Review of Quantitative Finance and Accounting, 26: 55–66, 2006
2006 Springer Science + Business Media, Inc. Manufactured in The Netherlands.
The GARCH Option Pricing Model: A Modiﬁcation
of Lattice Approach
Department of International Trade, Chung Yuan University, Chung-Li, Taiwan,
Tel: 886-3-2655217, Fax: 886-3-2655299,
Abstract. Ritchken and Trevor (1999) proposed a lattice approach for pricing American options under
discrete time-varying volatility GARCH frameworks. Even though the lattice approach worked well for the
pricing of the GARCH options, it was inappropriate when the option price was computed on the lattice using
standard backward recursive procedures, even if the concepts of Cakici and Topyan (2000) were incorporated.
This paper shows how to correct the deﬁciency and that with our adjustment, the lattice method performs
properly for option pricing under the GARCH process.
Key words: GARCH, American options, lattice algorithm, trinomial trees.
JEL Classiﬁcation: C10, C32, C51, F37, G12
Using an equilibrium argument, Duan (1995) illustrated that options can be priced when
the dynamics of the price of the underlying asset price follow a General Autoregressive
Conditionally Heteroskedastic (GARCH) process. Unfortunately, analytical solutions to
the price of options are generally not available and hence numerical procedures need to
be used. Ritchken and Trevor (1999) (hereafter, RT) introduced an efﬁcient numerical
procedure (a lattice approach) for pricing European and American options under discrete-
time GARCH processes. At almost the same time, Duan and Simonato (2001) proposed
another numerical algorithm (a Markov chain approach), which can also estimate the
The purpose of this paper is to provide constructive modiﬁcations to the methods
proposed by RT (1999), with Cakici and Topyan’s (2000) modiﬁcation incorporated.
The algorithm becomes more reasonable with some modiﬁcations.
2. The GARCH option pricing model
In RT (1999), they developed an efﬁcient lattice algorithm to price options under
NGARCH processes. It is assumed that the underlying security price satisﬁes the fol-
lowing diffusion process: