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

and Ghysels claim that other existing filtration schemes (extensions of Kalman filter) cannot accommodate
derivative security market information.

Unmodified MLE is, clearly, not feasible. To my best knowledge, attempts are being undertaken elsewhere
to reduce the computational burden of the Fourier inversion (e.g., by approximating the integrand function
with an easy-to-integrate function, by using quasi-Monte-Carlo integration schemes, etc.).

I take a different route by literally calibrating, rather than estimating, the parameters. This is done by
minimizing the sum of squared differences of actual XEO put prices and their corresponding predicted values
on June 30^th through July 8^th data (134 options). As the parameters are specific to the price evolution of
the underlying asset, there is no need to use OEX data at this stage.

Since volatility is changing from day to day, the objective function, SSQR, is treated as dependent on ten
parameters: u_tj0₁,1_t,02, ι_t,03, ι_t,04, ι_t,05, ι_t,06, ɑ, β, 7, p. Parameters a,β,7,p are common across all options.
Parameters v_t,## are “day-specific”, that is, u_tj0₁ is υ_t at market’s closingon June 30^th,..., t⅛_j06 is v_t at
market’s closing on July 8^th.

Previous research has shown that the objective function is difficult to minimize as local extrema abound.
Therefore, I use a relatively powerful, but resource consuming simulated annealing algorithm. The objective
function is set to a “penalty” value once the trial parameter set violates 7² ≤ 2α. The algorithm allows to
customize the lower and upper bounds for all parameters. For p, the bounds are set to theoretical —1 and 1.
For the remaining nine parameters, the bounds are practically unrestricted.

Optimal parameter values are presented in Table 3. A few facts are worth noting. First, in the optimum,
the restriction 7² ≤ 2a is non-binding. This important result guarantees that v_t ∈ (0, ∞) almost surely.
Second, estimated v_tj##’s are of the same order of magnitude as the long-run value of υ, ^. Third, calibrated
p < 0. Loosely speaking, this conforms to the empirical observation that the variance of log-returns is
inversely related to the initial price level. So, I conclude that the calibrated parameters make sense.

To make out-of-the-sample predictions, v_tj0r (at market’s closing time on July 9^th) is set to: v_tj0r =
v_t,06 + (« — βvt,06~) ʌt = 0.015848, where ʌ = ₃6¹₆.

6.3. Ch.F.

The shape of the integrand function, c ^ddl"'^r+^2^vτ) . ψ (ζ_ι,ζ₂), has important implications for FFT. As a
rough illustration, consider Figures 1 and 2. In these Figures, I plot the real part of the integrand function
with parameters as of June 30. In view of result (10), for simplicity, I look at a special case of sγ = vγ = 0,
that is, when the integrand is identically the ch.f. Also, I restrict attention to just 2 values of τ: a “small”
one, which corresponds to ∣ of the time span (in years) between June 30 and 3^rd Friday of July, and a “large”
one, which stands for ∣ of the time till 3^rd Friday of December.

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