  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 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.
