how the points on the efﬁcientfrontiercan be generated. The developmentof portfolio
theoryis dividedinto threegeneric problems, whichrecurinsubsequentchapters.
Chapter7 looks at (equity) asset pricing, starting with the single-index model and
the capital asset pricing model (CAPM)and concluding with Value-at-Risk (VaR). This
introduces the assumption that asset log returns follow a normal distribution, another
Chapter8 covers performance measurement, again ranging from single-parameter
representcurrentbestpractice.Weshow, forthe ﬁrsttime ina textbook, howconﬁdence
intervals canbe determined fortheassetweights fromstyleanalysis.
Chapter9introduces thesecondapplicationpart, thatdealingwithoptionsonequities.
of the hedge portfoliothatis thekey insight behindthe Black–Scholes option valuation
formula. The subsequent interpretation of the option value as the discounted expected
value ofthe optionpayoffina risk-neutralworldis alsointroduced.
Chapter10 looks at binomial trees, which can be viewed as a discrete approxima-
tion to the continuous normal distribution assumed for log equity prices. In practice,
binomial trees formthe backbone ofnumerical methods for option valuation since they
cancopewithearlyexerciseand hence thevaluation ofAmericanoptions. We illustrate
standardparameterchoices. Weuse anine-steptreeinourspreadsheetexamples,butthe
Chapter11 returns to the Black–Scholes formula and shows both its adaptability
(allowing options on assets such as currencies and commodities to be valued) and its
dependenceonthe asset priceassumptions.
Chapter12 covers two alternative ways of calculating the statistical expectation that
lies behind the Black–Scholes formula for European options. These are Monte Carlo
simulation and numerical integration. Although these perform less well for the simple
options we consider, eachofthese methods has avaluable role inthevaluation of more
shows howsuchdeviation(typicallythroughdifferingskewness andkurtosisparameters)
leads to the so-called volatility smile seen in the market prices of options. Efﬁcient
Chapter14 introduces the third application part, that dealing with options on bonds.
While bond prices have characteristics that are different from equity prices, there is a
lot ofcommonality in the mathematical problems and numerical methods used to value
options. Wedeﬁnethe term structure based ona series of zero-couponbondprices, and
show howthe short-terminterestratecan be modelledin abinomial tree as a means of
and Ross. We detailanalytic solutions forzero-couponbond prices and options on zero-
couponbonds togetherwithan iterative approach tothe valuationofoptions on coupon
Chapter16 shows how the short rate can be modelled in a binomial tree in order
to match a given term structure of zero-coupon bond prices. We build the popular