Toward an Efficient Hybrid Method for Pricing Barrier Options on Assets With Stochastic Volatility

Abstract

We combine the one-dimensional Monte Carlo simulation and the semi-analytical one-dimensional heat potential method to design an efficient technique for pricing barrier options on assets with correlated stochastic volatility. Our approach to barrier options valuation utilizes two loops. First we run the outer loop by generating volatility paths via the Monte Carlo method. Second, we condition the price dynamics on a given volatility path and apply the method of heat potentials to solve the conditional problem in closed-form in the inner loop. We illustrate the accuracy and efficacy of our semi-analytical approach by comparing it with the two-dimensional Monte Carlo simulation and a hybrid method, which combines the finite-difference technique for the inner loop and the Monte Carlo simulation for the outer loop.

We apply our method for computation of state probabilities (Green function), survival probabilities, and values of call options with barriers. Our approach provides better accuracy and is orders of magnitude faster than the existing methods. s a by-product of our analysis, we generalize Willards' (1997) conditioning formula for valuation of path-independent options to path-dependent options and derive a novel expression for the joint probability density for the value of drifted Brownian motion and its running minimum.

Keywords: barrier options, stochastic volatility, Heston model, heat potentials, semi-analytical solution, Volterra equation, Willards' formula