Topics in financial and computational mathematics

In this thesis, we consider two different aspects in financial option pricing. In the first part, we consider stochastic differential equations driven by general Lévy processes (SDEs) with finite and infinite activity and the re- lated, via the Feynman-Kac formula, Dirichlet problem for integro-partial differential equation (IPDE). We approximate the solution of IPDE using a numerical method for the SDEs. The method is based on three ingredients: (i) we approximate small jumps by a diffusion; (ii) we use restricted jump- adaptive time-stepping; and (iii) between the jumps we exploit a weak Euler approximation. We prove weak convergence of the considered algorithm and present an in-depth analysis of how its error and computational cost depend on the jump activity level. We present the results of a range of numerical experiments including application of the suggested numerical scheme in the context of Foreign Exchange (FX) options, where we present an example on barrier basket currency option pricing in a multi-dimensional setting. In the second part of the thesis, we suggest an intermediate currency approach that allows us to price options on all FX markets simultaneously under the same risk-neutral measure which ensures consistency of FX option prices across all markets. In particular, it is sufficient to calibrate a model to the volatility smile on the domestic market as, due to the consistency of pricing formulas, the model automatically reproduces the correct smile for the inverse pair (the foreign market). We first consider the case of two currencies and then the multi-currency setting. We illustrate the intermediate currency approach by applying it to the Heston and SABR stochastic volatility models, to the model in which exchange rates are described by an extended skewed normal distribution, and also to the model-free approach of option pricing