Weak Numerical Methods for Jump-Diffusion Processes with Applications in Finance

Several authors argue that the evolution of financial quantities exhibit jumps that cannot be captured by diffusion processes, see, for instance, [1] for foreign exchange and stock markets and [2] for short term interest rate movements. Therefore, models for the dynamics of financial quantities that incorporate jumps as risk sources have become increasingly popular, see for instance [3], [4] and [5]. Since the class of jump-diffusion stochastic differential equations (SDEs) that admits explicit solutions is rather limited it is important to construct discrete time approximations which can be applied in simulations. There are two types of numerical approximations of SDEs, the strong and the weak ones, see [6]. Strong schemes are needed for problems such as scenario simulation, filtering and hedge simulation, when a pathwhise approximation is sought. Weak schemes are appropriate when approximating, by Monte Carlo simulation, the expectation of a payoff function of the solution of the underlying SDE, such as moments, derivative prices or risk measures. In the current paper we analyze the weak approximation of SDEs driven by Wiener processes and Poisson random measures, while we refer to [7] for the strong approximation. The literature on the weak approximation of jump-diffusion SDEs is rather limited. In [8] a theorem for the weak convergence of jump-adapted weak Taylor schemes of any order β ∈ {1, 2, . . .}, which are based on time discretisations that include all the jump times, is presented. The paper [9] analyzes weak Taylor schemes of any weak order β ∈ {1, 2, . . .} which are based on time discretisations that do not include the jump times. A weak convergence theorem is given and for the Euler and the order 2.0 weak Taylor schemes the leading coefficients of their global error are derived and extrapolation methods are presented. In [10] the Euler scheme in the case of Hölder continuous coefficients is treated. The article [11] considers the convergence of the jump-adapted Euler scheme with some weaker assumptions on the jump coefficient. The current paper presents a survey and several new results on weak approximations for SDEs driven by Wiener processes and Poisson random measures. A new simplified Euler scheme, whose computational complexity is independent of the intensity of the Poisson measure, is proposed. Moreover, new implicit and predictor-corrector schemes are derived and their order of weak convergence is established. Finally, a numerical study on the accuracy of several weak schemes when applied to the Merton jump-diffusion model is presented.