Smart expansion and fast calibration for jump diffusion

TL;DR

This paper introduces an analytical formula for European option pricing in jump diffusion models using Malliavin calculus, enabling fast calibration and high accuracy, especially for small jumps and diffusions.

Contribution

It develops a novel asymptotic expansion approach for jump diffusion models, improving calibration speed and pricing accuracy for European options.

Findings

Error in implied volatility less than 2 basis points

Calibration process is significantly faster

High accuracy across various strikes and maturities

Abstract

Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black-Scholes volatilities for call option is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.