Anna Maria Gambaro

Università degli Studi del Piemonte Orientale

Title

Advancing Complex Step Approximation

Authors

Milan Kumar Das, Anna Maria Gambaro, Ioannis Kyriakou, and Gianluca Fusai

Abstract

In this work, we introduce an extended complex step approximation (ECSA) for computing derivatives of expectations of discontinuous performance functions, covering derivatives of arbitrary order. The n-th order derivative is represented as an infinite series, for which we establish convergence and provide an upper bound on the truncation error. This formulation extends the classical complex step approximation (CSA) by moving beyond first-term approximations toward a complete infinite-series representation, and by proposing a novel method to evaluate expectations of discontinuous performance functions at complex arguments. We further embed this framework into a Monte Carlo simulation setting, analyzing both the variance properties of the stochastic derivative estimator and its empirical performance.