###### 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 *B ^{Call}*() denotes Black’s (1976) pricing formula for call options on futures and

[9.39]

Based upon current and historical market data, assume ^{1}** R **~

^{0}

*N*

_{3}(

^{1|0}

**μ**,

^{ 1|0}

**Σ**) where

[9.40]

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

[9.41]

Realization ^{1}*r*^{[10]} is set equal to ^{1|0}**μ**. We choose to weight this three times the other realizations in the least squares analysis. This is accomplished by adding two redundant realizations ^{1}*r*^{[11]} = ^{1|0}**μ** and ^{1}*r*^{[12]} = ^{1|0}**μ**. The remaining nine realizations, ^{1}*r*^{[1]} through ^{1}*r*^{[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 α_{1} = .001 and α_{2} = 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 ^{1}*r*^{[k]}. These, with the three realizations ^{1}*r*^{[10]}, ^{1}*r*^{[11]}, and ^{1}*r*^{[12]} at ^{1|0}**μ**, are indicated in Exhibit 9.23. Corresponding portfolio values ^{1}*p*^{[k]} = θ(^{1}*r*^{[k]}) are also indicated.

^{1}

*r*^{[k]}are employed. The realization equal to

^{1|0}

**μ**is repeated three times so it will be weighted more heavily in the least squares analysis. Corresponding portfolio values

^{1}

*p*

^{[k]}= θ(

^{1}

*r*^{[k]}) are also indicated.

Applying ordinary least squares, we fit quadratic polynomial [9.41] to the points (^{1}*r*^{[k]}, ^{1}*p*^{[k]}), obtaining coefficient values

[9.42]

To assess how well approximates ^{1}*P*, we generate 1,000 pseudorandom realizations ^{1}*r*^{[k]} of ^{1}** R** based upon a conditional distribution for

^{1}

**. We determine corresponding realizations**

*R*^{1}

*p*

^{[k]}= θ(

^{1}

**r**^{[k]}) and = (

^{1}

**r**^{[k]}) for

^{1}

*P*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.

^{1}

*P*as some of the holdings remappings we considered earlier.