Finite-difference solution ansatz approach in least-squares Monte Carlo

Jiawei Huo

Abstract

This paper presents a simple but effective and efficient approach to improve the accuracy and stability of the least-squares Monte Carlo method. The key idea is to construct an ansatz for the conditional expected continuation payoff using the finite-difference solution from one dimension, to be used in linear regression. This approach acts as a bridge between solving backward partial differential equations and Monte Carlo simulation, aiming to achieve 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 in the final pricing error. We illustrate the technique with realistic examples including Bermudan options, worst-of issuer callable notes and the expected positive exposure on European options under valuation adjustments.