Quantitative Risk Management: A Practical Guide to Financial Risk, + Website

Appendix 8.2: Second Derivatives and the Parametric Approach

One of the biggest drawbacks with the parametric or linear estimation approach is that it cannot capture nonlinear instruments well. This is not usually a fatal flaw, as most portfolios will be at least locally linear and the parametric approach can provide useful information.⁵⁰ More problematic, however, is that with the standard approach, there is no way to confirm that nonlinearities are small or tell when they are large enough to make a difference (thus requiring an alternate approach).

I discuss in this section a way to estimate the effect of nonlinearity in the asset payoff using second derivative (gamma) information from the original assets or risk factors; a measure in particular that indicates when linearity fails to provide a good summary of the P&L distribution.⁵¹ Although this is more involved than the straightforward calculation of the portfolio variance, it is orders of magnitude less computationally intensive than Monte Carlo techniques. This measure provides a good indicator for the breakdown of linearity but not necessarily a good estimate for the size of the nonlinearity effect.

To lay out the ideas, consider first the univariate case with a single risk factor, where f represents risk factor changes. We assume that the risk factor changes are normal. Since we are particularly focused on whether and how nonlinearity in asset payoff translates into deviations from normality, assuming normality in the risk ...