The Euler–Maruyama approximation for the asset price in the mean-reverting-theta stochastic volatility model

Stochastic differential equations (SDEs) have been used to model an asset price and its volatility in finance. Lewis (2000) [10] developed the mean-reverting-theta processes which can not only model the volatility but also the asset price. In this paper, we will consider the following mean-reverting-theta stochastic volatility model dX(t)=α1(μ1−X(t))dt+σ1V(t)X(t)θdw1(t),dV(t)=α2(μ2−V(t))dt+σ2V(t)βdw2(t).dV(t)=α2(μ2−V(t))dt+σ2V(t)βdw2(t). We will first develop a technique to prove the non-negativity of solutions to the model. We will then show that the EM numerical solutions will converge to the true solution in probability. We will also show that the EM solutions can be used to compute some financial quantities related to the SDE model including the option value, for example.