Correction to Black-Scholes formula due to fractional stochastic volatility

TL;DR

This paper analyzes how fractional stochastic volatility models, specifically using fractional Ornstein-Uhlenbeck processes, affect the implied volatility's term structure, revealing a fractional power dependence on maturity.

Contribution

It provides a rigorous analysis of implied volatility in fractional Ornstein-Uhlenbeck volatility models, highlighting the fractional power law dependence on maturity.

Findings

Implied volatility exhibits a fractional power law dependence on maturity.

The model captures long-range dependence in volatility correlations.

Analytical results connect fractional Ornstein-Uhlenbeck processes to observed market phenomena.

Abstract

Empirical studies show that the volatility may exhibit correlations that decay as a fractional power of the time offset. The paper presents a rigorous analysis for the case when the stationary stochastic volatility model is constructed in terms of a fractional Ornstein Uhlenbeck process to have such correlations. It is shown how the associated implied volatility has a term structure that is a function of maturity to a fractional power.