On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model

Abstract

The rough Heston model is a form of a stochastic Volterra equation, which was proposed to model stock price volatility. It captures some important qualities that can be observed in the financial market—highly endogenous, statistical arbitrages prevention, liquidity asymmetry, and metaorders. Unlike stochastic differential equation, the stochastic Volterra equation is extremely computationally expensive to simulate. In other words, it is difficult to compute option prices under the rough Heston model by conventional Monte Carlo simulation. In this paper, we prove that Euler’s discretization method for the stochastic Volterra equation with non-Lipschitz diffusion coefficient E[|Vt−Vtn|p] is finitely bounded by an exponential function of t. Furthermore, the weak error |E[Vt−Vtn]| and convergence for the stochastic Volterra equation are proven at the rate of O(n−H). In addition, we propose a mixed Monte Carlo method, using the control variate and multilevel methods. The numerical experiments indicate that the proposed method is capable of achieving a substantial cost-adjusted variance reduction up to 17 times, and it is better than its predecessor individual methods in terms of cost-adjusted performance. Due to the cost-adjusted basis for our numerical experiment, the result also indicates a high possibility of potential use in practice.

Keywords

• rough Heston model
• weak convergence error rate
• Monte Carlo method
• control variate method
• multilevel Monte Carlo method