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Fast Hybrid Schemes for Fractional Riccati Equations (Rough is not so Tough)

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  • Callegaro Giorgia
  • Grasselli Martino
  • Pag`es Gilles

Abstract

We solve a family of fractional Riccati differential equations with constant (possibly complex) coefficients. These equations arise, e.g., in fractional Heston stochastic volatility models, that have received great attention in the recent financial literature thanks to their ability to reproduce a rough volatility behavior. We first consider the case of a zero initial value corresponding to the characteristic function of the log-price. Then we investigate the case of a general starting value associated to a transform also involving the volatility process. The solution to the fractional Riccati equation takes the form of power series, whose convergence domain is typically finite. This naturally suggests a hybrid numerical algorithm to explicitly obtain the solution also beyond the convergence domain of the power series representation. Our numerical tests show that the hybrid algorithm turns out to be extremely fast and stable. When applied to option pricing, our method largely outperforms the only available alternative in the literature, based on the Adams method.

Suggested Citation

  • Callegaro Giorgia & Grasselli Martino & Pag`es Gilles, 2018. "Fast Hybrid Schemes for Fractional Riccati Equations (Rough is not so Tough)," Papers 1805.12587, arXiv.org, revised Feb 2020.
    • Handle: RePEc:arx:papers:1805.12587
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    References listed on IDEAS

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    1. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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