Probability Density of Lognormal Fractional SABR Model

Jiro Akahori, Xiaoming Song and Tai-Ho Wang
Additional contact information
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

Risks, 2022, vol. 10, issue 8, 1-27

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.

Keywords: asymptotic expansion; lognormal fractional SABR model; mixed fractional Brownian motion; Malliavin calculus; bridge representation (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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Related works:
Working Paper: Probability density of lognormal fractional SABR model (2019)
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