Toward an efficient hybrid method for pricing barrier options on assets with stochastic volatility

Abstract

Alexander Lipton and Artur Sepp 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. The authors� approach to barrier options valuation utilizes two loops. First, the authors run the outer loop by generating volatility paths via the Monte Carlo method. Second, the authors 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. Lipton and Sepp illustrate the accuracy and efficacy of their 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. The authors apply their method for computation of state probabilities (Green's function), survival probabilities, and values of call options with barriers. The approach provides better accuracy and is orders of magnitude faster than the existing methods. As a by-product of their analysis, Lipton and Sepp generalize Willard's (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.