in Appendices E and F at the end of the book.
The Taylor Series expansions in Section 8.7 (explained in Appendix G) is an important
way of thinking about the Greek letters. It has implications for some of the material that
comes later in the book on short cuts to measuring VaR for a portfolio. It also explains
the relationship between theta and gamma for a delta-neutral portfolio (see equation 8.2).
Section 8.9 deals with static options replication. An understanding of this is not necessary
for subsequent material. As a result the material in this section can be covered or skipped
depending on the tastes of the instructor.
Any of Problems 8.15 to 8.19 can be used as hand-in assignments. 8.15 also works
well for class discussion.
Chapter 9: Interest Rate Risk
This chapter is similar to Chapter 8 in the third edition. The chapter starts with a
discussion of the management of net interest income. It then moves on the explain the
importance of LIBOR rates, swap rates, and OIS rates. The rest of the chapter is then
spent on duration, convexity and related issues. It is worth spending time on the risk-free
rate issue. (See Section 9.2.) Traders have traditionally used the LIBOR/swap zero curve
as a proxy for the risk-free yield curve but now use the OIS rate. (This edition displays the
LIBOR-OIS spread in Figure 9.1.) Students should understand the difference between the
five-year AA-borrowing rate and the five-year swap rate. The former is the fixed rate at
which a AA-rated company can borrow for five years. The latter is the rate earned when
a series of short-term loans are made to AA-rated companies.
Duration and convexity provide simple ways of hedging exposures to interest rate
movements. By matching the duration of assets and liabilities a bank is hedged against
small parallel shifts in the yield curve. By matching both duration and convexity it is
hedged against small and relatively large parallel shifts in the yield curve. The duration
measure can be extended to consider non-parallel yield curve shifts. One approach is to
define the duration with respect to the ith point, yi, on the yield curve as
−
1
P
∂P
∂yi
where Pis the value of the portfolio. This is sometimes referred to as the partial duration
approach. (See Figure 9.5.) Another approach is to consider the effect on the portfolio
value of commonly occurring non-parallel shifts in the yield curve. (Figure 9.6 shows a
rotation.) The convexity measure can be extended similarly.
Section 9.7 discusses four ways interest rate deltas can be calculated. One of these is
the DV01 approach that assumes parallel shifts and fits in with the traditional definition
of duration. Another is the partial duration approach mentioned earlier. A third involves
bucketing the zero curve. This is illustrated in Figure 9.7 and is commonly used in asset-
liability management. The fourth involves calculating partial derivatives with respect to
the instruments that will be used for hedging (which tend to be the same as those used
for constructing the zero curve) and is commonly used by traders.
Section 9.8 covers principal components analysis and an be regarded as an extension of
the duration approach where commonly occurring non-parallel shifts are considered. The
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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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