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

Samaneh Sojudi, Mahdieh Tahmasebi

TL;DR

This work develops a Malliavin-calculus-based sensitivity analysis for mean-field SDEs with jumps, establishing existence and Malliavin differentiability of solutions in Wiener-Poisson space and deriving a chaos-expansion-based Delta formula that does not rely on the chain rule. It provides explicit Malliavin weight constructions for computing Delta of path-dependent payoffs, including European calls and barrier options, and proves convergence of Euler-based Delta approximations. The approach yields variance reduction compared with finite-difference methods and demonstrates practical efficacy through numerical experiments on jump-diffusion McKean-Vlasov models. The results advance risk management and derivative pricing in complex systems where coefficients depend on both the state and its law, under discontinuous dynamics.

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.

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

TL;DR

This work develops a Malliavin-calculus-based sensitivity analysis for mean-field SDEs with jumps, establishing existence and Malliavin differentiability of solutions in Wiener-Poisson space and deriving a chaos-expansion-based Delta formula that does not rely on the chain rule. It provides explicit Malliavin weight constructions for computing Delta of path-dependent payoffs, including European calls and barrier options, and proves convergence of Euler-based Delta approximations. The approach yields variance reduction compared with finite-difference methods and demonstrates practical efficacy through numerical experiments on jump-diffusion McKean-Vlasov models. The results advance risk management and derivative pricing in complex systems where coefficients depend on both the state and its law, under discontinuous dynamics.

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.

Paper Structure

This paper contains 11 sections, 22 theorems, 177 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mean-Field SDEs with jump
  4. Stochastic flow and weak derivative of the solution
  5. Existence of stochastic flow and its expression
  6. Relationship between $\mathit{D_{t,z}}$ and flow in semi-linear case
  7. Applications
  8. Numerical Scheme and the convergence of Delta
  9. European call option
  10. Barrier Options
  11. Numerical experiments

Key Result

Theorem 2.1

Let $\mathit{F} \in {{\mathit{L}}^2}(\mathit{P})$ be a ${\mathcal{F}}_T$-measurable random variable. Then $\mathit{F}$ admits the representation via a unique sequence of elements ${{\mathit{f}}_n \in {\tilde{L}}^2 ((\lambda \times \mu)^n), n = 1, 2, ... .}$ Here we set ${{\mathit{I}}_0}({{\mathit{f}}_0}) := {\mathit{f}}_0$ for the constant values ${\mathit{f}}_0 \in {{\mathbb{R}}_0}$. Moreover, w

Figures (8)

  • Figure 1: On top parameters set to $a= b= c= 1$, $\lambda = 0.1$, $T=1$, Example \ref{['E1']}.
  • Figure 2: Comparison of Malliavin calculus with Central Finite Difference method for European Call Options. On top parameters set to $a= b= c= 1$, $T=1$ and h=0.001 for Central Finite Difference, Example \ref{['E1']}.
  • Figure 3: Comparison of Malliavin calculus with Central Finite Difference method for Barrier Up and Out Options. On top parameters set to $a= b = c= 1$, $T=1$ and h=0.001 for Central Finite Difference, Example \ref{['E1']}.
  • Figure 4: Comparison Convergence rate of Malliavin Method with Exact solution, Example \ref{['E1']}
  • Figure 5: On top parameters set to $a= -1, b=c=1$, $T=1$, Example \ref{['E2']}.
  • ...and 3 more figures

Theorems & Definitions (38)

  • Theorem 2.1: nunno2008malliavin Theorem 10.2
  • Proposition 2.2: nunno2008malliavin Theorem 12.7
  • Proposition 2.3: nunno2008malliavin Theorem 12.8
  • Lemma 2.4: nualart2018introduction Lemma 10.2.4
  • Proposition 2.5: nunno2008malliavin Theorem 12.10
  • Proposition 2.6: nunno2008malliavin Theorem 12.15
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • ...and 28 more