# 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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