9.3.9 Example: Cocoa Options
Assume today’s date is September 28, 2001. Consider a 1-week value-at-risk horizon. A cocoa merchant holds 39 call options struck at 1,050 on the December 2001 future, which is trading at 1,077 USD/ton. The options expire in 35 actual days. Each future is for 10 tons. A primary mapping is
[9.38]
where BCall() denotes Black’s (1976) pricing formula for call options on futures and
[9.39]
Based upon current and historical market data, assume 1R ~ 0N3(1|0μ, 1|0Σ) where
[9.40]
Let’s use ordinary least squares with 10 realizations 1r[k] to construct a quadratic remapping of the form
[9.41]
Realization 1r[10] is set equal to 1|0μ. We choose to weight this three times the other realizations in the least squares analysis. This is accomplished by adding two redundant realizations 1r[11] = 1|0μ and 1r[12] = 1|0μ. The remaining nine realizations, 1r[1] through 1r[9], are selected using the minimum energy approach described in Section 9.3.8. We generate 9 pseudorandom points on the unit sphere and iteratively shift these until they achieve a minimum energy configuration. Computations employ α1 = .001 and α2 = 1/180. Results are indicated in Exhibit 9.22.
The points are projected onto ellipsoid [9.30] with constant q = 1. The result is 9 realizations 1r[k]. These, with the three realizations 1r[10], 1r[11], and 1r[12] at 1|0μ, are indicated in Exhibit 9.23. Corresponding portfolio values 1p[k] = θ(1r[k]) are also indicated.
Applying ordinary least squares, we fit quadratic polynomial [9.41] to the points (1r[k], 1p[k]), obtaining coefficient values
[9.42]
To assess how well 
approximates 1P, we generate 1,000 pseudorandom realizations 1r[k] of 1R based upon a conditional distribution for 1R. We determine corresponding realizations 1p[k] = θ(1r[k]) and 
= 
(1r[k]) for 1P and . These are plotted in the scatter diagram of Exhibit 9.24. Although points cluster around a line passing at a 45° angle through the center of the graph, the fit is not as good as in previous examples. This behavior is typical of quadratic remappings.
