# Mathematical Finance

#### Computational methods

Even with the explosive growth in computing power available to each of us, a lot of time can be wasted in using the wrong algorithms with financial models. In some cases, the more powerful the model is, the more important it is to be careful in choosing the right computational tools.

We developed a range of efficient methods. Examples include multidimensional tree algorithms for swing options, spectral methods that price options on mean-reverting assets, and finite-element methods for multi-factor affine models. In each case we've aimed to prove the reliability and efficiency of the methods and to create software that implements them and demonstrates their effectiveness. Currently, we are working on adaptive wavelet-based methods for solving option pricing PDEs, multi-asset simulation codes calibrated to forward markets, and on methods for option pricing in the presence of jumps.

#### Exotic derivative pricing

It is clear that the Black-Scholes formula, despite its popularity and power, is not adequate for all situations. In the energy industry, the assumptions underlying the Black-Scholes model simply do not hold. One simple illustration of this is the presence of mean-reversion in many energy prices. We are interested in how to price options when the underlying price is mean-reverting and have developed models and pricing technologies for this situation. We are also interested in methods for pricing structured contracts - that cannot be regarded as combinations of simple puts and calls. One example of such a contract is the swing option, where the holder is given some freedom in the exercise strategy, but this freedom is constrained. The option can be viewed as a generalization of the American option, where the holder cannot exercise all of the option at one time, but must exercise with some constraints on the rate of exercise. One application of such options is to gas storage models.

#### Energy price modelling

Mean-reversion is just the tip of the iceberg when it comes to describing the awkward nature of energy price processes. Seasonality, discontinuity, long-range correlations, fat tails and many other features can be identified. We are interested in modelling these processes in a way that enables us to capture the qualitative nature of the way the prices behave while still being able to answer quantitative questions of interest to the investor or risk manager.

#### Portfolio optimisation

Simply put, the goal of most portfolio management is to maximise returns while simultaneously minimising risk. But how should risk be measured? Should we simply use the variance or should we use some other measure, such as Value at Risk (VaR)? What are the implications of using VaR to determine your portfolio strategy?

#### Simulation technologies

Monte-Carlo simulation is a standard industry workhorse for quantitative risk management. But it must be remembered that the numbers it gives are always random, and have only a given probability of being approximately right. How can that probability be improved? It is very expensive computationally to improve it simply by doing more simulations, and if you are doing a large-scale simulation with hundreds of independent random variables this may be impossible.

We are working both on more efficient Monte-Carlo simulation engines where the accuracy and reliability are improved, and on quasi Monte-Carlo methods as a means of getting all the advantages of Monte-Carlo without the drawbacks.

## Research Spotlight: Jinniao Qiu

My present research includes the following five main aspects:

(i) stochastic control, mathematical finance and stochastic partial differential equations (SPDEs);

(ii) stochastic partial differential variational inequalities and option pricing problems;

(iii) limit order books and optimal trading;

(iv) probabilistic representations and variational principles for nonlocal PDEs and SPDEs;

(v) machine/deep learning methods and applications.

**Stochastic Control, Mathematical Finance, and SPDEs**

Most of my research is related to stochastic control problems. For the non-Markovian problems of stochastic control and mathematical finance like the utility maximization with random coefficients, the (backward) SPDE arises naturally as the stochastic HJB equation. My interests include the applications in the area of energy, commodity and environmental finance, and the well-posedness of fully nonlinear stochastic HJB equations whose solvability is an open problem in the theory of stochastic optimal control.

**Stochastic Partial Differential Variational Inequality and Options Pricing Problems**

The stochastic partial differential variational inequality arises as the Hamilton-Jacobi-Bellman equation for the optimal stopping problem of stochastic differential equations with random coefficients. When dealing with the singular control problem of stochastic partial differential equations, it is also introduced as the adjoint equations for the maximum principle of Pontryagin type, and its solution can further characterize the value function of the optimal stopping problem of SPDEs. Interesting topics include the well-posedness and associated numerical approximations for certain pricing problems.

**Modeling Limit Order Books and Optimal Trading**

In modern financial markets, almost all transactions are settled through Limit Order Books (LOBs). From the mathematical point of view, LOBs can be viewed as high-dimensional complex priority queuing systems. Recently, we proposed a queuing-theoretic LOB model for the dynamics of the two-sided limit order book (LOB). For the joint dynamics of the 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 to a continuous-time limit when the order arrival rates tend to infinity, the impact of an individual order arrival on the book as well as the tick size tends to zero. Along with the model analysis, some of my research efforts will be devoted to the optimal trading problems in LOBs.

**Probabilistic Representations and Variational Principles for nonlocal PDEs and SPDEs**

It is a tradition to represent solutions of PDEs as the expected functionals of stochastic processes (with associated variational principles), which may also be done for stochastic PDEs; examples include the Navier-Stokes equation and the stochastic Keller-Segel equation. The concerned PDEs and SPDEs arise from areas including finance, economics, physics, biology, and even engineering. Besides, I am also interested in applying such probabilistic representations to numerical approximations for the solutions.