Sensitivity Analysis for Mean-Field SDEs With Jump By Malliavin Calculus: Chaos Expansion Approach

TL;DR

This paper develops an explicit sensitivity analysis method for jump-diffusion mean-field SDEs using Malliavin calculus and chaos expansion, improving computational efficiency for financial derivatives.

Contribution

It introduces a novel explicit extension formula for sensitivity analysis of jump-diffusion mean-field SDEs with a chaos approach, and proves convergence of the Euler method for Delta approximation.

Findings

Malliavin derivatives are effectively defined in a chaos framework.

The Euler method converges for approximating Delta Greek.

Malliavin Monte-Carlo method outperforms finite difference in efficiency.

Abstract

In this paper, we describe an explicit extension formula in sensitivity analysis regarding the Malliavin weight for jump-diffusion mean-field stochastic differential equations whose local Lipschitz drift coefficients are influenced by the product of the solution and its law. We state that these extended equations have unique Malliavin differentiable solutions in Wiener-Poisson space and establish the sensitivity analysis of path-dependent discontinuous payoff functions. It will be realized after finding a relation between the stochastic flow of the solutions and their derivatives. The Malliavin derivatives are defined in a chaos approach in which the chain rule is not held. The convergence of the Euler method to approximate Delta Greek is proved. The simulation experiment illustrates our results to compute the Delta, in the context of financial mathematics, and demonstrates that the Malliavin Monte-Carlo computations applied in our formula are more efficient than using the finite difference method directly.