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Probability density of lognormal fractional SABR model

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  • Jiro Akahori
  • Xiaoming Song
  • Tai-Ho Wang

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 to joint density at multiple times leads to a heuristic derivation of the large deviations principle for the joint density in small time. Approximation of implied volatility is readily obtained by applying the Laplace asymptotic formula to the call or put prices and comparing coefficients.

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  • Jiro Akahori & Xiaoming Song & Tai-Ho Wang, 2017. "Probability density of lognormal fractional SABR model," Papers 1702.08081, arXiv.org, revised Jan 2019.
    • Handle: RePEc:arx:papers:1702.08081
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    References listed on IDEAS

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    Cited by:

    1. 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).
    2. 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.
    3. Elisa Alos & Rupak Chatterjee & Sebastian Tudor & Tai-Ho Wang, 2018. "Target volatility option pricing in lognormal fractional SABR model," Papers 1801.08215, arXiv.org.

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