Efficient Counterfactual Estimation of Conditional Greeks via Malliavin-based Weak Derivatives

TL;DR

This paper introduces a Malliavin calculus-based method for efficiently estimating conditional Greeks in diffusion processes, especially in rare-event scenarios, overcoming limitations of traditional Monte Carlo and kernel smoothing techniques.

Contribution

The authors develop a novel kernel-free approach using weak derivatives and Skorohod integrals for efficient counterfactual gradient estimation in diffusion models.

Findings

Achieves constant variance in gradient estimates for rare events.

Provides a faster convergence rate compared to kernel smoothing methods.

Enables efficient counterfactual analysis in financial models.

Abstract

We study counterfactual gradient estimation of conditional loss functionals of diffusion processes. In quantitative finance, these gradients are known as conditional Greeks: the sensitivity of expected market values, conditioned on some event of interest. The difficulty is that when the conditioning event has vanishing or zero probability, naive Monte Carlo estimators are prohibitively inefficient; kernel smoothing, though common, suffers from slow convergence. We propose a two-stage kernel-free methodology. First, we show using Malliavin calculus that the conditional loss functional of a diffusion process admits an exact representation as a Skorohod integral, yielding classical Monte-Carlo estimator variance and convergence rates. Second, we establish that a weak derivative estimate of the conditional loss functional with respect to model parameters can be evaluated algorithmically with constant variance, in contrast to the widely used score function method whose variance grows linearly in the sample path length. Together, these results yield an efficient framework for counterfactual conditional stochastic gradient algorithms and financial Greek computations in rare-event regimes.