Asymptotic expansion for some local volatility models arising in finance

TL;DR

This paper develops small noise asymptotic expansions for local volatility models in finance, providing explicit coefficients, error estimates, and numerical validation through Monte Carlo and Polynomial Chaos methods.

Contribution

It introduces explicit formulas and error bounds for asymptotic expansions in local volatility models, along with comprehensive numerical comparisons.

Findings

Explicit coefficients for asymptotic expansions derived.

Accurate error estimates provided for the expansions.

Numerical analysis confirms the effectiveness of the methods.

Abstract

In this paper we study the small noise asymptotic expansions for certain classes of local volatility models arising in finance. We provide explicit expressions for the involved coefficients as well as accurate estimates on the remainders. Moreover, we perform a detailed numerical analysis, with accuracy comparisons, of the obtained results by mean of the standard Monte Carlo technique as well as exploiting the polynomial Chaos Expansion approach.