financial derivatives pricing applications and mathematics pdf

2.1 uncertainty, utility theory, and risk One of the most important concepts in finance is that of risk.. For example, we may characterize alluncertainty associated with our commute to w

Financial Derivatives

This book offers a succinct account of the principles of financial tives pricing The first chapter provides readers with an intuitive expo-sition of basic random calculus Concepts such as volatility and time,random walks, geometric Brownian motion, and It ˆo’s lemma are dis-cussed heuristically The second chapter develops generic pricing tech-niques for assets and derivatives, determining the notion of a stochasticdiscount factor or pricing kernel, and then uses this concept to priceconventional and exotic derivatives The third chapter applies the pric-ing concepts to the special case of interest rate markets, namely, bondsand swaps, and discusses factor models and term-structure-consistentmodels The fourth chapter deals with a variety of mathematical topicsthat underlie derivatives pricing and portfolio allocation decisions, such

deriva-as mean-reverting processes and jump processes, and discusses relatedtools of stochastic calculus, such as Kolmogorov equations, martingalestechniques, stochastic control, and partial differential equations

Jamil Baz is the chief investment strategist of GLG, a London-basedhedge fund Prior to holding this position, he was a portfolio managerwith PIMCO in London, a managing director in the Proprietary Trad-ing Group of Goldman Sachs, chief investment strategist of DeutscheBank, and executive director of Lehman Brothers fixed income researchdivision Dr Baz teaches financial economics at Oxford University Hehas degrees from the London School of Economics (M.Sc.), MIT (S.M.),and Harvard University (A.M., Ph.D.)

George Chacko is chief investment officer of Auda, a global asset agement firm, in New York He is also a professor at Santa Clara Uni-versity, California, where he teaches finance Dr Chacko previouslyserved for ten years as a professor at Harvard Business School in thefinance department Dr Chacko held managing directorships in fixedincome sales and trading at State Street Bank in Boston and in pensionasset management at IFL in New York He holds a B.S from MIT, anM.B.A from the University of Chicago, and an M.A and Ph.D fromHarvard University

www.cambridge.org Information on this title: www.cambridge.org/

© Jamil Baz and George Chacko 2004

This publication is in copyright Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without

the written permission of Cambridge University Press.

First published 2004 First paperback edition 2009

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication Data

Baz, Jamil Financial derivatives : pricing, applications, and mathematics / Jamil Baz,

2002041452

ISBN 978-0-521-81510-9 hardback ISBN 978-0-521-06679-2 paperback

Cambridge University Press has no responsibility for

the persistence or accuracy of URLs for external or

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and does not guarantee that any content on such

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9780521815109

To Maurice and Elena J.B.

1.4 Continuous Time: Brownian Motion; More

1.7.2 Paradox 2: The Stock, Free-Lunch Paradox 18

1.7.3 Paradox 3: The Skill Versus Luck Paradox 19

2.2 Risk and the Equilibrium Pricing of Securities 28

3.1 Interest Rate Derivatives: Not So Simple 78

3.2.2 Discount Factors, Zero-Coupon Rates, and

3.3.3 The Free Lunch in the Duration Model 104

3.4 An Overview of Interest Rate Derivatives 108

3.7.3 The Ho-Lee Model with Time-Varying

Contents ix

3.9 The Heath, Jarrow, and Morton Approach 172

4.2.1 Arithmetic Brownian Motion as a Limit of a

4.2.2 Moments of an Arithmetic Brownian Motion 196

4.2.3 Why Sample Paths Are Not Differentiable 198

4.2.4 Why Sample Paths Are Continuous 198

4.2.5 Extreme Values and Hitting Times 199

4.5.3 Calculations of Moments with the Dynkin

4.7.1 The Kolmogorov Forward Equation 234

4.7.3 The Kolmogorov Backward Equation 236

4.8.2 Some Useful Facts About Martingales 241

4.8.3 Martingales and Brownian Motion 242

x Contents

4.9.2 Optimal Control of It ˆo Processes:

4.9.3 Optimal Control of It ˆo Processes:

4.10.1 The Kolmogorov Forward Equation Revisited 253

4.10.4 Resolution of the Kolmogorov Forward

We are as ever in many people’s debt Both authors are lucky tohave worked with or been taught by eminent experts such as JohnCampbell, Sanjiv Das, Jerome Detemple, Ken Froot, Andrew Lo,Franco Modigliani, Vasant Naik, Michael Pascutti, Lester Seigel,Peter Tufano, Luis Viceira, and Jean-Luc Vila A list, by no means ex-haustive, of colleagues who have read or influenced this manuscript in-cludes Richard Bateson, Eric Briys, Robert Campbell, Marcel Cassard,Didier Cossin, Fran¸cois Degeorge, Lev Dynkin, David Folkerts-Landau, Vincent Koen, Ravi Mattu, Christine Miqueu-Baz, ArunMuralidhar, Prafulla Nabar, Brian Pinto, David Prieul, Vlad Putyatin,Nassim Taleb, Michele Toscani, Sadek Wahba, and Francis Yared.Special thanks are due to Tarek Nassar, Saurav Sen, Feng Li, and DeeLuther for diligent help with the manuscript The biggest debt claimant

to this work is undoubtedly Robert Merton, whose influence pervadesthis manuscript, including the footnotes; as such, because there is nofree lunch, he must take full responsibility for all serious mistakes,details of which should be forwarded directly to him

It is easy to see how this story is a metaphor of our lives We areshaped daily by doses of randomness This is where the providen-tial financial engineer intervenes The engineer’s thoughts are alongthe following lines: to confront all this randomness, one needs artifi-cial randomness of opposite sign, called derivative securities And theengineer calls the ratio of these two random quantities a hedge ratio.Financial engineering is about combining the Tinker Toys of capitalmarkets and financial institutions to create custom risk-return profilesfor economic agents An important element of the financial engineeringprocess is the valuation of the Tinker Toys; this is the central ingredientthis book provides.

We have written this book with a view to the following twoobjectives:

r to introduce readers with a modicum of mathematical background

to the valuation of derivatives

The target audience includes advanced undergraduates in ematics, economics, and finance; graduate students in quantitative fi-nance master’s programs as well as PhD students in the aforementioneddisciplines; and practitioners afflicted with an interest in derivativespricing and mathematical curiosity.

math-The book assumes elementary knowledge of finance at the level

of the Brealey and Myers corporate finance textbook Notions such

as discounting, net present value, spot and forward rates, and basicoption pricing in a binomial model should be familiar to the reader.However, very little knowledge of economics is assumed, as we developthe required utility theory from first principles

The level of mathematical preparation required to get through thisbook successfully comprises knowledge of differential and integral cal-culus, probability, and statistics In calculus, readers need to know ba-sic differentiation and integration rules and Taylor series expansions,and should have some familiarity with differential equations Readersshould have had the standard year-long sequence in probability andstatistics This includes conventional, discrete, and continuous proba-bility distributions and related notions, such as their moment generat-ing functions and characteristic functions

The outline runs as follows:

to understand the valuation concepts developed inChapters 2

and 3 It provides an intuitive exposition of basic random

Introduction 3

calculus Concepts such as volatility and time, random walks, ometric Brownian motion, and It ˆo’s lemma are exposed heuris-tically and given, where possible, an intuitive interpretation Thischapter also offers a few appetizers that we call paradoxes of fi-nance: these paradoxes explain why forward exchange rates arebiased predictors of future rates; why stock investing looks like

ge-a free lunch; ge-and why success in portfolio mge-ange-agement mighthave more to do with luck than with skill

derivatives The chapter starts from basic concepts of utilitytheory and builds on these concepts to derive the notion of

a stochastic discount factor, or pricing kernel Pricing kernelsare then used as the basis for the derivation of all subsequentpricing results, including the Black-Scholes/Merton model Wealso show how pricing kernels relate to the hedging, or dynamicreplication, approach that is the origin of all modern valuationprinciples The chapter concludes with several applications toequity derivatives to demonstrate the power of the tools thatare developed

in-terest rate markets; namely bonds, swaps, and other inin-terestrate derivatives It starts with elementary concepts such as yield-to-maturity, zero-coupon rates, and forward rates; then moves

on to na¨ıve measures of interest rate risk such as duration andconvexity and their underlying assumptions An overview of in-terest rate derivatives precedes pricing models for interest rateinstruments These models fall into two conventional families:factor models, to which the notion of price of risk is central, andterm-structure-consistent models, which are partial equilibriummodels of derivatives pricing The chapter ends with an inter-pretation of interest rates as options

underlie derivatives pricing and portfolio allocation decisions Itdescribes in some detail random processes such as random walks,arithmetic and geometric Brownian motion, mean-revertingprocesses and jump processes This chapter also includes an ex-position of the rules of It ˆo calculus and contrasts it with the

4 Introduction

competing Stratonovitch calculus Related tools of stochasticcalculus such as Kolmogorov equations and martingales are alsodiscussed The last two sections elaborate on techniques widelyused to solve portfolio choice and option pricing problems:dynamic programming and partial differential equations

We think that one virtue of the book is that the chapters arelargely independent Chapter 1 is essential to the understanding ofthe continuous-time sections inChapters 2and3.Chapter 4may beread independently, though previous chapters illuminate the conceptsdeveloped in each chapter much more completely

an almost aesthetic undertaking: Some finance experts think of ematics as a way to learn finance Our point of view is different Wefeel that the joy of learning is in the process and not in the outcome

math-We also feel that finance can be a great way to learn mathematics

Preliminary Mathematics

This chapter presents a brief overview of the technical language ofmodern finance In the apparatus that we shall use, expressions such as

Random Walk, Brownian Motion, and It ˆo Calculus may carry a shroud

of mystery in the readers’ minds In an attempt to lift this shroud, wewill be guilty of oversimplification and make no apology for that Topicsdiscussed in this chapter will be revisited and fleshed out in more detail

Picture a particle moving on a line Define X t as the position of the

particle at time t, with X0 = 0 The particle moves one step forward(+1) or backward (−1) with equal probability at each instant of time,

and successive steps are independent At t= 1

Pr [X1 = −1] = Pr [X1 = 1] = 1/2

Similarly at t= 2, the particle can be at the positions−2, 0, 2 with

probabilities 1/4, 1/2, and 1/4 respectively (seeFigure 1.2).At t= 3,

the values for X and their respective probabilities are

We can calculate the expected value, variance, and standard

devi-ation for X tas of time zero For example:

1.1 Random Walk 7

Using a similar logic, we can find the expected value, variance, and

standard deviation of X t , conditional on X0 = 0, for any t > 0:

We can calculate the mean, variance, and standard deviation of X1

as before We now have

Defineµ = (p − q) σ The variable µ is called the drift of X X is

said to follow a random walk with drift when p = q, and a driftless random walk when p = q = 1/2 In general we have

E [X n]= n (p − q) σ = nµ

Var [X n]= 4σ2npq

SD [X n]=Var [X n]= 2σ√npq

If the particle takes one step per unit of time then n = t, where t is

the number of units of time We see that the mean of a random walk isproportional to time, whereas the standard deviation is proportional tothe square root of time The latter result stems from the independence

of the increments in a random walk In a financial context, stock returns

are often modeled as random walks If R t −1,trepresents the return on

a stock between t − 1 and t, then the return over T periods is

R0,T = R0 ,1 + R1 ,2 + · · · + R T −1,T

8 Preliminary Mathematics

Returns in successive periods are assumed to be independent Thismeans that

Var [R0,T]= Var [R0 ,1 + R1 ,2 + · · · + R T −1,T]

= Var [R0 ,1]+ Var [R1 ,2]+ · · · + Var [R T −1,T]

Additionally, if the return in each period has a constant variance ofσ2,

We now give another perspective of volatility and time (Figure 1.3).X

follows a two-dimensional random walk The step size isσ The angle

θ i at step i is random After two steps, the distance D between the departure point X0and X2is given by

D2= (X0 C)2+ (X2C)2

x2σ

1.3 A First Glance at It ˆo’s Lemma 9

D2= σ2[(cosθ1+ cos θ2)2+ (sin θ1 + sin θ2)2]

= σ2[cos2θ1+ sin2θ1+ cos2θ2+ sin2θ2

+ 2(sin θ1sinθ2+ cos θ1cosθ2)]

Recall the equalities

cos2θ1+ sin2θ1= 1cosθ1cosθ2+ sin θ1sinθ2= cos(θ1 − θ2)Using these equalities yields

D2 = σ2[2+ 2 cos(θ1 − θ2)]

Because the cosine term equals zero on average, we get

E(D2)= 2σ2

SD(D) = σ√2

1.3 a first glance at it ˆo’s lemma

Recall the experiment discussed in the previous section (see

X as shown inFigure 1.5,whereµ ≡ (p − q)σ , we get the same drift

and volatility for each process

We now represent the binomial tree for a change in X given a time

interval t (seeFigure 1.6).The process for the binomial tree can bewritten as

X = X t − X0 = µ t + σ ε√ t

where the random variableε keeps the same properties as above Now,

how can the variation on a function f (X) be expressed? This is the

ques-tion solved by It ˆo’s lemma A simple Taylor expansion to the secondorder gives the result shown inFigure 1.7

For a small t, we may choose to neglect terms in ( t) n (with n > 1)

to get the outcome shown inFigure 1.8.In shorthand notation,

1.4 Continuous Time: Brownian Motion; It ˆo’s Lemma 11

This is the simplest expression of It ˆo’s lemma

more on it ˆo’s lemma

The smaller the time interval, the better It ˆo’s lemma approximates

f (X) In the process of this shrinkage, t gets partitioned into smaller

and smaller time intervals and a larger and larger number of binomialrealizations occur as a result.Chapter 4shows that in this passage tocontinuous time, the random variableε ends up converging toward a

standard normal variableφ ∼ N (0, 1) which has a normal distribution

with mean zero and variance one We call an infinitesimal time interval

dt The random process in continuous time can be written as

d X (t) = µdt + σ φ√dt, with X(0) = x0

The process X is said to follow an arithmetic Brownian motion (with

drift) The above expression is commonly written as

d X (t) = µdt + σ dW (t) where dW (t) ≡ φ√dt is called a Wiener increment Naturally

12 Preliminary Mathematics

Example 1.1 There is a 95% probability that φ is in the [−1.96, 1.96] interval With µ = 2%, σ = 1%, and dt = 1, dX will be in the

[0.04%; 3.96%] interval with 95% probability.

More generally, a variable X is said to follow an It ˆo process if

d X = µ(t, X )dt + σ (t, X )dW

where µ(t, X ) is the drift function and σ(t, X ) is the volatility for

an increment in X Of particular interest in finance is the so-called

geometric Brownian motion whereµ(t, X ) = µX and σ(t, X ) = σ X,

whereµ and σ are constants The process can be written as

d X = µXdt + σ XdW

This process is used to describe the return dynamics of a wide range

of assets For example, we can assume that a stock price follows ageometric Brownian motion This simply states that we assume the

returns to the stock price to follow a normal distribution Let S be the

stock price Then the process can be written as

dS

S = µdt + σdW

whereµdt is the deterministic drift term and σdW, or σφ√dt, is the

random term Forµ = 10%, dt = 1, and σ = 30%, the stock return dS/S is

dS

S = 0.1 + 0.3φ

whereφ ∼ N(0, 1) If φ turns out to be 1, then dS/S = 0.4 If φ = −0.5,

then dS /S = −0.05 From this it can be seen that dS/S has a normal

distribution with meanµdt, variance σ2dt, and volatility σ√dt This

means that as time elapses, the probability density function shifts andflattens With a positive drift term, the passage of time can transformthe density function of stock returns as shown inFigure 1.9

For general It ˆo processes of the form

d X = µ(t, X)dt + σ(t, X)dW

It ˆo’s lemma turns out to be similar to the primitive version in the

previous section For a function f (t , X) (where f is at least twice

1.4 Continuous Time: Brownian Motion; It ˆo’s Lemma 13

dt + σ

T

0

dW

Taking exponentials and noting that W(0) = 0, we have an expression

for S T as a function of S0for a geometric Brownian motion:

It is tempting to try solving the equation on the left by

observ-ing that dS /S = d log S, leading to the erroneous answer S T =

S0exp{µt + σ W} However, since S is stochastic (random), dS/S is not equal to d log S , and we have to use It ˆo’s lemma to get the correct

answer It ˆo’s lemma tells us that d log S=µ −1

2σ2

dt + σdW Since

this is “linear” in the derivatives, we can indeed integrate it as in normalNewtonian calculus to get the correct answer

Let asset prices X and Y follow geometric Brownian motions:

d X

X = µ X dt + σ X dW X dY

Y = µ Y dt + σ Y dW Y

Each asset return is characterized by its own drift termµ and

volatil-ity termσ Also note that the Wiener process is not the same in each

equation If dW X = dW Y , then both returns would be perfectly

corre-lated This is because d X / X would then be a deterministic function of dY/Y:

1.6 Bivariate It ˆo’s Lemma 15

Asset returns are generally not perfectly correlated (i.e., dW X=

1.6 bivariate it ˆo’s lemma

It ˆo’s lemma for the two-dimensional geometric Brownian motionabove is given by:

Y = µ Y dt + σ Y dW Y The correlation between returns is ρ What random process does the ratio f (X , Y) = X/Y follow?

dW f ≡ σ X dW X − σ Y dW Y

σ f

d f

f = µ f dt + σ f dW f with µ f = µ X − µ Y − ρσ X σ Y + σ2

Y

We illustrate the results discussed thus far with three paradoxes Theseparadoxes elaborate in turn upon three fields of finance: exchange rateforecasting, stock outperformance, and skill and luck in the money-management industry

1.7.1 Paradox 1: Siegel’s Paradox

We give a simple example to motivate our discussion of Siegel’s dox Let the€/$ exchange rate today be 1 For the sake of simplicity,assume the euro will be worth either $1.25 or $0.80 in one year, withequal probability for each outcome This means that the expected value

para-of the euro is (0.5∗1.25) + (0.5∗0.80) = $1.025 However, an

equiv-alent statement of the problem is that the dollar in one year will

be worth either €0.80 or €1.25 with equal probability for each come We face the paradoxical situation where one euro is expected

out-to be worth $1.025 and one dollar is expected out-to be worth€1.025 in

a year

1.7 Three Paradoxes of Finance 17

In the setting of a continuous-time process, if the€/$ exchange rate

η follows a geometric Brownian motion

dη

η = µdt + σ dW

then the $/€ exchange rate is f (η) = 1/η To write the process for f (η),

we use It ˆo’s lemma Note that f η = −1/η2and f ηη = 2/η3 f is not an

explicit function of time, and so f t= 0 A straightforward application

of It ˆo’s lemma gives

average, we haveE [dη/η] = µdt and E [df/f ] =σ2− µdt A trader

can assign the€/$ exchange rate and the $/€ exchange rate the sameexpected value ifσ2= 2µ.

We now discuss two implications of Siegel’s paradox:

1 Looking at the simple example in Paradox 1, one conclusion isthat investors need to keep part of their currency exposure un-hedged If an American holds€1, then the euro is expected to beworth $1.025 Simultaneously, a European owning $1 can expect

it to be worth€1.025 It appears there are gains for both investorskeeping their currency holdings (at least partly) unhedged

2 Forward exchange rates cannot be unbiased predictors of futurespot exchange rates To be an unbiased predictor, the forward

€/$ exchange rate at time t, F(t) needs to be by definition the

expected value of the €/$ exchange rate at t + 1 So, F(t) =

E [η t+1] If forwards were unbiased predictors of future spotrates, then this should also apply to the $/€ exchange rate:

1/F(t) = E [1/η t+1] But this can never happen in an uncertain

world This mathematical result:

1

E[η t+1] ≥ E 1

η t+1

18 Preliminary Mathematics

is known as Jensen’s inequality Viewed in our context, it statesthat forward currency trades can be “profitable” on average

1.7.2 Paradox 2: The Stock, Free-Lunch Paradox

Say you have to choose at time t between investing in a goverment bond

or in the stock market The question is: What is the probability that the

stock investment will outperform the bond investment by time T ? Say the value of the stock index S follows a geometric Brownian

motion:

dS

S = µdt + σdW

If invested in a zero-coupon bond, the amount S over time period

T − t will yield S(t) exp{R(T − t)}where R is the zero-coupon rate on

a bond maturing in T − t periods The probability p that the stock

investment outperforms the bond investment is

where is the cumulative standard normal distribution.

To illustrate the result, say the risk premium of a stock index overzero-coupon bonds, that is,µ − r, is 5.6% and index volatility is 20%.

Then

p = [0.18√T − t]

function of the holding period

The probability of outperformance goes to 100% as the holdingperiod goes to infinity However, if the investor wanted to buy in-surance against the risk of the stock investment underperforming thebond investment, as the reader will discover in the next chapter, thehedging investment is a put option on the stock index purchased at

1.7 Three Paradoxes of Finance 19

time t with strike price S(t) exp {R(T − t)} and expiring at T The

in-vestor would realize in dismay that the price of such insurance increaseswith the time to expiration! It appears, paradoxically, that the insur-ance market is assigning a higher and higher price to an event withprobability shrinking to zero This is just an illustration of the grow-ing insurance premium that hedgers are willing to pay as the hedginghorizon increases

1.7.3 Paradox 3: The Skill Versus Luck Paradox

Is outperformance versus an index a signal of managerial talent or theresult of market randomness? The question is more than academicconsidering the size of the money management industry today Thissection attempts to give no more than a stylized answer (see Ambarish

and Seigel, 1996) Consider a money manager whose portfolio value P

follows the motion

dW P and dW I have correlationρ With F(P, I) = R = P/I denoting

relative performance, it follows from Example 1.3 that

20 Preliminary Mathematics

Now defineµ R = µ P − µ I + σ2

I − σ I σ P ρ We therefore have dR

one can predict the above inequality For example, when φ = 1, the

Table 1.3 Time it Takes to Outperform Versus

Confidence Level ( σ P = 25%; σI= 15%;

1.7 Three Paradoxes of Finance 21

Alternatively, based on the above, the probability of beating theindex is

portfolio manager outperforms the index by 3% per year on averageand under the volatility and correlation conditions described above, itwould take 300 years for this manager to outperform the index with90% probability

Principles of Financial Valuation

This chapter introduces the fundamentals of security pricing It beginswith a discussion on utility theory and risk and introduces the stochas-tic discount factor, or pricing kernel, as the fundamental determinant

of all security prices Some basic applications are introduced as ples and to help develop martingale pricing principles This is initiallyaccomplished in a discrete-time setting and then taken to continuoustime The stochastic discount factor and martingale pricing are sub-sequently used to develop various option-pricing results Throughoutthe chapter, every effort is made to relate the mathematics of pricing

exam-to the underlying economic concepts

2.1 uncertainty, utility theory, and risk

One of the most important concepts in finance is that of risk The fact

that there is so much risk in the world is what makes finance a verycomplicated subject As we will see later, the existence of financial mar-kets and institutions can be explained by the need to control risk in ourlives Therefore, before we can dive into the theory of finance, we mustfirst understand what risk is and how it affects us The mathematicalcharacterization of risk is one of the main goals of economic theoryand the building block of modern finance theory In this section, wewill present the basics of this theory

Generally, when thinking of risk, we think of an uncertain tor affecting our “happiness” in a negative manner For example, a

fac-2.1 Uncertainty, Utility Theory, and Risk 23

homeowner may be very concerned about the risk of fire destroyinghis home A person commuting to work may be concerned about therisk of traffic conditions or the weather making him late for work.The two key elements that comprise risk are uncertainty and how thatuncertainty affects “happiness.”

Uncertainty refers to the fact that we do not know for certain theoutcome of a particular event For example, we do not know for cer-tain what the weather will be like tomorrow when we commute towork As another example, we do not know when lightning will strike

a house In economics, uncertainty is characterized using probabilitytheory We use random variables to characterize the uncertainty asso-ciated with an uncertain event For example, we may characterize alluncertainty associated with our commute to work by a random vari-

able x t, which represents the time in minutes it takes to commute to

work on a given day The subscript t on the random variable indicates

that repeated draws occur through time from the distribution

charac-terizing that random variable Therefore, if x t ∼ N(30, 4), that is, a

nor-mal random variable with mean of 30 and standard deviation of 2, wecould say that our expected commute time to work for any given day is

30 minutes and that there is a 95% chance that it will take between 26and 34 minutes to get to work on any given day Of course, a normal dis-tribution cannot be the true distribution for the commute time becausethis distribution allows for negative commute times In other words, thenormal distribution gives positive probabilities to negative commutetimes, which does not make physical sense Therefore, better distri-

butional assumptions for x t would be the lognormal or chi-squareddistributions However, the convenient properties of the normal distri-bution make it very attractive to use even with events where it may notmake physical sense to do so With the lightning strike example above,

we may want to characterize the event of a lightning strike as a drawfrom a Poisson distribution Alternatively, we may consider the timebetween lightning strikes to be approximately normally distributed

Of course, uncertainty by itself is not a sufficient description of risk.For example, we may be uncertain as to what the winning lottery num-ber will be tomorrow, but if we have not entered the lottery then this

is not a source of risk Therefore, for an uncertain event to representrisk, it must somehow affect our well-being or happiness In economics,

we use the expression utility to represent happiness Mathematically,

24 Principles of Financial Valuation

we can represent the happiness that we get from things such as sumption of goods by functions called utility functions, or preferencefunctions For example, the utility from purchasing (consuming) a tele-vision could be represented by the utility function

con-u(C t)= C2

where u(C t ) represents the level of utility at time t and C t represents

the number of televisions purchased at time t.

Now, it may seem as though we could use any arbitrary function torepresent a utility function, but this is not completely true Utility func-tions have to satisfy two basic mathematical properties First, a utilityfunction must be a monotonically increasing function Intuitively, thismeans that if we get utility from consuming a good, then the more ofthat good we consume the more utility we must get from it However,

it seems unreasonable to assume that we would get the same utilityfrom consumption of the tenth television as from the first television.For most of us, one or two televisions in the home is enough so that thetenth television in a home adds very little to our happiness This char-acteristic is captured by the concept of diminishing marginal utility.Marginal utility is the incremental utility that one gets from each ad-ditional unit of consumption, and it can be calculated by simply takingthe derivative of the utility function Therefore, for the utility functiongiven in(2.1)above, the marginal utility function is given by

The second mathematical property that utility functions must isfy is that they have diminishing marginal utility functions, in otherwords, marginal utility functions should be monotonically decreasingfunctions Notice that while the utility function in(2.1)satisfies the con-dition that utility functions be increasing functions, it does not satisfythe diminishing marginal utility condition

sat-A commonly used utility function that satisfies both conditions isthe log utility function given by

With the log utility function, utility is increasing in consumption, butmarginal utility is decreasing in consumption

2.1 Uncertainty, Utility Theory, and Risk 25

Now that we have discussed the concepts of uncertainty and utility,

we are ready to introduce the definition of risk Risk is simply tainty about anything that either directly or indirectly affects our utility.Another way of stating this is that something is considered risky if itmakes us uncertain about our level of utility in the future So if utility

uncer-is determined by consumption (as we will assume for the remainder ofthe book), and something makes future consumption uncertain, thenthat something represents risk to us For example, the performance

of the economy may represent risk to us if the economy affects ourlevel of consumption because our uncertainty about the performance

of the economy translates into uncertainty in our consumption level,and thus uncertainty about our utility However, the arrival time of

a flight arriving in Hong Kong does not represent risk to most of usbecause it does not make us more uncertain about our utility

Since we have a way of quantifying uncertainty and our preferences,there should be a way of quantifying the level of risk we face andour attitude toward risk We demonstrate how this is done through

an example Suppose that it is time t− 1 today and that our utility

function for time t consumption is given by(2.3).Assume, however,that we are uncertain about our level of consumption; therefore, weface risky consumption because we are uncertain about it and it affectsour utility Assume furthermore that the uncertainty in consumption

is characterized by a Bernoulli distribution

whereε represents a Bernoulli random variable that takes on the value

5 with probability 1/2 and the value 10 with probability 1/2.Figure 2.1

depicts how this uncertainty in consumption translates to uncertainty

in utility, or risk

Suppose we are in the “good” state and we have a consumptionlevel of 10 Then, our utility level is log 10 utils (for the remainder of

the book, we will refer to a unit of utility as a util) However, if we are

in the “bad” state, then we have a utility level of log 5 utils Therefore,our utility level is also characterized by a Bernoulli distribution Onenatural way of characterizing our level of utility under uncertainty is

by our expected level of utility Since the good and bad states occurwith probability 1/2 each, our expected level of utility at time t is given

26 Principles of Financial Valuation

of consumption Our expected level of consumption at time t is given

by

E[C t]=1

2(10)+1

Our utility level at this level of consumption is log 7.5 ≈ 2.015 Note

that this is higher than the expected level of utility Why? This is becausethe utility function is a nonlinear function, and for nonlinear functions

which states that expected utility of consumption is not equal to theutility of expected consumption More specifically, for concave func-tions (which all utility functions are due to the diminishing marginalutility condition)

This is known as Jensen’s inequality

Example 2.1 Suppose that someone were willing to sell us an insurance contract that allowed us to lock in, or hedge, the level of consumption

at time t at the level of consumption that we expect to have, 7.5 This contract takes out all the uncertainty from consumption Would we be willing to take this contract? Yes, because it would increase our expected level of utility from 1 956 to 2.015 This demonstrates precisely why