Convergence of Monte Carlo Simulations involving the Mean-Reverting Square Root Process∗

The mean-reverting square root process is a stochastic differential equation (SDE) that has found considerable use as a model for volatility, interest rate, and other financial quantities. The equation has no general, explicit, solution, although its transition density can be characterized. For valuing path-dependent options under this model, it is typically quicker and simpler to simulate the SDE directly than to compute with the exact transition density. Because the diffusion coefficient does not satisfy a global Lipschitz condition, there is currently a lack of theory to justify such simulations. We begin by showing that a natural Euler–Maruyama discretization provides qualitatively correct approximations to the first and second moments. We then derive explicitly computable bounds on the strong (pathwise) error over finite time intervals. These bounds imply strong convergence in the limit of the timestep tending to zero. The strong convergence result can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. We spell this out for a bond with interest rate given by the mean-reverting square root process, and for an up-and-out barrier option with asset price governed by the mean-reverting This manuscript appears as University of Strathclyde Mathematics Research Report 17 (2004). Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK. Supported by Research Fellowships from The Leverhulme Trust and The Royal Society of Edinburgh/Scottish Executive Education and Lifelong Learning Department. Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G1 1XH, UK.