Markovian Approximation of the Rough Bergomi Model for Monte Carlo Option Pricing

Qinwen Zhu,Wen Chen,Nicolas Langrené,Grégoire Loeper

doi:10.3390/math9050528

Abstract

The recently developed rough Bergomi (rBergomi) model is a rough fractional stochastic volatility (RFSV) model which can generate a more realistic term structure of at-the-money volatility skews compared with other RFSV models. However, its non-Markovianity brings mathematical and computational challenges for model calibration and simulation. To overcome these difficulties, we show that the rBergomi model can be well-approximated by the forward-variance Bergomi model with wisely chosen weights and mean-reversion speed parameters (aBergomi), which has the Markovian property. We establish an explicit bound on the L2-error between the respective kernels of these two models, which is explicitly controlled by the number of terms in the aBergomi model. We establish and describe the affine structure of the rBergomi model, and show the convergence of the affine structure of the aBergomi model to the one of the rBergomi model. We demonstrate the efficiency and accuracy of our method by implementing a classical Markovian Monte Carlo simulation scheme for the aBergomi model, which we compare to the hybrid scheme of the rBergomi model.

Highlights

The rough Bergomi model introduced by Bayer et al [1] has gained acceptance for stochastic volatility modelling due to its power-law at-the-money (ATM) volatility skew, which is consistent with empirical studies and with the effect of the no-arbitrage assumption on the market impact function
We proved the power-law behavior of the ATM volatility skew as time to maturity goes to zero of the rough Bergomi model, and proposed an approximate Bergomi model with a finite number of forward variance terms to approximate the rBergomi model
The approximation enables the adoption of classical pricing methods, while keeping the fractional feature of the model

Summary

Introduction

The rough Bergomi (rBergomi) model introduced by Bayer et al [1] has gained acceptance for stochastic volatility modelling due to its power-law at-the-money (ATM) volatility skew, which is consistent with empirical studies (see Forde and Zhang [2], Fukasawa [3], Gatheral et al [4]) and with the effect of the no-arbitrage assumption on the market impact function (see Jusselin and Rosenbaum [5]). Regarding the pricing of exotic options in the rBergomi model, Tomas [12] considered the pricing of Asian options, and Bayer et al [13]. The calibration of the rBergomi model is a challenge, for which Bayer et al [15], Zeron and Ruiz [16], and Horvath et al [17] propose to use deep learning methods. In spite of this number of recent efforts, the inherent challenges brought by the rBergomi model still prevent its widespread adoption in the industry

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