 # 9.3.10 Exercises

###### Exercises
9.3

A forward portfolio has primary mapping

[9.43]

where 1R1 is a forward price and 1R2 is an interest rate. Use gradient approximation [9.23] at point

[9.44]

to construct a linear remapping of [9.43].

9.4

Prove that realization [9.31] is on ellipsoid [9.30].

9.5

Repeat our cocoa options example using ordinary interpolation to construct a remapping of form [9.41]. Do so according to the following steps:

1. Based upon the coefficients required for form [9.41], select five points pk on the unit sphere centered at 0, as described in Section 9.3.8.
2. Use formula [9.31] with q = 2 to project the points from item (a) onto ellipsoid [9.30]. This will yield five realizations 1r[k]. Add to these realization 1r = 1|0μ for a total of six realizations.
3. Use [9.38] to value 1P at each realization to obtain six points for interpolation (1r[k], 1p[k]). Black’s (1976) pricing formula for call options on futures is

[9.45]

where

[9.46]

[9.47]

and

• s = underlying future’s price;
• x = strike price;
• y = time to expiration in years;
• r = maturity continuously compounded (actual/actual) interest rate;
• v = implied volatility for strike x and maturity y;
• Φ = CDF of the standard normal distribution.

We are assuming cash valuation. We use 1-month Libor in [9.45] out of convenience (despite the fact that USD Libor is for 2nd day settlement and the expiration for the option is only approximately 1 month). You do need to convert Libor rates from simple actual/360 to continuous actual/actual before using them in [9.45]. At time 1, there will be 28 actual days until the options’ expiration.

4. Apply ordinary interpolation to the six points (1r[k], 1p[k]) from part (c) to obtain a quadratic remapping of form [9.41].

9.6

Arrange 11 points in a minimum energy configuration on the surface of a sphere in four dimensions.