The Hybrid-Exponential Scheme for Stochastic Volterra Equations

Abstract

We propose a hybrid scheme for the simulation of stochastic Volterra equations. The scheme is a mix of the hybrid scheme for Brownian semistationary processes of Bennedsen et al. [Financ. Stoch., 21(4), 931-965, 2017] and then the multifactor approximations of Abi Jaber et al. [SIAM J. Finan. Math., 10(2), 309-349, 2019]. Merging the schemes allow us to both accurately capture any singularities and efficiently track the inherent path-dependence. For equations with deterministic drift conditional expectations are easily computable under the scheme and we show how this can be used to simulate the VIX index for several important volatility models of the Volterra type. Numerical experiments indicate good convergence for rough Bergomi type models as well as for the quadratic rough Heston model. Experiments on the (ordinary) rough Heston model, where values are non-negative and thus need to be truncated, in some cases, resulted in a significant positive bias.