math underlying principal components analysis is in Appendix I. Software for carrying out
a principal components analysis is on the author’s web site.
Any of Problems 9.16 to 9.20 can be used as hand-in assignment questions. 9.17 and
9.18 can be used as part of the classroom discussion of duration and convexity. Problem
9.19 can be used to illustrate the partial duration approach in class.
Chapter 10: Volatility
This chapter is an updated version of Chapter 10 of the third edition. The EWMA/GARCH
example toward the end of the chapter is new. The chapter requires two to three hours of
classroom time. It provides a formal definition of volatility and then moves on to discuss
how the volatility of a variable can be monitored by risk managers. One issue is whether
volatility should be considered to be a trading-day or calendar-day phenomenon. This is
discussed in Business Snapshot 10.1. Whatever the reason, volatility is much greater when
markets are open than when they are closed. It therefore makes sense to measure volatility
using trading days rather than calendar days. This is what traders and risk managers do.
Implied volatilities are explained in Section 10.2. Section 10.3 explores whether returns
are approximately normally distributed for exchange rates. It finds that they are not. This
leads on to a discussion of the power law that has been found to hold for a wide range
of financial variables. (The purpose of the material in this chapter is to introduce the
power law; extreme value theory and the theoretical underpinnings of the power law are in
Chapter 14.) A “quick and dirty” analysis shows that the α= 5.5 fits the exchange rate
data in Table 9.2 quite well. (The example has been made clearer in the fourth edition by
defining vas the number of standard deviations the exchange rate moves rather than as
the number of standard deviations it increases.)
The rest of the chapter covers exponentially weighted moving average (EWMA) and
GARCH (1,1) procedures for estimating the current level of a volatility. It explains max-
imum likelihood methods. Material is included on the implications of GARCH (1,1) for
forecasting option volatility and calculating vega. (See Section 10.10.) At the outset, it is
important to make sure students understand the notation. The variable σnis the volatility
estimated for day nat the end of day n−1; unis the realized return during day n. The
EWMA approach, although not as sophisticated as GARCH(1,1), is widely used and is a
useful lead-in to GARCH(1,1).
Although it is not difficult to find “black box” software for implementing GARCH
(1,1) I like students to develop their own Excel applications. By doing this they develop a
much better understanding of how maximum likelihood methods work. As indicated, the
Solver routine in Excel can be made to work reasonably well if used in such a way that all
the parameters being searched for are the same order of magnitude.
Any of Problems 10.18 to 10.23 can be used as hand-in assignment questions. Problems
10.20 and 10.22 are more challenging than the others.
Chapter 11: Correlations and Copulas
This chapter is similar to Chapter 11 of the third edition. It requires about two hours
of classroom time. It starts by defining correlation and explaining the difference between
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