A Fast Mean-Reverting Correction to Heston's Stochastic Volatility Model

TL;DR

This paper introduces a multi-scale stochastic volatility model that extends Heston's model with a fast mean-reverting factor, providing semi-analytic pricing formulas that improve fit to implied volatility surfaces without increasing computational complexity.

Contribution

A novel multi-scale stochastic volatility model with a fast mean-reverting factor built on Heston's model, offering semi-analytic pricing formulas and enhanced flexibility for fitting implied volatility.

Findings

Computational complexity similar to Heston model

Significant improvement in fitting implied volatility surface

Numerical and empirical validation of the model

Abstract

We propose a multi-scale stochastic volatility model in which a fast mean-reverting factor of volatility is built on top of the Heston stochastic volatility model. A singular pertubative expansion is then used to obtain an approximation for European option prices. The resulting pricing formulas are semi-analytic, in the sense that they can be expressed as integrals. Difficulties associated with the numerical evaluation of these integrals are discussed, and techniques for avoiding these difficulties are provided. Overall, it is shown that computational complexity for our model is comparable to the case of a pure Heston model, but our correction brings significant flexibility in terms of fitting to the implied volatility surface. This is illustrated numerically and with option data.