Mean-reversion is an important component of many financial models. When simulations are performed with numerical methods, it is therefore desirable to reproduce this qualitative property. Here, we study a square root process with jumps that has been used to model interest rates and volatilities, and we characterize the parameter regimes under which the first and second moments revert to steady state values. We then consider a class of implicit theta methods and investigate the same moment properties for the corresponding stochastic difference equation. We find that the theta method is unconditionally stable in first and second moment for theta values below a cutoff level. This cutoff level depends on the parameters governing the mean reversion and the jumps, but is always more favourable than the value of one half that arises in the deterministic setting. In the case of high jump intensity, large jump magnitude or slow mean reversion, it is even possible for the explicit Euler-Maruyama type method from this class to be unconditionally stable. We also establish upper and lower bounds for the second moment steady state that are close to that of the continuous-time process for small step-sizes. Numerical experiments are given to illustrate the results.
- interest rate
- Ito lemma
- monte carlo
- stochastic differential equation