Approximation and calibration of stochastic processes in finance

This thesis is a study of approximation and calibration of stochastic processes with applications in finance. It consists of an introduction and four research papers. The introduction is as an overview of the role of mathematics in certain areas of finance. It contains a brief introduction to the mathematical theory of option pricing, as well as a description of a mathematical model of a financial exchange. The introduction also includes summaries of the four research papers. In Paper I, Markov decision theory is applied to design algorithmic trading strategies in an order driven market. A high dimensional Markov chain is used to model the state and evolution of the limit order book. Trading strategies are formulated as optimal decision problems. Conditions that guarantee existence of optimal strategies are provided, as well as a value-iterative algorithm that enables numerical construction of optimal strategies. The results are illustrated with numerical experiments on high frequency data from a foreign exchange market. Paper II focuses on asset pricing with Lévy processes. The expected value E[g(XT )] is estimated using a Monte Carlo method, when Xt is a d-dimensional Lévy process having infinite jump activity and a smooth density. Approximating jumps smaller then a parameter ! > 0 by diffusion results in a weak approximation, Xt, of Xt. The main result of the paper is an estimate of the resulting model error E[g(XT )]− E[g(XT )], with a computable leading order term. Option prices in exponential Lévy models solve certain partial integro-differential equations (PIDEs). A finite difference scheme suitable for solving such PIDEs is studied in Paper III. The main results are estimates of the time and space discretization errors, with leading order terms in computable form. If the underlying Lévy process has infinite jump activity, the jumps smaller than some ! > 0 are replaced by diffusion. The size of this diffusion approximation is estimated, as well as its effect on the space and time discretization errors. Combined, the results of the paper are enough to determine how to jointly choose the grid size and the parameter !. In Paper IV it is demonstrated how optimal control can be used to calibrate a jump-diffusion process to quoted option prices. The calibration problem is formulated as an optimal control problem with the model parameter as a control variable. The corresponding regularized Hamiltonian system is solved with a symplectic Euler method.