Finite Difference Solution Ansatz approach in Least-Squares Monte Carlo

Abstract: This article presents a simple but effective and efficient approach to improve the accuracy and stability of Least-Squares Monte Carlo. The key idea is to construct the ansatz of conditional expected continuation payoff using the finite difference solution from one dimension, to be used in linear regression. This approach bridges between solving backward partial differential equations and Monte Carlo simulation, aiming at achieving the best of both worlds. In a general setting encompassing both local and stochastic volatility models, the ansatz is proven to act as a control variate, reducing the mean squared error, thereby leading to a reduction of the final pricing error. We illustrate the technique with realistic examples including Bermudan options, worst of issuer callable notes and expected positive exposure on European options under valuation adjustments.