Pricing Derivatives on Multiscale Diffusions: an Eigenfunction Expansion Approach

TL;DR

This paper introduces a spectral analysis-based method for analytically approximating derivative prices in complex multiscale stochastic volatility models, simplifying the computation to solving a single eigenvalue problem.

Contribution

It develops a systematic eigenfunction expansion approach for pricing derivatives with multiscale stochastic volatility and default features, enhancing analytical tractability.

Findings

Successfully applied to vanilla options, path-dependent options, and bonds.

Provides explicit formulas for derivative prices in complex models.

Demonstrates accuracy and efficiency of the eigenfunction expansion method.

Abstract

Using tools from spectral analysis, singular and regular perturbation theory, we develop a systematic method for analytically computing the approximate price of a derivative-asset. The payoff of the derivative-asset may be path-dependent. Additionally, the process underlying the derivative may exhibit killing (i.e. jump to default) as well as combined local/nonlocal stochastic volatility. The nonlocal component of volatility is multiscale, in the sense that it is driven by one fast-varying and one slow-varying factor. The flexibility of our modeling framework is contrasted by the simplicity of our method. We reduce the derivative pricing problem to that of solving a single eigenvalue equation. Once the eigenvalue equation is solved, the approximate price of a derivative can be calculated formulaically. To illustrate our method, we calculate the approximate price of three derivative-assets: a vanilla option on a defaultable stock, a path-dependent option on a non-defaultable stock, and a bond in a short-rate model.