23-RiskManagement Page 750 Wednesday, February 4, 2004 1:13 PM
750 The Mathematics of Financial Modeling and Investment ManagementA different way of measuring risk consists in computing different pos-
sible scenarios and defining risk as the maximum loss that can be incurred
in any of these scenarios. This technique is used in the SPAN system devel-
oped by the Chicago Mercantile Exchange which computes 16 scenarios,
two of which are extreme scenarios. Risk is the largest maximum loss in
the 14 scenarios or 35% of the loss in the two extreme scenarios.
The idea of analyzing risk under different scenarios is widely used in
practice, often together with quantile measures such as VaR. Extreme
scenarios can be computed in different ways, in particular with the use
of Extreme Value Theory (EVT), which we covered in Chapter 13. As
we noted in that chapter, and as we will see in the following sections,
the use of EVT is still in its infancy.
Risk measures can be seen, from a different point of view, as sensitivi-
ties to given factors. In this case, rather than capture the uncertainty of a
given distribution it captures the amount of fluctuation of a given quantity
as a function of the fluctuations of another quantity. We have already
encountered most of these measures. In the analysis of stock prices, the
coefficients of factor models, the betas, capture the sensitivity of returns to a
number of factors. As we have seen in Chapters 11 and 12 on financial
econometrics, sensitivities apply to a static as well as to a dynamic frame-
work. A dynamic framework is generally represented as a state-space model.
In the analysis of bond prices, duration captures the sensitivity of
bond prices to parallel shifts in the term structure of interest rates. Con-
vexity, which is defined as the first derivative of duration, captures the
sensitivity of bond prices to the curvature of the term structure.
In the analysis of derivative instruments, a number of sensitivities
are used to capture the sensitivity of their prices to changes in different
parameters. These sensitivities are usually indicated with specific Greek
letters. Hence, they are called the “Greeks.” The most common Greeks
are listed below:Vega Theta Delta GammaSensitivity to a Sensitivity to a Sensitivity to a Linearized rate
change in change in time change in the of change of
volatility remaining price of underlying deltaA concept related to risk measures is the Sharpe ratio developed by
William Sharpe.^12 Sharpe himself called this ratio the “Reward to Vari-(^12) William F. Sharpe, “Mutual Fund Performance,” Journal of Business (January
1966), pp. 119–138; and William F. Sharpe, “Adjusting for Risk in Portfolio Perfor-
mance Measurement,” Journal of Portfolio Management (Winter 1975), pp. 29–34.