An unbiased Monte Carlo estimator for derivatives. Application to CIR

TL;DR

This paper introduces an unbiased Monte Carlo method for estimating derivatives of expectations related to stochastic processes, specifically applied to the CIR model, improving accuracy and efficiency over traditional methods.

Contribution

The paper develops a novel unbiased estimator for derivatives of expectations in stochastic processes, with extensions to multi-dimensional cases and practical application to the CIR process.

Findings

The unbiased estimator accurately computes derivatives in the CIR model.

The method outperforms classical approximation procedures in numerical tests.

Bias control is exponentially small with respect to the truncation parameter.

Abstract

In this paper, we present extensions of the exact simulation algorithm introduced by Beskos et al. (2006). First, a modification in the order in which the simulation is done accelerates the algorithm. In addition, we propose a truncated version of the modified algorithm. We obtain a control of the bias of this last version, exponentially small in function of the truncation parameter. Then, we extend it to more general drift functions. Our main result is an unbiased algorithm to approximate the two first derivatives with respect to the initial condition \(x\) of quantities with the form \(\mathbb{E}\Psi(X_T^x)\). We describe it in details in dimension 1 and also discuss its multi-dimensional extensions for the evaluation of \(\mathbb{E}\Psi(X_T^x)\). Finally, we apply the algorithm to the CIR process and perform numerical tests to compare it with classical approximation procedures.