## Abstract

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

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.

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.

Original language | English |
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Pages (from-to) | 2209-2223 |

Number of pages | 15 |

Journal | Computers and Mathematics with Applications |

Volume | 64 |

Issue number | 7 |

DOIs | |

Publication status | Published - Oct 2012 |

## Keywords

- Euler-Maruyama method
- stochastic differential equation
- Brownian motion
- option value