Option pricing under fast-varying long-memory stochastic volatility

TL;DR

This paper develops an analytical model for European option pricing where the underlying volatility exhibits long-memory and fast mean-reversion, capturing slow decay correlations observed in markets.

Contribution

It introduces a novel approach modeling volatility as a fractional Ornstein-Uhlenbeck process, deriving explicit formulas for option prices and implied volatility in this context.

Findings

Derived an analytical expression for option prices under long-memory volatility.

Identified a fractional term structure in implied volatility.

Validated the model in the fast mean-reversion regime.

Abstract

Recent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long-range correlation properties in order to capture such a situation, and we consider European option pricing. This means that the volatility process is neither a Markov process nor a martingale. However, by exploiting the fact that the price process is still a semimartingale and accordingly using the martingale method, we can obtain an analytical expression for the option price in the regime where the volatility process is fast mean-reverting. The volatility process is modeled as a smooth and bounded function of a fractional Ornstein-Uhlenbeck process. We give the expression for the implied volatility, which has a fractional term structure.