A Monte Carlo method for estimating sensitivities of reflected diffusions in convex polyhedral domains

TL;DR

This paper introduces a Monte Carlo method based on infinitesimal perturbation analysis for efficiently estimating sensitivities of reflected diffusions in convex polyhedral domains, with applications to financial models.

Contribution

It develops a new unbiased estimator for sensitivities using Euler approximation and proves convergence, improving variance reduction over traditional likelihood ratio methods.

Findings

The proposed estimator has substantially lower variance than likelihood ratio estimators.

Numerical experiments demonstrate the estimator's effectiveness in financial diffusion models.

The method outperforms finite difference approaches in rank-based equity market models.

Abstract

In this work we develop an effective Monte Carlo method for estimating sensitivities, or gradients of expectations of sufficiently smooth functionals, of a reflected diffusion in a convex polyhedral domain with respect to its defining parameters --- namely, its initial condition, drift and diffusion coefficients, and directions of reflection. Our method, which falls into the class of infinitesimal perturbation analysis (IPA) methods, uses a probabilistic representation for such sensitivities as the expectation of a functional of the reflected diffusion and its associated derivative process. The latter process is the unique solution to a constrained linear stochastic differential equation with jumps whose coefficients, domain and directions of reflection are modulated by the reflected diffusion. We propose an asymptotically unbiased estimator for such sensitivities using an Euler approximation of the reflected diffusion and its associated derivative process. Proving that the Euler approximation converges is challenging because the derivative process jumps whenever the reflected diffusion hits the boundary (of the domain). A key step in the proof is establishing a continuity property of the related derivative map, which is of independent interest. We compare the performance of our IPA estimator to a standard likelihood ratio estimator (whenever the latter is applicable), and provide numerical evidence that the variance of the former is substantially smaller than that of the latter. We illustrate our method with an example of a rank-based interacting diffusion model of equity markets. Interestingly, we show that estimating certain sensitivities of the rank-based interacting diffusion model using our method for a reflected Brownian motion description of the model outperforms a finite difference method for a stochastic differential equation description of the model.