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

33 Pages Posted: 5 Apr 2022

See all articles by Alex Lipton

Alex Lipton

Hebrew University of Jerusalem; Massachusetts Institute of Technology (MIT)

Artur Sepp

Clearstar AG

Date Written: February 15, 2022


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

Suggested Citation

Lipton, Alex and Sepp, Artur, Toward an Efficient Hybrid Method for Pricing Barrier Options on Assets With Stochastic Volatility (February 15, 2022). Available at SSRN: or

Alex Lipton

Hebrew University of Jerusalem ( email )

Mount Scopus
Jerusalem, Jerusalem 91905

Massachusetts Institute of Technology (MIT) ( email )

77 Massachusetts Avenue
50 Memorial Drive
Cambridge, MA 02139-4307
United States

Artur Sepp (Contact Author)

Clearstar AG ( email )



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