Markovian approximations of stochastic Volterra equations with the fractional kernel

TL;DR

This paper introduces a new approximation method for rough stochastic volatility models with fractional kernels, enabling efficient simulation and accurate implied volatility computation.

Contribution

It develops an N-dimensional diffusion approximation for stochastic Volterra equations with fractional kernels, demonstrating strong convergence and practical application to option pricing.

Findings

Approximation converges strongly with superpolynomial rate in N.

Method effectively computes implied volatility smiles.

Applicable to rough Bergomi and rough Heston models.

Abstract

We consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low H\"older-regularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an $N$-dimensional diffusion process defined as solution to a system of ordinary stochastic differential equation. If the coefficients of the stochastic Volterra equation are Lipschitz continuous, we show that these approximations converge strongly with superpolynomial rate in $N$. Finally, we apply this approximation to compute the implied volatility smile of a European call option under the rough Bergomi and the rough Heston model.