Modeling Rough Stochastic Volatility with Fractional Brownian Motion

Summary

This article relaxes the constant volatility assumption in Black–Scholes by allowing the asset's volatility to vary over time. It models log volatility with a mean reverting Ornstein–Uhlenbeck style equation driven by a stochastic process. To represent volatility's rapidly varying behavior, it introduces fractional Brownian motion, whose Hurst parameter controls path roughness; values below one half produce rougher paths than standard Brownian motion.

The rough fractional stochastic volatility model uses this process to drive log volatility. The article points to historical Apple price and volatility plots as motivation, and cites empirical work that found the model consistent with observed volatility, with an estimated Hurst exponent around 0.1. This is an introduction to a model specification and its motivation, not a full option pricing derivation or calibration procedure. The empirical claim is attributed to one cited paper, so the article alone does not establish how well the model performs across assets or market regimes.

Key ideas

• The Black–Scholes constant volatility assumption can be relaxed by making volatility stochastic.
• The model represents log volatility with a mean reverting process.
• Fractional Brownian motion provides a way to model paths with varying degrees of roughness.
• A Hurst parameter below one half corresponds to rougher paths than standard Brownian motion.
• The cited empirical study reports that rough fractional stochastic volatility is consistent with observed volatility.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.