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A martingale method for option pricing under a CEV-based fast-varying fractional stochastic volatility model

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Abstract

Modeling the volatility smile and skew has been an active area of research in mathematical finance. This article proposes a hybrid stochastic–local volatility model which is built on the local volatility term of the CEV model multiplied by a stochastic volatility term driven by a fast-varying fractional Ornstein–Uhlenbeck process. We find that the Hurst exponent of the implied volatility is less than 1/2 usually but it is larger than 1/2 during an immediate period of recovery from the COVID-19 pandemic. We use a martingale method to obtain option price and implied volatility formulas in the both short- and long-memory volatility cases. As a result, the existing CEV implied volatility can be complemented to reflect implied volatility patterns (skewed smiles) that arise in pricing short time-to-maturity options in equity markets by incorporating convexity into it and controlling the downward slope of it at-the-money. We verify that one additional parameter of the CEV-based fractional stochastic volatility model contributes to a better qualitative agreement with market data than the Black–Scholes-based fractional stochastic volatility model or the CEV-based non-fractional stochastic volatility model.

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Data are available from the authors upon request.

Notes

  1. In many literature works, the fBm \((W_t^H)\) is defined as equation (3) divided by \((\Gamma (2\,H+1) \sin (\pi H))^{-1/2}\) so that \(\sigma _H\) becomes 1. This is essentially equivalent to the definition (3) introduced by Mandelbrot and Van Ness (1968) from which \(\sigma _H\) depends on H.

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Acknowledgements

We would like to thank the anonymous referee for valuable comments that improved the quality of the paper. The research of J.-H. Kim was supported by the National Research Foundation of Korea NRF2021R1A2C1004080.

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Authors and Affiliations

  1. Department of Financial Engineering, Ajou University, Suwon, 16499, Republic of Korea

    Hyun-Gyoon Kim

  2. Department of Mathematics, Yonsei University, Seoul, 03722, Republic of Korea

    So-Yoon Cho & Jeong-Hoon Kim

  3. Financial Supervisory Service, Seoul, 07321, Republic of Korea

    So-Yoon Cho

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  1. Hyun-Gyoon Kim
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Correspondence to Jeong-Hoon Kim.

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Kim, HG., Cho, SY. & Kim, JH. A martingale method for option pricing under a CEV-based fast-varying fractional stochastic volatility model. Comp. Appl. Math. 42, 296 (2023). https://doi.org/10.1007/s40314-023-02432-5

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  • DOI: https://doi.org/10.1007/s40314-023-02432-5

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