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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

    as
    1. Baillie, Richard T. & Bollerslev, Tim & Mikkelsen, Hans Ole, 1996. "Fractionally integrated generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 74(1), pages 3-30, September.
    2. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
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