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

6.5. Pricing OEX Options

Once the joint p.d.f.’s are obtained, it is straightforward to program pricing of OEX options by Richardson
extrapolation (13). The two corresponding options are: Di, the value of a Europeamstyle derivative security,
and D₂, the value of a “Bermudan” security with one intermittent exercise date at tɪ (half-way to expiration).
In turn, D₂ is calculated according to (12), where the integral, E [max {∙}], is approximated in the simplest
non-adaptive way on the equispaced and fixed grid of (s,υ).

On a Birch Linux cluster node, pricing of one option takes approximately 11 minutes (including the time
spent on reading-in a p.d.f. matrix from a corresponding saved file).

Consider Table 4, where I report signed pricing errors of all traded OEX options that expire in September.
Pricing error is the difference between the actual CBOE last sale value and its corresponding predicted value.

Clearly, a perfect fit can never be attained. Still, it is remarkable that with several layers of approximation,
pricing errors look more or less reasonable. For September-expiring options, instances when the absolute
pricing error is in excess of 2.00 are rare and most predicted option values do not deviate from CBOE last
sale values by more than 1.00. The method seems to underprice far out-of-the-money puts and deep-in-the-
money calls. Apart from that, it is hard to find a consistent trend in pricing errors across strikes. Out-of-
the-sample pricing errors (July 9^th ) are not particularly different from their in-the-sample counterparts (June
30^th through July 8^th ).

To assess how pricing accuracy varies with the time to expiration, examine Table 5, where I report root
MSE’s for almost all traded OEX options that mature in 2004, by trading day and expiration month. There
is no clear indication that pricing errors decrease on average for options with longer maturities. On the
contrary, with two exceptions (July 7^th and July 8^th), predictions for December-expiring derivatives are
relatively coarse. Probably, one should expect the precision of the linear Richarson extrapolation to degrade
as the time to maturity of an American-style security increases. Overall, ∖∕MSE’s are of sensible magnitudes.

7. Conclusion

In this paper, I consider valuation of the American-style derivative securities when the price of an underlying
asset follows the dynamics of the Heston model. As a feasible alternative to popular FD techniques, I employ
a version of the Geske-Johnson scheme with linear Richardson extrapolation. The method requires knowledge
of the joint p.d.f. of the future log-price, s, and squared volatility, v, to price the “Bermudan” option with
one intermittent early exercise date half-way to expiration.

The joint p.d.f. can be recovered by inverting the corresponding joint ch.f. Unfortunately, the pro-
nouncedly oscillatory nature of the integrand function makes accurate bivariate numerical integration pro-

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