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Probability Density of Lognormal Fractional SABR Model

Author

Listed:
  • Jiro Akahori

    (Department of Mathematical Sciences, Ritsumeikan University, Noji-Higashi 1-1-1, Kusatsu 525-8577, Shiga, Japan)

  • Xiaoming Song

    (Department of Mathematics, Drexel University, 32nd and Market Streets, Philadelphia, PA 19096, USA)

  • Tai-Ho Wang

    (Department of Mathematical Sciences, Ritsumeikan University, Noji-Higashi 1-1-1, Kusatsu 525-8577, Shiga, Japan
    Department of Mathematics, Baruch College, The City University of New York, 1 Bernard Baruch Way, New York, NY 10010, USA)

Abstract

Instantaneous volatility of logarithmic return in the lognormal fractional SABR model is driven by the exponentiation of a correlated fractional Brownian motion. Due to the mixed nature of driving Brownian and fractional Brownian motions, probability density for such a model is less studied in the literature. We show in this paper a bridge representation for the joint density of the lognormal fractional SABR model in a Fourier space. Evaluating the bridge representation along a properly chosen deterministic path yields a small time asymptotic expansion to the leading order for the probability density of the fractional SABR model. A direct generalization of the representation of joint density often leads to a heuristic derivation of the large deviations principle for joint density in a small time. Approximation of implied volatility is readily obtained by applying the Laplace asymptotic formula to the call or put prices and comparing coefficients.

Suggested Citation

  • Jiro Akahori & Xiaoming Song & Tai-Ho Wang, 2022. "Probability Density of Lognormal Fractional SABR Model," Risks, MDPI, vol. 10(8), pages 1-27, August.
    • Handle: RePEc:gam:jrisks:v:10:y:2022:i:8:p:156-:d:878430
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    References listed on IDEAS

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    Cited by:

    1. Elisa Alos & Rupak Chatterjee & Sebastian Tudor & Tai-Ho Wang, 2018. "Target volatility option pricing in lognormal fractional SABR model," Papers 1801.08215, arXiv.org.
    2. Tudor, Sebastian F. & Chatterjee, Rupak & Nguyen, Lac & Huang, Yuping, 2019. "Quantum systems for Monte Carlo methods and applications to fractional stochastic processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 534(C).
    3. Elisa Alòs & Kenichiro Shiraya, 2019. "Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach," Finance and Stochastics, Springer, vol. 23(2), pages 423-447, April.

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