9.3.10 Exercises

Exercises

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].
Solution

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

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 p_k 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^[6] = ^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 y 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 2^nd 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].

Solution

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