An analytical perturbative solution to the Merton Garman model using symmetries

TL;DR

This paper develops an analytical perturbative approach to solve the Merton Garman model by leveraging symmetries, demonstrating its effectiveness through comparison with Monte Carlo simulations and highlighting the role of symmetry in financial modeling.

Contribution

It introduces a novel perturbative solution to the Merton Garman model based on Galilean symmetry, expanding the analytical tools available for option pricing.

Findings

Perturbative solution matches Monte Carlo results across various parameters.

Symmetry-based methods can effectively construct option pricing models.

The approach offers a new analytical perspective in financial mathematics.

Abstract

In this paper, we introduce an analytical perturbative solution to the Merton Garman model. It is obtained by doing perturbation theory around the exact analytical solution of a model which possesses a two-dimensional Galilean symmetry. We compare our perturbative solution of the Merton Garman model to Monte Carlo simulations and find that our solutions performs surprisingly well for a wide range of parameters. We also show how to use symmetries to build option pricing models. Our results demonstrate that the concept of symmetry is important in mathematical finance.