An unbiased Monte Carlo estimator for derivatives. Application to CIR

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.