Exact simulation of the Hull and White stochastic volatility model

Abstract

We show how to simulate exactly the asset price and the variance under the Hull and White stochastic volatility model. We derive analytical formulas for the Laplace transform of the time integral of volatility conditional on the variance level at the endpoint of the time interval and the Laplace transform of integrated variance conditional on both integrated volatility and variance. Based on these results, we simulate the model through a nested-conditional factorization approach, where Laplace transforms are inverted through the (conditional) Fourier-cosine (COS) method. Under this model, our approach can be used to generate unbiased estimates for the price of derivatives instruments. We propose some variants of the exact simulation scheme for computing unbiased estimates of option prices and sensitivities, a difficult task in the Hull and White model. These variants also allow for a significant reduction in the Monte Carlo simulation estimator's variance (around 93-98%) and the computing time (around 22%) when pricing options. The performances of the proposed algorithms are compared with various benchmarks. Numerical results demonstrate the faster convergence rate of the error in our method, which achieves an $O\left(s^{-1/2}\right)$ convergence rate, where s is the total computational budget, largely outperforming the benchmark.