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

Derivative security D_n can be represented as a polynomial of infinite order in h_n :

D_n — ao + aɪhn + a2hn + ∙ ∙ ∙ ■

Clearly, D_œ — ao and its linear approximation is:

^D1  —  ^αo^{+ α}1^h1            h₂D₁ — h₁D₂    „   „

> D, = ɪɪ--√ ² — 2D₂ — D_v

D₂  =  ao + a1h₂               ^{h2 — h1}

As noted, D₁ can be obtained as the value of the corresponding Europeamstyle option. D2 can be calcu-
lated from recursion (12). This task is sufficiently difficult per se, so, quadratic or higher order approximations
for D_œ are not feasible.

6. Empirical Application

6.1. Data

The tools developed in the preceding sections are used to predict prices of S&P 100 index options.

The S&P U.S. 100, a subset of the S&P 500, is comprised of 100 leading U.S. stocks, which together
represent almost 45 percent of the market capitalization of the U.S. equity market. As of June 2004, the
5 largest companies included were: General Electric, Exxon Mobil Corp., Microsoft Corp., Pfizer Inc., and
Citigroup Inc. The S&P 100 index was originally developed by the CBOE and later transfered to Standard
& Poor’s for management.

The CBOE offers three distinct options on S&P 100. Two of these have relevance to this paper. The
most popular one is the American-style S&P 100 index option, commonly known by its ticker OEX. As a
rule, on each trading day, OEX puts and calls have high trading volumes and open interest for a wide range
of strikes.

Less popular, but still actively traded at CBOE is the European-style S&P 100 index option, ticker
XEO. Both OEX and XEO are cash-settled, and apart from the difference in the exercise style, share same
characteristics (exercise dates, minimal strike intervals, minimum ticks, etc.).

I collected closing CBOE prices of both OEX and XEO on 7 consecutive trading days: June 30^th - July
2^nd, July 6^th - July 9^th. Only options with positive trading volume and open interest are used in estimations.
July 9^th data is set aside for out-of-the-sample predictions.

As a proxy for the risk-free interest rate I use the T-bill rate. Rates for different times to maturity are
obtained by interpolation: linear OLS fit to the whole set of T-bill quotes reported in the Wall Street Journal

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