Probability density of lognormal fractional SABR model

TL;DR

This paper derives a small-time asymptotic expansion for the probability density of the lognormal fractional SABR model, addressing the complex interplay of Brownian and fractional Brownian motions in financial modeling.

Contribution

It introduces a bridge representation for the joint density in Fourier space and extends it to multiple times, providing new analytical tools for fractional SABR models.

Findings

Derived a bridge representation for the joint density in Fourier space.

Obtained a small-time asymptotic expansion for the probability density.

Provided a heuristic derivation of the large deviations principle for small-time joint density.

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.