The Bismut-Elworthy-Li formula for mean-field stochastic differential equations

TL;DR

This paper extends the Bismut-Elworthy-Li formula to mean-field stochastic differential equations with law-dependent coefficients, demonstrating its application in finance for computing Greeks more efficiently than finite difference methods.

Contribution

The paper introduces a generalized Bismut-Elworthy-Li formula for mean-field SDEs, enabling more efficient derivative computations in finance.

Findings

The generalized formula applies to law-dependent SDEs.

Simulation shows improved efficiency over finite difference methods.

Potential applications in financial derivative pricing.

Abstract

We generalise the so-called Bismut-Elworthy-Li formula to a class of stochastic differential equations whose coefficients might depend on the law of the solution. We give some examples of where this formula can be applied to in the context of finance and the computation of Greeks and provide with a simple but rather illustrative simulation experiment showing that the use of the Bismut-Elworthy-Li formula, also known as Malliavin method, is more efficient compared to the finite difference method.