9.3.4 Example: Gradient Approximations
In an example of Section 8.5, we constructed a primary mapping for an Australian trader’s foreign exchange portfolio, as indicated in Exhibit 8.16. We obtained primary mapping [8.66], which is presented again here
The primary mapping is quadratic, which makes taking its gradient easy. All we need is a value for 1|0μ = 0E(1R), and we can apply [9.23]. Assuming 0E(1R) = 0r, we have
where current discount factors are
and current exchange rates are
Taking the gradient of [9.24] at 1|0μ, and applying [9.23], we obtain a linear remapping = (1R):
To assess how well approximates 1P, we generate 1000 pseudorandom realizations 1r[k] of 1R based upon an assumed distribution for 1R. We determine corresponding realizations 1p[k] = θ(1r [k]) and = θ() for 1P and . These are plotted in the scatter diagram of Exhibit 9.15.
All points cluster near a line passing through the center of the chart at a 45° angle. The remapping provides an excellent approximation. Although primary mapping [9.24] is mathematically quadratic, its behavior is essentially linear over a region of likely realizations for 1R.