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.
Entry n.
4643