Quadratic remappings are used with portfolios that hold one or more “nonlinear” instruments—options, interest-rate caps or floors, mortgage-backed securities, callable bonds, various “exotic” derivative, most structured notes, etc. These remappings are appealing because they facilitate quadratic transformation procedures as well as certain Monte Carlo transformation procedures that employ variance reduction. Despite such appeal, quadratic remappings should be applied with care.

Exhibit 9.16 illustrates a quadratic polynomial approximation for the Black-Scholes (1973) option pricing formula. The value of a short option is graphed as a function of underlier value. A quadratic polynomial approximation is fit to this based upon the first and second derivatives of the option’s value. Over a region of underlier values, the approximation is excellent. Outside that region, the approximation is poor. Such behavior is typical of situations in which we might apply a quadratic remapping. The approximations are highly localized.

Exhibit 9.16: Quadratic remappings generally provide a good approximation over a certain region of values for 1R. Outside that region, the approximation can be poor. This is illustrated with a quadratic polynomial used to approximate the Black-Scholes market value of a short call option.

Before incorporating a quadratic remapping into a mapping procedure, we must assess whether that region of good approximation will be large enough for the needs of the value-at-risk measure. The decision is complicated by the fact that we don’t know in advance all the portfolios to which the value-at-risk measure will be applied. We also don’t know the distributions—especially their standard deviations—of applicable key factors. We want to be sure that a quadratic remapping will be reasonable for all portfolios and all distributions of 1R that may be encountered.

A region of good approximation will need to be larger for a 95% value-at-risk metric than for a 90% value-at-risk metric because the larger quantile encompasses more extreme market events. For reasons such as this, a quadratic remappings may be reasonable with one value-at-risk metric but not with another. Key-factor standard deviations tend to be larger for long horizons than for short horizons. A quadratic remapping’s region of good approximation will need to be larger for a 1-week value-at-risk horizon than for a 1-day horizon.

Quadratic remappings may poorly approximate discontinuous or non-smooth primary mappings. An obvious problem is at-the-money put or call options expiring at time 1. Even worse are expiring at-the-money digital options, which introduce a jump discontinuity into the primary mapping function. A small number of such instruments in a portfolio should not rule out the use of a quadratic remapping, but it does raise serious concerns.

Exhibit 9.17: Price functions for an expiring at-the-money call option and expiring at-the-money digital option. A quadratic polynomial could not reasonably approximate either.

A final issue is the purpose of a quadratic remapping. If a quadratic remapping is intended to replace a primary portfolio mapping, it needs to be a good approximation. In Chapter 10, we will discuss how quadratic remappings can complement primary mappings to facilitate variance reduction in a Monte Carlo transformation. When used in this manner, the quadratic remapping doesn’t have to be as good an approximation.