Closed form asymptotics for local volatility models

TL;DR

This paper introduces new closed-form formulas for pricing options under local volatility models using Dyson-Taylor methods, enabling efficient and accurate calculations for both short and long maturities.

Contribution

It develops a novel application of Dyson-Taylor commutator techniques to derive explicit asymptotic formulas for local volatility models, extending their validity to larger times.

Findings

Derived explicit formulas for local volatility option pricing

Validated accuracy through numerical error analysis

Compared results favorably with existing methods

Abstract

We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5, 6, 8] for short-time asymptotic expansions of heat kernels, and obtain a family of general closed-form approximate solutions for both the pricing kernel and derivative price. A bootstrap scheme allows us to extend our method to large time. We also perform analytic as well as a numerical error analysis, and compare our results to other known methods.