenarios involving higher dimensional data with rich structure. The practise of structured probabilistic modelling can simplify the task of developing models in which variables interact in both direct and indirect ways. This infers the development of a formal framework allowing models to have significantly fewer parameters and therefore be estimated reliably from less data. This presentation gives an overview of the basics of structured probabilistic modelling and talks briefly about how it can be adapted for use in applications of deep learning.
October 27, 2020
Speaker: Matthew Greenberg
TOPIC: Review of ‘Deep Learning’ book by I. Goodfellow, Y. Bengio and A. Courville, The MIT Press, 2016. Chapter 10: Sequence Modeling via Recurrent Neural Networks
Abstract: Recurrent neural networks (RNNs) are a family of neural networks for processing sequential data, e.g. time series, text. In contrast to simple feedforward or convolutional networks, the input size (length of the sequence) does not need to be known in advance. I’ll start the talk by describing the architecture of RNNs and how to train them using “backpropagation through time”. In the remaining time, I’ll discuss “gated” networks allowing for long-term memory and the encoder-decoder architecture
October 13, 2020
Speaker: Ehsan Fooladamoli
TOPIC: Review of ‘Deep Learning’ book by I. Goodfellow, Y. Bengio and A. Courville, The MIT Press, 2016. Chapter 11: Practical Methodology
Abstract: Most of the book under review is about different machine learning models, training algorithms, and objective functions. In practice, one can usually do much better with a correct application of a commonplace algorithm than by sloppily applying an obscure algorithm. Correct application of an algorithm depends on mastering some fairly simple methodology. Many of the recommendations in this Chapter 11 are adapted from Ng, A. (2015) paper “Advice for applying machine learning”. The book recommends the following practical design process: 1) determine your goals; 2) establish a working end-to-end pipeline as soon as possible, including the estimation of the appropriate performance metrics; 3) instrument the system well to determine bottlenecks in performance; 4) repeatedly make instrumental changes such as gathering new data, adjusting hyperparameters, or changing algorithms
September 29, 2020
Speaker: Qi Guo
TOPIC: Review of ‘Deep Learning’ book by I. Goodfellow, Y. Bengio and A. Courville, The MIT Press, 2016. Chapter 9: Convolutional Networks
Abstract: The goal of this chapter is to describe the kinds of tools that convolutional networks (CN) provide. CN are a specialized kind of neutral network for processing data that has a known grid-like topology. This chapter explains the motivation behind using convolution in a neural network, and describes the operation called pooling, which almost all convolutional networks employ. Several variants on the convolutional function that are widely used in practice for neural networks are introduced. It is also shown how convolution may be applied to many kinds of data, with different numbers of dimensions. (Part of this Chapter 9 was presented in Winter 2020).
September 15, 2020
Speaker: Qi Guo
Title: Review of ‘Deep Learning’ book by I. Goodfellow, Y. Bengio and A. Courville, The MIT Press, 2016 Chapter 7 (sec. 7.8-7.14): Regularization for Deep Learning
Abstract: A central problem in machine learning is how to make an algorithm that will perform well not just on the training data, but also on new inputs. Many strategies used in machine learning are explicitly designed to reduce the test error, possibly at the expense of increase training error. These strategies are known collectively as regularization. Developing more effective regularization strategies has been one of the major research efforts in the field. This Chapter 7 describes regularization in more details than previous Chapter 5, focusing on regularization strategies for deep models or models that may be used as building blocks to form deep models.
WINTER 2020
March 10, 2020
Speaker: Ebrahim Ghaderpour
Title: Review of ‘Deep Learning’ book by I. Goodfellow, Y. Bengio and A. Courville, The MIT Press, 2016: Chapter 9: Convolutional Networks
Abstract: The goal of this chapter is to describe the kinds of tools that convolutional networks (CN) provide. CN are a specialized kind of neutral network for processing data that has a known grid-like topology. This chapter explains the motivation behind using convolution in a neural network, and describes the operation called pooling, which almost all convolutional networks employ. Several variants on the convolutional function that are widely used in practice for neural networks are introduced. It is also shown how convolution may be applied to many kinds of data, with different numbers of dimensions.
March 3, 2020
Speaker: Professor Dr. Christian Bayer (Weierstrass Institute, Berlin, Germany)
Title: Pricing American Options by Exercise Rate Optimization
Abstract: We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black--Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black--Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model.
February 25th, 2020
Speaker: Weiliang Lu
Abstract: This second talk will finalize the Chapter 8 by reviewing Sections 8.4 (Parameter Initialization Strategies ), 8.5 (Algorithms with Adaptive Learning Rates), 8.6. (Approximate Second-Order Methods) and 8.7 (Optimization Strategies and Meta-Algorithms).
February 11th, 2020
Speaker: Weiliang Lu
Abstract: Of all the many optimization problems involved in deep learning, the most difficult is neural network training. Because this problem is so important and so expensive, a specialized set of optimization techniques have been developed for solving it. Chapter 8 presents these optimization techniques for neural network training. It focuses on one particular case of optimization: finding the parameters \theta of a neural network that significantly reduce a cost function J(\theta), which typically includes a performance measure evaluated on the entire training set as well as additional regularization terms.
January 30th, 2020
Speaker: Clarence Simard (UQAM, Montreal, QC, Canada)
Title: Martingale representation with nonlinear stochastic integrals
Abstract: The Kunita-Watanabe decomposition is the decomposition of the L^2 space into orthogonal martingales. This decomposition is at the heart of the delta hedging in continuous time. A more general decomposition, the Follmër-Schweizer decomposition, is fundamental in the theory of quadratic hedging. In this talk, I will present a generalisation of the Kunita-Watanabe decomposition using nonlinear stochastic integrals. I will begin the talk by giving an intuitive presentation of the orthogonal decomposition of a linear space.
January 21th, 2020
Speaker: Anatoliy Swishchuk
Abstract: In this talk, I’ll quickly go through probability theory's description (Sec. 3.1-3.9, 3.11-3.12), and concentrate more on information theory (Sec. 3.10 and 3.13) and structured probabilistic models’ ideas (3.14). In the context of machine or deep learning, we can apply information theory to characterize probability distributions or to quantify similarity between probability distributions.
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FALL 2019
December 3rd, 2019
Speaker: Qi Guo
Abstract: Chapter 7, sec. 7.1-7.7, is devoted to the parameter norm penalties (7.1), norm penalties as constrained optimization (7.2), regularization and under-constrained problems (7.3), dataset augmentation (7.4), noise robustness (7.5), semi-supervised learning (7.6) and multitask learning (7.7).
November 26th, 2019
Speaker: Matthew Greenberg
Title: Deep Learning with Tensorflow
Abstract: In this talk, I will introduce Tensorflow, currently the most popular library for computations involving neural networks. I will explain the essential ideas underlying Tensorflow and its most important capabilities. We will interact with these capabilities both "by hand" as well as through the higher level Keras front-end. My hope is that the audience acquires sufficient skill to experiment independently with simple feedforward and convolutional neural networks in Tensorflow.
November 19th, 2019
Speaker: Ana Karen Roldan
Abstract: Section 6.4-6.6. are devoted to Architecture Design (6.4), Back-Propagation and Other Differentiation (6.5) and Historical Notes (6.6). Code can be found here.
November 19th, 2019
Speaker: Ana Karen Roldan
Abstract: Section 6.4-6.6. are devoted to Architecture Design (6.4), Back-Propagation and Other Differentiation (6.5) and Historical Notes (6.6). Code can be found here.
October 29th, 2019
Speaker: Anatoliy Swishchuk
Abstract: The sections 5.10-5.11 are devoted to the Building a Machine Learning Algorithms (MLA) (sec. 5.10) and Challenges Motivating Deep Learning (sec. 5.11).
October 22nd, 2019
Speaker: Ana Karen Roldan
Abstract: The sections 5.8-5.9 are devoted to the Unsupervised Learning Algorithms (ULA) (sec. 5.8) and Stochastic Gradient Descent (SGD) (sec. 5.9).
October 15th, 2019
Speaker: Weiliang Lu
Abstract: The sections 5.5-5.7 are devoted to the Maximum Likelihood Estimation (MLE) (sec. 5.5), Bayesian Statistics (sec. 5.6) and Supervised Learning Algorithms (sec. 5.7).
October 8th, 2019
Speaker: Qi Guo
Abstract: The sections 5.2-5.4 are devoted to Capacity, Overfitting and Underfitting topics (sec. 5.2), as well as Hyperparameters and Validation Sets (sec. 5.3), and Estimators, Bias and Variance (sec. 5.4).
October 1st, 2019
Speaker: Anatoliy Swishchuk
Abstract: This academic year our ‘Lunch at the Lab’ mathematical finance seminar will be devoted to the deep and machine learning, in particular, we will be reviewing ‘Deep Learning’ book by I. Goodfellow, Y. Bengio and A. Courville, The MIT Press, 2016. All are welcome to review the book!
September 17th, 2019
Speaker: Lisa Ponti and Gerald Bueshel (GARP (Global Association of Risk Professionals))
Abstract: Lisa and Gerald will speak about the ERP and FRM certification programs, and how they might build upon their current skill sets and how they might fit into their future career plans.
SPRING/SUMMER 2019
May 28, 2019
Risk and Trading Operations: Recent Research
Speakers:
- Alexis Arrigoni (Energy and Fuel-switching Pricing using Levy Processes-An Application to Albertan Data)
- Qi Guo (Application of Hawkes Processes in High Frequency Trading)
- Yi (Ivy) Zhang (Option Pricing with Semi-Markov Switching Levy Processes)
- Qiyue He (Application of Hawkes Processes in High Frequency Trading)
- Aiden Huffman (Application of Hawkes Processes in High Frequency Trading)
Bios:
Alexis Arrigoni: I obtained my Bachelor of Science in Economics at the University of Geneva in 2015 and continued my Master studies at the same university graduating in 2017 with a MSc in Economics with specialization in Monetary and Financial Economics. During that time, I spent a semester abroad and attended Stockholm University. In 2017, I entered the Math and Stats program at the University of Calgary focusing on Mathematical Finance and graduated this April. I have also been working as a quantitative analyst in Calgary for the past year.
Qi Guo: Qi Guo is a Ph.D. student in applied mathematics program at the University of Calgary supervised by Anatoliy Swishchuk. His research interest is the mathematical modelling for high-frequency trading and he is conducting research about multivariate Hawkes processes. He got his master’s degree in applied mathematics at the University of Saskatchewan with thesis “logistic operator equation and the induced stochastic process for complex system modelling” in 2018 and he finished his bachelor’s degree at Beijing Institute of Technology in 2016. He also worked for Daimler as a strategy intern for half a year to conduct statistical analysis and time series modelling.
Yi (Ivy) Zhang: As a PhD student under the supervision of Dr. Anatoliy Swishchuk, my research area focuses on pricing of financial derivatives with a selection of stochastic models. My academic and working experiences also provide me with a strong foundation upon which I have built my quantitative modeling, risk assessment and data analysis skills.
Qiyue He: She is a first-year master student in Statistics at the Department of Mathematics and Statistics, University of Calgary, Calgary, Canada. She got her bachelor’s degree in Mathematics and Applied Mathematics at Northeastern University, Shenyang, China. During her undergraduate study, she attended various mathematical modelling contests and won outstanding prizes. She is also a co-author of the paper named Deep Learning for Computer-aided Diagnosis of Brain Diseases Through MRI Multi-classification, which is published on 2017 International Conference on Medicine Sciences and Bioengineering Proceedings. (ICMSB 2017). Recently, she changed her research interest into financial mathematics and works with Dr. Anatoliy Swishchuk on the Applications of Hawkes Process in Limit Order Book. Her research areas include financial mathematics, stochastic process, data mining and statistical computing.
Aiden Huffman: I am an undergraduate at the University of Calgary, about to pursue a masters in Applied Mathematics at Waterloo. I've done research in Quantum Information Theory, Fluid Mechanics and Mathematical Finance. My favorite pastimes include hiking, skiing and learning guita
Abstract: Quantitative finance plays a key role in risk management, particularly in the areas of price analysis, position taking and market understanding. Join us for three presentations by University of Calgary graduate students on how leading edge research can be applied to business operations and provide valuable insight to Alberta companies.
Topics to be addressed in three deep-dive discussion include:
- Energy- and Fuel-Switching Pricing: an Approach to Carbon Pricing
- Applying Hawkes Processes in High Frequency Trading
- Option Pricing with Semi-Markov Switching Levy Processes
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WINTER 2019
April 2th, 2019
Speaker: Qiyue He
Title: Mid-Price Modelling by General Compound Hawkes Process and Its Diffusive Limit
Abstract: In this talk, I will start from basic introduction about Limit Order Book (LOB) and the widely used LOBSTER data. Then, I will move forward to the analytical framework, reviewing some papers about One-Dimensional (Non-linear) Hawkes Process, Non-Linear Compound Hawkes Process with n-state Dependent Orders (NLCHPnSDO), General Compound Hawkes Process with special cases (GCHPnSDO, GCHP2SDO, GCHPDO). After that, I will also introduce the diffusive limit for NLCHPnSDO, GCHPnSDO, GCHP2SDO, and GCHPDO. Finally, I will focus more on simulations where I did lots of my work. I will tell you the main ideas of the mid-price modelling, introduce my dataset which is different from LOBSTER Data, and show you my results and quantitative analysis step-by-step.
March 19th, 2019
Speaker: Zuming Sun
Title: Polynomial processes for energy markets
Abstract: Energy market prices are notable for seasonality, mean reversion and high volatility, often marked by “short-lived” spikes. Polynomial processes are characterized by the property that expectations of polynomial functions of the future state of the process conditional on the current state are given by polynomials of the current state. In this talk, we will explore the application of polynomial processes for spark spread and natural gas price. I will start from increasing cubic polynomial maps on OU process, then extend to fifth-order polynomial and discuss more about the general construction of polynomial maps and inhomogeneous geometric Brownian motion(iGBM). Maximum log-likelihood estimation (MLE) is applied to the calibration of our model. And I’m also going to present the preliminary results and the future work.
March 19th, 2019
Speaker: Junchi Ma
Title: Credit Risk Pricing Based on Epstein-Zin Preference
Abstract: I will present a consumption based equilibrium framework for credit risk pricing in an Epstein-Zin setting. The default time is modelled as the first hitting time of a default boundary. And bond investors have imperfect information about the firm value which is unobservable. The state variables; consumption and volatility are modelled as affine diffusion processes. Using the Epstein-Zin equilibrium solution as the pricing kernel, the price of a zero-coupon bond is expressed as the solution of a system of second order parabolic partial differential equation which is solved numerically. And the price under the imperfect information is solved as a solution of a stochastic partial differential equation. Finally, I will give some comments for the results and see the relationship between the results and parameters. This is joint work with Mobolaji Ogunsolu, Jinniao Qiu and Deniz Sezer.
March 7th, 2019
Speaker: Zhen-Qing Chen
Title: Anomalous diffusions and fractional order differential equations
Abstract: Anomalous diffusion phenomenon has been observed in many natural systems, from the signaling of biological cells, to the foraging behavior of animals, to the travel times of contaminants in groundwater. In this talk, I will first discuss the interplay between anomalous sub-diffusions and time-fractional differential equations, including how they arise naturally from limit theorems for random walks. I will then present some recent results in the study of these two topics. No prior knowledge in these two subjects is assumed.
February 26th, 2019
Speaker: Yi (Ivy) Zhang
Title: Option Pricing with Semi-Markov Switching Levy Processes
Abstract: In this talk, I will first introduce the semi-Markov switching Levy processes, and use Variance Gamma as an example to generate NDX index option prices. We use the fractional fast Fourier transform algorithm as the method for pricing options. I am also going to talk briefly about the calibration method of Gradient Descent, which is the optimization method we use to estimate the parameters.
February 15th, 2019
Speaker: Dr. Hui Huang
Title: On the mean-field limit for interacting (stochastic) particle systems
Abstract: Systems of interacting particles are widely used to establish different mathematical models in physics, biological science and social science. Such particle systems are called microscopic (discrete) models, since we simply observe the motion of many agents. On the other hand, at the macroscopic (continuous) level, we observe a mass of individuals moving in a coherent manner, which can be represented as a partial differential equation. In this talk I will give a brief introduction of the mean¬-field limit theory, which aims at connecting those two descriptions mentioned above.
February 12th, 2019
Speaker: Karen Roldan Contreras
Title: An Optimal Control Problem in Limit Order Books
Abstract: In this talk, I will review an optimal solution to a high frequency trading optimization problem from Fodra and Pham (2015) paper: “High frequency trading and asymptotic for small risk aversion in a Markov renewal model”. First, I will talk about the existing literature on high frequency trading. Next, the model of the stock price, which is based on a Markov renewal process, will be reviewed. Then, I will introduce the model of the market order flow, show the results by solving the market optimization problem (using stochastic optimal control techniques), and explain their financial interpretation. Finally, some potential directions in this research will be described.
February 5th, 2019
Speaker: Weiliang Lu
Abstract: In my presentation, I’ll review some relevant papers in this area of research. Then I’ll present in more details the following main paper ‘Estimation of Levy-driven Ornstein-Uhlenbeck (OU) processes: application to modelling of CO2 and fuel-switching’ written by Chevallier, J. and Goutte S. (2017). It focuses on modelling of switching the valuation between EEX Coal and Natural Gas prices. Then I'll introduce the Levy-driven Ornstein-Uhlenbeck process assuming that the Levy process in the model follows a Normal Inverse Gaussian distribution (NIG). I'll give the basic properties of NIG distribution, such as moments, log likelihood, etc., and estimate the parameters of the model based on the properties. Finally, I will also talk about Alberta energy data, and propose some future work on Markov switching Levy-driven OU process.
January 22th, 2019
Speaker: Qi Guo
Title: Multivariate Hawkes process and their Applications
Abstract: In this talk, we mainly focus on the multivariate Hawkes process and their applications. First, we give a brief introduction about the one-dimensional Hawkes process. Next, we conduct a paper review for the multivariate Hawkes process. The review is based on the paper ‘Some limit theorems for Hawkes processes and application to financial statistics’ by E. Bacry et al. (2013). It includes two important limit theorems for the multivariate Hawkes process: The Law of Large Numbers (LLN) and the Functional Central Limit Theorems (FCLT). Then, we briefly present some results about the compound Hawkes process from the paper 'General Compound Hawkes Processes in Limit Order Books' by A. Swishchuk (2017). In the end, we introduce the multivariate compound Hawkes process and propose some future work.
January 15th, 2019
Speaker: Vakhtang Poutkaradze
Title: Integrability and Chaos in Figure Skating
Abstract: We derive and analyze a three dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly nontrivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories. We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of real-life skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice. (Join work with Vaughn Gzenda (UofA))
January 7th, 2019
Speaker: Lisha Lin
Title: The Pricing of Options with Delay and Bayesian Methods
Abstract: This talk gives a short summary of my Ph.D. research project. The ultimate goal of my research is to price options on underlying assets with delays by using Bayesian methods. However, it’s not an easy work for us. So, we divided this project into several parts to realize it. At first, we did some extension work. We discussed the pricing of European options on two underlying assets with delays and examined the impact of delays on option pricing and obtain sufficient conditions for the robustness of delays for certain two-asset options, where the two underlying assets involved are modelled by stochastic delay differential equations. Secondly, we developed Bayesian methods to carry out inference for the pricing of Quanto options in a Black-Scholes framework and showed how to compute the price of Quanto options with different types of payoff functions using Bayesian prediction techniques, which provide insights about how option prices are distributed. Finally, we presented a Bayesian method to analyze the stochastic delay differential equation (SDDE) model, and attempted to apply it to the problem of option pricing. Numerical simulations are provided to verify our theoretical results.
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FALL 2018
December 4th, 2018
Speaker: Alexis Arrigoni
Title: An introduction to electricity and fuel-switching pricing under stochastic approaches
Abstract: Electricity prices are characterised by sudden high spikes and jumps due to government intervention and macroeconomic events. A recent example is the introduction of the carbon tax in Alberta whose goal is to limit the use of coal in favour of natural gas. In this talk, I review how electricity prices are modelled and introduce the notion of fuel-switching, which determines when an electricity company should produce electricity with coal rather than natural gas or vice-versa. Finally, I present preliminary results of my current research about Alberta.
November 21th, 2018
Speaker: Aiden Huffman
Title: General compound Hawes processes in limit order books
Abstract: In this talk, we introduce various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regards to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional Central Limit Theorems (FCLT) for several specific variations. We apply several of these FCLTs to limit order books to study the link between price volatility and order flow, where the volatility in mid-price changes is expressed in terms of parameters describing the arrival rates and mid-price process. Numerical examples and quantitative analysis will be presented for LOBster data on June 21st, 2012. (The talk is based on joint paper with A. Swishchuk)
October 23th, 2018
Speaker: Maciej Augustyniak
Title: Effectiveness of local and global quadratic hedging
Abstract: Local and global quadratic hedging are alternatives to delta hedging that more appropriately address the hedging problem in incomplete markets. The effectiveness of these strategies is investigated experimentally and empirically. The analysis centers on three important practical issues: (i) the value added of global over local quadratic hedging, (ii) the importance of the choice of measure (physical or risk-neutral) when implementing quadratic hedging, and (iii) the robustness of quadratic hedging to model mis-specification. We find that a global approach to quadratic hedging significantly reduces the risk of hedging European call and put options with long-term maturities (one year or more), provided that it is implemented under the physical probability measure. Moreover, a modified global quadratic hedging strategy is proposed that is more profitable on average to the hedger without substantially increasing his downside hedging risk, if at all. We prove mathematically that the expected terminal hedging gain of our modified strategy is greater than that of the global quadratic hedging strategy.
October 11th, 2018
Speaker: Jinniao Qiu
Title: Viscosity Solutions of Stochastic Hamilton-Jacobi-Bellman Equations
Abstract: We shall talk about the stochastic Hamilton-Jacobi-Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and the value function of the optimal stochastic control problem is proved to be the unique viscosity solution of the associated stochastic HJB equation. Applications in mathematical finance and other fields may be discussed as well.
October 11th, 2018 (This talk was cancelled and is postponed for a later time.)
Speaker: Zhen-qing Chen
Title: Anomalous diffusions and fractional order differential equations
Abstract: Anomalous diffusion phenomenon has been observed in many natural systems, from the signaling of biological cells, to the foraging behavior of animals, to the travel times of contaminants in groundwater. In this talk, I will first discuss the interplay between anomalous sub-diffusions and time-fractional differential equations, including how they arise naturally from limit theorems for random walks. I will then present some recent results in the study of these two topics. No prior knowledge in these two subjects is assumed.
September 25th, 2018
Speaker: Gabriela Zeller
Title: Hawkes Processes in Insurance: Modelling, Applications to Empirical Data and Optimal Investment
Abstract: We propose a risk model based on general compound Hawkes process (claims arrive w.r.t. Hawkes process and claim sizes are modelled as an N-state Markov chain). The Law of Large Numbers (LLN) and Functional Central Limit Theorem (FCLT) are presented for this model, and the latter is used to construct a diffusion approximation which leads to closed formulas for finite and infinite horizon ruin probabilities. We show that this model is appropriate by fitting it to an empirical insurance dataset, and verifying the result using simulation and well-known goodness of fit measures. Finally, we explain how to study the mean-variance efficient investment strategy for an insurer, whose random risk process follows a Brownian motion with drift in an incomplete market, for the case of the risk model with Hawkes process.
September 11th, 2018
Speaker: Ciprian Necula
Title: The Equivalent European Payoff of the American Put Option
Abstract: Is the American put option simply an incognito European one? The talk is focused on presenting a numerical procedure in the context of the Black-Scholes model, to approximate the payoff of a European type option that generates prices that are equal to the prices of the American put option in the continuation region. The resulting equivalent European payoff is a sum of power payoffs and therefore the price and the hedging indicators can be computed in closed form. For a given set of model parameters (interest rate, dividend rate and volatility) the computation of the equivalent European payoff reduces to solving a linear optimization problem. We conduct a numerical experiment spanning a wide range of model parameters and contract characteristics and, overall, the method produces American option prices with a relative RMSE less than 0.01% compared to a benchmark. We also discuss how the numerical procedure could be extended to other option pricing models.
SPRING/SUMMER 2018
May 16th, 2018
Presentation 1
Speaker: Erik Provencher
Title: Intrinsic Valuation of theCross-field Natural Gas Storage Facility
Abstract: Natural gas is a critical component of Alberta’s energy economy as it is used for industrial production, residential heating, and electricity production. In high demand in the winter, consumption greatly outweighs the province’s production capacity and results in increased prices. To ensure there are no shortages (and make a profit), storage facilities are needed to bank the excess production in the summer months, when demand and prices are low. In this presentation, we will discuss the different types of storage facilities and how these are regulated in Alberta. Next, we will discuss the intrinsic value of such facilities, which comes from the ability to take advantage of the futures market to make a profit. Finally, we will investigate the intrinsic value of the Crossfield facility north of Calgary, and relate it to the $ 210 million TransCanada paid for it in 2012.
Presentation 2
Speaker: Alexis Arrigoni
Title: Do Speculators Dominate Agricultural Futures Markets?
Abstract: As financial stock markets were hit by the 2008 crisis, prices in the agricultural sector increased drastically. What is now commonly referred to as the agricultural crisis of 2008 has led us to discuss extensively the role of speculation in the commodity market. Speculation in commodities is the holding of a financial position, for gain, and not as a normal incident to operating a producing, merchandising, or processing business (Working, 1960). The aim of this paper is to determine to what extent speculation is to blame. Eight commodities are under investigation: Corn, Cotton, Oats, Rice, Soybeans, Soybeans Oil, Sugar and Wheat. The empirical inquiry is split in two parts. The first step determines whether prices were distorted. Price behavior should be unpredictable and be independent of past prices, hence they should be similar to a random walk. The analysis tests the memoryless property of prices under two approaches: variance ratio and run analysis. The second step tries to define the potential factor distorting prices using a GARCH model and focuses on speculation. The empirical results tend to show an impact of both oil and speculation.
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WINTER 2018
May 9th, 2018
Speaker: Kris Vasudevan
Title: Non-linear Hawkes Processes III: Interaction of Neuronal Populations
Abstract: In Part I of the 3-part seminar series, I considered non-linear Hawkes processes in relation to interacting neuronal populations. In Part II, I introduced the Izhikevich two-dimensional neuronal model to simulate various types of observed neuron activity and showed what steps were taken to establish how well generalized linear regression models (GLMs) captured both the variability and the structure of different neuronal behaviour. In the concluding part of this 3-part seminar series, I would like to explore the stochastic optimal control of single neuron spike trains to suggest solutions to regulate epileptic seizures. To this end, first, I will introduce the elements of optimal control theory. Following this, I will consider how the Hamilton-Jacobi-Bellman (HJB) equation arises in optimal control problems. In connection with solving such problems, I will discuss several aspects such as closed-loop and open-loop control, maximum principle, value function, and dynamic programming. Finally, I will treat the stochastic HJB equation for a neuronal spiking problem. In this regard, I will consider an appropriate neuronal model, numerical methods to solve the stochastic HJB, and how neural activity could be regulated or controlled.
April 25th, 2018
Speaker: Matthias Fengler
Title: Textual Sentiment, Option Characteristics, and Stock Return Predictability
Abstract: A growing literature shows a predictability of stock returns based on sentiment proxies. However, also variables implied from single stock options markets carry predictive content for future equity returns, which is attributed to private information implicit in option markets. How do these empirical facts line up?
To look in to this question we distill sentiment from NASDAQ news articles and examine their predictive power on stock returns and option market characteristics. We find that options markets react to sentiment from NASDAQ articles and that options variables predict stock returns. Moreover, option characteristics are still highly significant predictors, both in the presence of sentiment variables and after singling out sentiment related information in option data. Results from a trading strategy confirm these findings, underling the informational content of option data for price discovery.
April 18th, 2018
Speaker: Yilan Luo
Title: Forecasting of Wind Energy Generation in Alberta
Abstract: In this work, our goal is to build a model for the future wind power generation of Alberta, as Alberta's wind power capacity is growing, and new wind farms are expected to be built in the near future. An important feature of the wind power data is spatial and temporal correlation. To capture this, we model the wind power generation in Alberta as a spatiotemporal process. We apply the method of Gaussian random fields to analyze the wind power time series of 20 wind farms of Alberta. Following the work of Tilmann Gneiting et al. (Tilmann Gneiting, Marc G.Genton and Peter Guttorp (2005) Geostatistical Space-Time Models, Stationary, Separability and Full Symmetry), we build several spatiotemporal covariance function estimates with increasing complexity: separable, non-separable symmetric, and non-separable, non symmetric. We compare the performance of the models using simple kriging. We also use kriging to demonstrate the performance of the models to forecast the future wind generation for both an existing wind farm and a new farm in Alberta.
April 1st, 2018
Speaker: Tony Ware
Title: Polynomial processes for energy commodity prices
Abstract: Energy commodities are notorious for the extreme dynamics exhibited by prices in spot and futures markets. Models for such prices typically include mean reversion, seasonality, as well as jumps and regime switching components. Polynomial processes have the property that conditional expectations of polynomial functions of future values are themselves given by polynomial functions of the current value. In this talk we will explore how polynomial processes can be combined with Itô diffusions to produce models that are able to effectively capture the rich dynamics seen in energy prices.
March 28th, 2018
Speaker: Yi Zhang
Title: A Continuous Time Semi-Markov Switching Process with Applications in Finance
Abstract: In this talk we first introduce a continuous-time finite state semi-Markov process and present a semi-martingale representation for the semi-Markov process. Then, we put our focus on constructing a continuous-time semi-Markov switching model where the holding times are Weibull distributed. Since our financial market is incomplete as there is an infinite number of equivalent local martingale measures, we need to find an equivalent martingale measure that can preserve the Weibull distributed semi-Markov regime switching structure. Then we can discuss the Markov property under the equivalent martingale measure. For the purpose of parameter estimation, we will introduce a technique called Markov Chain Monte Carlo method, which is based on the past values of underlying asset up to time T, not on any observation of the switching process, meaning that we treat the semi-Markov process as hidden. We will also talk about our future work on model calibration to select parameters that is as consistent as possible with market observations.
March 14th, 2018
Speaker: Mohamed Badaoui
Title: An Optimal Investment Problem in Incomplete Markets
Abstract: In this talk we consider the problem of an insurance company where the wealth of the insurer is described by a Cramér-Lundberg process. The insurer is allowed to invest in a risky asset with stochastic volatility subject to the influence of an economic factor and the remaining surplus in a bank account. The price of the risky asset and the economic factor are modeled by a system of correlated stochastic differential equations. In a finite horizon framework and assuming that the market is incomplete we study the problem of maximizing the expected utility of terminal wealth. In order to show that any solution to the Hamilton-Jacobi-Bellman equation solves the optimization problem, we prove a verification theorem as well as an existence and uniqueness theorem. The optimal strategy and the value function have been produced in closed form. In addition and in order to show the connection between the insurer’s decision and the correlation coefficient we present some numerical results.
March 1st, 2018
Speaker: Jianfeng Zhang
Title: Some thoughts about time inconsistency problems
Abstract: Time inconsistency means that an optimal strategy found today does not remain optimal anymore tomorrow. This roughly means that the "value function" violates the dynamic programming principle. This issue is extensively studied in the economics literature. There are typically two approaches. One is the backward approach, in the sense that you make today's decision by expecting tomorrow's behavior would be the optimal one for tomorrow's problem. The other is to study only today's problem, ignoring the fact that you may "regret" tomorrow. These two approaches lead to different strategies and different values, and the latter approach has no dynamic theory available. We propose to study the dynamic problem in the line of the latter approach. In this talk we shall argue that in many applications the backward approach, while beautiful in mathematics, is not appropriate. Our main idea is to use the so called "forward utility", and thus our approach can be called a forward approach.
February 1st, 2018
Speaker: Anatoliy Swishchuk
Title: Compound Hawkes Processes in Limit Order Books
Abstract: In this talk we introduce two new Hawkes processes, namely, compound and regime-switching compound Hawkes processes, to model the price processes in limit order books. We prove Law of Large Numbers and Functional Central Limit Theorems (FCLT) for both processes. The two FCLTs are applied to limit order books: we use these asymptotic methods to study the link between price volatility and order flow in our two models by using the diffusion limits of these price processes. The volatilities of price changes are expressed in terms of parameters describing the arrival rates and price changes. We also present some numerical examples. (This talk is based on our joint paper ‘Compound Hawkes Processes in Limit Order Books’ by J. Chavez-Casillas, R. Elliott, B. Remillard and A. Swishchuk).
January 31st, 2018
Speaker: Kris Vasudevan
Title: Non-linear Hawkes Processes II: Point Processes and Generalized Linear Regression Models in Neuronal Dynamics
Abstract: In Part I of the 3-part seminar series, I considered non-linear Hawkes processes in relation to interacting neuronal populations. In Part II, I will revisit point processes and describe how generalized linear regression models (GLMs) are used to examine complex neuronal data structures such as neuronal spike trains. I will discuss how goodness-of-fit tests from theory dealing with point processes are applied to GLMs. I will introduce the Izhikevich two-dimensional neuronal model to simulate various types of observed neuron activity and show what steps are taken to establish how well GLMs capture both the variability and the structure of different neuronal behaviour. Finally, I will consider some of the challenges observed in the use of GLMs to non-linear Hawkes processes.
January 17th, 2018
Speaker: Wenning Wei
Title: Maximum principle for optimal control of neutral stochastic functional differential systems
Abstract: This talk is concerned with optimal control of neutral stochastic functional differential equations (NSFDEs). The Pontryagin maximum principle is proved for optimal control, where the adjoint equation is a linear neutral backward stochastic functional equation of Volterra type (VNBSFE). The existence and uniqueness of the solution are proved for the general nonlinear VNBSFEs. Under the convexity assumption of the Hamiltonian function, a sufficient condition for the optimality is addressed as well.
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FALL 2017
November 29th, 2017
Speaker: Mohamed Badaoui
Title: An optimal Investment Strategy and Its Ruin Probability
Abstract: In this talk we consider the Cramér-Lundberg model with the possibility of investment in both a bank account and a risky asset described by stochastic volatility models. By using stochastic control techniques and assuming that the insurer preferences are exponential, we obtain an optimal investment strategy maximizing the expected utility of terminal wealth as well as an upper bound for the ruin probability.
November 15th, 2017
Speaker: Jinniao Qiu
Title: A functional limit theorem for limit order books
Abstract: We shall consider a stochastic model for the dynamics of the two-sided limit order book (LOB). The model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges in distribution to a fully coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics. The talk is based on a joint work with Ulrich Horst and Christian Bayer.
November 1st, 2017
Speaker: Kris Vasudevan
Title: Non-linear Hawkes Processes: Earthquake occurrences to collective dynamics in neuronal ensembles
Abstract: Stochastic modelling of non-linear processes such as earthquake sequencing and collective dynamics of spiking neurons has been shown to be useful to study their spatio-temporal complexity. One such model I consider here is non-linear Hawkes process. By definition, non-linear Hawkes processes are point processes with self- and mutually-exciting patterns with long memory and clustering characteristics. The sequencing of the patterns in them is complicated both in earthquake occurrences and neuronal spiking dynamics. In the case of earthquake occurrences, a point process refers to successive times at which an earthquake occurs. For spiking neurons, the successive times correspond to times when each neuron discharges or fires an action potential or a spike. In both cases, Hawkes processes are known to provide good models to understand the structure of the intensity processes associated with the point processes. In this presentation, I would like to focus on multivariate and interacting Hawkes processes. First, I would like to express point processes in terms of stochastic conditional intensity functions. Second, I would like to estimate multivariate point processes within the context of a statistical framework. Third, I would like to explore how an extension of the methodologies developed in the first two steps to interacting Hawkes processes is possible.
October 18th, 2017
Speaker: Anatoliy Swishchuk
Title: The Hawkes Processes and their Applications in Finance and Insurance
Abstract: The Hawkes process is a self-exciting simple point process first introduced by A. Hawkes in 1971. The future evolution of a self-exciting point process is influenced by the timing of past events. The process is non-Markovian except for some very special cases. Thus, the Hawkes process depends on the entire past history and has a long memory. The Hawkes process has wide applications in neuroscience, seismology, genome analysis, finance, insurance, and many other fields. The present talk is devoted to the introduction to the Hawkes process and their applications in finance and insurance.
October 4th, 2017
Speaker: Mohamed Badaoui
Title: Bounds of Ruin Probabilities for Insurance Companies
Abstract: In this talk we consider the classical Cramér-Lundberg model with the possibility of investment in both a bank account and a risky asset. The risky asset is modelled by geometric Brownian motion with stochastic volatility that depends on an external factor described by a diffusion process. By using martingale theory and Itô's formula for jump-diffusion processes we obtain an upper and lower bounds for the ruin probabilities. Finally, we show that our approach can recover the known bounds for constant volatility models.
September 20th, 2017
Speaker: Jinniao Qiu
Title: An Introduction to Stochastic HJB Equations
Abstract: This talk is concerned with the Stochastic Hamilton-Jacobi-Bellman (HJB) equation that is a type of backward stochastic partial differential equations (SPDEs). We will start with a simple introduction of both forward and backward SPDEs, and then derive the stochastic HJB equations from theory of optimal stochastic controls. Examples in finance and economics would be discussed as well.
WINTER 2017
April 5th, 2017
Speaker: Jonathan Chavez-Casillas
Book Review: ‘Algorithmic and High-frequency Trading’ by A. Cartea, S. Jaimungal and J. Penalva, Cambridge University Press (2015);
(Chapter 12: Order Imbalance)
Abstract: In many of the previous chapters, the agent made trading decisions based on three key ingredients:
1) the mid price,
2) the arrival of incoming market orders
3) the agent’s own inventory.
In some cases, these state variables were supplemented by observables such as order flow, short-term alpha and co-integration of prices. In this Chapter 11, they investigate the role that another important state variable plays: the quoted volume order imbalance (or simply order imbalance). This is a measure of the buy versus sell pressure on an asset, and it contains predictive power on both the arrival rates of market orders, and the direction and size of future price movements.
March 29th, 2017
Speaker: Jonathan Chavez-Casillas
Book Review: ‘Algorithmic and High-frequency Trading’ by A. Cartea, S. Jaimungal and J. Penalva, Cambridge University Press (2015);
(Chapter 11: Pair Trading and Statistical Arbitrage Strategies)
Abstract: The success of many trading algorithms depends on the quality of the predictions of stock price movements. Predictions of the price of a single are generally less accurate than predictions of a portfolio stocks. A classical strategy which makes the most of the predictability of the joint, rather than the individual, behaviour of two assets is pairs trading where a portfolio consisting of a linear combination of two assets is traded. At the heart of the strategy is how the two assets co-move. The class of strategies for such pairs trading is called statistical arbitrage (or StatArb for short). They are not true arbitrage but rather are strategies which bet off of the typical behaviour of asset prices, and hence are not risk-free. Thus, Chapter 11 is devoted to statistical arbitrage and pairs trading.
March 8th, 2017
Speakers: Jonathan Chavez-Casillas & Anatoliy Swishchuk
Book Review: ‘Algorithmic and High-frequency Trading’ by A. Cartea, S. Jaimungal and J. Penalva, Cambridge University Press (2015);
(Chapter 10: Market Making - sec.10.1-10.2-Jonathan, sec. 10.3-10.4-Anatoliy)
Abstract: Chapter 10 show how market makers choose where to post limit orders in the book. The models that are developed look at how the strategies depend on different factors including the market maker’s aversion to inventory risk, adverse selection, and short-term lived trends in the dynamics of the mid price.
March 1st, 2017
Speaker: Speaker: Jonathan Chavez-Casillas
Book Review: ‘Algorithmic and High-Frequency Trading’ by A. Cartea, S. Jaimungal and J. Penalva (2015), Cambridge University Press (Chapter 9: Targeting Volume)
Abstract: Chapter 9 deals with execution algorithms that target volume-based schedules. The strategies for investors, who wish to track the overall volume trade in the market by targeting percentage of volume, percentage of cumulative volume, and volume weighted average price (VWAP), were developed.
February 15th, 2017
Speaker: Jonathan Chavez-Casillas
Book Review: ‘Algorithmic and High-Frequency Trading’ by A. Cartea, S. Jaimungal and J. Penalva (2015), Cambridge University Press (Chapter 8: Optimal Execition with Limit and Market Orders)
Abstract: In the previous two Chapters 6 and 7 on optimal control for limit orders the authors focused on execution strategies which relied on Market Orders only. This Chapter 8 looks at optimal execution problems when the agent employs Limit Orders and possibly also Market Orders. The investor’s objective is to execute a large position over a trading window, but she employs only Limit Orders, or uses both Limit and Market Orders.
February 8th, 2017
Speaker: Zijia Wang
Paper Review: ‘Optimal Execution of Portfolio Transactions’ by R. Almgren and N. Chriss (2000)
Abstract: The paper considers the execution of portfolio transactions with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact. In the second part of the two presentations we will be reviewing Section 3 (interpretation of the efficient frontier in terms of utility function and value-at-risk) and Section 4 (consideration the value of possible additional information on future stock price motion) of the paper.
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FALL 2016
November 30th, 2016
Speaker: Zijia Wang
Title: 'Modelling of Variance and Volatility Swaps with Stochastic Volatility and Jumps'
Abstract: In this presentation, we will introduce the financial derivatives-variance and volatility swaps. A general analytic approach for pricing variance and volatility swaps under Merton's jump model and Heston’s stochastic volatility model will be presented. Two different methods: convexity correction and Laplace transform for evaluating fair volatility strike from fair variance strike will be discussed. The closed-form pricing formulas for variance swap under exponential Lévy model and Lévy-based Heston model will also be discussed, and we will investigate the effect of asset price jumps on fair swaps strikes.
November 23rd, 2016
Speaker: Robert Elliott
Title: Heston Stochastic Volatility with a Markov Switching Regime
Abstract: In this talk we introduce a Markov switching regime to the Heston-type stochastic volatility model to price options. The characteristic function of the log price contains solutions of a first order linear matrix ODE with time-dependent coefficients. Option prices calculated with analytic formulae tend to be lower (resp. higher) than those by Monte Carlo simulations in the low (resp. high) regime. The generalization does not lose numerical tractability while reflecting the stylized facts.
November 16th, 2016
Speaker: Yingying Lai
Title: 'Linear-Quadratic Mean Field Stackelberg Games with State and Control Delays'
Abstract: In this article, we consider a linear-quadratic mean field game between a leader (dominating player) and among a group of followers (agents) under the Stackelberg game setting, so that the evolution of each individual follower is now also subjected to delay effects from both their state and control variables, as well as those of the leader. The overall Stackelberg game is solved by tackling three sub-problems hierarchically. By first regarding the mean field term and the delay influence of the leader as exogenous, we use the adjoint equation approach to solve for the optimal control of each follower. Next, we utilize the fixed point property to get the desired mean field equilibrium. Finally, we solve for the optimal control of the leader and conclude that its presence would not interfere the original existence of the equilibrium of the community.
November 9th, 2016
Speaker: Yi (Ivy) Zhang
Title: Semi-martingale Representations for Semi-Markov Chains with Applications
Abstract: This talk is devoted to the semi-martingale representations for discrete- and continuous-time semi-Markov chains, and their applications to semi-Markov regime-switching models in finance. (Joint talk with Robert Elliott and Anatoliy Swishchuk).
November 2nd, 2016
Speaker: Anatoliy Swishchuk
Book Review: 'Algorithmic and High-frequency Trading', by A. Cartea, S. Jaimungal and J. Penalva, Cambridge University Press, 2015. (Chapter 6: Optimal Execution with Continuous Trading I)
Abstract: This Chapter deals with the modelling of algorithmic trading strategies, and is concerned with optimal execution strategies where the agent must liquidate or acquire a large portion over a pre-specified window and trades continuously using only market orders. The Chapter covers the classical execution problem when the investor's trades impact the price of the asset and also adjusts the level of urgency with which he desires to execute the programme.
October 26th, 2016
Speaker: Joshua Novak
Book Review: 'Algorithmic and High-frequency Trading', by A. Cartea, S. Jaimungal and J. Penalva, Cambridge University Press, 2015. (Chapter 5)
Abstract: This chapter discusses the tools from optimal control that are used in finance to solve many problems, such as optimal investment-consumption, maximizing an agent’s terminal utility by trading a risky asset (stock) and a riskless asset (cash), payouts, entry/exit, and indifference pricing. The main tool for solving these problems is the Dynamic Programming Principle and the related nonlinear PDE known as the Hamilton-Jacobi-Bellman Equation.
October 12th, 2016
Speaker: Jonathan Chavez-Casillas
Book Review: 'Algorithmic and High-frequency Trading', by A. Cartea, S. Jaimungal and J. Penalva, Cambridge University Press, 2015. -Chapter 3.
Abstract: Chapter 3 is devoted to the description of the data in the Limit Order Books, and contains empirical analysis of different aspects of trading: prices, returns, spreads, volume, etc., using primarily millisecond stamped data.
October 5th, 2016
Speaker: Bruno Remillard
Title: 'Price Dynamics in a General Markovian Limit Order Book'
Abstract: We propose a simple stochastic model for the dynamics of a limit order book, extending the recent work of Cont and de Larrard (2013), where the price dynamics are endogenous, resulting from market transactions. We also show that the diffusion limit of the price process is the so-called Brownian meander.
September 21st, 2016
Speakers: Katharina Cera & Julia Schmidt
Title: 'General Semi-Markov Model for Limit Order Books: Theory, Implementation and Numerics '
Abstract: Our talk gives an introduction to Limit Order Books and summarizes the model proposed by Cont and de Larrard in their paper “Price Dynamics in a Markovian Limit Order Market” (SIAM J. Finan. Math (2013)). We present the generalizations done by Swishchuk and Vadori in “A Semi-Markovian Modeling of Limit Order Markets” (e.g., arXiv (2016)) and the evidence found in our data that justifies their approach. The talk closes with highlighting our own work on extending this model to the general semi-Markov model for stock price process in the Limit Order Book including gained numerical results for diffusion limits.
WINTER 2014
March 28th, 2014
Speaker: Dr. ir. Hans J.H. Tuenter
Title: The Modeling of Wind Energy
Abstract: We discuss the mathematical and statistical models that are needed to model wind energy, and discuss our experiences with the wind-energy forecasting system that was developed and built in-house. As the share of renewables in power generation is growing, we show how that it is no longer demand, but net load that is driving the dispatch of the other generation sources, and hence makes accurate forecasting of wind (and solar) energy a necessity.
March 26th, 2014
Speaker: Kaijie Cui
Book Review: 'Weather Derivatives. Modeling & Pricing Weather-related Risk' by A. Alexandridis & A. Zapranis, Springer, 2013. PART II.
Abstract: This talk is the second part of the book review "Weather Derivatives. Modeling & Pricing Weather-related Risk". It includes pricing temperature derivatives, using meteorological forecasts for pricing, effects of geographical and basis risk, pricing of wind and precipitations.
March 12th, 2014
Speaker: Tony Ware
Title: 'A stochastic dynamic programming approach to quantifying reservoir reliability'
Abstract: The recent floods in southern Alberta have highlighted the importance of being able to assess and minimize the risks inherent in our water management systems. Indeed, effective water management has been a vital concern throughout human civilization. In this talk we describe a stochastic dynamic programming approach to the quantification and maximization of reservoir reliability - the ability to avoid exceeding the upper safety level, and to maintain at least minimum supply to downstream water users. The key insight we will exploit is that the risks of failure are integrated with the release strategies designed to minimize these risks. The dynamic programming approach allows us to determine both together.
February 26th, 2014
Speaker: Nelson Vadori
Title: 'Limit Theorems for Inhomogeneous Semi-Markov Processes, and Applications to Finance and Insurance.'
Abstract: In this talk, we will show how we can obtain some limit theorems for inhomogeneous Semi-Markov processes. The results that will be presented are new, to the best of the speaker's knowledge: in the homogeneous case, there exists well-known such results but not in the inhomogeneous case. We will also discuss some applications to finance and insurance.
February 5th, 2014
Speaker: Kaijie Cui
Book Review: 'Weather Derivatives. Modeling & Pricing Weather-related Risk' by A. Alexandridis & A. Zapranis, Springer, 2013. PART I.
Abstract: This talk is the first part of the book review "Weather Derivatives. Modeling & Pricing Weather-related Risk". It includes data handling, pricing approaches of temperature derivatives and modeling the daily average temperature.
January 15th, 2014
Speaker: Anatoliy Swishchuk
Book Review: 'Weather Derivatives. Modeling & Pricing Weather-related Risk' by A. Alexandridis & A. Zapranis, Springer, 2013.
Abstract: I'll overview the above mentioned book which is devoted to study analytically and in depth the financial products that are traded in the weather markets. (FYI: Weather derivatives are financial instruments that can be used by organizations or individuals as part of a risk management strategy to reduce risk associated with adverse or unexpected weather conditions. Reported from Weather Risk Management Association (WRMA), an industry body that represents the weather market, the total notional value of the global weather risk market has reached $11.8 billion in last year).
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FALL 2013
December 4th, 2013
Speaker: Tony Ware
Title: 'Commodities, Energy and Environmental Finance: PART II'
Abstract: In August this year the Fields Institute hosted a focus programme on commodities, energy and environmental finance (http://www.fields.utoronto.ca/programs/scientific/13-14/envirofinance/), and I attended two workshops as part of this programme. In this talk I will give a birds-eye view of the range of topics that featured in the workshops, and I will also give brief introductions to some of the many interesting ideas I encountered there. These include ambit field models for electricity futures, assessing model risk for energy markets, modelling wind, modelling emissions markets, using game theory to analyze production and R&D decisions in energy and commodity markets, valuing hydro power plants with environmental ramping restrictions, and more. (This is a continuation of my previous talk on October 30, 2013).
November 13th, 2013
Speaker: Zachary Moyer
Title: 'GMRP-modulated Stocks and Pricing of Perpetual Options'
Abstract: The pricing of options, derivatives of a stock listed in a stock market, is of fundamental importance in advanced market strategies. A classical model of a stock price is presented, based on the binomial model, as are formulae for calculating option prices. We introduce the GMRP (Geometric Markov Renewal Process) as a model for a stock price. This is considered a good model for energy market stocks, as it allows for consideration of large time-scale market behaviors often exhibited by these stocks. We derive results on the GMRP model, and show a way to calculate the price of European calls and puts as well as a new result for the pricing of perpetual options under GMRP market assumptions.
October 30th, 2013
Speaker: Tony Ware
Title: 'Commodities, Energy and Environmental Finance'
Abstract: In August this year the Fields Institute hosted a focus programme on commodities, energy and environmental finance, and I attended two workshops as part of this programme. In this talk I will give a birds-eye view of the range of topics that featured in the workshops, and I will also give brief introductions to some of the many interesting ideas I encountered there. These include ambit field models for electricity futures, assessing model risk for energy markets, modelling wind, modelling emissions markets, using game theory to analyze production and R&D decisions in energy and commodity markets, valuing hydro power plants with environmental ramping restrictions, and more.
October 16th, 2013
Speaker: Maksym Tertychnyi
Paper review: 'Markov-modulated jump-diffusions for currency option pricing' by Lijun Bo, Yongjin Wang, Xuewei Yang, 2010.
Abstract: This paper introduces dynamic models for the spot foreign exchange rate with capturing both rare events and the time-inhomogenity in the fluctuating currency market. For the rare events, authors use the compound Poisson process with log-normal jump amplitude to describe the jumps. As for the time-inhomogenity in the market dynamics, they particularly stress the strong dependence of domestic/foreign interest rates, the appreciation rate and the volatility of the foreign currency on the time-varying sovereign ratings in the currency market. The time-varying ratings are formulated by continuous-time finite-state Markov chain. Based on such a spot foreign exchange rate dynamics, they study the pricing of some currency options. Here, they adopt a regime-switching Esscher transform to identify a risk-neutral martingale measure. By determining the regime-switching Esscher parameters authors then get an integral expression on the prices of European style currency options. Numerical illustrations are also provided.
October 9th, 2013
Speaker: Ilnaz Asadzadeh
Paper review: 'Properly designed emissions trading schemes do work!' by R. Carmona, M. Fehr & J. Hinz, 2009.
Abstract: Emissions trading markets have been touted as the most efficient mechanism to achieve environmental goals at least cost. Whether in the form of voluntary markets or in a mandatory framework like in the first phase of the European Union (EU) Emission Trading Scheme (ETS), the regulator sets a cap on the emissions which can occur without penalty, and provides emissions allowances accordingly. The recipients are free to use these emission certificates to cover their emissions, or to sell them to the firms which are expected to emit more than what they can cover with their original allocations. As observed in most existing programs, cap-and-trade systems can fail to reach their emission targets as too generous an allocation of pollution permits serves as a disincentive for emissions reductions and deflates pollution prices. Moreover, the implementation of the first phase of the EU-ETS has been widely criticized on one more sensitive account: providing significant (some went as far as calling them obscene) windfall profits for power producers. Here we weight on this debate with the results of a rigorous quantitative modeling undertaking, providing insight into what went wrong in the first phase of the EU-ETS, and proposing alternative reduction schemes with provable advantages. Using market equilibrium models and numerical tools, we demonstrate that properly designed market- based pollution reduction mechanisms can reach pre-assigned emissions targets at low reduction cost and windfall profits, while being flexible enough to promote clean technologies. In the present article, we illustrate our claims with the results of a hypothetical cap-and- trade scheme for the Japanese electricity market.
September 25th, 2013
Speaker: Kaijie Cui
Book Review: 'Modelling & Pricing in Financial Markets for Weather Derivatives' by F.E. Benth & J.S. Benth, World Scientific, 2013: Introduction and Data.
Abstract: This talk will include introduction to weather markets and weather derivatives, data description from the book, and also literature review for stochastic weather modelling.
September 18th, 2013
Speaker: Anatoliy Swishchuk
Title: 'Modelling & Pricing in Financial Markets for Weather Derivatives' by F.E. Benth & J.S. Benth, World Scientific, 2013.
Abstract: I'll overview the above mentioned book which is devoted to an integrated approach to weather derivatives. (FYI: Weather derivatives are financial instruments that can be used by organizations or individuals as part of a risk management strategy to reduce risk associated with adverse or unexpected weather conditions. Reported from Weather Risk Management Association (WRMA), an industry body that represents the weather market, the total notional value of the global weather risk market has reached $11.8 billion in last year).
SPRING/SUMMER 2013
July 23rd, 2013
Speaker: Alexander Melnikov (University of Alberta)
Title: "Partial Hedging via Conditional VaR and Related Questions"
Abstract: Hedging of options is one of the basic and comprehensive problems of mathematical finance which has very interesting insurance applications. The most visible developments in this area during the last decades were done by using the notion of partial or imperfect hedging. We formulate this problem as a possibility to create a terminal capital which is close enough to given contingent claim in some probabilistic sense. Such understanding of the problem explains clearly why a reasonable statistical technique properly works here creating new types of hedging like quantile, efficient etc. Due to new developments in the theory of risk measures, hedging problem became a new insight. We investigate the partial hedging problem using the most applicable risk measure CVaR-Conditional Value at Risk. We develop the CVaR optimization technique along with the fundamental Neuman-Pearson lemma to solve hedging problem by minimizing CVaR under initial budget constraints. Explicit solutions will be derived in the framework of the Black-Scholes and regime-switching market models. Applications for pricing of equity-linked life insurance contracts and for valuation of regulatory capital requirements will be given. Besides that an efficient technique for CVaR estimation will be provided based on the so-called path-wise comparison theorem for strong solutions of stochastic differential equations. Theoretical findings will be numerically illustrated.
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WINTER 2013
April 22nd, 2013
Speaker: Sam Cohen (Oxford University, UK)
Title: "Uniformly Uniformly-Ergodic Markov Chains and applications"
Abstract: If one starts with a uniformly ergodic Markov chain on countable states, what sort of perturbation can one make to the transition rates and still retain uniform ergodicity? In this talk, we will consider a class of perturbations that can be simply described, where a uniform estimate on convergence to an ergodic distribution can be obtained. We shall see how this is related to Ergodic BSDEs and BSDEs up to stopping times in this setting and outline some novel applications of this approach.
March 20th, 2013
Speaker: Kaijie Cui
Title: "Introduction to Regime Switching Models and Application to Weather Derivatives"
Abstract: Markov state switching models are a type of specification which allows for the transition of states as an intrinsic property of the time series model. Such property of statistical representations is well known and utilized in different problems in the field of economics and finance. In this talk, we will give a brief introduction to the regime switching models and their possible applications in the field of weather derivatives modeling and pricing.
March 6th, 2013
Speaker: Lazman, Mark (Numerix LLC)
Title: 'Commodity models: formulation, analysis and implementation'
Abstract: We present the formulation, analysis and implementation of commodity models within the hybrid modelling framework. The models for commodity price represent the generalization of the benchmark stochastic models for application within the hybrid model framework. The models are fitted to the forward prices and allow calibration to the current market observables, such as options on spot, futures, spread options and basket options. We analyze the calibration of our models to the market observables and discuss the robust calibration initialization. We discuss the seasonality adjustment and seasonality impact on the pricing and calibration of commodity models. We analyze the estimation of commodity models by historical data and present the estimation algorithm based on Kalman filter maximum likelihood methodology. We discuss the applications to the energy / commodity domain.
February 27th, 2013
Speaker: Anatoliy Swishchuk
Title: 'Quantitative, Energy and Environmental Finance: Overview'
Abstract: In this talk, I shall focus on the history of quantitative finance and give an introduction to the very new areas of finance:
-environmental finance
-carbon trading finance
-weather derivatives
-energy finance.
February 6th, 2013
Speaker: Yuriy Shkolnikov
Title: 'Analytic price approximation for American options under time-dependent settings, proportional and discrete dividends. Calibration from American options'
Abstract: We present a super-fast and parallelizable analytic approximation for a price of an American option on a log-normal underlying with discretely time-dependent parameters (volatility, IR) with proportional or absolute discrete dividends. The following state and time Greeks are also analytically approximated and calculated at the time of price computation. Results and computational times are compared to the ones achieved on a trinomial tree with time-dependent input parameters. We also present the Decoupled Volatility Model which consistently extends time-dependent log-normal pricing settings to non-log-normal underlying assets and results in an efficiently solvable inverse problem of computing the estimated time-dependent instantaneous underlying volatility and set of time-dependent implied volatilities of off-market European contracts from exchange-traded American calls and puts at multiple strikes and multiple maturities. Due to being based on the introduced pricing method, the inverse problem is also fast and parallelizable. The applicable underlying asset classes include proportional dividends assets (FX, ETF, Commodities) and discrete dividend assets (Equities). The intended application areas for both direct and inverse problems are listed and OTC option pricing, UHF trading (including market making and/or statistical arbitrage), option portfolio management, risk valuation (various approaches).
January 23rd, 2013
Speaker: Babacar Seck
Title: 'Computational Dynamic Market Risk in Discrete Time Models'
Abstract: Different approaches to defining dynamic market risk are available in the literature. Most are focused or derived from probability theory, economic behaviour or dynamic programming. Here, we propose an approach to define and implement dynamic market risk measures based on recursion and state economy representation. The proposed approach is to be implementable and to inherit properties from static market risk measures.
Quick Links
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FALL 2012
December 11th, 2012
Speaker: Chao Qiu
Title: 'Option Pricing and Hedging under Discrete Time Regime Switching Models'
Abstract: This talk is to explore the option pricing and hedging in a discrete time regime-switching environment. If the regime risk cannot be hedged away, then the Black-Scholes pricing and hedging framework no longer generates a unique pricing and hedging measure. We also compared several measures for pricing and hedging, with a focus on the risk neutral measure generated by applying Esscher Transforms to the real world regime-switching asset process. I will also briefly present other works in my research.
November 30th, 2012
BP University: Opportunity to learn, network and experience a day as an Energy Trader, Energy Market Analyst and Marketer
November 27th, 2012
Speaker: Anthony Ware
Title: 'Splitting Methods in Computational Finance'
October 30th, 2012
-BP Canada Trading Competition Annoucement
-Bloomberg Assesment Test Annoucement
Speaker: Daniel Leonhardt
Title: 'Modeling Multidimensional Futures Prices in a Cointegrated Geometric Model'
September 25th, 2012
Speakers: Anatoliy Swishchuk & Giovanni Salvi
Title: 'Covariance and Correlation Swaps for Markov-modulated Volatility'
September 11th, 2012
Speakers: Martin Hiller & Anatoliy Swishchuk
Title: 'Option pricing in a Black-76 framework with semi-Markov modulated volatility'
SPRING/SUMMER 2012
August 3rd, 2012
Speaker: Ross Pinsky
Title: 'Asymptotics for Exit Problem and Principal Eigenvalue for a Class of Non-local Elliptic Operators Related to Diffusion Processes with Jumps'
May 17th, 2012
Speaker: Nikolaos Limnios
Title: 'Discrete-time Semi-Markov Random Evolutions and their Applications'
Abstract: This talk introduces discrete-time semi-Markov random evolutions (DTSMRE) and studies asymptotic properties, namely, averaging, diffusion approximation and diffusion approximation with equilibrium by martingale weak convergence method. The applications are given to the additive functionals (AF), geometric Markov renewal processes (GMRP) and dynamical systems in discrete-time. The rates of convergence in the limit theorems for DTSMRE and AF and GMRP are also presented.
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WINTER 2012
May 8th, 2012
Grad Students in Math Finance Research Presentations to the Calgary Business Community
April 12th, 2012
Grad Students in Math Finance Research Presentations to the PRMIA Calgary Chapter steering committee
April 5th, 2012
Speaker: Akbar Shahmoradi
Title: 'US North East and Ontario Power'
March 29th, 2012
Speaker: Elham Negahdary
Title: 'Taxonomy of Power Models'
Abstract: This presentation focus on identifying, classifying and characterizing the diversity of trends in electricity market modelling. The pure price processes entail substantial difficulties when are used to model the evolution of power prices and fail to address the unique features of electricity market such as instantaneous balance of supply and demand. In this survey of the most recent publications regarding power modelling, the objective, advantage and disadvantage of each approach is evaluated in context of its application. Finally, the most suitable approaches are identified to answer the question of what causes the price to move. That 'best practice' market-data aware model allows us to capture the evolution of the most salient, primary variables that describe the movement of price.
March 22nd, 2012
Speaker: Anatoliy Swishchuk
Title: 'Stochastic Processes with Independent Increments: Ideas, Results, History'
Abstract: We give an overview on stochastic processes with independent increments that are now becoming a very popular models in energy and related markets, such as electricity, natural gas, temperature markets, etc. These processes are more general than Levy processes in a way that the increments are independent, but not necessary stationary. We also present a short history of these processes.
March 15th, 2012
Speaker: Giovanni Salvi
Title: 'Backward Time Multivariate semi-Markov Process for Couterparty Credit Risk'
Abstract: We start from the work of Ching et al. on the multivariate Markov chain and we generalize it by allowing any kind of sojourn time distribution, or in other term we introduce a multivariate semi- Markov process. We derive an explicit expression for the transition probability of this multivariate semi-Markov process in the discrete time case. We apply this multivariate model to the study of the counterparty credit risk, with regard to correlation in a CDS contract. The financial crisis has stressed the importance of the study of the correlation in the financial market. In this regard, the study of the risk of default of the counterparty in any financial contract has be- come crucial in the credit risk. Many works has been done to trying to describe the counterparty risk in a CDS contract, but all this work are based on the Markovian approach to risk. In the our opinion this kind of model are too restrictive, because they require that the distribution function of the waiting times has to be exponential or geometric, for discrete time. In the our model, we describe the evolution of credit rating of the financial subjects like a multivariate semi-Markov model, so we allow for arbitrarily distributed sojourn time. The age state dependency, typical of the semi-Markov environment, allow us to insert the correlation in a dynamical way. In particular, suppose that A is a default-free bondholder and C is the relative firm. The bondholder buy protection against C's default by another defaultable subject, say B the protection seller. Our model describe the evolution of the credit rating of the couple B and C. We admit for simultaneous default of C and B, the single default of C or single default of B.
March 8th, 2012
Speaker: Kaijie Cui
Title: 'Weather Derivatives with Applications to Canadian Data'
Abstract: Modelling of Daily average temperature variations of Canadian Data by a mean-reverting Ornstein-Uhlenbeck process driven by general Levy Process is proposed. The process also contains seasonal mean and volatility. It is empirically proved that the proposed dynamics fit Calgary and Toronto temperature data successfully. The model is also applied to derive an explicit price of CAT futures, and numerical prices of CDD and HDD futures using fast Fourier transform are also included.
March 1st, 2012
Speaker: LiFeng Zhang
Title: 'Geometric Markov Renewal Processes and Their Applications in Finance'
Abstract: First we give some basic concepts and properties on Markov and Semi-Markov Processes and Chains, along with Wiener process and Levy process, all of which are prepared for the next Generalized Geometric Markov Renewal Processes. Next, we introduce Cox-Ross-Rubinstein binomial model and Aase model, and from these cases, generalized GMRP models. Then, we consider its approximation in the geometric Markov renewal processes as model for a security market and also study the processes in a diffusion approximation and normal deviation schemes. As an application, we consider the case of two ergodic classes. We present European call option pricing formulas in the case of ergodic, double-averaged, and merged diffusion geometric Markov renewal processes. Finally, we introduce Poisson averaging scheme for the geometric Markov renewal processes obtain compound Poisson process with deterministic drift and derive its option price under risk-neutral measure. European call option pricing formulas for GMRP are presented.
February 16th, 2012
Speaker: Babacar Seck
Title: 'Taking Market Risk into Account in Portfolio Optimization'
Abstract: We provide an economic interpretation of the practice consisting in incorporating risk measures as constraints in portfolio optimization problem. For what we call the infimum of expectations class of risk measures, we show that if the decision maker maximizes the expectation of a random return under constraint that the risk measure is bounded above, he then behaves as a 'generalized expected utility maximizer'. As an application, we make the link between a portfolio maximization problem, subject to Conditional Value-at-Risk being less than a threshold value, and a non-expected utility formulation involving 'loss aversion'-type utility functions.
February 9th, 2012
Speaker: Nelson Vadori
Title: 'Smiling for the Delayed Volatility Swaps'
Abstract: Using change of time method, we derive a closed-form formula for the volatility swap in an adjusted version of the Heston model with stochastic volatility with delay. The numerical result is presented for underlying EURUSD on September 30th 2011. The novelty of the result is two-fold: application of change of time method to the delayed Heston model and calculation of the volatility swap for this model.
February 3rd, 2012
Speaker: Tom Hurd
Title: 'Modelling Financial Networks and Systematic Risk'
Abstract: The study of contagion in financial systems is very topical in light of recent events in the global markets. "Contagion" refers to the spread of defaults through a system of financial institutions, with each successive default causing increasing pressure on the remaining components of the system. The term "systemic risk" refers to the contagion-induced threat to the financial system as a whole, due to the default of one (or more) of its constituent institutions. The ultimate question for me is how mathematical models can help us understand systemic risk. In this talk I will explore some of the background concepts, then look at certain "deliberately simplified models of systemic risk" to see what they may say about the problem.
January 26th, 2012
Speaker: Gordon Sick
Title: 'Using Streaming Market Data from Bloomberg and Thomson Reuters Terminals'
Abstract: The Haskayne School has installed a lab of 18 student stations and one instructor station with streaming data feeds that are popular in industry: Bloomberg (10 stations) and Thomson Reuters Eikon (8 stations). Access to the lab is by ID Card, and students and Faculty can apply for use of the Lab. Access policy hasn't been settled, but there is intent that the lab should be available to Mathematical Finance students, through referral from Tony Ware or Anatoliy Swischuk. These streaming data machines provide screens of market news and commentary (filtered by topic of interest), quotes in a broad variety of exchange-traded (TSX, NYSE, CME, NYMEX, NASDAQ, CBOE) and OTC markets (FX), and some analytics, such as implied volatility surfaces. Both Bloomberg and Eikon feeds set up an Excel menu that allows one to structure live DDE links to live data and historic data. One can download high-frequency tick data (bid, ask, size and trade), as well as daily data for a variety of markets.
This workshop will review:
--how to log into these services
--what data is available
--how to build spreadsheets that harvest the data.
January 19th, 2012
Speaker: Anatoliy Swishchuk
Title: Organizational meeting: schedule, announcements, information, etc.
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FALL 2011
December 7th, 2011
Speaker: Yang Liu
Title: 'Affine General Equilibrium Models'
November 30th, 2011
Speaker: Dimbi Ramarimbahoaka
Title: 'A stochastic discount function modeled by a finite state Markov chain and related asset pricing'
Abstract: Robert J.Elliott and John van der Hoek in 2010 investigated the theory of asset pricing using a stochastic discount function process where uncertainties in the economy are modeled by a Markov chain. Stock price models, futures pricing etc were derived. In a later paper (2011), in the same framework, they discussed finite maturity American options where prices are obtained as solutions of a finite dimensional variational inequality which is expressed in terms of a system of ordinary differential equation. We also give a discussion on the perpetual American option case, a recent work done by Robert J.Elliott and myself.
November 23rd, 2011
Speaker: Matthew Couch
Title: 'Variance Swap pricing with GARCH Models'
Abstract: A closed form variance swap strike price formula for generalized GARCH models will be presented. The generalized GARCH framework considered allows for non-normal innovation distributions and flexible GARCH volatility specifications. We consider numerical examples with normal inverse Gaussian and normal conditional return distributions. We further compare our results with the existing result for a continuous time GARCH limit.
November 16th, 2011
Speaker: Paul Obour
Title: 'Hedging Strategies for Canada's Oil -An Application of Currency Translated Options Driven by a Copula'
Abstract: We study pricing and hedging of Currency Translated Options (CTOs). Canadian oil producers wish to hedge (- an investment to limit loss) their production risk with futures contracts denominated in United States dollars (USD). Evidence of this has motivated this study. CTOs are denominated in Canadian dollars (CAD) and provide various levels of currency protection depending on the payoff structure chosen. We examine the performance of linearly delta hedging a Quanto option and discuss the pricing implication of the more efficient hedge driven by a Levy process / Copulas.
November 9th, 2011
Presentation 1
Speaker: Lea Steinrucke
Title: 'The LIBOR Market Model - The Seminal Papers and Extensions'
Abstract: Since it was first introduced by Miltersen et al. (1997), Brace et al. (1997) and Jamshidian (1997), the LIBOR market model (LMM) has continuously gained in importance and popularity. In contrast to previous approaches, the LMM focuses on modeling effective simple rates instead of continuously compounded spot or forward interest rates. The talk will give an introduction to this seminal idea and approach of these original papers and present extensions and generalizations to the model that have been made over the last 14 years. After a short literature review, we will in particular concentrate on models that incorporate regime-switching techniques and/or the incorporation of Markov Switching Renewal processes. Finally, we will examine how default risk can be incorporated into the LMM.
Presentation 2
Speaker: Bo Wang
Title: 'Capital Requirements and Optimal Investment for Insurance Companies'
Abstract: Asset allocation is one of the central issues in banking, finance and insurance industries. Mean, Variance, Value at Risk (VaR) and Conditional Value at Risk (CVaR) modeling were three approaches investigated towards this issue. After introducing ruin probability and expected loss at ruin as risk measures, the optimization results became more accurate and feasible. Using this new method, we also investigated the Solvency II problem.
November 2nd, 2011
Speaker: Nelson Vadori
Title: 'My Experience as a Quant at Deloitte Paris'
Abstract: 'I worked as a Quantitative Analyst at Deloitte Paris from April 2008 to August 2011. I was a member of the Quantitative team of the Risk Advisory department (which is included in Deloitte Consulting). My job focused on three main aspects: i)implementation of equity/FX/interest rate models for derivative pricing (with special focus on FX and equity markets); ii) theoretical review of the models used in major banks/corporates; iii) derivative pricing for major banks/corporates. I also did some missions outside the financial derivative pricing area: Monte Carlo VaR model review (for a broker), Capital allocation methods (for a bank) or Liquidity Gap model review (for an insurance company).'
October 26th, 2011
Speaker: Azamed Gezahagne
Title: 'Bayesian Estimation of a Natural Gas Forward Market Models'
Abstract: In this talk, Bayesian estimation technique on Finance, the idea of Principal Component Analysis and Factor analysis for calibration of the Natural Gas Forward Curve will be discussed.
October 19th, 2011
Speaker: BinBin Wang
Title: 'Fourier Transform Methods in Mathematical Finance'
Abstract: I will give an overview on methodology and applications of Fourier Transform in Finance. The basic idea of derivative pricing using Fast Fourier Transform will be explained and some selected papers will be introduced. I will also give a rough "family tree" of the development of this subject.
October 12th, 2011
Speaker: Anatoliy Swishchuk
Title: 'Levy Processes: History, Idesas, Applications'
Abstract: This talk is devoted to the rich history, definitions, examples and many applications (e.g., finance, number and relativity theories, etc.) of Levy processes (stochastically continuous processes with independent and stationary increments). We'll consider in details applications of Levy processes in finance.
October 5th, 2011
Speaker: Ke Zhao
Title: 'Generalization of the Black-76 Formula: Markov-modulated Volatility'
Abstract: In this talk, literature on Black-76 formula and Markov-modulated models will be given first. We then invoke the Markov-modulated volatility and apply it to generalize Black-76 formula. Black formulas for Markov-modulated markets with and without jumps will be showed for two states Markov chain. Application is given using Nordpool weekly electricity forward prices.
September 28th, 2011
Speaker: Anatoliy Swishchuk
Title: 'Math Finance Grad Students' Research and Presentations'
Abstract: We'll be discussing math finance graduate students research projects and planning their presentations for this Fall 2011. The information about the PRMIA Calgary Chapter Grad Students Presentations next year (April, 2012) will be discussed as well.
September 21st, 2011
Speaker: Juan-Pablo Ortega
Title: 'Hedging of discrete time auto-regressive stochastic volatility options'
Abstract: Numerous empirical proofs indicate the adequacy of the time discrete auto-regressive stochastic volatility models introduced by Taylor [1986, 2005] in the description of the log-returns of financiall assets. The pricing and hedging of contingent products that use these models for their underlying assets is a non-trivial exercise due to the incomplete nature of the corresponding market. In this paper we apply two volatility estimation techniques available in the literature for these models, namely Kalman filtering and the hierarchical-likelihood approach, in order to implement various pricing and dynamical hedging strategies. Our study shows that the local risk minimization scheme developed by Follmer, Schweizer, and Sondermann is particularly appropriate in this setup, especially for at and in the money options or for low hedging frequencies.
September 16th, 2011
Speaker: Eckhard Platen
Title: 'Numerical Solutions of Stochastic Differential Equations with Jumps in Finance'
Abstract: In financial and actuarial modeling and other areas of application, stochastic differential equations with jumps have been employed to describe the dynamics of various state variables. The numerical solution of such equations is more complex than that of those only driven by Wiener processes. The presentation builds on the recent monograph of the presenter co-authored with Bruti-Liberati. It provides some background on the benchmark approach for jump-diffusion markets and presents a survey and new results on higher-order methods for scenario and Monte-Carlo simulation. Literature: E. Platen and N. Bruti-Liberati: Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer 2010.
September 14th, 2011
Presentation 1
Speaker: Wes Devauld
Title: 'Independent Price Verification of Options on Forwards'
Abstract: Better understanding and measuring of default risk at a company level is an essential. Energy credit risk differs from that of bank credit risk in that the nature of exposures is quite different. Unlike loans in banks, an energy company's credit risk deals with the fulfillment of derivative deals by exchanges, and the receipt of payment for delivered goods. In Direct Energy credit risk project, potential future exposure was simulated by correlated risk factors. Based on survival of correlated counterparties, distribution of loss was simulated to estimate expected loss, economic capital and other credit risk metrics. Automating various risk measures gives flexibility to the model in generating scenario analyses. Concentration of risk analysis was studied based on Direct Energy's exposure and risk limit.
Presentation 2
Speaker: Elham Negahdary
Title: 'Independent Price Verification of Options on Forwards'
Abstract: The procedure of verifying an internal volatility surface with a market consensus surface constructed through a co-operative of market participants is detailed in the talk. Using Black's model to calculate implied volatility and Delaunay triangulation to construct a surface, differences in volatility for open positions are applied to Vegas to determine mark to market profit or loss.
SPRING/SUMMER 2011
August 25th, 2011
Speaker: Rudi Zagst
Title: 'Mathematical Finance at TUM'
Abstract: TUM is one of only six Elite-Universities in Germany and was honored for excellence in education in 2009. The Chair of Mathematical Finance is part of the Department of Mathematics at TUM and is one of the leading research centers for applied mathematical finance in Germany. The research focus lies on financial engineering, pricing of complex derivatives, risk and asset management. Teaching activities concentrate on the Master program 'Mathematical Finance and Actuarial Science' as well as the elite graduate program 'Finance and Information Management (FIM)'. In the first part of the talk, an overview on these two programs will be given with a special focus on FIM and the international student research exchange. In the second part, a selection of current research topics at the Chair of Mathematical Finance will be presented.
July 28th, 2011
Speaker: Christoph Reisinger
Title: 'Penalty Methods for the Numerical Solution of Hamilton-Jacobi-Bellmann Equations in Finance'
Abstract: Hamilton-Jacobi-Bellmann (HJB) Equations arise when applying Bellman's dynamic programming principle to stochastic optimisation problems. We outline the common structure of a number of applications arising in financial engineering, e.g. from European and American option valuation in incomplete financial markets. Penalty methods have been recognised as a conceptually appealing and computationally efficient method for valuing early exercise options. In this talk, we present a penalty method for HJB equations, analyse its convergence properties, and highlight the relation to state-of-the-art policy iteration methods.
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WINTER 2011
April 7th, 2011
Speaker: Anatoliy Swishchuk
Title: 'Variance and Volatility Swaps in Energy Markets'
March 31st, 2011
Speaker: Gordana Dmitrasinovic-Vidovic & Tony Ware
Title: 'Modern Portfolio Theory-Part II'
March 24th, 2011
Speaker: Ghashang Piroozfar
Paper Review: 'Games with Exhaustible Resources' by Chris Harris, Sam Howison and Ronnie Sircar
Abstract: What we study here is the problem of declining oil reserves, and its consequences for energy supplies and prices. There exists different point of views for confronting this problem. One way is the assumption of the few number of competitors and firms over energy market. The other way, which we are interested in the current article is the game theoretical techniques for constructing the outcome of competition, while there exists too many various market choices. The kind of games we will analyze, are dynamic games, since exhaustibility leads to the importance of anticipation of changing resource impacts on prices and production.
March 17th, 2011
Speaker: Anatoliy Swishchuk
Title: 'Modern Portfolio Theory-Part I'
Abstract: In this talk we start with the well-known, discrete time Markowiz portfolio problem, and define self-financing portfolios, tangency portfolio, and efficient frontier. We then proceed with continuous time portfolio problem and utility based optimization. We define utility and consumption functions, and standard market characteristics such are the market price of risk and risk premium in the Black Scholes setting. We then state the Merton's portfolio problem in which an investor must choose how much to consume, and must allocate his wealth between stocks and a risk-free asset to maximize his expected lifetime utility. Finally, we present one of the major results of portfolio optimization theory, i.e.the two-fund separation theorem which states that, under appropriate conditions, every investor's optimal portfolio is a weighted average of the market portfolio and a bond.
March 10th, 2011
Speaker: Tony Ware
Book Review: 'The Volatility Surface' by Jim Gatheral, (Chapter 11: 'Volatility Derivatives') - Continued
Abstract: The second part of this talk is devoted to the valuing of volatility swaps and quadratic-variation based securities.
March 3rd, 2011
Speaker: Tony Ware
Book Review: 'The Volatility Surface' by Jim Gatheral, (Chapter 11: 'Volatility Derivatives')
Abstract: This chapter focuses on the pricing and hedging of claims whose underlying is quadratic variation and presents some of the most elegant and robust results in financial mathematics, thereby explaining in part why the market in volatility derivatives is suprisingly active and liquid. The fisrt part of this was is devoted to the spanning generalized European payoffs and valuing of variance swaps.
February 17th, 2011
Speaker: Anatoliy Swishchuk
Title: 'Variance Swap for Local Levy based Stochastic Volatility with Delay'
Abstract: The valuation of the variance swaps for local Levy based stochastic volatility with delay (LLBSVD) is discussed in this talk. We provide some analytical closed forms for the expectation of the realized variance for the LLBSVD. As an applications of our analytical solutions, we fit our model to 10 years of S&P500 data (2000-01-01--2009-12-31) with variance gamma model and apply the obtained analytical solutions to price the variance swap. (This is a joint work with Kevin Malenfant).
February 10th, 2011
Speaker: Deniz Sezer
Book Review: 'The Volatility Surface' by Jim Gatheral, (Chapter 10: Exotic Cliquets)
Abstract: This chapter studies in detail three actual exotic cliquet transactions that happen to have matured so that one can explore both pricing and ex post performance. Specifically, it studies a locally capped and globally floored cliquet, a reverse cliquet, and a Napoleon.
February 3rd, 2011
Speaker: Akbar Sahmoradi
Title: 'Monte Carlo Simulation of NYMEX Crude Oil Option Prices Under A Generalized Hyperbolic Distribution Function'
Abstract: The empirical studies indicate that financial data usually does not follow typical normal distribution. Our empirical investigation of front month NYMEX crude oil prices over the period from 1983:01:04 to 2010:12:14 show that the crude returns significantly deviate from normal distribution, as they show fat tails. To address this issue, a generalized hyperbolic distribution is used and its parameters are calibrated using Multi-cycle Expectation Conditional Maximization. Monte Carlo estimate of the crude European Call options is calculated. Also, the sensitivity of option prices to key parameters are investigated.
January 27th, 2011
Speaker: Ke Zhao
Book Review: 'The Volatility Surface' by Jim Gatheral, (Chapter 9: Barrier Options)
Abstract: This chapter presents various types of barrier option and show how intuition may be developed for these by studying two simple limiting cases.
January 20th, 2011
Speaker: Anatoliy Swishchuk
Book Review: 'The Volatility Surface' by Jim Gatheral, (Chapter 8: 'Dynamics of the Volatility Surface')
Abstract: This chapter shows how the dynamics of volatility can be deduced from the time series properties of volatility surface.
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FALL 2010
December 7th, 2010
Speaker: Philippe Dovoedo
Book Review: 'The Volatility Surface' by Jim Gatheral, (Chapter 7: 'Volatility Surface Asymptotics')
Abstract: This chapter examines the asymptotic properties of the Volatility Surface showing that all models with SV and jumps generate VS that are roughly the same shape.
November 30th, 2010
Speaker: Kaijie Cui
Paper Review: Platen E. and West J. 'A fair pricing approach to weather drivatives' Asian-Pac. Financial Markets, (2005), 11(1), 23-53.
November 23rd, 2010
Speaker: Azamed Gezahagne
Book Review : 'The Volatility Surface' by Jim Gatheral, (Chapter 6: 'Modelling Default Risk')
Abstract: This hapter applies the work on jumps (Chapter 5) to Merton's jump-to-ruin model of default; it also explains the Credit Grades model.
November 16th, 2010
Speaker: Deniz Sezer
Book Review : 'The Volatility Surface' by Jim Gatheral, (Chapter 5: 'Adding jumps')
Abstract: This chapter explores the modelling of jumps showing first why jumps are required; introduces then characteristic function techniques and applies these to the computation of IV in models with jumps; concludes by showing that the SVJ (SV with jumps in the stock price) is capable of generating a volatility surface that has most of the features of the empirical surface.
November 9th, 2010
Speaker: Tony Ware
Title: Accurate semi-Lagrangian time stepping for gas storage problems
Abstract: Stochastic dynamic programming approaches for the valuation of natural gas storage, and the determination of the optimal continuous-time injection/withdrawal strategy, give rise to HJB P(I)DEs which are typically solved using finite differences [Thompson et. al., 2009]. A semi-Lagrangian discretization was analyzed by [Chen and Forsyth, 2007], who demonstrated first-order convergence to the viscosity solution.This talk will show how a semi-Lagrangian approach for such problems can be formulated in such a way that it generates a second-order accurate discretization in time. Combined with a hybrid Fourier/finite difference discretization in the remaining dimensions, the resulting method can provide efficiency gains over existing approaches.
November 2nd, 2010
Speaker: Anatoliy Swishchuk
Book Review : 'The Volatility Surface' by Jim Gatheral, (Chapter 4: 'The Heston-Nandi Model' )
Abstract: In this chapter the author chooses specific numerical values for the parameters of the Heston (1993) model, $\rho=-1$ as originally studied by Heston and Nandi (1998) and demonstrates that an approximate formula for implied volatility derived in Chapter 3 works particularly well in this limit. As a result, they are able to find parameters of LV and SV models that generate almost identical European option prices.
October 26th, 2010
Presentation 1
Speaker: Alex Badescu
Book Review : 'The Volatility Surface' by Jim Gatheral, (Chapter 3: 'The Implied Volatility Surface')
Abstract: In Chapter 3, author derives a powerful representation for implied volatility (IV) in terms of local volatility and applies this to build intuition and derive some properties of the implied volatility surface (VS) generated by the Heston model and compare with the empirically observed SPX surface; deduces that SV cannot be the whole story.
Presentation 2
Speaker: Anatoliy Swishchuk
Title: 'Approximations of Security Markets by Geometric Markov Renewal Processes (GMRP)' . Part II: Diffusion Approximation, Normal Deviations and Poisson Approximation of GMRP
October 20th, 2010
Speaker: Sebastian Sager
Title: 'Optimization with Differential Equations: Algorithms and Applications'
Abstract: Scientific computing with its core components mathematical modeling, simulation, and optimization has developed into a key technology for understanding and mastering challenges in science and engineering. Problems as diverse as the design and operation of chemical production plants, economic decision making, the understanding of the dynamics of cancer, or the optimal control of autonomous cars all require strong cross-disciplinary efforts supported by mathematical and computational methods. The generically interdisciplinary approach of scientific computing is generally considered a third pillar of science, complementary to experiment and theory. It has already become a standard in physics, chemistry and engineering and is currently entering research in the life sciences and economics. We will survey recent developments in the optimization with differential equations and discuss parameter estimation, optimum experimental design, and optimal control, with a focus on integer control functions. We present fast and reliable deterministic algorithms that are based on derivative information. We will present several applications from different application fields.
October 19th, 2010
Speaker: Alex Badescu
Book Review : 'The Volatility Surface' by Jim Gatheral, (Chapter 3: 'The Implied Volatility Surface')
Abstract: In Chapter 3, author derives a powerful representation for implied volatility (IV) in terms of local volatility and applies this to build intuition and derive some properties of the implied volatility surface (VS) generated by the Heston model and compare with the empirically observed SPX surface; deduces that SV cannot be the whole story.
October 12th, 2010
Speaker: Anatoliy Swishchuk
Title: 'Approximations of Security Markets by Geometric Markov Renewal Processes (GMRP)' . Part I: Definition of GMRP, Martingale Properties and Averaging of GMRP
Abstract: This talk is devoted to the study of discrete Markov-modulated (B,S)-security markets which are described by geometric Markov renewal processes (GMRP). We study their martingale properties and derive Markov renewal equation for expectation.We also study GMRP in series scheme. We state averaging, merging, diffusion approximation, normal deviations and Poisson approximation results for such models. These limit models can be used for approximations of regime-switching security markets. We state the option pricing formula for GMRP as well.
October 5th, 2010
Speaker: Tony Ware
Book Review: 'The Volatility Surface. A Practitioner's Guide' by Jim Gatheral (Chapter 2: The Heston Model)
September 28th, 2010
Speaker: Matthew Couch
Title: Variance and Volatility Swaps in stochastic volatility models based on Levy processes.
Abstract: We present a brief review of Levy process based asset price modeling and some of the better known stochastic volatility models based on Levy processes. The quadratic variation process is considered as a measure of asset price volatility in non-Gaussian models. We show how the fair strike price for Variance and Volatility swaps may be calculated in such models through the quadratic variation process.
September 21st, 2010
Speaker: Anatoliy Swishchuk
Book Review: 'The Volatility Surface. A Practitioner's Guide' by Jim Gatheral (Chapter 1: Stochastic Volatility and Local Volatility)
SPRING/SUMMER 2010
June 17th, 2010
Speaker: Samuel Cohen
Title: 'BSDEs in Discrete time and their extensions '
Abstract: BSDEs (Backward Stochastic Differential Equations) are increasingly important equations in many areas of mathematical finance and stochastic control. In this talk we shall explore BSDEs in discrete time finite state systems, where many strong results can be obtained very simply. Using this, we shall give a representation of all time-consistent nonlinear expectations in this context. We shall then discuss extensions of this theory to infinitely many outcomes, and to general probability spaces in continuous time.
June 10th, 2010
Speaker: Marianito Rodrigo
Title: 'American options with time-varying parameters via Mellin transforms'
Abstract: We use a Mellin transform approach to address the American option valuation problem under a time-dependent Black-Scholes modeling framework. The value of the put is calculated first, and then we establish a quasi put-call parity relation for American options to determine the value of the corresponding call. Since the integral equation for the free boundary is not solvable, we provide an approximating ordinary differential equation satisfied by the optimal exercise price. For the constant-parameter case, the ordinary differential equation is analytically tractable. An examination of the delta and pricing errors in our numerical experiments reveals that the proposed approach is remarkably robust and accurate.
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WINTER 2010
April 1st, 2010
Speaker: Anatoliy Swishchuk
Title: 'Modeling and Pricing of Variance and Volatility Swaps for Stochastic Volatilities Driven by Fractional Brownian Motion'
Abstract: In this talk, we study financial markets with stochastic volatilities driven by fractional Brownian motion with Hurst index H>1/2. Our models include fractional versions of Ornstein-Uhlenbeck, Vasicek, geometric Brownian motion and continuous-time GARCH models. We price variance and volatility swaps for above-mentioned models. (Joint work with Yu. Mishura)
March 25th, 2010
Speaker: Gordana Vidovich-Dmitrasinovich & Tony Ware
Title: 'Portfolio Optimization Under Downside Risk Measures'
Abstract: We give an overview of our work on portfolio optimization with respect to various downside risk measures, including Value at Risk, Capital at Risk, and Conditional Capital at Risk (or Expected Shortfall). In some cases we are able to give explicit formulae for the optimal investment strategy. We consider portfolios of lognormal assets, as well as portfolios containing mean-reverting assets.
March 18th, 2010
Presentation 1
Speaker: Kevin Malenfant
Book Review: 'Stochastic Modelling of Electricity and Related Markets' (Chapter 10, sec. 10.1-10.2: 'Analysis of Temperature Derivatives')
Presentation 2
Speaker: Anatoliy Swishchuk
Book Review: 'Stochastic Modelling of Electricity and Related Markets' (Chapter 10, sec. 10.3-10.4: 'Analysis of Temperature Derivatives')
March 11th, 2010
Speaker: Anatoliy Swishcuk
Book Review: 'Stochastic Modelling of Electricity and Related Markets'
-'Modelling of Electricity Futures Markets' (Chapter 8, sec. 8.5-8.8)
-'Analysis of Temperature Derivatives' (Chapter 10, sec. 10.1-10.2)
March 4th, 2010
Speaker: Tony Ware
Book Review: 'Stochastic Modelling of Electricity and Related Markets' (Chapter 9: 'Pricing and Hedging of Energy Market')
February 25th, 2010
Speaker: Anatoliy Swishchuk
Book Review: 'Stochastic Modelling of Electricity and Related Markets' (Chapter 8: 'Modelling of Electricity Futures Markets')
February 11th, 2010
Speaker: Ke Zhao
Title: Three Papers' Review by F. Benth et al. (2005-2007)
February 4th, 2010
Speaker: Anatoliy Swishchuk
Title: Pricing of Variance Swaps for Local Stochastic Volatilities with Delay and Jumps'
Abstract: The valuation of the variance swaps for local stochastic volatility with delay and jumps is discussed in this talk. We provide some analytical closed forms for the expectation of the realized variance for the stochastic volatility with delay and jumps. Besides, we also present a lower bound for delay as a measure of risk. As applications of our analytical solutions, numerical examples using S&P60 Canada Index (1998-2002) and S&P500 Index (1990-1993) are then provided to price variance swaps with delay and jumps.
January 28th, 2010
Speaker: Tony Ware
Title: Book Review: 'Stochastic Modelling of Electricity and Related Markets' (Chapter 7: 'Constructing Smooth Forward Curves in Electricity Markets')
Abstract: When applying the HJM approach (Chapter 6) to electricity markets, one may base the electricity futures price dynamics on a model for non-traded forwards. To estimate such models, one needs to derive forward data from the observed electricity futures prices. An algorithm for the derivation of smooth forward curves in electricity markets is presented in Chapter 7. The algorithm may be applied to gas market as well. We demonstrate the algorithm at work on Nord Pool electricity futures data, and further apply it to study the term structure of volatility of electricity.
January 21st, 2010
Speaker: Anatoliy Swishchuk
Title: Book Review: 'Stochastic Modelling of Electricity and Related Markets' (Chapter 6: 'Modelling Forwards and Swaps Using the Heath-Jarrow-Morton (HJM)
Abstract: The HJM approach to the modelling of forward and swap prices is presented in this Chapter 6. The different modelling issues regarding forward prices and swaps are investigated in detail, along with theincorporation of jump processes. As we show, the no-arbitrage condition for the term structure dynamics of the swap price rules out most of the relevant models. To resolve this issue, we introduce market models for the swaps, much in the spirit of LIBOR models for fixed income markets.
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FALL 2009
December 9th, 2009
Speaker: Vladimir Surkov
Title: 'Pricing and Hedging of Commodity Derivatives using the Fast Fourier Transform'
Abstract: Energy commodities, such as oil, gas and electricity, exhibit high volatilities, have sudden price jumps and tend to revert to a long run equilibrium. This talk develops a Fast Fourier Transform-based method for valuing and hedging of contingent claims written on mean-reverting processes with jumps. The Mean-Reverting Fourier Space Time-stepping (mrFST) method developed in this talk solves the option pricing partial integro-differential equation (PIDE) by applying the Fourier transform to obtain an explicitly solvable linear system of ordinary differential equations. Solving the PIDE in Fourier space allows for the integral term to be handled efficiently and avoids the asymmetrical treatment of diffusion and integral terms, common in the finite difference schemes found in the literature. For path-independent options, prices can be obtained for a range of spot prices in one iteration of the algorithm. For exotic, path-dependent options, a time-stepping methodology is developed to handle free boundaries and exercise policies. Finally, an efficient methodology for computing the various option Greeks is developed for use in conjunction with dynamic and static hedging in the presence of jumps.
December 2nd, 2009
Speaker: Yuriy Zenchenko
Title: 'Optimization in financial engineering'
Abstract: We will survey the use of optimization techniques in the context of financial engineering. In particular, we will discuss the so-called portfolio optimization problem that corresponds to optimal asset allocation, and the problem of pricing derivative securities. No optimization background will be assumed.
November 25th, 2009
Speaker: Tony Ware
Book Review: 'Stochastic Modeling of Electricity and Related Markets'
by F. Benth, J, Benth and S. Koekebakker, 2008, World, Sci. Publ.; Chapter 5: 'Applications to the Gas Markets'
November 18th, 2009
Speaker: Anatoliy Swishchuk
Book Review: 'Stochastic Modeling of Electricity and Related Markets'
by F. Benth, J. Benth and S. Koekebakker (Chapter 4: 'Pricing of Forwards and Swaps Based on the Spot Price', Sec. 4.3-4.4)
November 4th, 2009
Speaker: Thomas Nedunthally
Title: 'A new approach to modelling the Natural Gas futures curve and Levy based models'
Abstract: In this talk, we discuss modelling the natural gas futures curve by first using a regression equation to seperate the seasonality and the underlying curve. A two factor model based on Pilipovic, Xu with spot prices and a long run mean is used to compute the futures price. The long run mean, which also tells us if the curve is in contango or backwardation, is revealed through the underlying futures curve using a procedure we shall discuss. This allows for more accurate simulations of the gas spot price through the two factor model, and lets us capture the dynamics of the futures curve. Levy-based one factor OU type models using alpha stable and NIG processes are also introduced. The calibration of these models are also discussed.
October 28th, 2009
Speaker: Anatoliy Swishchuk
Book Review: 'Stochastic Modeling of Electricity and Related Markets' by F. Benth, J. Benth and S. Koekebakker (Chapter 4: 'Pricing of Forwards and Swaps Based on the Spot Price', Sec. 4.1-4.2)
October 21st, 2009
Speaker: Matthew Couch
Title: 'Variance and Volatility Swaps for the COGARCH(1,1) Model'
Abstract: In this talk, we present variance and volatility swaps valuations for the COGARCH (1,1) model intriduced by Kluppelberg, Lindner and Maller (2005). We consider two numerical examples: for compound Poisson COGARCH(1,1) and for variance gamma COGARCH(1,1) processes. Also, we demonstrate two different situations for the volatility swaps: with and without convexity adjustment to show the difference in price values.
October 14th, 2009
Speaker: Tony Ware
Book Review: 'Stochastic Modeling of Electricity and Related Markets' by F. Benth, J, Benth and S. Koekebakker (Chapter 3: 'Stochastic Models for Energy Spot Price Dynamics')
October 7th, 2009
Speaker: Anatoliy Swishchuk
Title: 'Pricing of Variance and Volatility Swaps with Semi-Markov Volatilities'
Abstract: In this talk, we introduce a general class of semi-Markov processes and model a stock price with stochastic volatility (SV) that depends on the semi-Markov process (we call it semi-Markov volatility). We price variance and volatility swaps for the SV driven by the semi-Markov process. We also discuss some extensions of the obtained results such as local semi-Markov volatility (LSMV), Dupire formula for LSMV and residual risk associated with the swap pricing with LSMV.
September 30th, 2009
Speaker: Anatoliy Swishchuk
Book Review: 'Stochastic Modeling of Electricity and Related Markets' by F. Benth, J, Benth and S. Koekebakker (Chapter 2: 'Stochastic Analysis for Independent Increment Processes')
September 23rd, 2009
Speaker: Alexandru Badescu
Title: 'Bond Valuation Under Discrete-Time Regime-Switching Term-Structure Models'
Abstract: We propose a discrete-time, Markov, regime-switching, affine term structure model for valuing bonds and other interest-rate securities. The proposed model incorporates the impact of structural changes in (macro)-economic conditions on interest rate dynamics. It also has some econometric advantages compared to its continuous-time counterpart. The market in the proposed model is, in general, incomplete. We introduce a modified version of the Esscher transform, namely, a double Esscher transform, to specify a price kernel so that both market and economic risks are taken into account. We provide a simple and streamlined way to derive exponential-affine forms of bond prices using backward induction. (Joint work with Robert J. Elliott and Tak Kuen Siu)
September 16th, 2009
Speaker: Tony Ware
Book Review: 'Stochastic Modeling of Electricity and Related Markets' by F. Benth, J, Benth and S. Koekebakker, (Chapter 1 'A Survey of Electricity and Related Markets')
SPRING/SUMMER 2009
July 16th, 2009
Speaker: Yuliya Mishura
Title: 'Long-range dependence and non-semimartingale models in finance'
Abstract: Financial markets fairly often have a long memory and it is a natural idea to model them with the help of fractional Brownian motion (fBm) or some of its modifications. However, it is not so straightforward to implement because the market model is appropriate when it does not admit arbitrage and the models involving fractional Brownian motion are not arbitrage-free. The talk is devoted to some methods of construction of the long-memory arbitrage-free models and to the discussion of different approaches to this problem. In particular, we introduce the mixed Brownian-fractional-Brownian model and establish conditions that ensure the absence of arbitrage in such a model. Also we consider a fractional version of the Black-Scholes equation for the mixed Brownian-fractional-Brownian model which contains pathwise integrals w.r.t. fBm, discuss possible applications of Wick products in fractional financial models and produce Black-Scholes equation for the fractional model involving Wick product w.r.t. fBm.
References:
[1] Biagini, F., Hu, Y., Oksendal, B., Zhang T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Probability and Its Applications, Springer (2008).
[2] Mishura, Yu. S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics 1929, Springer (2008).
June 29th, 2009
Speaker: Alok Gupta
Title: Calibration Using Consistent Bayesian Estimators
Abstract: The general calibration problem in financial models is considered. We reformulate the problem into a Bayesian framework to attain posterior distributions for calibration parameters. We show that, for any continuous and bounded loss function, the corresponding Bayesian estimatoris consistent. Finally we work through numerical examples to clarify theconstruction of Bayesian posteriors and its uses. The main focus is on the local volatility model.
May 19th, 2009
Speaker: Sebastian Jaimungal
Title: 'Multi-Factor Levy Processes and Regime Switching for Commodities'
Abstract: Energy commodities, such as oil, gas and electricity, lack the liquidity of equity markets, have large costs associated with storage, exhibit high volatilities and can have significant spikes in prices. Furthermore, and possibly most importantly, commodities tend to revert to long run equilibrium prices. Many complex commodity contingent claims exist in the markets, such as swing and interruptible options; however, the current method of valuation relies heavily on Monte Carlo simulations and tree based methods. In this talk, I will describe a new model of cointegrated prices containing mean-reverting jumps and diffusions as well as a new valuation framework by working in Fourier space. The method is based on the Fourier space time-stepping algorithm of Jackson, Jaimungal, and Surkov (2008), but is tailored for mean-reverting models. I will demonstrate the utility of the method by applying it to the valuation of European, American, Spread and swing options. In addition, I will discuss the real option to invest in an oil field where the volume is stochastic but is discovered as time evolves.[ This is based on joint work with Vladimir Surkov, Ph.D. candidate, Dept. Computer Science, U. Toronto]
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WINTER 2009
April 30th, 2009
Speaker: Rossitsa Yalamova
Title: 'Explaining What Leads Up to Stock Market Crashes: A Phase Transition Model and Scalability Dynamic'
Abstract: The market crash as phase transition in Johansen and Sornette (1999) points at the analogy between the three states of a physical system (solid, liquid and gas) and stock market dynamics at a "microscopic" level, where the individual trader has only three possible actions: selling, buying or waiting. According to this model at the critical point order prevails in the market as all traders have the same opinion sell which leads to 'significant drawdowns'. We do not offer an alternative to EMH/CAPM but extend the existing framework to accommodate situations with higher information complexity, interactions with positive feedback, and extreme events that cannot be simply explained by presuming independent-additive data point, and normal distributions.
April 2nd, 2009
Speaker: Thomas Salisbury
Title: 'Equity guarantees and retirement'
Abstract: A new generation of retirement savings products offer the upside associated with equity returns, coupled with the downside protection of annuities. I will describe the changing demographics of retirement planning, and some of the products created to serve this market. Also some of the mathematical problems associated with hedging, pricing, and managing portfolios that incorporate these guaranteed living benefits. This talk describes joint work with Huaxiong Huang and Moshe Milevsky.
March 26th, 2009
Speaker: Clifford Kitchen
Title: 'Applications of the Normal Inverse Gaussian (NIG) Processes in Mathematical Finance'
Abstract: Many of the papers we have discussed in this lab use L\'{e}vy driven stochastic process rather than the usual Gaussian type. These models are typically complex and empirical results are shown with little detail of how to calibrate and implement the model. The intent here is give the introductory steps required to complete this work. I will show how to incorporate a Levy process by modeling, calibrating and pricing European options using a basic NIG process.
March 19th, 2009
Speaker: Thomas Nedunthally
Title: Stochastic Models of Natural Gas Prices and Applications to Natural Gas Storage Valuation (Chen and Forsyth, 2007)
Abstract: Two one factor models with regime switching are introduced in this paper to mimic the behaviour of two-factor models that have been previously developed to model natural gas spot prices. Calibration results for the regime switching model show the ability to capture both the long term and short term dynamics of market futures prices. The calibrated models are then used to price a natural gas storage facility.
March 5th, 2009
Speaker: Matthew Couch
Title: 'Regime-switching models with applications in finance'
Abstract: We review selected sections of early papers developing Markov chain based regime switching time series models. We also review the paper 'Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching' by Robert J. Elliott, Tak Kuen Siu, Leunglung Chan, 2005. In this paper a model is developed for pricing volatility derivatives, such as variance swaps and volatility swaps under a continuous-time Markov-modulated version of the stochastic volatility (SV) model developed by Heston.
February 26th, 2009
Speaker: John Sheriff
Title: 'The Use of Evolutionary Algorithms to Estimate Jump-Diffusion Models of Equity Markets'
Abstract: The use of Levy processes and related models is an attractive option in mathematical finance due to their ability to more accurately capture observed market behavior. However, this comes at the price of greater complexity and the associated challenge of fitting a more complex model to market data. One approach to meeting this challenge is to employ evolutionary algorithms, an iterative procedure that relies upon mutation, propagation, and selection to estimate important model parameters. The talk will address the use of such algorithms in the context of fitting jump-diffusion models to market data.
February 12th, 2009
Speaker: Kevin Malenfant
Title: 'Analysis of Valuation Formulae and Applications to Exotic Options in Levy Models' by E. Eberlein, K. Glau & A. Papapantoleon
Abstract: This paper discusses the valuation problem for a broad spectrum of plain vanilla and path-dependent options in a general framework, and specifically for Levy driven models. Among the derivatives which paper considers are digitals, double digitals, asset-or-nothing options, self-quantos, lookback and one-touch options.
February 5th, 2009
Speaker: Miro Powojowski
Title: 'Some observations on implied volatility'
Abstract: The observed departures of real world markets from the Black-Scholes model have generated much research activity in academic circles and many practical attempts at handling the problem in industry. With some exceptions, the two groups chose very different approaches to the problem. While academics have been building more complex models based on more general processes for underlying assets, practitioners have focused on building ad-hoc corrections to the Black-Scholes models. Both have been adding parameters to the basic model, the academics preferring internal model consistency over the ease of parameter estimation, while finance professionals making the exact opposite tradeoffs. In this talk I will present some theoretical arguments justifying the industry practice of using implied volatilities as a basis for pricing and hedging options.
January 29th, 2009
Speaker: Anatoliy Swishchuk
Title: 'Multi-Factor Levy Models II: Pricing of Financial and Energy Derivatives'
Abstract: This talk is devoted to the multi-factor Levy models and their applications in financial and energy derivatives' pricing. The first part 'Multi-Factor Levy Models I: Alpha-Stable Levy Processes' (Jan 22) introduced and described alpha-stable Levy processes and based on them multi-factor Levy models. The second part 'Multi-Factor Levy Models II: Pricing of Financial and Energy Derivatives' (Jan 29) will be devoted to the applications of multi-factor Levy models in financial and energy derivatives' pricing. We'll consider swaps, interest rate derivatives, forward and futures pricing. The approach is based on change of time for alpha-stable Levy processes.
January 22nd, 2009
Speaker: Anatoliy Swishchuk
Title: 'Multi-Factor Levy Models I: Symmetric Alpha-Stable (SaS) Levy Processes'
Abstract: This talk is devoted to the multi-factor Levy models and their applications in financial and energy derivatives' pricing, and consists of two parts. The first part 'Multi-Factor Levy Models I: Alpha-Stable Levy Processes' introduces and describes alpha-stable Levy processes and based on them multi-factor Levy models. The second part 'Multi-Factor Levy Models II: Pricing of Financial and Energy Derivatives' (next seminar) will be devoted to the applications of multi-factor Levy models in financial and energy derivatives' pricing.
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FALL 2008
November 12th, 2008
Speaker: Tony Ware
Title: 'A Fourier transform method for pricing options on mean-reverting Levy-driven assets'
Abstract: Fourier transform methods are well-suited to the computation of expectations of functions of Levy processes (such as Poisson jump-diffusion, or Variance Gamma processes), and thus they often form the basis of methods for numerical option pricing when the underlying asset follows a Levy process. In this talk I will review these methods, and show how, by use of semi-Lagrangian time-stepping and the non equally-spaced fast Fourier transform (NFFT), they can be extended to mean-reverting and other processes with Levy random shocks. Numerical examples will be provided. (Joint work with Li Xu).
November 5th, 2008
Speaker: Thomas Nedunthally
Title: 'Spot Convenience Yield Models for the Energy Markets' by Rene Carmona and Michael Ludkovski
Abstract: This paper reviews the literature of spot convenience yield models, and analyzes in detail two new extensions. First, discussion a variant of the Gibson-Schwartz model with time-dependent parameters. Second, description a new three-factor affine model with stochastic convenience yield and stochastic market price of risk.
October 29th, 2008
Speaker: Anatoliy Swishchuk
Title: 'Levy-based Interest Rate Derivatives. Part II: PIDEs.'
Abstract: In the second part of this talk we describe the second approach in pricing of Levy-based interest rate derivatives based on partial integro-differential equations (PIDEs). We show how to price zero-coupon bonds and bond options. Also, we present PIDEs for pricing of swaps, caps, floors and options on them, swaptions, captions and floortions, respectively. (In the first part of this talk we described the first approach in pricing of Levy-based interest rate derivatives based on change of time method for alpha-stable Levy processes).
October 22nd, 2008
Speaker: Anatoliy Swishchuk
Title: 'Levy-based Interest Rate Derivatives: Part I: Change of Time Method'
Abstract: We describe two approaches in pricing of Levy-based interest rate derivatives. The first approach is based on change of time method for alpha-stable Levy processes. The second approach is based on partial integro-differential equations (PIDEs). We show how to price zero-coupon bonds and bond options. Also, we present PIDEs for pricing of swaps, caps, floors and options on them, swaptions, captions and floortions.
October 15th, 2008
Speaker: Deniz Sezer
Title: 'Quantitative bounds for Markov Chain convergence: Wasserstein and Total variation distances'
Abstract: In this talk I will present recent results on Markov Chain convergence based on joint work with Neal Madras. Let P_n^x, and \pi be respectively the n-step transition probability kernel and the stationary distribution of a Markov chain. In many applications it is desirable to have a quantitative bound for convergence of P_n^x to \pi, i.e. a bound of the form d(P_n^x,\pi)<g(x,n) where d is a metric on the space of probability measures and g is a function which can be computed explicitly. In continuous state spaces one way to obtain a quantitative bound is formulating the Markov chain as an iterated system of random maps and applying David Steinsaltz's local contractivity convergence theorem. If the conditions are satisfied, this theorem yields a quantitative bound in terms of Wasserstein distance. We first develop a systematic framework to check for the conditions of Steinsaltz's theorem, and then show how one can obtain a quantitative bound in terms of total variation distance from a quantitative bound in terms of Wasserstein distance.
October 8th, 2008
Speaker: Kevin Malenfant
Title: 'An Introduction to Levy Processes with Applications in Finance' by Antonis Papapantoleon
Abstract: This paper aims at introducing Levy processes in an informal and intuitive way, accessible to non-specialists in the field.
October 1st, 2008
Speaker: Sivia Mayoral
Title: 'Applications of the Method of Maximum Entropy in Mean to Finance'
September 24th, 2008
Speaker: Matt Lyle
Title: A 'Simple' Hybrid Model for Power Derivatives
Abstract: This paper presents a method for valuing power derivatives using a supply-demand approach. Our method extends work in the field by incorporating randomness into the base load portion of the supply stack function and equating it with a noisy demand process. We obtain closed form solutions for European option prices considering two different supply models: a mean-reverting model and a Markov chain model. The results are extensions of the classic Black-Scholes equation. The model provides a relatively simple approach to describe the complicated price behaviour observed in electricity spot markets and also allows for computationally efficient derivatives pricing.
September 17th, 2008
Speaker: Deniz Sezer
Title: 'An information reduction model for credit risk based on level crossings of a diffusion'
Abstract: I will talk about a reduced information model for credit risk. In this model, the time when a company claims bankruptcy is the hitting time of the asset value process of the company, denoted by X_t, to a default threshold. The market can not observe X_t prior to bankruptcy, however it can observe R(X_t) , where R(x)=i , if x_i<x<x_{i+1}, where x_1,...x_N are certain thresholds. I will explain how we derive zero coupon bond prices and default intensities when the X process is a diffusion. In the time remaining I will discuss open questions and future directions related to this model. (Based on joint work with Robert Jarrow and Philip Protter).
SPRING/SUMMER 2008
June 19th, 2008
Speaker: Anatoliy Malyarenko
Title: 'Analytical Finance Package'
Abstract: We describe the Java package afp that contains a collection of applets in the area of analytical finance. The user of the package is able to price different financial instruments using Monte Carlo simulation, finite difference methods etc.
June 12th, 2008
Speaker: Thomas Neduthally
Title: 'Gas Storage Valuation Using a Monte Carlo Method' by A. Boogert & C. de Jong (2006)
May 22nd, 2008
Speaker: Dmitrii Silvestrov
Title: 'Optimal Pricing of American Type Options for Modulated Price Processes'
Abstract: This lecture presents a survey of the latest results on option optimal pricing for modulated price processes achieved by the author and his collaborators. These results are: discovery of multi-threshold structure of optimal stopping strategies for option models with general convex payoffs and formulation of conditions, which implicate multi- and one-threshold structures for optimal stopping strategies; introduction and investigation of new models of pricing processes modulated by semi-Markov market indices; obtaining of skeleton approximations, uniform with respect to a perturbation parameter, for continuous- and discrete-time option pricing models; finding of new effective general conditions for convergence of option reward functions; constraction of effective Monte Carlo algorithms for pricing of options based on information about structure of optimal stopping domains, experimental software for pricing of options, and the latest achievements are connected with stochastic models for reselling of options.
May 15th, 2008
Speaker: Anatoliy Swishchuk
Title: 'Levy Processes in Finance. Pricing Financial Derivatives' (Chapter 10 'Interest-Rate Models')
May 2nd, 2008
'Lunch at the Lab' in conjunction with North/South Dialog Meeting (Friday-Saturday, May2-3, U of C): Mathematical Finance Session
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WINTER 2008
April 24th, 2008
Speaker: Tony Ware
Title: 'Levy Processes in Finance. Pricing Financial Derivatives' (Chapter 8 'Simulation Techniques', sec.8.4 'Simulation of Particular Processes')
April 17th, 2008
Speaker: Alexandru Badescu
Title: 'Levy Processes in Finance. Pricing Financial Derivatives' ('Simulation of Generalized Hyperbolic Processes')
April 10th, 2008
Speaker: Anatoliy Swishchuk
Title: 'Levy Processes in Finance. Pricing Financial Derivatives' (Chapter 8: 'Simulation Techniques', Sections 8.1-8.3.)
April 3rd, 2008
Speaker: Anatoliy Swishchuk
Title: 'Levy Processes in Finance. Pricing Financial Derivatives' (Chapter 7: 'Levy Models with Stochastic Volatility')
March 27th, 2008
Speaker: James Pang
Title:
-Value, Trading Strategies and Financial Investment of Natural Gas
-Storage Assets
March 20th, 2008
Speaker: Anatoliy Swishchuk
Title: 'Levy Processes in Finance. Pricing Financial Derivatives', (Section 5.4.-5.5.: 'Adding an Additional Term' and 'Examples of OU Processes'& Chapter 6: 'Stock Price Models Driven by Levy Processes')
March 13th, 2008
Speaker: Andrei Badescu
Title: 'Return Probabilities of Stochastic Fluid Flows and Their Use in Collective Risk Theory'
March 6th, 2008
Speaker: Anatoliy Swishchuk
Title: 'Levy Processes in Finance. Pricing Financial Derivatives' (Chapter 5: 'Levy Processes and OU Processes', section 5.3. Examples of Levy proceses )
February 28th, 2008
Speaker: Matt Lyle
Book Review: 'Levy Processes in Finance. Pricing Financial Derivatives' (Chapter 4: "Imperfections of the Black-Scholes Model")
Paper Review: Empirical properties of asset returns: stylized facts and statistical issues (Quant. Finance, 2001, 1, pp.1-14).
February 14th, 2008
Speaker: Tony Ware
Title: 'Levy Processes in Finance. Pricing Financial Derivatives' (Chapter 5: Levy Processes and OU Proceses)
February 7th, 2008
Speaker: Anatoliy Swishchuk
Title: 'Levy Processes in Finance. Pricing Financial Derivatives' (Chapter 3: The Black-Scholes Model)
January 31st, 2008
Speaker: Alexandru Badescu
Title: 'Levy Processes in Finance. Pricing Financial Derivatives' (Chapter II: Financial Mathematics in Continuous Time)
January 24th, 2008
Speaker: Anatoliy Swishchuk
Title: 'Levy Processes in Finance. Pricing Financial Derivatives' (Chapter I: Introduction)
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FALL 2007
November 22nd, 2007
Speaker: Greg Orosi
November 15th, 2007
Speaker: Li Xu
Paper Review
November 8th, 2007
Speaker: Hong Miao
Title: 'Investment timing under regime switching'
November 1st, 2007
Speaker: Hua Li
Title: 'Application of fuzzy sets in finance'
October 25th, 2007
Speaker: Anatoliy Swischuk
Title: 'Pricing of variance swaps for stochastic volatilites with delay and jumps '
October 18th, 2007
Speaker: Hua Li
Paper Review:
-'Pricing and hedging derivative securities in markets with uncertain volatilites' by Avellaneda M., Levy A. and Paras A. (Appl. Math. Finance, 1995)
-'Uncertain parameters, an empirical stochastic volatility model and confidence limits' by Wilmott P. and Oztukel A. (1998)
October 11th, 2007 - PRMIA Students Research Presentations
Presentation 1
Speaker: Scott MacDonald
Presentation 2
Speaker: Hong Miao
Presentation 3
Speaker: Greg Orosi
October 4th, 2007
Speaker: Tony Ware
Title: 'Modelling Natural Gas Markets II'
September 27th, 2007
Speaker: Alexandru Badescu
Title: 'Risk neutral measures for GARCH option pricing with normal variance-mean mixture examples'
September 20th, 2007
Speaker: Tony Ware
Title: 'Modelling Natural Gas Markets'
SPRING/SUMMER 2007
August 8th, 2007
Speaker: Lu Zhao
Title: 'Modelling and Pricing of Variance and Volatility Swaps for Stochastic Volatilities with Jumps'
July 18th, 2007
Speaker: Hua Li
Paper Review:
- "Pricing European options based on the fuzzy pattern of Black-Scholes formula" (Wu 2004).
-"Using fuzzy sets theory and Black-Scholes formula to generate pricing boundaries of European options" (Wu 2007).
July 11th, 2007
Speaker: Yuyuan Ouyang
Title: 'European and Swing Option Pricing under mean-reverting jump diffusion models'
June 20th, 2007
Speaker: Matt Lyle
Title:
-'A Brief Highlight of the "Mathematics of Electricity Supply & Pricing" Workshop in Surfers Paradise, Australia, 2007'
-'The Decomposition of Electricity Prices: A Master of Science Thesis Preview'
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WINTER 2007
April 1st, 2007
Presentation 1
Speaker: Yuyuan Ouyang
Title: 'Swing Option Pricing under Jump-Diffusion Models'
Presentation 2
Speaker: Greg Orosi
Title: 'Are Options Mispriced?'
Presentation 3
Speaker: Leunglung Chan
Title: 'Option Pricing for GARCH Models with Markov Switching'
Presentation 4
Speaker: Hong Miao
Title: 'VaR and CVaR: A Non-Normal Regime Switching Framework'
March 28th, 2007
Presentation 1
Speaker: Li Xu
Title: 'Pricing Variance Swaps for Stochastic Volatilities with Delay and Jumps'
Presentation 2
Speaker: Lu Zhao
Title: 'Variance Swaps for Mean -Reverting Jump-Diffusion Models'
Presentation 3
Speaker: Scott MacDonald
Title: 'Time Scale Decomposition of Economic Relationships Using Wavelet Analysis'
March 21st, 2007
Speaker: Greg Orosi
Title: 'Energy Derivatives' by Clewlow and Strickland, 2000. (Chapter 11: 'Credit Risk in Energy Markets')
March 14th, 2007
Speaker: Li Xu
Title: 'Energy Derivatives' by Clewlow and Strickland, 2000. (Chapter 10: 'Value at Risk')
March 7th, 2007
Speaker: Lu Zhao
Title: 'Energy Derivatives' by Clewlow and Strickland, 2000. (Chapter 9: 'Risk Management of Energy Derivatives')
February 28th, 2007
Speaker: Matt Lyle
Title: 'Energy Derivatives' by Clewlow and Strickland, 2000. (Chapter 8: Forward Curve Models)
February 14th, 2007
Speaker: Yuyuan Ouyang
Title: 'Energy Derivatives' by Clewlow and Strickland, 2000. (Chapter 7: Spot Price Models: Pricing Path Dependence and American Style Options)
February 7th, 2007
Speaker: Leunglung Chan
Title: 'Regime-Switching GARCH Models'
January 31st, 2007
Speaker: Anatoliy Swishchuk
Title: 'Energy Derivatives' by Clewlow and Strickland, 2000. (Chapter 6: Spot Price Models and Pricing Standard Instruments)
January 24th, 2007
Speaker: Lu Zhao
Title: 'Energy Derivatives' by Clewlow and Strickland, 2000. (Chapter 5: Energy Derivatives: Structures and Applications)
January 12th, 2007
Speaker: Mahmoud Hamada
Title: 'Real Options Theory and Electricity Forwards'
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FALL 2006
December 5th, 2006
Speaker: Xu Li
Title: Book Review: 'Energy Derivatives' by Clewlow and Strickland, 2000. (Chapter 4: 'Energy Forward Curves')
November 28th, 2006
Speaker: Anatoliy Swischuk
Title: 'Energy Derivatives' by Clewlow and Strickland, 2000. (Chapter 3: 'Volatility Estimation in Energy Markets')
November 21st, 2006
Speaker: Yuyuan Ouyang
Title: 'Energy Derivatives: Pricing and Risk Management' by Clewlow and Strickland, 2000. (Chapter 2: 'Understanding and Analysing Spot) Pricing'
November 7th, 2006
Speaker: Anatoliy Swishchuk
Title: 'Energy Derivatives: Pricing and Risk Management' by Clewlow and Strickland, 2000. (Chapter 1: 'Introduction to Energy Derivatives and Fundamentals of Modelling and Pricing')
October 31st, 2006
Speaker: Greg Orosi
Title: 'Survey of Local Volatility Models'
Abstract: In my talk I'll go over a survey of local volatility models (polynomial, spline, penalized spline). Also, I'll touch on the volatility smile problem and calibration of models to current option prices.
October 24th, 2006
Speaker: Lu Zhao
Title: "A Benchmark Approach to Finance", Math. Finance, vol 16, N1 (Jan 2006)
Abstract: This paper derives a unified framework for portfolio optimization, derivative pricing, financial modelling, and risk management.
October 17th, 2006
Speaker: Anatoliy Swischuk
Title: Stochastic Volatilities with Delay (SVD): Modelling and Pricing of Variance Swaps for MFSVD" (Part II).
Abstract: Variance swaps for financial markets with underlying asset and multi-factor stochastic volatilities with delay are modelled and priced in this talk. We obtain some analytical closed forms for the expectations and variances of the realized continuously sampled variances for multi-factor stochastic volatilities with delay. As applications, we provide numerical examples using the S&P60 Canada Index (1998-2002) to price variance swaps with delay for all these models.
October 10th, 2006
Speaker: Anatoliy Swishchuk
Title: "Stochastic Volatilities with Delay (SVD): Modelling and Pricing of Variance Swaps for SVD" (Part I).
Abstract: Modelling and pricing of variance swaps for financial markets with underlying asset and stochastic volatilities with delay are discussed in this talk. We found some analytical close forms for expectation and variance of the realized continuously sampled variance for stochastic volatility with delay both in stationary regime and in general case. The key features of the stochastic volatility model with delay are the following: i) continuous-time analogue of discrete-time GARCH model; ii) mean-reversion; iii) contains the same source of randomness as stock price; iv) market is complete; v) incorporates the expectation of log-return. As applications, we provide two numerical examples using $S\&P60$ Canada Index (1998-2002) and $S\&P500$ Index (1990-1993) to price variance swaps with delay.
October 3rd, 2006
Presentation 1
Speaker: Matt Lyle
Title: "Three electricity spot price models: Evidence from the PJM and Alberta markets" (Part II).
Presentation 2
Speaker: Lu Zhao
Journal Review:
-Mathematical Finance (Jan/April/July/Oct 2006).
-Journal of Quantitative Finance (Aug/Oct/Dec 2005, Feb/April/June/Aug/Oct 2006).
September 26th, 2006
Presentation 1
Speaker: Matt Lyle
Title: "Cycle Detection and Removal in Electricity Prices" (Part 1)
Presentation 2
Speaker: Yuyuan Ouyang
Journal Review:
-Finance & Stochastics (Volume 10, Jan/Apr/Sep 2006)
-International Journal of Theoretical & Applied Finance (Volume 9, Feb/Mar/May/Jun/Aug/Sep 2006)
SPRING/SUMMER 2006
July 7th, 2006
Speaker: Tony Ware
Title: "Commodity Swaptions, Swing Contracts and Real Options in the Energy Industry" (Chapter 5 of Helyette Geman's book "Commodities and commodity derivatives")
June 23rd, 2006
Speaker: Anatoliy Swischuk
Title: "Spot and Forward Electgricity Market" (Chapter 11 of Helyette Geman's book "Commodities and commodity derivatives")
June 9th, 2006
Speaker: Tony Ware
Title: "The Gas Market" (Chapter 10 of Helyette Geman's book "Commodities and commodity derivatives")
June 2nd, 2006
Speaker: Anatoliy Swishchuk
Title: "The Oil Market as a World Market" (Chapter 9 of Helyette Geman's book "Commodities and commodity derivatives")
May 26th, 2006
Speaker: Alex David
Title: "Agricaltural Commody Markets and The Structure of Metal Markets and Metal Prices" (Chapter 7-8 of Helyette Geman's book "Commodities and commodity derivatives")
May 19th, 2006
Speaker: Tony Ware
Title: "Monte Carlo Simulations and Analytical Formulae for Asian, Barrier and Quanto Options" (Chapter 6 of Helyette Geman's book "Commodities and commodity derivatives")
May 12th, 2006
Speaker: Anatoliy Swishchuk
Title: "Risk-Neutral Valuation of Plain-Vanilla Options" (Chapter 5 of Helyette Geman's book "Commodities and commodity derivatives").
May 5th, 2006
Speaker: Traian A. Pirvu
Title: "Maximizing portfolio growth rate under risk constraints."
Abstract: This work studies the problem of optimal investment subject to risk constraints: Value-at-Risk, Tail Value-at-Risk and Limited Expected Loss. We get closed-form solutions for this problem, and find that the optimal policy is a projection of the optimal portfolio of an unconstrained log agent (the Merton proportion) onto the constraint set, with respect to the inner product induced by the variance-covariance volatilities matrix of the risky assets. In the more complicated situation of constraint sets depending on the current wealth level, we maximize the growth rate of portfolio subject to these risk constraints. We extend the analysis to a market with random coefficients, which is not necessarily complete. We also perform a robust control analysis. We find that a trader subject to Value-at-Risk and Tail Value-at-Risk is allowed to incur some risk. A trader faced with the Limited Expected Loss constraint behaves more conservatively and does not exhibit the above behavior. This is a joint work with Steven Shreve and Gordan Zitkovic.
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WINTER 2006
April 28th, 2006
Speaker: Tony Ware
Title: "Option pricing: from stocks to commodities" (Chapter 4 of Helyette Geman's book "Commodities and commodity derivatives").
April 21st, 2006
Speaker: Anatoliy Swishchuk
Title: "Stochastic modeling of commodity price processes" (Chapter 3 of Helyette Geman'sbook "Commodities and commodity derivatives").
April 7th, 2006
Speaker: Tony Ware
Title: "Commodity and commodity derivatives" (Chapter 2 of Helyette Geman's book "Commodities and commodity derivatives").
March 31st, 2006
Speaker: Greg Orosi
Title: "Retrieving the implied volatility surface using splines."
Abstract: Since its publication, the Black- Scholes option pricing formula has been widely used in the industry. However, there is much empirical evidence that the model is too simplistic because of the constant volatility assumption. In this talk I'll describe an improvement over the Black-Scholes pricing framework that uses cubic splines to estimate the implied volatility surface from option prices. I'll also describe an optimization method called genetic algorithm and how this can be applied to the above problem.
March 24th, 2006
Speaker: Jennie La
Title: "Pricing of asian options."
Abstract: Option pricing still remains an important problem for researchers, in particular, the options being considered may not have closed-form expressions, so it is difficult to price these options analytically. For this reason, numerical techniques such as simulations are often used for option pricing. This talk will introduce the type of simulation methods used, in particular, for pricing Asian options. In addition, this talk will show how simulation methods can be improved upon by introducing variance reduction techniques, specifically importance sampling. By applying variance reduction techniques to simulation methods, the price of the option can be accurately estimated and the computation time can be reduced.
March 17th, 2006
Speaker: Anatoliy Swischuk
Title: "Fundamentals of commodity spot and futures markets: instruments, exchanges and strategies". (Chapter 1 of Helyette Geman's book "Commodities and commodity derivatives"
March 3rd, 2006
Speaker: Matt Davison
Title: "Success and failure (in modelling) deregulated electricity markets."
Abstract: Deregulated Energy Markets provide many opportunities and challenges - for mathematician and maker of public policy alike. In this talk I will describe the electricity markets work done in my group at UWO over the last six years, while placing this work in its broader economic and political context. The energy markets work we have done at UWO sits at the intersection of financial mathematics, operational research, and engineering. Our work has two main threads. We have developed a discrete - time model for spot electricity prices sitting between the old-fashioned "stack"-based models of regulated electricity markets and a fully econometric model appropriate to mature financial markets. I will describe the resulting "hybrid" model and some of its lessons. I will also describe our second, continuous-time, approach to energy markets. There we use classical dynamic programming tools to study the optimal control of electricity generating assets. In each case I will discuss obvious next steps as well as existing results. I will conclude my talk by discussing some crucial (though non-mathematical) aspects of electricity finance. These include lessons of Ontario's largely failed deregulation experiment not only for energy modelers but also for public policy wonks. I will also discuss some promising technological and policy developments which suggest that, for electricity deregulation, "better luck next time" might be more than just empty words.
February 17th, 2006
Speaker: Gordana Dmitrasinovic-Vidovic and Tony Ware
Title: Portfolio optimisation for log-normal and mean-reverting assets with respect to downside risk measures.
Abstract: We consider managed portfolios of log- normal and mean-reverting assets, optimised with reference to various risk measures, but most notably quantile-based risk measures such as Capital-at- Risk and Value-at-Risk. These risk measures focus attention on the downside tail of the distribution of future portfolio values and are important for regulatory purposes. We give formulae for constructing optimal portfolios for specific choices of risk measure and explore some implications. Some of this work can be found in `Asymptotic behaviour of mean-quantile efficient portfolios' (D-V & W), to appear in Finance and Stochastics.
February 10th, 2006
Speaker: Anatoliy Swishchuk
Paper Review: "On the pricing and hedging of volatility derivatives" by S. Howison, A. Rafailidis and H. Rasmussen (2004).
February 3rd, 2006
Speaker: Anatoliy Swishchuk
Paper Review: "Parameter estimation in a stochastic drift hidden Markov model with a cap" by J. Hernandez, D. Suanders and L. Seco (2005).
January 27th, 2006
Speaker: Matthew Zhao
Title: Variance and volatility swaps for markets with jumps.
January 20th, 2006
Speaker: Lance Ouyang
Title: Mean-reverting models in finance with jumps.
Abstract: Mean-reverting models are important in finance, and are widely used in energy markets. I will give anpresentation on mean-reverting models, mainly one-factor and two-factor Pilipovic models and their applications to obtaining explicit European option pricing formulae. A one-factor mean-reverting model with jumps will also be introduced.
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FALL 2005
6th December, 2005
Speaker: Anatoliy Swishchuk
Title: Girsanov's Theorem: from game theory to finance.
29th November, 2005
Speaker: Tsung-lin Cheng
Title: Statistical aspects of the GARCH model.
22nd November, 2005
Speaker: Andrew Royal
Title: Utility maximization in incomplete markets.
15th November, 2005
Speaker: Hong Miao
Title: A volatility model with Markov switching.
8th November, 2005
Speaker: Anatoliy Swishchuk
Title: Explicit option pricing formula for mean-reverting assets.
1st November, 2005
Speaker: Tony Ware
Title: Options with continuous exercise.
Abstract: I will present an alternative formulation of the american option as a limiting case of continuously-exercisable options. This class of options can also be used to model gas-storage contracts and also includes swing options as a special case. I will show that the value of such an option may be found by solving a semilinear PDE (c.f. Benth et. al., Finance and Stochastics, 2003 for the american option case), and I will illustrate some numerical solutions of such equations.
25th October, 2005
Speaker: Matthew Lu
Paper review: "A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets" by D. Heath, E. Platen and M. Schweizer (Math. Finance, vol. 11, N.4, 2001)
18th October, 2005
Speaker: Anatoliy Swishchuk
Title: Change of time method: Applications in mathematical finance
Abstract: I am going to give an introduction to the change of time method in the martingale and SDE settings and to show how it works for different kind of models and problems arising in mathematical finance. The first part contains applications to Black-Scholes model and Heston model. The second part will contain application to the mean-reverting model and option pricing.
11th October, 2005
Speaker: Lance Ouyuang.
Paper review: "Convergence of Monte-Carlo Simulations Involving the Mean-Reverting Square Root Process" by D. J. Higham and X. Mao (J. Computational Finance, 2005).
4th October, 2005
Speaker: Cody Hyndman
Title: Forward-Backward Stochastic Differential Equations and the Cox-Ingersoll-Ross Model.
27th September, 2005
Speaker: Yuyuan (Lance) Ouyuang and Zhao (Matthew) Lu
Abstract: A review recent issues of Finance and Stochastics, the Journal of Computational Finance, Mathematical Finance and the Journal of Quantitative Finance.
20th September, 2005
Speaker: Leunglung Chan
Title: Pricing Volatility Swaps Under Heston's Volatility Model with Regime Switching (this talk is based on joint paper with Robert Elliott and Tak Kuen Siu).
SPRING/SUMMER 2005
This summer the Finance Lab will read select chapters from Foundations of Modern Probability (Springer, 2002, 2nd ed.) by Olav Kallenberg.
July 21st, 2005
Chapter 26: Semimartingales and General Stochastic Integration.
Speaker: Anatoliy Swishchuk
July 14th, 2005
Chapter 24: Connections with PDEs and Potential Theory.
Speaker: Ka Chun Cheung
July 7th, 2005
Chapter 23: One-Dimensional Stochastic Differential Equations and Diffusions.
Speaker: Anatoliy Swishchuk
June 30th, 2005
Chapter 21: Stochastic Differential Equations and Martingale Problems
Speaker: Ka Chun Cheung
June 23rd, 2005
Chapter 19: Feller processes and semigroups.
Speaker: Anatoliy Swishchuk
June 16th, 2005
Chapter 18: Continuous martingales and Brownian motion
Speaker: Ka Chun Cheung
June 9th, 2005
Chapter 17: Stochastic integrals and quadratic variation.
Speaker: Anatoliy Swishchuk
June 2nd, 2005
Chapter 13: Gaussian Processes and Brownian Motion.
Speaker: Ka Chun Cheung
May 26th, 2005
Chapter 12: Poisson and pure jump-type Markov processes.
Speaker: Anatoliy Swishchuk
May 18th, 2005
Chapter 7: Martingales and optional times.
Speaker: Ka Chun Cheung
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WINTER 2005
April 21st, 2005
Speaker: Miro Powojowski
Title: How to calculate the vega without working too hard.
April 14th, 2005
Speaker: Anatoliy Swishchuk
Title: Yet one more derivation of Black-Scholes formula: change of time method.
Abstract: I am going to present yet one more derivation of the well-known Black-Scholes formula using a change-of-time method.
April 7th, 2005
Speaker: Robert Elliott
Title: Cutting the hedge.
Abstract: We shall describe a short empirical way of calculating the delta.
March 31st, 2005
Speaker: Anatoliy Swishchuk
Title: Mean-Reverting Models in Financial and Energy Markets.
March 24th, 2005
Speaker: Hua Li
Title: Numerical Methods for Parabolic Integro-Differential Equations (PIDEs).
Abstract: Parabolic integro-differential equations(PIDEs) have been used in option pricing when the underlying price process has jumps. In this talk, we introduce various numerical methods for solving them that have appeared in the literature, and summarize the corresponding advantages and drawbacks. We will conclude that the wavelet-Galerkin (or wavelet-Petrov-Galerkin) method is preferable to finite-difference and finite- element approaches.
March 17th, 2005
Speaker: Cheung, Ka Chun
Title: Ordering optimal proportions in the asset allocation problem with dependent default risks.
March 10th, 2005
Speaker: Anatoliy Swishchuk
Title: Explicit Option Pricing Formula for Mean-Reverting Asset.
Abstract: Unlike stock price, some commodity prices (i.e., oil and gas) exibit mean-reversion, i.e., they tend over time to return to some long-term mean. We consider a risky asset following a mean-reverting stochastic process S(t) described by the following stochastic differential equation dS(t)=a(L-S(t))dt+\sigma S(t)dW(t), where W is a standard Wiener process, \sigma>0 is the volatility, constant L is callled the 'long-term mean' of the process, to which it reverts over time, and a>0 measures the 'strength' of mean reversion.
Using change of time method we find the explicit solution to this equation and using this solution we are able to find the explicit option pricing formula.
We are going to apply our solution to the calculation of the values of a European call option on the price of a daily natural gas contract, using futures prices for the AECO Natural Gas Index for the period 1 May 1998 to 30 April 1999.
March 3rd, 2005
Speaker: Leung Leung Chan
Title: Option Pricing and Esscher Transform under Regime Switching.
February 17th, 2005
Speaker: Tony Ware
Title: Pricing options on mean-reverting assets using the finite element method.
February 10th, 2005
Speaker: Anatoliy Swishchuk
Title: Paper Review: "New Insight into Smile, Mispricing, and Value at Risk: The Hyperbolic Model" by E. Eberlein, U. Keller and K. Prause (1998).
February 1st, 2005
Speaker: Anatoliy Swishchuk
Title: Levy Processes-From Probability to Finance.
January 27rd, 2005
Speaker: Graham Weir
Title: The Valuation of Petroleum Lease Contracts as Real Options.
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FALL 2004
December 16th, 2004
Speaker: Gordon Sick
Title: Calibrating mean-reverting models to NYMEX oil and gas futures and options.
December 9th, 2004
Speaker: Guanghui Quan
Title: Nine Ways to implement the binomial method for option valuation in MATLAB (based on Higham's SIAM review paper).
December 2nd, 2004
Speaker: Gordana Dmitrasinovic-Vidovic
Title: Portfolio optimization under downside risk measures.
November 25th, 2004
12pm
Speaker: Anatoliy Swishchuk
Title: Stability of Financial Models
Abstract: Stochastic stability of financial models will be considered in this talk. In particular, stochastic stability of interest rates (including Vasicek, Cox-Ingersoll-Ross, etc.) and their generalization on the case of models with jumps will be discussed .
1pm
Speaker: Paul Malcolm
Title: New Gaussian Mixture Techniques For Filtering and Smoothing Of Discrete-Time Gauss-Markov Jump Markov Systems.
November 18th, 2004
Speaker: Gergely Orosi
Title: Neural networks in finance.
November 4th, 2004 - Finance Research Seminar
Speaker: Matt Spiegel
Title: Improved forecasting of mutual fund alphas and betas.
October 28th, 2004
Speaker: Anatoliy Swishchuk
Title: Financial Markets with Stochastic Volatilities.
October 21st, 2004 - Finance Research Seminar
Speaker: Lisa Kramer
Title: Investing Confidence in the Ex Ante Equity Premium: A New Methodology and a Narrower Range of Estimates.
October 14th, 2004
Speaker: Lei Xiong
Title: Calibration of energy price processes using jump-diffusion models
October 7th, 2004
Speaker: Gordon Sick
Title: Calibrating mean-reverting models to NYMEX oil and gas futures and options.
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