Conditional Leibniz Derivative Estimation with an Application to American Call Min-Options

2026-06-25

AI Summary

This work addresses the high variance of parameter gradient estimators in stochastic models with discontinuous samples—a variance that grows linearly with input dimensionality. To mitigate this issue, the authors propose a novel approach that integrates recursive conditioning within the Leibniz rule–based differentiation framework. The resulting conditional Leibniz estimator eliminates the likelihood ratio term, thereby removing the estimator’s dependence on input dimension and substantially reducing variance while retaining a simple, implementable form. Empirical evaluation on a simulated American call min-option model demonstrates that the proposed method achieves markedly lower variance and highly accurate gradient estimates compared to existing techniques.

Abstract

Leibniz derivative estimation is a Monte Carlo technique for estimating derivatives of a discontinuous sample performance in stochastic models with respect to parameters of interest. By combining the push-out likelihood ratio (LR) method with Leibniz integral rules, it generalizes a broad class of existing LR-based derivative estimators. However, as an LR-based method, its variance is often higher than that of perturbation analysis-based methods and may grow linearly with the dimension of the stochastic input whose distribution depends on the parameter. In this paper, we propose a recursive conditioning approach and combine it with the Leibniz derivative estimation framework. The resulting conditional Leibniz estimator does not involve LR terms and therefore is not subject to variance growth with the input dimension. It also has a simple form and is easy to implement. We apply the method to an American call min-option model, and simulation results show its effectiveness and low-variance performance.

Problem

• Leibniz derivative estimation
• variance reduction
• discontinuous performance
• likelihood ratio
• stochastic models

Innovation

• Conditional Leibniz Estimator
• Recursive Conditioning
• Variance Reduction
• Monte Carlo Derivative Estimation
• American Min-Option