A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model ^*

Luca Vincenzo Ballestra
Dipartimento di Scienze Sociali ``D. Serrani'',
Università Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy,
Ph. N. 071-2207251, FAX N. 071-2207150, E-mail: ballestra@posta.econ.unian.it
Graziella Pacelli
Dipartimento di Scienze Sociali ``D. Serrani'',
Università Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy,
Ph. N. 071-2207050, FAX N. 071-2207150, E-mail: g.pacelli@univpm.it
Francesco Zirilli
Dipartimento di Matematica ``G. Castelnuovo'',
Università di Roma ``La Sapienza'', Piazzale Aldo Moro 2, 00185 Roma, Italy,
Ph. N. 06-49913282, FAX N. 06-44701007, E-mail: f.zirilli@caspur.it

Abstract

We consider the problem of pricing European exotic path-dependent derivatives on an underlying described by the Heston stochastic volatility model. Lipton has found a closed form integral representation of the joint transition probability density function of underlying price and variance in the Heston model. We give a convenient numerical approximation of this formula and we use the obtained approximated transition probability density function to price discrete path-dependent options as discounted expectations. The expected value of the payoff is calculated evaluating an integral with the Monte Carlo method, using a variance reduction technique based on a suitable approximation of the transition probability density function of the Heston model. As a test case, we evaluate the price of a discrete arithmetic average Asian option, when the average over n = 12 prices is considered, that is when the integral to evaluate is a 2n = 24 dimensional integral. We show that the method proposed is computationally efficient and gives accurate results.

JEL Classification Codes: G13, C63.
Key Words: stochastic volatility, Heston model, path-dependent options, Monte Carlo integration.

* A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model, Journal of Banking and Finance 31 (2007) 3420-3437.

The numerical experience reported in this paper has been obtained using the computing grid of Enea (Roma, Italy). The support and sponsorship of Enea is gratefully acknowledged.

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