9.3.9 Example: Cocoa Options

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


where BCall() denotes Black’s (1976) pricing formula for call options on futures and


Based upon current and historical market data, assume 1R ~ 0N3(1|0μ, 1|0Σ) where


Let’s use ordinary least squares with 10 realizations 1r[k] to construct a quadratic remapping of the form


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.

Exhibit 9.22: Nine points distributed in a minimum energy configuration on the unit sphere.

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.

Exhibit 9.23: Ten distinct realizations 1r[k] are employed. The realization equal to 1|0μ is repeated three times so it will be weighted more heavily in the least squares analysis. 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


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.

Exhibit 9.24: A scatter plot indicates that our quadratic remapping does not provide as good an approximation for 1P as some of the holdings remappings we considered earlier.