9.3.4 Example: Gradient Approximations

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.

Exhibit 9.15: A scatter diagram illustrates that the linear remapping of our example provides an excellent approximation for 1P.

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.