WIAS Preprint No. 2868, (2021)

Markovian approximations of stochastic Volterra equations with the fractional kernel


Authors

  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Breneis, Simon

2020 Mathematics Subject Classification

  • 91G60 60G22 60H35

Keywords

  • Stochastic Volterra equation, fractional kernel, rough volatility model, Markovian approximation, strong error

DOI

10.20347/WIAS.PREPRINT.2868

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 Hölder-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.

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