Chebyshev Greeks: Smoothing Gamma without Bias

TL;DR

This paper introduces a Chebyshev interpolation method to improve the stability and accuracy of second order Greeks, like Gamma, in Monte Carlo simulations, addressing a key challenge in financial risk management.

Contribution

It presents a novel application of Chebyshev interpolation to enhance the computation of Greeks, especially Gamma, providing a simple and general alternative to finite differences.

Findings

Improved stability of Greeks computation using Chebyshev techniques

Enhanced accuracy of Gamma estimates in Monte Carlo simulations

Applicable to various real-world financial payoffs

Abstract

The computation of Greeks is a fundamental task for risk managing of financial instruments. The standard approach to their numerical evaluation is via finite differences. Most exotic derivatives are priced via Monte Carlo simulation: in these cases, it is hard to find a fast and accurate approximation of Greeks, mainly because of the need of a tradeoff between bias and variance. Recent improvements in Greeks computation, such as Adjoint Algorithmic Differentiation, are unfortunately uneffective on second order Greeks (such as Gamma), which are plagued by the most significant instabilities, so that a viable alternative to standard finite differences is still lacking. We apply Chebyshev interpolation techniques to the computation of spot Greeks, showing how to improve the stability of finite difference Greeks of arbitrary order, in a simple and general way. The increased performance of the proposed technique is analyzed for a number of real payoffs commonly traded by financial institutions.