Topics in Financial Mathematics

This abstract contains summaries of two projects in mathematical finance. The first topic is about methods of extracting inflation expectations from market price data. The second topic concerns the numerical simulation of SDEs and Monte Carlo methods which have applications in option pricing. Estimation of risk-neutral densities of future inflation rates: Inflation expectations play a crucial role in monetary policy. As such, there have been several papers on methods to extract expectations from market information. From options written on inflation rates, we can extract a probability density function of future inflation rates. Moreover, by modelling a dependence structure between inflation rates over different time intervals, we can extract density functions of rates on which options are not written. For example, the 5y/5y inflation rate can be inferred from options written on both the 5-year inflation rate and the 10-year inflation rate as well as the dependence between the two rates. A key question is how should the dependence between different rates be modelled. One solution is to use copulas which can be fitted to additional data. This, however, raises problems such as: choosing the appropriate copula; and choosing the appropriate data on which to fit the copula. There might be the possibility of imposing a dependence structure in a 'model-free' way. This would require additional year-on-year option price data. Since current year-on-year options are not particularly liquid, observed prices are noisy and often violate no-arbitrage conditions. Hence, the starting point of this project would be to generate synthetic year-on-year option price data and develop a method of extracting correlations between different rates. Variance reduction methods for diffusions: When approximating SDEs, it is often the case that we only require that the expected value of the approximation is close to the expected value of the true solution. In such cases, we can rely on weak methods, which guarantee convergence in this sense. When simulating SDEs in the weak sense, the discretisation error decreases relatively quickly and the error can be approximated via the Talay-Tubaro expansion. The other source of error, resulting from the Monte Carlo approximation of the expectation, decreases slowly. As a consequence of the slow convergence of Monte Carlo method, variance reduction techniques have been developed to decrease the overall error. While these techniques do not increase the convergence rate, they decrease computational time by reducing the coefficient of the error. Simulation of SDEs has applications for solving PDEs, since parabolic PDEs have a probabilistic representation given by the Feynman-Kac formula. This has particular importance in high-dimensional settings, where traditional numerical methods of solving PDEs suffer from the 'curse of dimensionality'. Methods of variance reduction include importance sampling or control variates, or a combination of the two. In theory, the variance can be reduced to zero but it requires knowledge of the full solution of the PDE, which is not practical. This motivates the construction of practical methods of variance reduction, where the full solution is approximated. An advantage of these approaches is that the initial approximation of the solution requires no guarantee of accuracy, since these approximations do not bias the Monte Carlo approximation. Therefore, deep learning methods could be employed. An initial area to explore is whether deep learning approaches can compete with linear regression methods, in terms of computational cost. Another potential area to explore is Dirichlet boundary value problems, which have also have a probabilistic representation and can be subject to variance reduction techniques. Solving the Dirichlet problem has applications in the pricing of barrier options.