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.
Highlights
We propose a mixed estimator that uses the idea of the control variate method and the multilevel Monte Carlo method
This paper mainly contributes to the Monte Carlo computation for option prices under the rough Heston model
We have proposed a Control Variate estimator for variance reduction purpose and a mixed Monte Carlo method, which consists of the control variate method and multilevel method
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Rough volatility has recently emerged as an impressive tool to model an asset’s volatility in the financial market. The empirical study [1] by Gatheral and his co-authors have made all this possible. Rough volatility is a type of fractional Brownian motion (fBm) with a Hurst parameter of 0 < H < 0.5, or in terms of alpha notation, 0.5 < α = H + 0.5 < 1. FBm with 0 < H < 0.5 has rougher fluctuation than the ordinary Brownian motion, whereas fBm with 0.5 < H < 1 has a smaller/lesser fluctuation than the ordinary Brownian motion. Before continuing the discussion on rough volatility, we would like to mention some of the work that made this discovery possible
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