On the numerical stability of simulation methods for SDEs under multiplicative noise in finance

Abstract

When simulating discrete-time approximations of solutions of stochastic differential equations (SDEs), in particular martingales, numerical stability is clearly more important than some higher order of convergence. Discrete-time approximations of solutions of SDEs with multiplicative noise, similar to the Black-Scholes model, are widely used in simulation in finance. The stability criterion presented in this paper is designed to handle both scenario simulation and Monte Carlo simulation, i.e. both strong and weak approximations. Methods are identified that have the potential to overcome some of the numerical instabilities experienced when using the explicit Euler scheme. This is of particular importance in finance, where martingale dynamics arise frequently and the diffusion coefficients are often multiplicative. Stability regions for a range of schemes are visualized and analysed to provide a methodology for a better understanding of the numerical stability issues that arise from time to time in practice. The result being that schemes that have implicitness in the approximations of both the drift and the diffusion terms exhibit the largest stability regions. Most importantly, it is shown that by refining the time step size one can leave a stability region and may face numerical instabilities, which is not what one is used to experiencing in deterministic numerical analysis.

Strong Predictor-Corrector Euler-Maruyama Methods for Stochastic Differential Equations with Markovian Switching

“…In this section we consider numerical stability issues, extending the analysis in Platen and Shi [12] to the Markovian switching case. When simulating discrete time approximations of solutions of SDEwMSs, numerical stability is clearly as important as numerical efficiency.…”

“…When simulating discrete time approximations of solutions of SDEwMSs, numerical stability is clearly as important as numerical efficiency. There have been various efforts made in the literature trying to study numerical stability for a given scheme approximating solutions of SDEs, see, for instance, Hofmann and Platen [5], Higham [4], Bruti-Liberati and Platen [3] and Platen and Shi [12]. Generally, for analyzing numerical stability, some specifically designed test equations are necessary to be introduced, the test SDEs used in the above literatures are linear SDEs with multiplicative noise defined as…”

“…As a unified criterion, Platen and Shi [12] proposed the concept of pstability criterion, which means that a process is p-stable if in the long run its pth moment vanishes. Hence, for p > 0, λ < 0, p-stable in equation ( 5.1) may be characterized by…”

“…which is called the transfer function of the approximation Y t at time t n , Platen and Shi [12] derive the following useful result which can determine the stability regions for given schemes by the following theorem.…”

“…In the spirit of Platen and Shi [12], stability regions for a range of schemes of SDEwMSs are discussed in this paper. Here we consider the test process X r(t) = {X t,r(t) , t ≥ 0} satisfies the linear SDEwMSs with multiplicative noise…”