Numerical Solution of Jump-diffusion Stochastic Differential Equations

Download or read book Numerical Solution of Jump-diffusion Stochastic Differential Equations written by Gerald Teng and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Jump-diffusion processes are widely used in finance, economics, and other areas. They serve as models for asset, commodity and energy prices, interest and exchange rates, and the timing of corporate and sovereign defaults. The distributions of jump-diffusions are rarely analytically tractable, so Monte Carlo simulation methods are often used to treat the pricing, risk management, and statistical estimation problems arising in applications of jump-diffusion models. The first chapter is based on a paper that is joint work with Yexiang Wei. The chapter develops, analyzes and tests a discretization scheme for jump-diffusion processes with general state-dependent drift, volatility, jump intensity, and jump size. The scheme allows for an unbounded jump intensity, a feature of many standard jump-diffusion models in finance, economics, and other disciplines. It constructs the jump times as time-changed Poisson arrival times, and generates the process between the jump epochs using Euler discretization. Under technical conditions on the coefficient functions of the jump-diffusion, the convergence of the discretization error is proved to be of weak order arbitrarily close to one. The second chapter develops, analyzes and tests several methods for improving the computational efficiency of simulating jump-diffusions. The methods are applicable to simulation algorithms that discretize the Brownian component while using a standard Poisson process to generate the jump times, and whose weak order of convergence for the discretization error is known. We propose variance reduction methods based on nested simulation and antithetic variates, as well as methods for improving the efficiency of Richardson extrapolation techniques. We also investigate simulation efficiency improvements based on multilevel Monte Carlo methods. Numerical experiments demonstrate the methods give significant improvements to simulation efficiency.