Spectral Decomposition of Option Prices in Fast Mean-Reverting Stochastic Volatility Models

TL;DR

This paper introduces a spectral decomposition approach combined with singular perturbation theory to efficiently approximate prices of various options in fast mean-reverting stochastic volatility models, demonstrating versatility across multiple option types.

Contribution

The paper develops a systematic spectral decomposition method for option pricing in fast mean-reverting stochastic volatility models, extending previous approaches to a broader class of options.

Findings

Accurate approximations for European options matching existing methods

Versatile application to up-and-out and double-barrier options

Effective pricing for rebate options upon hitting boundaries

Abstract

Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of options in a fast mean-reverting stochastic volatility setting. Four examples are provided in order to demonstrate the versatility of our method. These include: European options, up-and-out options, double-barrier knock-out options, and options which pay a rebate upon hitting a boundary. For European options, our method is shown to produce option price approximations which are equivalent to those developed in [5]. [5] Jean-Pierre Fouque, George Papanicolaou, and Sircar Ronnie. Derivatives in Financial Markets with Stochas- tic Volatility. Cambridge University Press, 2000.