Pricing American-style Derivatives under the Heston Model Dynamics: A Fast Fourier Transformation in the Geske–Johnson Scheme



1. Introduction

Theoretical research on option valuation tends to focus on pricing the plain-vanilla European-Style derivatives.
Duffie, Pan, and Singleton (2000) showed that such options can be priced by transform methods whenever
the state vector (which includes functions of asset prices, unobserved volatilities, etc.) follows a multivariate
Gaussian-Poisson affine jump-diffusion. As a result, for a wide class of pricing problems a general solution
method has been found.

Contrastingly, no universal and analytically attractive approach is yet available for the American-style
derivatives. Still, most traded equity and FX-rate derivatives are the American-style ones. Accurate and
efficient pricing of such options is of a significant practical value.

Stochastic volatility models have been proposed in the hope of remedying the strike-price biases in option
valuation by the Black-Scholes formula. A model due to Heston (1993) has received considerable attention
in the literature. Heston’s original method has been modified and simplified by other scholars to deliver a
very efficient formula for the European-style puts. To price the American-style derivatives in the two-state-
variable setting of the model, authoritative sources (for instance, Wilmott, 2000) strongly suggest using the
finite difference (FD) schemes. FD methods, in which the partial differential equation (p.d.e.) in the value
function of a derivative security is approximated and solved for the initial option price numerically, are very
popular among practitioners and in academia. Applications of FD schemes for the Heston dynamics are
available (e.g., Winkler, 2001).

A number of efficient non-FD methods to price the American-style options have been proposed in the
context of the Black-Scholes model. A technique due to Broadie and Detemple (1996) is a smoothed binomial
scheme. The MacMillan-Barone-Adesi-Whaley approach relies on decomposing the value of an American-
style derivative into the value of a corresponding European-style option and early exercise premium. The
premium follows the fundamental p.d.e., which can be approximated by the 2nd-order ordinary differential
equation that is solved analytically. The Geske-Johnson scheme (1984) exploits the fact that an American-
style option is the limit of a sequence of “Bermudan” derivatives. The latter ones can be priced recursively
according to a simple formula.

In this paper, I adapt the Geske-Johnson method to the dynamics of the Heston model. As an empirical
test of the numerical accuracy of this approach, I consider pricing of the American-style S&P 100 index
options (OEX).

The rest of the paper is organized as follows. In Section 2, I state the assumption of the model, which are
used in Section 3, to derive the p.d.e. in the value function of a derivative security. Section 4 proceeds at a slow
pace from an analytical solution for the joint characteristic function (ch.f.) of log-price and squared volatility



More intriguing information

1. The name is absent
2. The name is absent
3. The name is absent
4. The name is absent
5. Family, social security and social insurance: General remarks and the present discussion in Germany as a case study
6. A Location Game On Disjoint Circles
7. The name is absent
8. Handling the measurement error problem by means of panel data: Moment methods applied on firm data
9. A Note on Productivity Change in European Co-operative Banks: The Luenberger Indicator Approach
10. Placenta ingestion by rats enhances y- and n-opioid antinociception, but suppresses A-opioid antinociception
11. I nnovative Surgical Technique in the Management of Vallecular Cyst
12. Death as a Fateful Moment? The Reflexive Individual and Scottish Funeral Practices
13. The name is absent
14. The name is absent
15. INTERACTION EFFECTS OF PROMOTION, RESEARCH, AND PRICE SUPPORT PROGRAMS FOR U.S. COTTON
16. Strategic Planning on the Local Level As a Factor of Rural Development in the Republic of Serbia
17. The name is absent
18. Whatever happened to competition in space agency procurement? The case of NASA
19. Pursuit of Competitive Advantages for Entrepreneurship: Development of Enterprise as a Learning Organization. International and Russian Experience
20. The name is absent