ch1-4

Mathematics of finance and valuation A. Ostaszewski 2790043 2007

Undergraduate study in Economics, Management, Finance and the Social Sciences

This guide was prepared for the University of London External Programme by: A. Ostaszewski, Department of Mathematics, The London School of Economics and Political Science.

This is one of a series of subject guides published by the University. We regret that due to pressure of work the author is unable to enter into any correspondence relating to, or arising from, the guide. If you have any comments on this subject guide, favourable or unfavourable, please use the form at the back of this guide.

The subject guide is for the use of University of London External students registered for programmes in the fields of Economics, Management, Finance and the Social Sciences (as applicable). The programmes currently available in these subject areas are: Access Route Diploma in Economics Diploma in Social Sciences Diplomas for Graduates BSc Accounting and Finance BSc Accounting with Law/Law with Accounting BSc Banking and Finance BSc Business BSc Development and Economics BSc Economics BSc Economics and Management BSc Economics and Finance BSc Geography and Environment BSc Information Systems and Management BSc International Relations BSc Management BSc Management with Law/Law with Management BSc Mathematics and Economics BSc Politics BSc Politics and International Relations BSc Sociology BSc Sociology with Law The External Programme Publications Office University of London 32 Russell Square London WC1B 5DN United Kingdom www.londonexternal.ac.uk Published by University of London Press © University of London 2007 Printed by: Central Printing Service, University of London, England

Contents

Preface i

1 Introduction 11.1 Relationship to previous mathematics units . . . . . . . . 11.2 Aims of the unit . . . . . . . . . . . . . . . . . . . . . . 21.3 Learning objectives . . . . . . . . . . . . . . . . . . . . 21.4 Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Organisation of the subject guide . . . . . . . . . . . . . 41.6 Reading advice . . . . . . . . . . . . . . . . . . . . . . . 61.7 Using the subject guide . . . . . . . . . . . . . . . . . . 91.8 Examination advice . . . . . . . . . . . . . . . . . . . . 9

2 A vector space approach to uncertainty 112.1 Introduction and aims of the chapter . . . . . . . . . . . 112.2 Learning objectives . . . . . . . . . . . . . . . . . . . . 112.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . 112.4 Concepts from probability and statistics . . . . . . . . . 11

2.4.1 Activities referring to basic statistics . . . . . . . 122.4.2 Concepts of probability requiring calculus . . . . 12

2.5 Linear algebra: the language of vectors . . . . . . . . . . 152.5.1 States of nature . . . . . . . . . . . . . . . . . . 152.5.2 Compound events . . . . . . . . . . . . . . . . . 172.5.3 Portfolios . . . . . . . . . . . . . . . . . . . . . 172.5.4 The portfolio space: replication and completeness 182.5.5 The riskless asset . . . . . . . . . . . . . . . . . 192.5.6 The call option: replication and linear span . . . 192.5.7 Arbitrage and linearity . . . . . . . . . . . . . . 232.5.8 Valuation if a linear transformation is an expectation 24

2.6 A geometric view of probability . . . . . . . . . . . . . . 262.7 Solutions to selected activities . . . . . . . . . . . . . . 31

3 The �nancial environment: a preliminary discussion 333.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . 333.2 Learning objectives . . . . . . . . . . . . . . . . . . . . 333.3 Suitable reading for this chapter . . . . . . . . . . . . . 343.4 Basic deterministic models . . . . . . . . . . . . . . . . 34

3.4.1 Savings and loans: compounding . . . . . . . . . 343.4.2 Continuous compounding . . . . . . . . . . . . . 373.4.3 Deposit rates . . . . . . . . . . . . . . . . . . . 383.4.4 Variable rates . . . . . . . . . . . . . . . . . . . 393.4.5 Loan accounts . . . . . . . . . . . . . . . . . . . 40

3.5 Present value of future income: discounting . . . . . . . 403.6 Payments under uncertainty: the paradigm . . . . . . . . 433.7 Contracts with the Stock Exchange . . . . . . . . . . . . 443.8 Some �nancial instruments . . . . . . . . . . . . . . . . 453.9 A �rst look at futures and forwards . . . . . . . . . . . . 50

1

43 Mathematics of �nance and valuation

3.10 Optimal hedges . . . . . . . . . . . . . . . . . . . . . . 523.11 The �duration�-based hedge: a �rst-order hedge . . . . . 533.12 Arbitrage arguments . . . . . . . . . . . . . . . . . . . . 553.13 Forward contracts written on an index . . . . . . . . . . 563.14 Beta hedging: a perfect hedge . . . . . . . . . . . . . . 573.15 E¢ cient market hypothesis . . . . . . . . . . . . . . . . 58

4 One period, one risky asset, and two states 614.1 Learning objectives . . . . . . . . . . . . . . . . . . . . 614.2 Essential reading . . . . . . . . . . . . . . . . . . . . . . 614.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . 614.4 Forward and future contracts . . . . . . . . . . . . . . . 63

4.4.1 Forward contracts . . . . . . . . . . . . . . . . . 634.4.2 Choosing the forward price . . . . . . . . . . . . 634.4.3 Valuing the contract . . . . . . . . . . . . . . . 644.4.4 Future contracts . . . . . . . . . . . . . . . . . . 65

4.5 A call option struck at the low SH . . . . . . . . . . . . 664.6 Valuation using replication . . . . . . . . . . . . . . . . 684.7 Valuation using expected value . . . . . . . . . . . . . . 68

4.7.1 Synthetic probabilities for the call and the forward 684.7.2 General claims . . . . . . . . . . . . . . . . . . . 71

4.8 Ball park �gures: on picking representative prices . . . . 724.9 An equilibrium interpretation . . . . . . . . . . . . . . . 734.10 Solutions to selected activities . . . . . . . . . . . . . 754.11 Sample examination questions . . . . . . . . . . . . . . 76

5 One period, many assets and many states 795.1 Learning objectives . . . . . . . . . . . . . . . . . . . . 795.2 Essential reading . . . . . . . . . . . . . . . . . . . . . . 795.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 795.4 Sure-thing arbitrage and arbitrage opportunity . . . . . . 805.5 Visualizing arbitrage opportunities . . . . . . . . . . . . 845.6 No-arbitrage theorem: two states of nature . . . . . . . 865.7 A Separating Hyperplane Theorem . . . . . . . . . . . . 875.8 No-arbitrage theorem: �nitely many states . . . . . . . . 885.9 Solutions to selected activities . . . . . . . . . . . . . . 905.10 Sample examination questions . . . . . . . . . . . . . . 94

6 Multi-period models 976.1 Learning objectives . . . . . . . . . . . . . . . . . . . . 976.2 Essential reading . . . . . . . . . . . . . . . . . . . . . . 976.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 976.4 Where to now? . . . . . . . . . . . . . . . . . . . . . . 996.5 The Information tree . . . . . . . . . . . . . . . . . . . 100

6.5.1 Labelling nodes . . . . . . . . . . . . . . . . . . 1026.6 Partitions, �ltrations (a preliminary de�nition) and trees . 1026.7 Price processes . . . . . . . . . . . . . . . . . . . . . . . 1046.8 Where do these prices come from? . . . . . . . . . . . . 105

6.8.1 A special learning activity . . . . . . . . . . . . . 1076.9 Partitions: di¤erent states of knowledge . . . . . . . . . 1096.10 Value of a European call option . . . . . . . . . . . . . . 110

6.10.1 Points to watch out for . . . . . . . . . . . . . . 1136.10.2 Summary . . . . . . . . . . . . . . . . . . . . . 114

6.11 Models and submodels . . . . . . . . . . . . . . . . . . 1156.12 Conditional probabilities: edge probabilities . . . . . . . 1156.13 Conditional expectations . . . . . . . . . . . . . . . . . 116

6.13.1 Identi�cation with a random variable . . . . . . . 1176.13.2 Iterated expectation �a special case . . . . . . . 118

6.14 Learning activities . . . . . . . . . . . . . . . . . . . . . 118

2

Contents

6.15 Trading strategies and arbitrage opportunities . . . . . . 1196.16 Other options . . . . . . . . . . . . . . . . . . . . . . . 1226.17 American call option . . . . . . . . . . . . . . . . . . . . 124

6.17.1 An example with dividends . . . . . . . . . . . . 1256.17.2 Valuation of the call by replication . . . . . . . . 1256.17.3 Summary . . . . . . . . . . . . . . . . . . . . . 126

6.18 Stopping times . . . . . . . . . . . . . . . . . . . . . . . 1276.18.1 Discussion . . . . . . . . . . . . . . . . . . . . . 1276.18.2 De�nition of stopping time . . . . . . . . . . . . 1286.18.3 Algebras of compound events and �ltrations . . . 1296.18.4 Application to American options . . . . . . . . . 1316.18.5 Dynamic programming and the Snell Envelope . . 132

6.19 Arbitrage opportunities: multi-period de�nition . . . . . 1336.20 Risk-neutral measure: multi-period model . . . . . . . . 1336.21 No-arbitrage theorem: statement . . . . . . . . . . . . . 1356.22 Arbitrage opportunities from a one-period model . . . . 1356.23 Example: Arbitrage opportunities in a two-period model 136

6.23.1 Submodel of time t = 0 . . . . . . . . . . . . . . 1376.23.2 Submodel of time t = 1 . . . . . . . . . . . . . . 137

6.24 Part 1: Existence of a risk-neutral measure ��the productformula works� . . . . . . . . . . . . . . . . . . . . . . . 1386.24.1 Conditional expectation of prices of time t = 2 . 1396.24.2 Unconditional expectation of prices of time t = 1 1396.24.3 Unconditional expectation of prices of time t = 2 139

6.25 Part 2: The absence of arbitrage . . . . . . . . . . . . . 1406.25.1 Values, gains and an identity . . . . . . . . . . . 1406.25.2 Establishing the stronger de�nition of a martingale 1406.25.3 The Martingale Transform Lemma . . . . . . . . 1416.25.4 No-arbitrage theorem: the �nal step . . . . . . . 1426.25.5 Comment on the explicit modelling of futures con-

tracts . . . . . . . . . . . . . . . . . . . . . . . 1426.26 Solutions to selected learning activities . . . . . . . . . . 1446.27 Sample examination questions . . . . . . . . . . . . . . 152

7 The binomial model 1557.1 Learning objectives . . . . . . . . . . . . . . . . . . . . 1557.2 Essential reading . . . . . . . . . . . . . . . . . . . . . . 1557.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1557.4 The T-period binomial model . . . . . . . . . . . . . . . 156

7.4.1 Selection of U and D . . . . . . . . . . . . . . . 1577.5 Valuing a European call-option . . . . . . . . . . . . . . 1587.6 Some asymptotic formulas . . . . . . . . . . . . . . . . 159

7.6.1 The risk-neutral probability of up-movement . . . 1597.6.2 Signi�cance . . . . . . . . . . . . . . . . . . . . 161

7.7 Look-back options . . . . . . . . . . . . . . . . . . . . . 1627.7.1 A �re�ection�argument . . . . . . . . . . . . . . 163

7.8 Solutions to selected activities . . . . . . . . . . . . . . 1647.9 Sample examination questions . . . . . . . . . . . . . . 165

8 Continuous-time modelling 1678.1 Learning objectives . . . . . . . . . . . . . . . . . . . . 1678.2 Essential reading . . . . . . . . . . . . . . . . . . . . . . 1678.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1678.4 The Discrete Random walk . . . . . . . . . . . . . . . . 1688.5 Brownian motion �a Limiting Random Walk . . . . . . 1718.6 A Stochastic di¤erential equation . . . . . . . . . . . . . 1738.7 Itô�s Formula � intuitive approach . . . . . . . . . . . . 175

8.7.1 The increment df = f(zt+�t)� f(zt) . . . . . . 1758.7.2 The increment df = f(pt+�t)� f(pt) . . . . . . 177

3


8.7.3 Itô�s Formula . . . . . . . . . . . . . . . . . . . 1778.8 Examples on Itô�s Formula . . . . . . . . . . . . . . . . 1788.9 Further activities on Itô�s Formula . . . . . . . . . . . . 1828.10 Solutions to selected activities . . . . . . . . . . . . . . 1858.11 Sample examination questions . . . . . . . . . . . . . . 188

9 The Black-Scholes model 1899.1 Learning objectives . . . . . . . . . . . . . . . . . . . . 1899.2 Essential reading . . . . . . . . . . . . . . . . . . . . . . 1899.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1899.4 The Black-Scholes model . . . . . . . . . . . . . . . . . 190

9.4.1 Two comments . . . . . . . . . . . . . . . . . . 1919.5 The Black-Scholes formula . . . . . . . . . . . . . . . . 1929.6 The Black-Scholes equation as a replication condition . . 194

9.6.1 Comment . . . . . . . . . . . . . . . . . . . . . 1969.6.2 Some assumptions underlying the derivation . . . 197

9.7 The historic derivations . . . . . . . . . . . . . . . . . . 1979.8 The Feynman-Kac Formula . . . . . . . . . . . . . . . . 199

9.8.1 The risk-neutral measure . . . . . . . . . . . . . 2009.8.2 A shift in the drift: towards Girsanov�s Theorem . 2009.8.3 Relation between two measures . . . . . . . . . . 2019.8.4 Enter the Radon-Nikodym derivative . . . . . . . 2029.8.5 Girsanov�s Theorem . . . . . . . . . . . . . . . . 2039.8.6 The Dynkin Equation . . . . . . . . . . . . . . . 203

9.9 Inclusion of dividend payments . . . . . . . . . . . . . . 2039.10 What can we learn from the PDE? . . . . . . . . . . . . 205

9.10.1 The heat equation . . . . . . . . . . . . . . . . . 2059.10.2 Laplace Transform method . . . . . . . . . . . . 2069.10.3 Lessons from other approaches to the PDE . . . 2079.10.4 Some other transformations . . . . . . . . . . . . 209

9.11 Solutions to selected activities . . . . . . . . . . . . . . 2109.12 Sample examination questions . . . . . . . . . . . . . . 218

10 Perpetual options 21910.1 Learning objectives . . . . . . . . . . . . . . . . . . . . 21910.2 Essential reading . . . . . . . . . . . . . . . . . . . . . . 21910.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 21910.4 Perpetual options: a �naive�view . . . . . . . . . . . . . 22010.5 Generalities . . . . . . . . . . . . . . . . . . . . . . . . 222

10.5.1 Present-value formulation . . . . . . . . . . . . . 22210.5.2 Time homogeneity: current-value formulation . . 22210.5.3 Conclusion: the Cauchy-Euler Equation . . . . . 223

10.6 The American perpetual put option . . . . . . . . . . . . 22310.6.1 Dynkin Equation . . . . . . . . . . . . . . . . . 22410.6.2 Boundary behaviour . . . . . . . . . . . . . . . . 22410.6.3 Value maximization . . . . . . . . . . . . . . . . 22510.6.4 Smooth-pasting . . . . . . . . . . . . . . . . . . 225

10.7 Further learning activities . . . . . . . . . . . . . . . . . 22610.8 Solutions to selected activities . . . . . . . . . . . . . . 22810.9 Sample examination questions . . . . . . . . . . . . . . 232

11 Sample examination paper 235

12 Guidance on answering the sample examination paper 241

4

Preface

This subject guide is not a course text. It sets out the logical sequencein which to study the topics in the unit. Where coverage in the maintexts is weak, it provides some additional background material. Furtherreading is essential.

i


ii

Chapter 1

Introduction

1.1 Relationship to previous mathematics units

The pre-requisite Mathematics for this unit is 05a Mathematics 1 and05B Mathematics 2 (or their equivalents if you have been awarded anexemption) and 116 Abstract mathematics.

You should therefore have knowledge and understanding of basic linearalgebra and computational aspects of calculus with regard both tosingle variable integration and to the basic essentials ofmultiple-variable di¤erential calculus with optimisation in mind,including the use of Lagrange Multipliers. At some critical points laterin the unit you will need to rely on a facility to perform calculationswith integrals, so we highlight the level of di¢ culty in some learningactivities in Chapter 2. Chapter 8 relies on knowing how to de�ne theRiemann integral, a matter that is covered in either of the units 116Abstract mathematics or 117 Advanced calculus. Elementarynotions of Probability or Statistics are also depended upon.

In most of the unit 43 Mathematics of �nance and valuation we usethe language of Linear Algebra. This does of course mean that we willalso be guided by the usual geometric arguments that are associatedwith ideas from Linear Algebra. That is so even when the argumentsare motivated by probability. Note that the approach taken in this unitdoes not rely on any use of measure theory.

We will think of the component of a vector as describing variouspossible sums of money payable according to future prevailingcircumstances. That is, if x = (x1; :::; xn) is a vector in Rn; we willwant to think of that vector as referring to a model of future eventsarising as n di¤erent possible circumstances, labelled 1; 2; :::; n andcalled �the states of nature�. Then the component xi is interpreted as asum of money payable when the state of nature is i: Thus x modelswhat is known in Probability Theory as a �random variable�X with thestates of nature described by the sample space = f1; 2; :::; ng andthe realization of X when the state of nature is i is xi:

In Chapter 2 we help you to line up your assumed knowledge of LinearAlgebra with your assumed knowledge of Basic Probability andStatistics. This is an important step which enables us to use geometricintuition to solve problems originating in the valuation of �nancialcontracts.

By �nancial contract we mean reasonably standard �nancialarrangements such as insurance policies, or a contract for the executionof building works which may involve foreseeable but currently unknowncomplications. In all these we ask how much to pay �today�for acontractually speci�ed payment (albeit uncertain sum of money) to bereceived at some speci�ed future date such as the proverbial �same

1


time next year�.

The aim of this unit is to identify the fundamental concepts andmethods of Financial Mathematics. We will thus learn two approachesto representing the uncertain evolution of asset prices, �rst in discretetime and then in continuous time. We will formulate a basis for valuingwell-de�ned future payments that depend on one of a number ofspeci�ed circumstances occurring when it is not known in advancewhich of these circumstances will arise.

The mathematical arguments in discrete time will be conductedrigorously, meaning that terms will be precisely de�ned and results willbe proved. Of course plenty of motivation and informal explanationswill be given.

We will rely on the rigorous development in discrete time to support amuch more informal approach to the continuous time approach. In thesecond part (the continuous time context) the emphasis will be onusing calculus to obtain valuations of what are called European call andput options.

So, for this subject, you will not only have to solve problems: you willhave to be able to reason abstractly, and be able to prove or justifythings. In most of this subject, we need to work with precisede�nitions, and you will need to know these. Not only will you need toknow these, but you will have to understand them, and be able(through the use of them) to demonstrate that you understand them.Simply learning the de�nitions without understanding what they meanis not going to be adequate. One hopes that these words of warningwon�t discourage you, but it�s important to make it clear that this is asubject at a higher level than some of its prerequisites.

1.2 Aims of the unit

This subject is intended to enable students to:

acquire fundamental skills in the techniques of mathematical�nance

understand the principles underlying the subject of mathematical�nance

prepare for further units in mathematical �nance and/or relateddisciplines (e.g. economics, actuarial science).

1.3 Learning objectives

Having followed this unit, students should have:

knowledge, understanding and formulation of the principles ofrisk-neutral valuation including some versions of the No-ArbitrageTheorem

knowledge of replication and pricing of contingent claims in certainsimple models (discrete and continuous)

knowledge of the derivation of the Black-Scholes equation, itssolution in special cases, the Black-Scholes formula.

2

Syllabus

Having completed this unit, and the relevant readings, you should beable to:

demonstrate knowledge of the subject matter, terminology,techniques and conventions covered in the subject

demonstrate an understanding of the underlying principles of thesubject

demonstrate the ability to solve problems involving understandingof the concepts.

1.4 Syllabus

This is an introduction to an exciting and relatively new area ofmathematical application. It is concerned with the valuation (i.e.pricing) of ��nancial derivatives�. These are contracts which are boughtor sold in exchange for the promise of some kind of payment in thefuture, usually contingent upon a share-price then prevailing (of aspeci�ed share) or alternatively the level achieved by a share index, i.e.by a certain weighted average of share prices. They are called�derivatives�because they are derived from some underlying �nancialasset such as a share.

The unit reviews the �nancial environment and some of the �nancialderivatives traded on the market. It then introduces the mathematicaltools which enable the modelling of the �uctuations in share prices.Inevitably these are modelled by equations containing a random term.It is this term which introduces risk; it is shown how to counterbalancethe risks by putting together portfolios of shares and derivatives so thatrisks temporarily cancel each other out and then this strategy isrepeated over time. As this procedure resembles hedging a bet � i.e.betting both ways �one talks of dynamic hedging. A very intuitivevaluation argument (albeit now regarded as only of historical interest,see Section 9.7 for details) is based on �hedging�and begins like this:the yield of a temporarily riskless portfolio must equal the rate of returno¤ered by a safe deposit bank account (i.e. a riskless bank rate). Thelatter, of course, needs to be assumed to exist. This equation assumesthat the market which values shares and derivatives actually is inequilibrium and hence eliminates the opportunities of �arbitrage�(suchas making a sure-thing pro�t from, say, buying cheap and selling dear).

The �riskless hedge�argument, just mentioned, implies in thecontinuous-time model that the price of a derivative is the solution of adi¤erential equation. One may either attempt to solve the di¤erentialequation by standard means such as numerical techniques or viaLaplace transforms, though this is not always easy or feasible. However,there is an alternative route which may provide the answer: acalculation of the expected payment to be obtained from the contractby using what is known as the synthetic probability (or the risk-neutralprobability). One proves that, regardless of what an investor believesthe expected growth rate in the share price will be, the dynamichedging has the e¤ect that he might as well believe the growth rate tobe the riskless growth rate. Though this may seem obvious inretrospect it does require some careful reasoning to justify.

The unit considers two approaches to risk-neutral calculation, usingdiscrete time and using continuous time. Continuous time requires theestablishment of a second-order volatility correction term when using

3


standard �rst-order approximation from calculus. This leads to what isknown as the Itô Rule/Formula. Finite time arguments need someapparatus from Linear Algebra such as the Separating HyperplaneTheorem. We enter the subject from the discrete-time model for aneasier discussion of the main issues.

1.5 Organisation of the subject guide

After this introductory chapter, in which we also discuss the readinglist, the next chapter recalls some basic background in mathematics.Chapter 3 introduces you to some �nancial ideas with which the latermathematics will be concerned.

The unit proper begins, in e¤ect, with Chapter 4, which is dedicated toa study of a very simple framework. There is just one risky asset andmoney (�cash�to be precise), just one future date, and only twopossible �states of nature�that might occur. One state is such that aneconomic agent may regard it as �favourable�for himself and anotherthat he regards as �unfavourable�. Of course another economic agentmay view these two states the other way about. What matters is thatthere are only two states. The aim is to explain two central ideas. Oneis �replication�of a claim by means of a portfolio, which enables theclaim to be valued by the cost of purchasing the portfolio. This meansthat once this cost is incurred the holder of the portfolio is not exposedto any risks associated with settling his future liabilities. He is madeneutral to risk. The other idea is valuation using expected values withreference to a �purpose-made�probability measure called a �risk-neutral�measure which is sometimes referred to as the �synthetic probability�(for an emphasis of its speci�c purpose). Thus the expected value ofthe claim under this measure is to agree with the purchase cost of thereplicating portfolio.

Once understood in the simplest concept, we re-establish in Chapter 5the two central ideas in a one-period model with several assets andseveral states. We prove the Fundamental Theorem of Asset Pricing,namely that a risk-neutral measure exists if and only if there are noarbitrage opportunities.

In Chapter 6 the two ideas are extended to cover multiple periods aidedby the idea of a �self-�nancing�trading strategy. This is intellectuallythe hardest part of the unit as it uses ideas from several quarters. Themain tool is conditional expectation and this requires quite a lot oftechnical apparatus to de�ne rigorously what information is known atcertain points in time.

Whereas Chapter 6 assumes a quite general framework for theevolution of asset price, we discuss in Chapter 7 a speci�c approach tomodelling asset price, the Binomial model. In this model, price changesare generated at each period in a uniform way by two constant factors,one factor inducing a possible price move up, the other an alternativeprice move down. This uniform behaviour permits easy formulas forvaluation of European calls and puts. Furthermore it o¤ers easycomputation procedures for evaluating American puts. The signi�canceof the model comes from limit considerations. In an appropriatelyconstructed limit (as the number of periods tends to in�nity and thetime between periods tends to zero) the model�s European call and putvaluations tend to the corresponding valuations in the Black-Scholes

4

Organisation of the subject guide

continuous-time model.

The theme of Chapter 8 is modelling price uncertainty in continuoustime. As its base the model assumes an anticipated (constant) rate ofgrowth � for the price of an asset and a noise of constant �amplitude��: The growth rate is modelled along the lines of the deterministicinterest rate reviewed in Chapter 3. Thus the anticipated percentagegrowth in price over a time span of �t is ��t: The further ingredientin the model is �standard�noise as generated by a �stochastic process�,that is by a family of random variables indexed by time, denoted herezt: The standard amount of noise over the time span �t starting attime t is �zt; which is de�ned to be zt+�t � zt: It is assumed that �ztis normally distributed with variance �t: The noise added to theanticipated growth is then ��zt; so that the �standard noise�isampli�ed or attenuated by a factor �: Thus the noise element hasvariance �2�t; which is thus proportional to the time span �t. Thecentral tool developed in this Chapter is Itô�s Formula. This isconcerned with functions of price and allows the computation of theincrement in a di¤erentiable function when the stochastic price changesover a time interval. The Formula gives a stochastic form of Taylor�sTheorem.

The techniques of Chapter 8 are put to use in the Chapter 9. Time isdiscretized so that the time span between periods is �xed but arbitrarilysmall, along lines familiar in units 116 Abstract mathematics or 117Advanced calculus. The Itô Formula is used to approximateincrements in value. Valuation proceeds by the familiar route ofreplication over the individual periods between the discrete time points.The emphasis is on deriving in the limit (as the time spans tend tozero) a valuation of a European option contract written on some asset.This will be a deterministic function which identi�es the option value interms of two free variables: the current asset price, and the time left tothe expiry date of the contract. The value function may thus beinterpreted as the conditional expected value of the option pay-o¤under a risk-neutral probability with conditioning on the current assetprice and the time to expiry. We are able to identify what thisrisk-neutral measure is once we derive the celebrated Black-Scholesequation. This is the partial di¤erential equation satis�ed by the valuefunction, and we obtain it from a consideration of increments in value.So we are �nally able to identify the risk-neutral measure. Armed withthe risk-neutral measure we can value a European call option.

The �nal chapter is concerned with American long-dated options, thatis, ones which have a relatively long life left (their expiry date is �far�into the future). For the purposes of a tractable analysis of suchoptions we place their expiration date at in�nity. Options which expireat in�nity are called perpetual. They satisfy an ordinary di¤erentialequation version of the Black-Scholes equation and the equation maybe solved explicitly. Thus we are able to trace the graphs of perpetualoptions and have an indication of the value of a long-dated Americanoption.

Finally, an epilogue sums up, in slightly more technical jargon, what wehope you will have achieved by the end of this unit.

Not all chapters of the guide are the same length. It should not bethought that you should spend the same amount of time on eachchapter. I will not try to specify how much relative time should bespent on each: that will vary from person to person, and we do not

5


want to be prescriptive. I can however indicate roughly what proportionof time might be spent by a lecturer teaching the material in thisguide. A lecturer would judiciously pick which examples to include in alecture, and which to leave to the students to study on their own.Though the mathematics of subsequent chapters is not any harder inthis unit than in any other, the table below articulates the signi�cantneed to spend time in Chapter 3 on building up an understanding ofthe �nancial environment.

Chapter 2 Vector space approach 5%Chapter 3 Financial environment 20%Chapter 4 One risky asset and two states 10%Chapter 5 One period many assets 10%Chapter 6 Multi-period models 20%Chapter 7 The binomial model 15%Chapter 8 Continuous-time modelling 5%Chapter 9 Black-Scholes model 10%Chapter 10 Perpetual options 5%

1.6 Reading advice

Notes on the reading lists

Most topics in this unit are covered in great detail by a large number ofbooks. For that reason, we have resisted the temptation to specifyessential reading in each chapter of the guide. What is true, however,is that textbook reading is essential. Textbooks will provide morein-depth explanations than you will �nd in this guide (which is explicitlynot meant to be a textbook), and they will also provide many moreexamples to study, and many more exercises to work through.

The following books are the ones I have referred to in this guide, listedroughly in order of usefulness and grouped according to their level. Inthe �rst group, the �rst text, by Pliska, is one of the earliest textbookson the �undergraduate market�. Our treatment in Chapters 5 and 6 isinspired in part by Pliska. The �rst Shreve text is a very accessiblealternative for the core material of Chapters 4�6. We follow the Notation variant to Pliska�snotation established in Pliska with two exceptions. Firstly, whenconsidering a trading strategy or portfolio H we write Vt(H) for itstime t value rather than Vt. Secondly, we refer to Pliska�s �dominantstrategies�as �sure-thing arbitrage strategies�. We strongly recommendthat the student reads through the book by Hull to understand theinstitutional arrangements governing stock-exchanges and the contractsthat they issue. A very good second text for this material is Cvitanicand Zapatero. It is hard to be quite as prescriptive in relation to thematerial of Chapters 7�10 because our approach is informal � thus wedescribe, rather than de�ne, stochastic integrals as limiting sumsavoiding the requisite formalities. Hull contains an account of theinformal theory. Luenberger is exquisite in clarity. Baxter and Rennie isadmirable on this front, though written for the �practitioners�in the�nance houses. That has been replaced by its �classroom�version, amore technically demanding text, by Alison Etheridge aimed atuniversity students with measure theory as a prerequisite. The secondShreve text is accessible but more formal. Finally, for a goodbackground for linear algebra and advanced calculus see the book byOstaszewski (which is the recommended text for unit 117), or Binmore

6

Reading advice

and Davies.

The second group of books on the reading list is meant for a veryspecial reader: students should know by now that any unit such as thisis an introduction to the collective knowledge amassed by an army ofmathematicians, so these books are a pointer for ambitious studentslooking for further developments. The books in the third group arepopularisations and provide either anecdotal background or a historicalperspective.1 1Earlier editions than those

listed here are equally useful.

Recommended reading

Shreve, S. Stochastic Calculus for Finance I, The Binomial asset pricingmodel. (Springer, 2004) [ISBN 978-0387249681].

Shreve, S. Stochastic Calculus for Finance II, Continuous-time models.(Springer, 2004) [ISBN 978-0387401010 ].

Pliska, S.R. Introduction to Mathematical Finance - Discrete TimeModels. (Blackwell, 1998) [ISBN 978-1557869456].

Hull, J.C. Options, Futures and other Derivative Securities. (PrenticeHall, 2005) sixth edition [ISBN 978-0131499089]. See also the URL:http://www.rotman.utoronto.ca/~hull/

Cvitanic, J. and F. Zapatero Introduction to the Economics andMathematics of Financial Markets. (MIT, 2004) [ISBN978-0262033206, 978-0262532594 (solutions manual)].

Luenberger, D. Investment Science. (Oxford University Press, 1997)[ISBN13: 9780195108095].

Roman, S.Introduction to Mathematics of Finance. UndergraduateTexts in Mathematics. (Springer, 2004) [ISBN 978-0387213644].

Baxter, M. and A. Rennie Financial calculus. (Cambridge UniversityPress: Cambridge, 1996) [ISBN 978-0521552899 ].

Etheridge, A. A Course in Financial Calculus. (Cambridge UniversityPress: Cambridge, 2002) [ISBN 978-0521890779].

Ostaszewski, A. Advanced Mathematical Methods. (CambridgeUniversity Press: Cambridge, UK, 1991) [ISBN 978-0521289641].

Binmore, K. and J. Davies Calculus: Concepts and Methods.(Cambridge University Press: Cambridge, UK, 2001) [ISBN978-0521775410].

Intermediate reading

Bingham, N.H. and R. Kiesel Risk-Neutral Valuation: Pricing andHedging of Financial Derivatives. (Springer, 1998) �rst edition [ISBN1852330015] second edition, 2004 [ISBN 978-1852334581].

Campbell, J.Y., A.W. Lo and A.C. MacKinlay The econometrics of�nancial markets. (Princeton University Press) [ISBN 978-0691043012].

Janson, S. Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics

7


129 (Cambridge University Press, 1997) [ISBN 978-0521561280 (hbk)]in connection with the geometric view of probability sketched inChapter 2.

Williams, D. Probability with martingales. (Cambridge, 1991) [ISBN978-0521406055].

Øksendal, B. Stochastic di¤erential equations. (Springer, 1998) [ISBN978-3540047582].

Wilmott, P., S. Howison and J. Dewynne Mathematics of FinancialDerivatives. (Cambridge University Press: Cambridge, 1995) [ISBN978-0521497893 ].

Whittaker, E.T. and G.N. Watson A course of modern analysis.(Cambridge University Press, 1984) [ISBN 978-0521588072].

Durrett, R. Stochastic Calculus - A Practical Introduction. (CRC Press,1996) [ISBN 978-0849380716]. An excellent though moderatelydi¢ cult account of Itô integration, properties of Brownian motion,solution to stochastic di¤erential equations.

Evans, L.C. Partial Di¤erential Equations. (American MathematicalSociety, Providence, 1998) [ISBN 978-0821807729].

Schuss, Z.Theory and Applications of Stochastic Di¤erential Equations.(J. Wiley, 1980) [ISBN 978-0471043942]. A very down-to-earth textbut directed at applications in physics. Discusses Stratonovichintegration as well.

Merton, R.C. Continuous-time Finance. (Blackwell, 1996) [ISBN978-0631185086].

Samuelson,P. �Proof that properly anticipated prices �uctuaterandomly�, Industrial Management Review, 6(2) 1965, pp.41�49.

Further reading (historical, anecdotal, or populartexts)

Bernstein, P. Against the Odds. (J. Wiley, 1998) [ISBN 0471295639].

Bernstein, P. Capital Ideas - The Improbable origins of modern WallStreet. (J. Wiley, 1998) [ISBN 978-0029030127]. A wonderful historyof the growth of ideas in this area - a very good read.

Davis, M. and A. Etheridge Louis Bachelier�s Theory of Speculation:the origins of modern �nance. Translated and with commentary(Princeton, 2006) [ISBN 978-0691117522].

Dunbar, N. Inventing Money: The Story of Long-Term CapitalManagement and the Legends Behind It. (J. Wiley, 2001) [ISBN978-0471498117].

Kay, J. Foundations of Corporate Success. (Oxford University Press,1995) [ISBN 978-0198289883].

Kay, J. The truth about markets. (Allen Lane, Penguin Press 2003)[ISBN 978-0140296723]. (�Everything you wanted to know abouteconomics, but were afraid to ask��Mervyn King, Governor, Bank of

8

Examination advice

England).

Lewis, M. Liar�s Poker. (Coronet Books, 1989) [ISBN978-0340767009].

Lowenstein, R. When Genius Failed: The Rise and Fall of Long-TermCapital Management. (Random House, 2000) [ISBN 978-0375758256].

Smithers, A. Valuing Wall Street: Protecting Wealth in TurbulentTimes. (McGraw-Hill, 2002) [ISBN 978-0071354615 (hbk)9780071387835 (pbk)].

Bachelier, L. Louis Bachelier�s Theory of Speculation: the origin ofmodern �nance. Translated and with commentary by Davis, M and A.Etheridge (Princeton: Princeton University Press, 2006) [ISBN978-0691117522].

1.7 Using the subject guide

I have already mentioned that this guide is not a textbook. It isimportant that you read textbooks in conjunction with the guide andthat you try problems from the textbooks. The learning activitiesthroughout the guide, and the sample questions at the end of thechapters, are a very useful resource. You should try them once youthink you have mastered a particular chapter. Do really try them: don�tjust simply read the solutions where provided. Make a serious attemptbefore consulting the solutions. Note that the solutions are often justsketch solutions, to indicate to you how to answer the questions, but inthe examination, you must show all your calculations. It is vital thatyou develop and enhance your problem-solving skills and the only wayto do this is to try lots of examples.

1.8 Examination advice

Important: Please note that subject guides may be used for severalyears. Because of this we strongly advise you to always check both thecurrent Regulations for relevant information about the examination,and the current Examiners�reports where you should be advised of anyforthcoming changes. You should also carefully check therubric/instructions on the paper you actually sit and follow thoseinstructions.

A sample examination paper is given as an appendix to this guide.There are no optional topics in this syllabus: you should do them all.

Please be aware that a few sections in this guide have beenstarred (*) to indicate that the material of the section is notexaminable. The material has been included in the guide eitherfor its interest value or in order to help you to see the connectionwith a wider view of the subject.

Please do not assume that the questions in a real examination willnecessarily be very similar to these sample questions. An examination isdesigned (by de�nition) to test you. You will get examination questionsthat are unlike any questions in this guide. The whole point ofexamining is to see whether you can apply knowledge in both familiar

9


and unfamiliar settings. The Examiners have an obligation to surpriseyou! For this reason, it is important that you try as many examples aspossible: from the guide and from the textbooks. This is not so thatyou can cover any possible type of question the examiners can think of!It�s so that you get used to confronting unfamiliar questions, grapplingwith them, and �nally coming up with the solution.

Do not panic if you cannot completely solve an examination question.There are many marks to be awarded for using the correct approach ormethod.

A �nal point about the examination for this unit is to reassure you thatexamination questions will not assume familiarity with the contents ofany of the starred sections in this subject guide. The purpose of suchexamples is to motivate the mathematics and to persuade you that themathematics you are learning has important uses.

Acknowledgement

This guide is based on an established lecture course at LSE of the samename. I am also very grateful to my colleagues Nick Bingham andMichalis Zervos for a careful reading of the manuscript and numeroussuggestions.

Adam Ostaszewski

10

Chapter 2

A vector space approach touncertainty

2.1 Introduction and aims of the chapter

This chapter begins with some activities helping to remind you of somebasic concepts from Probability such as mean, variance, momentgenerating function and the standard normal distribution. In particularyou are guided to calculate a few important integrals here, ahead oftheir natural appearance towards the end of the unit. This should helpyou to check on the kind of calculus that will be required. The mainmatter, however, is concerned with how to employ the language ofvector spaces to talk about random variables; we will use them todescribe possible future prices of �assets�and actual �portfolios�, that iscollections of assets held by an economic �agent�.


At the end of this chapter and the relevant readings, you should beable to:

compute expectations (integrals) using �log-normal densities�

be prepared to use vectors to study random variables, asset pricesand portfolios of assets.

2.3 Essential reading

There is no speci�c essential reading for this chapter. It is essentialthat you do some reading, but the topics discussed in this chapter areadequately covered in so many texts on the �applications of calculus�that it would be arti�cial and unnecessarily limiting to specify precisepassages from precise texts. The list below gives examples of relevantreading. (For full publication details, see Chapter 1.)

Ostaszewski, A. Advanced Mathematical Methods. Chapter 21.

2.4 Concepts from probability and statistics

The activities in the �rst subsection are meant to make you aware ofthe practical di¢ culties in applying some of the ideas in theBlack-Scholes model of Chapter 9.

11


2.4.1 Activities referring to basic statistics

Learning activity 2.1

Let X1; :::; Xn be independent random variables. For each i assumethat E[Xi] = � and var[Xi] = �2. De�ne

�X =1

n(X1 + :::+Xn);

V =1

n� 1 [(X1 � �X)2 + :::+ (Xn � �X)2]:

Check that(i) E[ �X] = � and so E[(X1 � �X)2] = var(X1 � �X);(ii) cov(X1; �X) =

1nvar(X1);

(iii) var[ �X] = 1n�

2.

Show that var(X1 � �X) = (n�1)�2n and so deduce that

E[V ] = �2:

[Hint: var(X1 � �X) = var(X1) + var( �X)� 2cov(X1; �X).]


Show that when the variables of the last question Xi are normallydistributed

var(V ) =2�4

n� 1 :

Thus the standard deviation is small when the sample is large.

Comments on the activity

Suppose Xi are observed rates of return on investment per period of a Mean blur.single year, there being a largish number n of periods (perhaps months,or days). Then S = nA will be the annual rate (at least to a good �rstapproximation) and E[S] = n�: The corresponding variance isvar[S] = n�2 and so the standard deviation for the year is �S = �

pn.

Hence�=� = (�S=

pn)=(E[S]=n) =

pn � (�S=E[S]):

Suppose the annual rate is reasonably accurate, as measured by theratio of its standard deviation to the expected value, for instance�S=E[S] might be of order 1 or 2. However, this ratio scales upwardshugely (by a factor of

pn) when the annual rates are scaled down to

per-period rates. See Luenberger pages 214�216.

2.4.2 Concepts of probability requiring calculus

The activities in this subsection revise the kind of use of integralcalculus that is required in this unit. Reference is made to the

12

Concepts from probability and statistics

probability density function of the standard normal variable denoted

'(x) =e�

12x

2

p2�

and the corresponding cumulative distribution function

�(t) =

Z t

�1'(x)dx =

Z t

�1e�

12x

2 dxp2�:


The moment generating function is de�ned as M(t) = E[etX ]:

When X is a standard normal variable show that M(t) = e12 t

2

. [Hint:

E[etX ] =

Z 1

�1etx'(x)dx =

1p2�

Z 1

�1e�

12x

2+txdx

and complete the square.]


Using the identity E[etX ] = e12 t

2

; show that for a standard normalrandom variable

E[X4] = 3:

[Hint: Expand both sides of the identity as series, thus:Z 1

�1

e�12x

2

p2�

f1+ tx+ t2x2

2+t3x3

6+t4x4

24+ :::gdx = 1+ 1

2t2+

1

8t4+ :::;

and compare both sides.]


If Z is a log-normally distributed random variable with mean m and Moment generating functionvariance s show that E[Z] = eme

12 s

2

: [Hint: The random variableX = (lnZ �m)=s is normally distributed with zero mean and unitvariance (i.e. is a standard normal random variable). UseE[Z] = E[esX+m] and the previous activity.]


Show that if Z is a random variable such that

lnZ = m+ aU + bV

where U and V are independent random variables with standardnormal distributions then

E[Z] = em+12 (a

2+b2):

13


Hint: E[Z] = E[em+aU+bV ] = emE[eaU ]E[ebV ]: Why?


The next activity shows how to deal with lnZ when the two normalrandom variables are correlated. The trick is to regard the randomvariables as two vectors and to replace them with two orthogonal oneswith the same linear span (recall the Gram-Schmidt process in LinearAlgebra). In this Gaussian variables context orthogonality has the samemeaning as zero correlation.


If Z is a random variable such that

lnZ = m+X + Y;

where X and Y are correlated random variables withCov(X;Y ) = ��X�Y normal distributions both with mean zero, thenthe trick is to replace X by rescaling it to a new random variable Uwith variance unity and to rewrite Y as a sum of U and anotherstandard normal variable V: Writing

Y = �U + �V;

where X = U; show that

= �X ; � = ��Y ; � =p1� �2�Y :

Deduce thatE[Z] = em+

12 (�

2X+�

2Y +2��Y �Y ):


Find the expected value of the function of a random variable X givenby g(X) = maxfX; 0g, when the probability density function of X,here denoted f(x), is log-normal. That is, evaluate

E[g(X)] =

Z 1

k

(x� k)f(x)dx =Z 1

k

xf(x)dx� kZ 1

k

f(x)dx:

This calculation is the basis of the Black-Scholes formula for valuing a A Black-Scholes calculation�call-option�.[Hint. Make a change of variable by putting

w = w(x) =lnx�m

�;

and write the two integrals in terms of the cumulative normaldistribution function �: It may be helpful to refer to �sectionalexpectation�, E[g(X);X > k]; as de�ned below, and how that changesunder a change of variable:

E[g(X);X > k] =def

Z 1

k

g(x)f(x)dx = E[g(e�W+m);W > w(k)]

=

Z 1

w(k)

g(e�w+m)'(w)dw:

14

Linear algebra: the language of vectors


The function Erf is de�ned in Mathematica as

Erf(x) =2p�

Z x

0

e�t2

dt:

Check that�(x) = (1 + Erf(x=

p2))=2:

2.5 Linear algebra: the language of vectors

In investment analysis one studies the possible prices an asset mighthave at some future point in time. The term �asset�is wide rangingand might be a share in a company, but it can also be the pay-outo¤ered by some form of insurance contract. Since in both cases thefuture value is a¤ected by �circumstances�we can draw attention tosuch dependence, if the need arises, by using the more general term�contingent claim�. This term is used interchangeably with �asset�.

Contingent claims most often considered in this unit take the form of a Primary and �derivative�assetspayment in an amount which depends on the value of some speci�ed

asset where the evolution in time of the value of the speci�ed asset isgiven. Since the value of the speci�ed asset is a �given�, the asset issaid to be a �primary asset�. The contingent claim is then said to be aderivative asset, or just a �derivative�.

It may be necessary to contemplate various plans of action now, or inthe future, and the action could well depend on the realized futureprice. In order to be speci�c about various future scenarios it becomesnecessary to list what possible prices should be considered in this kindof forward planning. Say the prices might be X1; X2; :::; Xn. We willinitially think of these numbers as being positive (or, sometimes, justnon-negative). It is therefore possible to think about the informationcurrently available concerning the future of an asset, or the informationcurrently assumed about the future, as being described by the �vectorof prices�(X1; X2; :::; Xn)

t:

2.5.1 States of nature

In order to discuss more complicated scenarios, say involving severalassets at once, it is more helpful as a starting point to describe thefuture as just a set listing all �possible future atomic events ofinterest�. Any one such event is typically denoted ! and nospeci�cation at this stage is given as to what is to be an �atomic event�.These events, in whatever way selected, will become �atomic�by sheerdint of de�nition only. Of course in any particular discussion of futureevents we would need to be speci�c about the list : For instance, ifthere are only two assets of interest and only one future date, anatomic event ! would be de�ned by: �the �rst asset will take the price

15


Xi and the other asset the price Yj�. Here, as above, X1; X2; :::; Xn

lists all possible prices of the �rst asset and Y1; Y2; :::; Ym those of thesecond. In fact we could identify ! with the pair of values (Xi; Yj):This example explains why the set is called the sample space: it isthe set of all possible data samples that may occur. It is also referredto as a �population�from which a sample of data may be drawn.

It is customary to refer to the atomic events also as �states of nature�.The original asset under discussion is now thought of as having priceX(!) when the future is given by the event !: The asset is thus bestdescribed by correspondence to the states of nature, that is by thevector (X(!1); :::; X(!m))t if = f!1; :::; !mg: When the contextimplies the use of we refer also to this vector as the �vector ofpossible prices�and denote it by X: A word of warning here: it issometimes convenient, but really very careless (clearly not a habit to beencouraged), to speak of �asset X�; but it is very important todistinguish conceptually between the name of an asset and the vectorof its prices at some given point in time. Assets, like any otherproducts in the market, though di¤erent in name could in principlehave equal vectors of possible prices; when there is danger of a mix-upwe will be more careful with the notation.

Example 1 (Potential gains and losses as vectors). An investorwith M dollars of cash in his account is considering whether to buy anasset now when its unit price is 1 dollar or to wait to the end of the daywhen he thinks the price is likely to go up or down by at most one centwith equal probability. He can think of this as a coin-tossing game witha fair coin. The result of the toss is either !1 = Heads or !2 = Tailswith Heads indicating a rise. Thus = f!1; !2g. He models the unitdollar-price of the asset with the vector X where X(!1) = 1:01 andX(!2) = 0:99: If the investor decides to wait to the close of business,the unit-price in dollars of the asset in his account �namely his cash �is modelled by B, where B(!1) = 1 and B(!2) = 1, i.e. B = (1; 1)since irrespective of the toss the denomination of the currency isunchanged. If he bought the asset his gain G from waiting is thuseither G(!1) = +0:01M or G(!2) = �0:01M: In vector form the gainG satis�es G =M � (X �B): The investor reckons he has a 50%chance of winning by waiting (but also a 50% chance of losing).

Example 2 (E¤ects of diversi�cation). Our investor has in mind tosplit his cash evenly between two assets, X;Y both currently selling at1 dollar per unit. He thinks that either of the two assets willindependently rise or fall by one cent by the end of the day. This timeit is a two-coin tossing game and so the outcomes will be!1 = (Heads;Heads); !2 = (Heads; Tails);!3 = (Tails;Heads); !4 = (Tails; Tails) with Heads indicating arise. As before, we have a vector of prices for the �rst asset, listedaccording to state, given byX = (X1; X2; X3; X4) = (1:01; 1:01; 0:99; 0:99); and a vector for thesecond asset given by Y = (Y1; Y2; Y3; Y4) = (1:01; 0:099; 1:01; 0:99):This time the investor�s gain will beG = M

2 (X + Y )�M �B =M(0:01; 0; 0;�0:01): By diversifying hisacquisitions he will lose only in one state of nature; the chance of thishe assesses as 25%. But then he also narrows his chances of a win.

We refrain from going through any more complicated examples of atthis stage. Instead we merely let name the members of the set ofpossible simple events; we suppose that there are at most m possiblesimple events and we label them !1; :::; !m: Thus = f!1; :::; !mg:

16


Although we are interested here in asset prices, the more generalsituation is that of modelling some numerical property, call it X, whichis associated with every member of a �population�: Thus a member! of drawn from the population displays a value X(!): The vector ofpotential values obtained by drawing an individual member from thepopulation is thus given by (X(!1); :::X(!m))t: In this context such avector is called a �random variable�. The term owes its name to twofacts: that a numerical property may in general be variable over thepopulation; and, that, when the population is large, rather thanconduct a census in order to learn its values, it may be moreeconomical to observe the values on a representative sample ofindividuals drawn at �random�. A simple example is that of �height�ofpeople in a population. We will not discuss appropriate ways to select asample of individuals at random, which is a matter for statisticstextbooks. But we will show how mathematical ideas may be appliedpro�tably to random variables. Most often we will be concerned withthe random variable model of asset prices.

2.5.2 Compound events

The notation adopted in this section does imply that the componentsof X need not be distinct (as in Example 2). So a future event such as�the price of the asset will be x�need no longer be a simple event. Thisis because one and the same price x may occur in several states ofnature. The event that the price is x is a �compound event�; we denotethat event by [X = x]. The de�nition is

[X = x] := f! 2 : X(!) = xg:

Evidently for distinct x the compound events [X = x] are disjoint and,as x varies through all the possible prices, these events togetherexhaust all the states of nature. We therefore say that the asset valuespartition : In Example 2 of the last subsection we have

[X = 1:01] = f!1; !2g and [X = 0:99] = f!3; !4g:

On the other hand notice that by referring to the language we haveadopted allows us to describe any other asset Y by the vector of itsprices (Y (!1); :::; Y (!m)). This is a vector with the same number ofcomponents as X:

2.5.3 Portfolios

If someone holds u units of asset X the value of the quantity u of theasset in state !i will be

uX(!i):

If u is negative, this number will be negative and may be interpreted as Long and short positionsthe value of a debt resulting from having initially borrowed juj units ofthe asset. When u is negative the holder is said to be in a positionwhich is short of juj units of the asset X, or more conveniently �shortjuj units X�. If u is positive the holder is correspondingly said to belong u units in X: Whilst you may regard as natural the use ofnegative quantities in this context, you should be aware that this use ofmathematics coincides with the system of �double-entry�book-keepingpromoted during the Italian Renaissance. Its �rst exposition, due to the

17


mathematician Pacioli in 1494, was in a printed book, at a time whenprinting was expensive (a mere quarter of a century after Guttenberg�sinvention of metal type). It is one of the foundation stones ofcapitalism. Double entry book-keeping

Consider now holding u units of asset X and v units of asset Y ; this isreferred to in �nancial terms as a portfolio of u units of X and v unitsof Y: The portfolio may be neatly represented by the vector (u; v)t, itbeing understood that a �xed order of listing the assets is beingreferred to. With a two-asset list X;Y the portfolio (u; v)t lies in R2though the assets themselves lie in Rm: The value at the future date ofthis portfolio in state !i will be

uX(!i) + vY (!i)

and this is simply the i-th co-ordinate of the vector uX + vY in Rm:

Example 3. A person who borrows, without interest, the M dollarsfrom our investor of Example 1 and exchanges them for asset X whenthe unit price is 1 dollar holds the portfolio which is M units long inasset X (so u =M ) and short M units (i.e. v = �M units) in assetB (the cash which he exchanged for the asset). Thus uX + vBrepresents his gain that day at close of business. This equals the vectorG considered earlier.

2.5.4 The portfolio space: replication and completeness

If we allow both long and short positions in the assets X and Y , theset of pay-o¤s uX + vY of all portfolios (u; v)t is simply the subspacein Rm spanned by the two vectors X;Y: This is called the portfoliospace of the assets X and Y: We will be concerned in future exampleswith several assets S1; :::; SN and with the assessment of the currentvalue of portfolios (u1; :::; uN )t: The portfolios are vectors in RN andrepresent u1 units of asset S1; u2 units of asset S2; :::; uN units of assetSN : The portfolio has payo¤ vector

u1S1 + :::+ uNSN :

When = f!1; :::; !mg this vector is in Rm. Clearly the vector lies inLinfS1; :::; SNg; a subspace of Rm, to be called the portfolio spaceof S1; :::; SN : The assets S1; :::; SN are not necessarily assumed to belinearly independent as vectors in Rm.

It is easy to confuse the distinction between the portfolio (u1; :::; uN )t

which is in RN and its pay-o¤ in Rm, so the reader should beware.

We will refer to S1; :::; SN as the spanning assets, or even basis assets(if the vectors happen to be linearly independent), and to a generalmember of the portfolio space LinfS1; :::; SNg as an asset X when

X = u1S1 + :::+ uNSN :

We can write this last line as

X = Su:

Here u = (u1; :::; uN )t and S is the m�N matrix2664S1(!1) S2(!1) ::: SN (!1)S1(!2) S2(!2) ::: SN (!2)::: ::: ::: :::S1(!m) S2(!m) ::: SN (!m)

3775 :18


When X in Rm is in the portfolio space, and X = Su, for someportfolio u; we say that u replicates the claim. If each vector X in Rmmay be replicated by some portfolio we say that the portfolio space iscomplete.

We will return to this point in Section 5.

2.5.5 The riskless asset

We let 1 denote the vector (1; 1; :::; 1)t in Rm having a one in eachco-ordinate. Thus u1 represents an asset which has price u in eachstate of nature. Since there is no risk regarding the value later in time,an asset represented by u1 is termed a riskless asset. By contrast anyvector not of this form is said to be risky. A dollar deposited in a bankwhen the interest rate is �xed in advance provides an example of ariskless asset, since the dollar together with interest accrual is paidback by the bank no matter what the state of nature. (This is trueprovided we believe the bank is safe from failure, which we dothroughout this unit.)

Of course the portfolio space spanned by the set of assets fS1; :::; SNgmight contain 1 but need not Figure 2.1. The riskless asset in atwo-state model.

0.5 1 1.5 2

0.250.5

0.751

1.251.5

1.752

Figure 2.1: The riskless asset in a two- state model.

2.5.6 The call option: replication and linear span

A �call option�is a �nancial contract issued with reference to a Strike/exercise pricespeci�ed asset, called the �underlying�asset, and a price known as the�exercise price�or �strike price�, which we denote by K; it is issued(often by an exchange) at some initial date, here taken to be t = 0,and it promises at a later date, here taken to be t = 1, a paymentwhich when the underlying asset has price S(1) is equal to S(1)�K,provided this is positive, but is zero otherwise. Thus the payment hasvalue C = maxfS(1)�K; 0g, the larger of S(1)�K and zero.

A related option is the �put option�. This again is issued (often by anexchange) at some initial date, which we take to be t = 0; and itpromises at a later date, which we take as t = 1; a payment equal toK � S(1) provided this is positive (as before the underlying asset has

19


price S(1)) and zero otherwise. Thus the payment has valueP = maxfK � S(1); 0g, the larger of K � S(1) and zero.

We will use the notation

(x)+ = maxfx; 0g:

Thus the call option pay-o¤ is (S(1)�K)+: The corresponding putoption pay-o¤ is (K � S(1))+:


Plot the following pay-o¤ functions in Mathematica obtained byconsidering combinations of puts and calls:

(x� 1)+; (1� x)+, (x� 1)+ � (1� x)+, (x� 1)+ � (x� 2)+,(1� x)+ � (2� x)+, (x� 1)+ � 2 � (x� 1:5)+ +(x� 2)+; (x� 1)+ + (1� x)+,(x� 1)+ + (2� x)+, (x� 1)+ + 2(1� x)+, 2(x� 1)+ + (x� 1)+

Say in your own in words what are the circumstances under whichpro�ts arise to the holder of these contracts.


Reading from left to right, these represent pay-o¤s to contractsdescribed by the following names:on the �rst line: a long call, a long put, forward contract, bull spread,on the middle line: bear spread, butter�y spread, bottom straddle,on the bottom line: strangle, strip, strap.Notes:

1. The function (x)+ = maxfx; 0g may be de�ned in Mathematica by:plus[x_] := If[x<=0,0,x].

2. The pay-o¤ for a long call option with exercise price k is given by(x� k)+, since this pays x� k when x > k and zero otherwise.

What is the use of the call option? Whenever the asset has a terminalprice S(1) above K; the call, by way of a cash compensation (paid outwhenever the contract is exercised), e¤ectively gives the holder theright to buy a unit of the asset from the exchange at the lower price K:(The holder �calls�for a sale at price K:) If the asset price is below Kthe holder need not exercise the contract. It is intuitively clear that theoption is a valuable form of insurance. But what should its fair pricebe?

We pause to consider how to illustrate the contract, as this will help usvalue it.

There are three natural approaches to illustrating the payment scheme.

The �rst approach is to plot payment against all values that we couldassign to S(1); as in the �rst �gure below. This picture helps us to see

20


how the payment formula works.

00.5 1 1.5 2

1

0.8

0.6

0.4

0.2

0

Fig. 2.2: Payment from a call option with strike price K = 1 againstall values of S(1):

The second approach recognises that, at least for the time being, wewant to have only a �nite number of states of nature to deal with. Soone way to illustrate this �niteness is to indicate on the price axis, saywith dots as on Figure 2.3, just those future prices which may be ofinterest to us in some forward planning exercise and then to graph onlythe corresponding option values. For instance, we might conceive oftwo economic scenarios, a good one in which the underlying asset has ahigh price SH , and a poor one in which the asset has a low priceSL,with SL < SH . What are the good and poor economic scenariosde�ned by? Just as you please; for instance according to whether thechosen asset has a high or low price (for example, the asset might be aportfolio of shares such as makes up a market indicator, like aFTSE-100 index).

Fig. 2.3: Payment from a call option with strike priceK against two �reference values�for S(1).

The third approach stresses the fact that we want S(1) to berepresented as a vector, with its components corresponding to states.So in the third �gure, Figure 2.4, we suppose that there are only twostates of nature !1 and !2 (corresponding to the poor and goodeconomic scenarios referred to above), and we put up two axes one foreach of these two states. The idea is that the two values in the twostates arising from any payment scheme X can be shown as vector(x1; x2), with x1 = X(!1), and x2 = X(!2). This makes the x1 axiscorrespond to state !1 and x2 axis correspond to state !2: Now thepayment from a sale of one unit of the underlying asset is represented

21


by a vector with two components (SL; SH): The prices paid in the twostates can then be read o¤ according to state from one or other axis. InFigure 2.4 this is illustrated with the vector S = (SL; SH) = ( 12 ; 2):Theriskless asset is also shown in the diagram as 1 = (1; 1):

1

0 . 5

2

1 . 5

1

0 . 5

0 . 5 0 . 5

Fig. 2.4: Replicating the call C = (0; 1:5) using theriskless vector 1 = (1; 1) and the risky vector

S(1) = (:5; 2) as spanning vectors.

On the same diagram we show a call option. It pays C(!) in state !where C(!) = maxfS(1; !)�K; 0g: For our illustration we tookK = 1

2 . Thus K here satis�es SL � K < SH : In the poor economicstate C(!1) = 0, and in the good economic stateC(!2) = SH �K = 1

2 ; thus C lies along the !2-axis.

This third diagram represents how we intend to model contingentclaims at their simplest. We take to consist of two states. We takefor our portfolio space the span of two assets: the riskless bank deposit1 and a �risky�underlying asset with price vector S, i.e. one with pricesSL < SH : Thus S =2 Linf1g.

What is special about this model? First of all, we choose these twoassets, because we know their value (price) at time t = 0 on themarket. The market trades asset S for S(0) and the bank o¤ers theriskless asset 1 for 1=(1 + r): Secondly LinfS;1g = R2; so the modelis complete; this means that any claim can be expressed as a sum ofthe two spanning assets in appropriate amounts. We hope to be able toprice the claim correspondingly, as the cost the two summands. Inother words, we use completeness to price claims.

Our �rst step is to see about spanning the call.

In this context it is natural to relabel the two states so that!H = High and !L = Low corresponding to the risky asset beingvalued high or low. So the asset price is SH = S(!H) or SL = S(!L)with SL < SH : As above, assume SL � K < SH ; so then

C(!H) = SH �K; C(!L) = 0:

We �nd a portfolio (u; v) so that the call is a combination of 1 and S:

C = u1+ vS:

We say the portfolio �replicates�the contract. Comparing both sides of

22


this equation, we see that for replication to hold we need

u+ vSH = SH �K;u+ vSL = 0:

Sov(SH � SL) = SH �K;

hence

u = �SL �SH �KSH � SL

; v =SH �KSH � SL

:

2.5.7 Arbitrage and linearity

Suppose the assets in their totality are given a de�nite numbering1; 2; :::; k (so that their future prices of assets X1; :::; Xk arerepresented by the vectors X1; :::; Xk). We can denote a portfolio ofthese assets by a vector whose i-th component gives the quantity of�asset�Xi held by an investor. We will denote this vector byH = (h1; h2; :::; hk): The notation appears to imply that we aretreating X1; :::; Xk as though they formed a basis and the questionarises whether we are entitled to do so. This in turn depends on whatassumptions we make about the initial prices of the assets. Denote theinitial prices respectively by p(Xi): We now show that when the assetmarket is in equilibrium these initial prices are connected to the futureprices.

Suppose for the sake of argument that

X1 = X2 +X3;

i.e. in all states of nature the price of asset X1 is the sum of the pricesof the other two assets. This says that the portfolio H0 = (1; 0; 0; : : :)and H 0

0 = (0; 1; 1; 0; :::0) are of equal value in one period�s time. Onemight expect that in this case the current market value of bothportfolios is also the same, i.e.

p(X1) = p(X2) + p(X3):

Suppose, however, that in fact, for some reasonp(X1) > p(X2) + p(X3), i.e. D = p(X1)� p(X2)� p(X3) > 0: Thisinequality suggest a method for making money, called arbitrage, basedon the idea of buying cheap and selling dear and pocketing thedi¤erence D. This idea is very simple to comprehend if done�instantaneously�. However, here we must wait for a period of time toelapse and so must worry about interest, which is either paid onborrowed money (and tied-up in stock bought), or earned on a deposit.

To explain this point consider again the riskless asset 1. If the interestrate over the period is r, then in exchange for 1 dollar depositedinitially, an amount 1 + r will be received one period later; equivalently,an amount 1=(1 + r) needs to be deposited initially to receive oneperiod later 1 dollar in all states of nature. Thus the price to be paidfor receiving the asset 1; denoted p(1), satis�es

p(1) =1

1 + r:

Let Xk+1 denote the riskless asset 1. Thus H = (h1; :::; hk+1) 2 Rk+1:

Armed with this notation, here is the idea of arbitrage across time.

23


First �nd someone who owns a unit of X1 and has no intention ofselling it during the following period. Borrow this unit of X1 and sell itfor p(X1): This amounts to borrowing the portfolio H0, de�ned above,in order to sell it. Next buy (for less) the portfolio H 0

0 comprising oneunit each of assets X2 and X3 for the total cost p(X2) + p(X3) anddeposit the di¤erence D in a bank. At the end of the period, when thestate of nature is !, sell H 0

0, which gives an amount of moneyX2(!) +X3(!); this equals X1(!) and so enables a unit of the assetX1 to be bought back, and given back to the lender. The overall e¤ectis that this behaviour leaves the enterprising borrower with D(1 + r) inhis bank account at the end of the period.

We draw attention to the fact that the model of market securities inwhich this argument works must include either a riskless security, or aportfolio of securities with pay-o¤ equal to 1.

A similar strategy exists if p(X1) < p(X2) + p(X3), i.e.D = p(X2) + p(X3)� p(X1) > 0, but based this time on borrowingH 00, selling it, acquiring H0, depositing the di¤erence D in the bank

and waiting the one period. (Figure this out for yourself.)

We conclude that if our market example is to have no arbitrage, wemust have

p(X1) = p(X2) + p(X3):

A second important conclusion is that, in the absence of an arbitrageas just discussed, a contingent claim which pays nothing in some statesand a positive amount in at least one state is prevented from having aninitial price of zero. (Otherwise you could buy for nothing a chance ofpositive pro�t with no risk.)

This argument clearly generalizes and we may conclude as follows.

Conclusion. If the assets X1; :::; Xk do not give rise to an arbitrageand an asset X is linearly dependent on the given assets with

X =kXi=1

uiXi

then its price satis�es

p(X) =kXi=1

uip(Xi):

Thus in particular

p(X

hiXi) =X

hip(Xi):

Moreover, for any portfolio H; we have

V0(H) =X

hip(Xi):

If a model is free of arbitrage the price functional p is thus a lineartransformation.

2.5.8 Valuation if a linear transformation is an expectation

We return to the point of view that portfolios form a vector space(compare with Section 2.3.8).

24


If we think of the states of nature !i occurring with probability pi thenthe expected value of asset X is

Ep[X] =Xi

piX(!i):

This is also referred to as the mean value of X, denoted mX :

Evidently the portfolio holding u units of X and v units of Y de�nesan asset Z = uX + vY , the expected value of which is

E[Z] =Xi

pifuX(!i) + vY (!i)g = uE[X] + vE[Y g;

mZ = u �mX + v �mY :

Thus, unsurprisingly, expectation is a linear transformation from theportfolio space to the reals R.

Now we set about establishing a converse: that valuation is anappropriate expectation.

Suppose that the portfolio space LinfS1; :::; SNg is complete, i.e.coincides with Rm (where m is the number of states of nature). Theneach of the natural base vectors e1 = (1; 0; 0; :::)t; e2 = (1; 0; 0; :::)t; :::represents a claim. For example, e1 pays one dollar if and only if stateone occurs, but is otherwise worthless. Note that in the absence ofarbitrage opportunities the price p1 of the claim e1 is positive andsimilarly p2; p3; ::: are all positive. These claims are known asArrow-Debreu securities.

Relative to the basis of Arrow-Debreu securities, an arbitrary claimX = (x1; :::; xm)

t may be represented in the form

X = x1e1 + x2e2 + :::+ xmem:

In the absence of arbitrage, the pricing function p is a lineartransformation from Rm to R, and so we may write

p(X) = x1p(e1) + x2p(e2) + :::+ xmp(em);

i.e.

p(X) = (p1; p2; :::; pm)

2664x1x2:::xm

3775 ;and the matrix representing matrix is just the row vector (p1; :::; pm):

Notice that if the interest rate is zero then p(1) = 1 and, since

1 = e1 + e2 + :::+ em;

we have

1 = p(1)

= p(e1) + p(e2) + :::+ p(em)

= p1 + p2 + :::+ pm:

So p = (p1; :::; pm)t is a probability vector and

Ep[X] =Xi

piX(!i)

= x1p(e1) + x2p(e2) + :::+ xmp(em)

= p(X):

Thus p is a �probability measure�on the states under which the marketprice of a claim agrees with its expected value.

25


2.6 A geometric view of probability

In linear algebra courses one is taught to think geometrically aboutvector spaces. We have seen above that vector space language is anatural language for describing uncertainty when working with a �nitenumber of states of nature. We will be reaping the bene�ts of thinkingabout probability geometrically in a considerable part of the course �especially so in Chapter 6. We see in this section, as a more modestapplication, why correlation is to be viewed through the spectacle oforthogonality and how hyperplanes correspond to a �xed expectedmonetary outcome. Later, in Chapter 6 in a converse kind of wayhyperplanes take a central role in constructing probabilities.

Let p = (p1; :::; pn)t be a probability vector such that pi > 0 for each i:Thus p1 + :::+ pn = 1:

Example 1. The formula

hX;Y ip = Ep[XY ] =nXi=1

piXiYi;

where X;Y are in Rn, de�nes an inner-product. Here we think ofX = (X1; :::; Xn)

t as describing the possible pay-o¤s Xi = X(!i),according to the possible outcomes !1; :::; !n, with associatedprobabilities of the outcomes given by (p1; :::; pn)t.

Since all the probabilities pi are positive, we have

hX;Xi = 0 if and only if X = 0;

so that jjXjjp =phX;Xi de�nes a norm.

Example 2. Find jj1jjp where 1 = (1; :::; 1)t is the riskless pay-o¤ ofone dollar in all states of nature.

Solution. We have

jj1jj2p =X

pi � 12 = 1:

Example 3. Relate the inner-product hX;Y ip whenp1 = p2 = ::: = pn = 1=n to the usual Pythagorean inner-producthX;Y i2.

Solution. We have

Ep[XY ] =Xi

1

nXiYi =

1

nhX;Y i2;

jjXjjp =1

njjXjj2:

Example 4. Relate the norm jjXjjp and inner product hX;Y ip to thePythagorean inner product hX;Y i2 and norm jjXjj2 by considering thetransform Z = Z(X) of the random variable X given by

Z = S1=2X;

26

A geometric view of probability

where S1=2 denotes the diagonal matrix diagfpp1;pp2; :::g; i.e. a

matrix with non-zero entries only along its diagonal. Thus

Zi =ppi �Xi:

Solution. This transformation amounts to a scale-change in theindependent directions. We �rst note thatS = S1=2S1=2 = diagfp1; p2; :::g and that S1=2 is symmetric, i.e.(S1=2)t = S1=2: Now we see that

hX;Y ip =nXi=1

piXiYi = XtSY;

so XtY = hX;Y i2 = hX;S�1Y ip: Thus a hyperplane fX : XtY = 1gwith normal Y translates to the hyperplane fX : hX;S�1Y ip = 1gwith normal S�1Y: Of course S�1 = diagf1=p1; 1=p2; :::g: Thus thehyperplane of portfolios X of initial value one, Xt1 = 1; translates tothe hyperplane with normal S�11 =(1=p1; 1=p2; :::).

Lastly, we compute the Pythagorean norm of jjZjj2 to be

jjZjj2 = XtS1=2S1=2X = XtSX = jjXjjp:

Of course this just says

jjZjj2 =X

piX2i = jjXjjp:

Obviously the transform of the riskless vector is

S1=21 = (pp1; :::;

pp1)

t:

From this last example it should come as no surprise that purelygeometric facts concerned with norms and inner-products translate intoprobabilistic statements and, conversely, that probabilistic statementsabout covariances and variances yield under specialization standardgeometric facts.

Application of Example 1. This inner product is used to measure theco-dependence of two vectors relative to the possible outcomes ! in :Indeed if the distinct values taken by X are xi for i = 1; :::; k and thedistinct values taken by Y are yj for j = 1; ::; k0 then

Ep[XY ] =Xi;j

xiyj � P [X = xi&Y = yj ];

Here [X = xi & Y = yj ] denotes the compound event

Eij = f! : X(!) = xi & Y (!) = yjg;

which is equal to

f! : X(!) = xig \ f! : Y (!) = yjg:

Its probability is just X!2Eij

p(!).

Stretching conventions to allow the summation operationPto be

applied to a set of real numbers one might have written this moreelegantly as X

fp(!) : X(!) = xi & Y (!) = yjg:

27


We de�ne X and Y to be independent if for each i and j

P [X = xi & Y = yj ] = P [X = xi] � P [Y = yj ]:

This corresponds to the intuitive notion that when dealing with twoevents involving independent properties one multiplies theirprobabilities.

Comment. A more careful interpretation is this. Suppose we partition up according to what value X takes. Then for any value x the�contour�of de�ned by the function X; namely

[X = x] = f! : X(!) = xg;

may be regarded as a population in its own right, denoted x: Then onx we may de�ne the probability

px(!) =p(!)

P [X = x]:

We may extend the de�nition so that px(!) = 0 for ! not in x: Thisis known as the �conditional probability�(conditional on the premiseX = x): Recall that X takes the distinct values x1; :::xk: If x = xithen under px the random variable Y takes the value yj with aprobability equal to

px[Y = yj ] =X!2Eij

px(!)

since

f! : ! 2 x & Y (!) = yjg = f! : X(!) = xi & Y (!) = yjg:

But X!2Eij

px(!) =X!2Eij

p(!)

P [X = x]=

1

P [X = x]

X!2Eij

p(!):

=P [X = x & Y = yj ]

P [X = x]:

So

px[Y = yj ] =P [X = x & Y = yj ]

P [X = x]:

If the number px[Y = yj ] does not depend on the choice of contour[X = x] and agrees with the value P [Y = yj ] (i.e. the probability thatY = yj on the whole population); we have every reason to say that Xand Y vary independently of each other over :

Example 5. Multiplication Theorem. Show that if X and Y areindependent then

Ep[XY ] = Ep[X]Ep[Y ]:

Solution. We have

Ep[XY ] =Xi;j

xiyj � P [X = xi&Y = yj ];

=Xi;j

xiyj � P [X = xi] � P [Y = yj ];

=Xi

xi � P [X = xi] �Xj

yj � P [Y = yj ]

= Ep[X]Ep[Y ]:

28

A geometric view of probability

Any equation immediately o¤ers a way for measuring the validity of theassumptions leading to it by a comparison of its two sides. So here,when X;Y are arbitrary, the discrepancy Ep[XY ]� Ep[X]Ep[Y ] isused to measure the amount of dependence between X and Y: In fact,as we shall see in a moment, this discrepancy can be rewritten as aninner-product as follows:

Ep[(X � Ep[X])(Y � Ep[Y ])];

i.e. using h; ip. This last expression is all the more rewarding by reasonof its geometric interpretation as measuring the cosine of the anglebetween the vectors X 0 = X �Ep[X] and Y 0 = Y �Ep[Y ]. When therandom variables X;Y are independent, the cosine of the anglebetween X 0 and Y 0 is zero, and so the latter two vectors are at rightangles. By contrast, when the angle between them is zero, the vectorsare collinear, and so for some scalar �

X � Ep[X] = �(Y � Ep[Y ]):

We reconsider this matter at the end of the current Section.

De�nition. The formula

cov(X;Y ) = Ep[(X � Ep[X])(Y � Ep[Y ])];

where X;Y are in Rn de�nes the covariance of X and Y: Thus ifX;Y are independent then cov(X;Y ) = 0: Note thatvar(X) = cov(X;X) is the variance of X; namely Ep[(X �Ep[X])2]:

Example 6. Verify that

E[XY ] = E[X]E[Y ] + cov(X;Y ):

Solution. Clearly we have

E[(X � E[X])(Y � E[Y ])] = E[XY ]� E[XE[Y ]]� E[Y E[X]] + E[X]E[Y ]= E[XY ]� E[X]E[Y ]:

We note that by expanding cov(X + Y;X + Y ) we may obtain theidentity

var(X + Y ) = var(X) + var(Y ) + 2cov(X;Y ):

(See the Learning activities.) Thus if X and Y are independent, asde�ned above, cov(X;Y ) = 0 and we have

var(X + Y ) = var(X) + var(Y ):

We will need this result in Chapter 8.

When pi > 0 for all states i, it follows from the de�nition that

cov(X;X) = 0 if and only if X = E[X] � 1;

i.e. the variance of X is zero if and only if Xi = E[X] for all i makingX a constant.

We de�ne the standard deviation of X to be

�X =pcov(X;X):

29


Unfortunately the formula

jjXjj = �X =pcov(X;X)

does not in general de�ne a norm. But, as the remaining normproperties listed in Section 2.4 of Ostaszewski do hold, it is said to be apseudo-norm.

All is not lost, however: the covariance formula does de�ne a normwhen it is applied to vectors X lying in any subspace of Rn that doesnot include Linf1g. We note that the largest dimensional suchsubspace has dimension n� 1: The most obvious candidate is the setof random variables

fX : Ep[X] = 0g:

This is indeed a vector subspace, indeed it is the null space (or kernel)of a linear transformation: expectation is after all a lineartransformation.

Comment. There is an alternative approach to covariance which allowsus to regard �X as a norm. We simply have to be prepared to identify(i.e. regard as equal) two vectors when their di¤erence lies in Linf1g.

Such an identi�cation involves a subtlety. Properly speaking, in thecircumstances envisaged, if X � Y 2 Linf1g; one ought to say that Xand Y are not so much �equal�as �equivalent�, written in symbolsX � Y: Reason: under this de�nition if X � Y then Y is in the a¢ neset X + Linf1g consisting of all vectors of the form X + Z whereZ 2 Linf1g; i.e. a line through X parallel to the subspace/lineLinf1g (for parallel a¢ ne subspaces refer to Section 2.1).

+ Lin { }

11

01

X

X 1

Fig. 2.5: The line through X parallel to Lin{1}.

As noted above, instead of working in Rn we can drop down by onedimension and consider any n� 1 dimensional subspace W not passingthrough 1. Then �W is a norm for W. Evidently such a subspace W isa hyperplane (see end of Section 2.6). One can select for any vector Xa unique vector W (X) in W equivalent to X namely the uniqueintersection point of the line X + Linf1g and W: (See the Learningactivities.) In the obvious sense W (X) is the �projection�of X onto Wparallel to Linf1g: We devote the whole of a later chapter to thisimportant idea.

30

Solutions to selected activities

It is helpful to consider the case W = fX : Ep[X] = 0g: Being thekernel of a linear mapping with rank one it is a subspace of dimensionn� 1. As Ep[1] = 1 the subspace W does not pass through 1. We�nd a formula for W (X): Since W (X) is in X + Linf1g; putW (X) = X + �1 with � a scalar. Now Ep[W (X)] = 0; soEp[X] + � = 0 i.e. � = �Ep[X]; and hence

W (X) = X � Ep[X]1:

In this notation cov(X;Y ) = Ep[W (X)W (Y )]; and so covariancemeasures the angle between the representatives of X and Y: When Xand Y are independent their representatives are at right angles.

X 0

1

WW(X)

Fig. 2.6: Selecting W (X) in W equivalent to X:

2.7 Solutions to selected activities

Solution to learning activity 2.1

var[ �X] =1

n2

Xvar[Xi] =

n�2

n2=�2

n:

By independence,

cov(X1; �X) =1

ncov(X1; X1 + :::+Xn) =

1

nvar(X1) =

�2

n:

Since E[X1 � �X] = 0 we have, using the hint,

E[(X1 � �X)2] = E[X1 � �X]E[X1 � �X] + var(X1 � �X)

= 0 + �2 +�2

n� 2�

2

n=(n� 1)�2

n:


Thus V is a combination of X and Y independent of U: We have

�2X = var(X) = 2var(U) = 2;

i.e. = �X : Moreover,

��X�Y = cov(X;Y )

= < U; �U + �V >

= � < U;U > +� < U; V >

= � ;

31


i.e. � = ��Y : Hence� =

p1� �2�Y ;

since �2Y =< �U + �V; �U + �V >= �2 + �2:

The idea of rescaling X and replacing Y is taken straight from theGram-Schmidt orthogonalization procedure. Indeed it is enough to viewjjXjj = var(X) and < X;Y >= cov(X;Y ) as the norm andinner-product on the space of zero-mean random variables. Here weapply the fact that the span of X;Y is the same as the span of U; V:

Finally, since X + Y = (�+ )U + �V we have

E[Z] = em+12 (

2+2� +�2+�2) = em+12 (�

2X+�

2Y +2��Y �Y ):

A reminder of your learning outcomes


compute expectations (integrals) using �log-normal densities�

be prepared to use vectors to study random variables, asset pricesand portfolios of assets.

32

Chapter 3

The �nancial environment: apreliminary discussion

3.1 Aims of the chapter

The purpose of this chapter is twofold. Firstly, we revise the basic ideasof interest rates and discounting within a deterministic framework. Youshould be aware of these ideas from earlier courses, so this is arefresher section. Next we describe informally the �nancial transactionsfor which we will later create formal mathematical models. By�modelling�is meant something quite di¤erent from calculating ordrawing mathematical inferences. The action of describing someaspects of reality in mathematical terms is in itself a separate creativeact. To be able to move backwards and forwards between somemathematical calculations (resembling quite often either some linearalgebra, or some calculus) and a description of �nancial markets is anacquired skill. We thus urge the reader to take time to understand the�nancial environment which is modelled by the mathematics in thiscourse. For completeness, we give a rapid account here of the featuresthat will be studied in the course. But the student is advised to readmore plentifully. At the end of this chapter you will be able to answer aquestion such as this: A leitmotif for this chapter

�A zero-coupon treasury-bond with face value £ 100 is a documentwhich may be exchanged for £ 100 at the Bank of England after a givenperiod of time , at which time it is said to mature. If a trader is willingto sell a bond maturing in a quarter of a year currently for £ 97.5, whatdoes this imply about the trader�s belief about the bank interest rateapplicable for the current quarter? The trader o¤ers two similargovernment bonds each with face value £ 100 one with a maturity ofone year the other with a maturity of two years. He sells the �rst at aprice as though he believes the interest rate applicable for the next yearwill be 10% per annum. What is the price? He sells the second bondas though the interest were constant at 10.5% per annum. What doeshe therefore believe the bank interest rate will be for the second year?�



compute the terminal value in a deposit account of an initialdeposit and of a deposit stream

compute the present value of a single future payment and of anincome stream

33


compute the expected value of a defaulting income stream.

3.3 Suitable reading for this chapter

Cvitanic, J. and F. Zapatero Introduction to the Economics andMathematics of Financial Markets. Chapter 1.

Hull, J.C. Options, Futures and other Derivatives. Chapters 1�4.

Luenberger, D. Investment Science. Chapters 1, 2, 3 and 7.

3.4 Basic deterministic models

3.4.1 Savings and loans: compounding

One of the services o¤ered by a bank is the use of a deposit account.This o¤ers individuals the opportunity to lend money to the bank.Implicitly this is a contract, in which the bank announces that it willperiodically �add money to the account�, an activity known as interestaccrual. The interest period length is of course announced as part ofthe contract, and a simple proportionality-rule for interest accrual isapplied on the assumption that no withdrawals occur in the period. Tobe precise: the bank speci�es (i) what the interest period length is (or,equivalently, the compounding frequency with which in each year itadds interest to the deposit account) and (ii) a rate of interest accrual,denoted by r; measured per annum (i.e. the unit of time is a year).Thus when the interest is added once a year and the interest rate isryear a sum D deposited into the account for one period grows toD(1 + ryear): It is assumed that the depositor cannot withdraw thedeposit during the year.

Banks do, of course, o¤er various kinds of deposit contract, in the formof di¤erently named deposit accounts. These account will thus varyboth in relation to the frequency of interest payments and interest ratesapplied to the account. For example, the bank may o¤er on one Annual and semi-annual

accruals: ryear and rsemi.account an interest rate ryear per annum with one interest payment perannum, and on another account an interest rate rsemi per annum withinterest paid semi-annually. In the second case an amount D depositedfor half a year grows to D

�1 + rsemi

12

�since the annual rate is rsemi

and the amount of time (measured in years) is 12 . After one year the

deposit therefore grows to D�1 + rsemi

2

�2:Thus a depositor willing to

deposit D dollars for the duration of one year will compare how he isrewarded by the bank with the two potential balances

D (1 + ryear) and D�1 +

rsemi2

�2:

The two rates will o¤er an equivalent reward if and only if

1 + ryear = 1 + rsemi +r2semi4

:

In general one expects the semi-annually paying account to give a lowerreward, because the bank permits the depositor to withdraw from thecontract after the �rst six months. Put another way, the bank o¤ers ahigher reward because the deposit may not be withdrawn earlier.

34

Basic deterministic models

As a part of the contract the bank may announce for how long the rateis to be held �xed into the future, or it will identify at what points intime it will vary the interest rate.

In this subsection we will examine the case of a �xed annuallycompounded rate which we denote by r = ryear: This means that weare considering deposit-making activity over a time horizon in whichthe rate stays at r and compounding occurs just once a year. Seesubsection 3.4.4 of this section to see how to adapt the argumentswhen rates are allowed to change.

One further comment is in order. The calculations that follow assumethat the bank does not fail to honour its contract. In reality some banksdo fail. The discussion here has to be interpreted as referring to a safebank, or at worst a time interval over which the bank remains safe!

Terminal deposit account value

At the end of N years the account holds

D(1 + r)N ;

where r = ryear:

Let�s measure time so that beginning of the �rst period is denoted byt = 0 and the end of the N -th year is denoted t = N: Suppose theamount D is deposited at some time t = k (with 0 � k � N): Thenthe terminal value (i.e. at time t = N) of the amount in the depositaccount is instead

D(1 + r)N�k:

More generally, suppose that at the start of each of the consecutiveperiods numbered t = 0; 1; 2; 3; : : : an amount Dt is deposited. Thelast of these deposits occurs at time t = N; and so is made at the endof the N -th period. In this case the terminal amount in the account,i.e. at the end of N periods, is

D0(1 + r)N +D1(1 + r)

N�1 + :::+DN�1(1 + r) +DN :

The t-th term for t = 0; 1:::; N is

Dt(1 + r)N�t:

Higher frequency interest payments

The period length may vary. Recall that it is normal to measure theperiod in units of time with one year as the unit. Similarly, to enablecomparison it is normal to quote the rate not per period but per unit oftime. Thus if a rate of rquart per annum is quoted and the period is a Quarterly and monthly

accruals: rquart and rmonth.quarter of the year, then the interest paid per dollar per period isrquart=4:

Thus with quarterly payments a sum D deposited into the account fort years grows to

D�1 +

rquart4

�4t;

since the number of periods, N; is 4t as there are 4t quarters in t years.

35


It is important to understand what happens to interest payments as theperiod is shortened, i.e. as the frequency of payments in one year isincreased. If the period is a month and the rate o¤ered is rmonth theformula changes to

D�1 +

rmonth12

�12t;

as there are 12 months in a year. Similarly if the period is a day and Daily accruals: rdaily.the rate o¤ered is rday the formula alters to

D�1 +

rday365

�365t:

In general if there are n periods in a year let us denote the rate o¤eredby the bank by rn: Thus r1 denotes ryear and r2 denotes rsemi: Theformula in the general case becomes

D�1 +

rnn

�nt:

Special cases are always easier to understand. Let us consider the casern = 1: We ask what happens to the expression�

1 +1

n

�n; (3.1)

as n increases without bound. This is apparently an academic question,but it has an interesting answer. We look at this case despite the factthat one�s �rst reaction is that you would not expect a bank to giveone and the same rate for a range of contracts with increasingcompounding frequency. Unsurprisingly, it turns out that the expression(3.1) is increasing in n (more interest is being paid!), however, theexpression remains bounded and approaches the number e = 2:7 : : : So,after all, if the bank �generously�did not alter the contractual rate, itwould still not be giving much away.

Why the number e

To see how the number e arises, consider the expansion�1 +

1

n

�n= 1 + n

1

n+n(n� 1)2!

�1

n

�2+n(n� 1)(n� 2)

3!

�1

n

�3+ :::

= 1 + 1 +1

2

�1� 1

n

�+

1

2 � 3

�1� 1

n

��1� 2

n

�+ :::

The (k + 1) term for k = 1; 2::: is

1

k!

�1� 1

n

��1� 2

n

�:::

�1� k � 1

n

�:

As n increases unboundedly this term tends to the value

1

k!:

It is plausible now that�1 +

1

n

�n! 1 + 1 +

1

2+

1

2 � 3 +1

2 � 3 � 4 + ::: = e:

Now suppose that a bank considers using a �xed deposit rate � andcontemplates di¤erent compounding frequencies n which it intends to

36


be �large�. That is, in the notation of the last subsection, suppose thatrn = � without �xing n: Using the substitution m = n=�; we have�

1 +�

n

�nt=

�1 +

1

m

�m�t=

��1 +

1

m

�m��t! e�t;

as n, (or, equivalently, as m) increases without bound.

In particular, a dollar deposited for a year would grow to nearly e� whenn is large enough. Thus the bank can select � to control just how muchit pays its depositors in this high-frequency compounding account.

We can revisit the problem a bank faces when it considers a range ofcontracts with compounding frequency n and corresponding rates rnthat in fact vary with n: Consider the more realistic situation of rn Instantaneous accruals:

rinst.varying with n in such a way that rn ! rinst. Thus rinst takes its placeas a limiting �instantaneous�rate at the end of our hierarchy of, yearly,monthly, and daily rates. Now it is true, with r denoting rinst; that�

1 +rnn

�n! er if rn ! r: (3.2)

Not wishing to labour the point, let�s note that a proper proof relies onthe observation that, for any " and all large enough n we shall haver � " � rn � r + ": In consequence�

1 +r � "n

�n��1 +

rnn

�n� lim

n

�1 +

r + "

n

�n:

But, we now know that

er+" = limn

�1 +

r + "

n

�nand er�" = lim

n

�1 +

r � "n

�n:

From here it is possible to deduce the result (3.2).

3.4.2 Continuous compounding

By the last section of work, since a good approximation to a veryfrequent compounding of interest account employing a �xed annualrate � is given by the formula

De�t;

we introduce a new mathematical model for high-frequency interestcompounding.

We will say that a bank account pays interest with a constant Continuous compoundingrate: �.continuous compounding rate � per annum, if after t units of time

(measured in years) an initial deposit of D dollars causes the balanceon the account to grow to

De�t:

Note that if the deposited amount D is made at a time t with0 < t < T; rather than at time t = 0; the deposit account�s terminalvalue will instead be

De�(T�t):

37


Comment. Even if a bank does not quote a continuous compoundingrate �; one can always compute an equivalent continuous rate. Indeed,if the bank quotes a rate ryear per annum and adds interest once a yearwe can compute an equivalent rate � such that

D(1 + ryear) = De�:

Solving for �; we get� = log(1 + ryear):

In conclusion, there are a number of possible interpretations of thephrase �a rate of interest r�. Interest may be added at discretemoments so that r refers to the rate denoted by rn in the earlier workfor some n. Alternatively, the rate may be applied according to thecontinuous compounding model, and then r stands for the rate whichwe have just denoted by �. The former interpretation will be applied inthe discrete models of Chapters 3�7. The latter will be applied in thecontinuous-time models of Chapters 8�10.

3.4.3 Deposit rates

The idea behind the de�nition of the last subsection was to model whathappens in the limit as the interest-period becomes increasingly smaller,that is, as the number of periods, n, in a year increases inde�nitely.

We will want to study deposit making over time rather than a singlesolitary deposit payment.

In a context where the number of periods in T years, namely nT ,becomes huge it makes more sense to describe deposit activity as arate of payment, just like interest payment activity. So, instead ofquoting the amount deposited in dollars, we will describe the deposit asoccurring at a rate of so many dollars per annum.

We assume that interest is compounded at a frequency of n and that arate r is applied to the account (i.e in the earlier notation the rate is rnper annum and here r = rn).

Thus when the deposit rate is quoted as q dollars per annum, theamount actually deposited at the beginning of a period is in fact

D = q � 1n;

since there are n interest periods in a year.

If the deposit rate is constant throughout the time of deposit making,the amount deposited at time t will always be

Dt =q

n:

Although the deposited amount is increasingly small as n increases,nevertheless the total amount deposited by the end of T years is the�nite amount qT:

In the presence of interest accruing in intervals of length 1=n, theterminal value of the account is the sum

nTXj=0

q

n

�1 +

r

n

�nT�j;

38


where j moves in integer steps: Alternatively, put j=n = t and with tmoving in steps of size 1=n the sum may be re-expressed as

TXt=0

q

n

�1 +

r

n

�n(T�t):

Note that with the substitution m = n=r we have�1 +

r

n

�n(T�t)=

�1 +

1

m

�mr(T�t)=

��1 +

1

m

�m�r(T�t)� er(T�t):

An explanation is given in Chapter 6 why the terminal value in thedeposit account tends to the limitZ T

0

qer(T�t)dt

and this of course is very simply computed to be�qer(T�t)

�r

�T0

=q

r(erT � 1):

Whilst very plausible, it is less obvious why if the deposit rate were todepend on time, and so take the form q(t); the corresponding answerfor the terminal value would beZ T

0

q(t)er(T�t)dt:

To see the di¢ culty, note that when q(t) is not constant it is no longerobvious what sense to make of the amount deposited at the the time tat the beginning of a time interval of length 1=n.

If q(t) is a continuous function, the amount may in fact be interpretedas any number q(s)=n where t � s � t+ 1

n , without a¤ecting thelimiting value of the sums. Not an obvious fact: see Chapter 6 formotivation.

One is thus at liberty to opt for q(t)=n as the interpretation.

3.4.4 Variable rates

Suppose D is deposited at time t = 0 and that a continuouslycompounded rate is held �xed at r until time t = T1, whereupon therate changes to s: If no withdrawals are made, what is the amount inthe deposit account at time t = T > T1? The answer is computed intwo steps. The �rst step is to note that by time t = T1 the amount inthe deposit account is

DerT1 :

This is now regarded as being deposited for the remainder of the time,namely for a length of time T � T1; during which the rate s operates.The terminal value of the deposit is thus�

DerT1�es(T�T1):

39


What is the e¤ective average rate if T = 2T1? The amount in thedeposit account is now

DerT1esT1 = De(r+s)T1 = De12 (r+s)T :

Thus, the e¤ective rate is the average of the two rates, i.e.

1

2(r + s):

This is a particularly appealing feature of continuous compounding.

3.4.5 Loan accounts

We can adapt the results of the preceding subsections by reversing therole of the deposit-maker and the deposit-taker. (Recall that until nowthe deposit-taker was the bank, but that now changes.) The reversal ofroles changes the direction of the �ow of money and now it is the newdeposit-taker, o¢ cially the borrower, who pays interest to the bank.The interest rate is likely to be di¤erent, ultimately re�ecting thehigher risk due to the fact that an individual may fail to honour hiscontractual commitments. (Compare this with the opening commenton safe banks.)

We may nevertheless still denote the rate here by r. In this case all ofthe formulas developed in earlier sections may be interpreted asmovements of money in the reverse direction, on the assumption thatthe borrower does not fail to honour his contractual obligations.

We note that di¤erential rates applied according to the assessed riskclass of customers is but one way in which banks may make money forthemselves. Inevitably, they also make money on the di¤erence betweenthe borrowing and the lending rates.

3.5 Present value of future income: discounting

The fundamental problem in mathematical �nance is to attach ameaningful value to a contract specifying a future payment of money.The �rst basic message is that value changes over time and so onelooks to mathematics for tools with which to describe such changes.The second message is that at any time there are various notions ofvalue: there may be valuations that are made individually (perhapsusing private information, or re�ecting the individual�s taste for risk)and there are valuations that are public, being the recorded priceachieved in a market transaction. In this section we concentrate on thelatter.

The valuation problem is complex and intellectually challenging. It isbeset with a variety of di¢ culties all of which mathematics seeks tomodel. For instance, details of the payment can vary: the future bankinterest rates may be unknown (a¤ecting the value of a depositaccount), the date of payment may be certain or uncertain (e.g. aninsurance policy claim for payment), likewise the amount for payment,especially if the amount is stated not as a number of currency units,but in terms of a quantity of assets whose price remains uncertain. Inall cases the party supplying the asset or the party required to make apayment may default on its contractual obligations.

40

Present value of future income: discounting

The public valuations are arrived at on markets where these contractsare themselves the subject of trade. In principle, therefore, in any givensituation one needs �only�to look up the reported value of the nearestequivalent contract in the market. This often provides a useful basisupon which to value other contracts.

But the reader needs to be warned that this approach avoids a basicquestion: how do the markets themselves (i.e. the market�sparticipants) arrive at these valuations?

To answer this question, the �rst point to make is that the price atwhich such contracts trade is a result of the way in which the marketmechanism responds to the supply and demand for these contracts bythe market�s participants. The participants�actions, in turn, depend onwhat may loosely be described as the �information�they hold. Thatinformation includes knowledge and beliefs about the factors that mayin�uence future trades in assets, including the valuation proceduresfollowed by other participants.

Single future payment

In this section we consider the simplest situation: the amount isspeci�ed as a known number of currency units that is certain to besupplied at a known future date. Let the amount be F and the date beT years ahead.

The valuation problem can be turned into a contract. The contract wehave in mind is to o¤er the future payment of F to the bank inexchange for a lump sum of money L at the present moment.

The bank views the contract as a standard loan. It o¤ers a loan L andrequires it to be paid back together with interest. We model interest asbeing compounded continuously. It thus requires the loan-taker to payLerT and so if this is to be paid o¤ by the sum to be received, namelyF; then

F = LerT :

We conclude that the present value of receiving F in T years�time isgiven by

L = Fe�rT :

It should be remembered that such a statement is just a limiting formof the situation in which the bank requires interest payments at the endof each period of length 1=n (measured in years), so that the presentvalue of F received at time t must equate the repayment of loan withinterest equal to

L�1 +

r

n

�nt= L

��1 +

1

m

�m�rt;

so that

L = F

��1 +

1

m

�m��rt:

Comment. This reference to a bank loan is really a beginner�stextbook approach towards valuation. In the real world companies andgovernments issue �bonds�, i.e. they write contracts promising a futuresum F and sell these contracts on the bond market. If the only

41


contractual payment o¤ered in the bond is the payment of F at thetime T; and the market participants pay a price P for the bond, onemay compute a notional interest rate at which a loan-based valuationwould yield the price P: This is called the yield rate, y; and we have

P = Fe�yT :

Future income stream

When we discuss loans in the continuous-time framework of Chapter 8we �nd that we need to use rates of interest payment to describe whathappens in the limit as the compounding of interest occurs withprogressively higher frequency. In much the same way, if we want todescribe income payments in continuous time we need to refer topayment rates.

Thus if a payment stream is asserted to be at a constant rate p; theinterpretation is that in an interval of length �t the actual payment isp�t: Any such payment has present value

p�te�rt;

and so the present value of the entire payment stream is the limitingsum of the present values of all of these contributions, namelyZ T

0

pe�rtdt =p

r(1� e�rT ):

Of course this is an idealized way of thinking about what would happenin reality. That is, this integral is the limiting outcome from calculatingthe more complicated-looking sum:

TXt=0

p

n

��1 +

r

n

�n��t=

TXt=0

p

n

��1 +

1

m

�m��rt;

where t moves in steps of size 1=n in both summations.


An investor owns an asset whose cash value, V (t), at time t he believeswill be

V (t) = 1 + t:

He also believes that for the next 20 years, thecontinuously-compounded interest rate will on average be rinitial = 5%;but will thereafter average at r�nal = 4%. Explain why the presentvalue of a sale of the asset at time t is

P (t) =

�V (t)e�t=20; if t < 20;

V (t)e�1e�(t�20)=25 if t � 20:

When should he sell the asset? [Note that the instantaneous interestrate is either rinitial or r�nal, unless t = 20.]If the investor changes his mind and now believes that the higherinterest rate regime will in fact last 22 years, will his plans alter? Youshould justify your answer.

42

Payments under uncertainty: the paradigm

3.6 Payments under uncertainty: the paradigm

Under uncertainty, one way to value a claim is to regard it as one of alarge number of independently arising claims that all have the sameprobability of arising. Then it is possible to invoke the law of averages(more accurately described as the law of large numbers) which suggeststhat, in practice, the average payout will be close to the �expectedpay-out�of a typical claim. This is all very well if the valuer is aninsurer who insures a very large number of similar claims. Indeed, evenif the insurer does not have a large customer base, he may seekso-called re-insurance which is obtained by creating a large enough poolfrom the total market for these claims.

Note that this approach to valuation is meaningless if there is just oneclaim. In such a case the law of large numbers says nothing in supportof the proposed valuation. Such a valuation is a gamble and thegambler can lose out.

Here is an argument about a debt repayment default which is based onthe law of averages approach. Here again we will model interest ratesas continuously compounded at a rate r:

Suppose a borrower of a loan, L; is required to repay the loan byrepaying at a constant rate q from time t = 0 to perpetuity.

Thus, in the absence of default, we should have

L =

Z 1

0

qe�rtdt =q

r;

so that

q = Lr:

Assuming the borrower pays only until time t = T the repayment isworth Z T

0

qe�rt =q

r(1� e�rT ):

Now suppose that the borrower may go broke at some time T and failsto pay anything after time T:

Suppose he is more likely to go broke earlier than later, and we assumethat the probability density of the failure occurring at time � isp(�) = �e�� ; where � is some constant: This is supposed to mean A model of default.that failure occurs in the interval [� ; � +�� ] with probability p(�)�� :Notice that Z 1

0

p(�)d� =

Z 1

0

�e��d� = 1:

The bank may compute an expected repayment as follows. Withprobability p(�) the bank receives payment up to time � only, and soreceives a value

Q(�) =

Z �

0

qe�rtdt =q

r(1� e�r� ):

The repayment it is to receive is thus valued in expectation as

43


Z T

0

Q(�)p(�)d� =

Z 1

0

q

r(1� e�r� )�e��d�

=q�

r

Z 1

0

(e�� e�(r+�)� )d�

=q�

r

�1

�� 1

r + �

�=

q

(r + �):

If the expected repayment is to equal L; then the bank requires that

q = L(r + �);

i.e. as though the interest rate were r + �:

Clearly this is a higher repayment rate than the safe rate r; if the bank Discriminatory lending rates.chooses to apply a larger value of the parameter � (known as a loadingfactor) when agreeing the loan, then this choice re�ects the bank�sbelief of a greater likelihood of an early default arising on the part ofthe borrower. Such a choice penalizes a particular class of borrowersmore heavily, so that the bank may recoup the entire repayment of allloans at an average rate r:

3.7 Contracts with the Stock Exchange

We shall be concerned with the valuation of some basic �nancialcontracts (also referred to as �nancial instruments) which can beentered into at stock exchanges or by �nance houses. (However, we donot get to consider such �nancial instruments as for instancemortgages.) All of these are contracts for the supply by one party toanother of some kind of asset, that is something capable of having�value�, say in money terms (see below). So long as the terms of such acontract are honoured (i.e. the counterparty does not default),possession of such a contract may therefore itself be regarded as havingvalue, or being equivalent to possessing a sum of money. The asset inthe contract could thus be not only a commodity, or some form ofmoney (possibly foreign), but also another contract for an agreedpayment at an agreed date (see later). A quotable literary example isin Flaubert�s Madame Bovary, where the heroine gets into debt, andher creditor sells the debt. These contracts may and do change hands,so that though the liability for the supply rests with the original partythe recipient may change. Of course the party liable for the supply canbalance its liability with an opposite contract (agreed perhaps even atthe same time, or at a later date which is prior to that in the �rstcontract) for delivery with another party (or even the same party)usually on di¤erent terms. This balancing procedure is known as Entering and closing out

contracts.closing out. Thus payments for entry into various contracts may bemade but no supply need ever occur.

Why all this continual play between swings and roundabouts? In A method for riskmanagement.essence these contracts and counter-contracts are made as a means of

�managing risk�� forms of insurance against loss of, say, savings orinvestment capital, or of income. For instance, a construction �rm mayquote terms for a contract to build an air liquefaction plant. Thecontract may be agreed during the period of the quote�s validity, yetthe cost of parts supplied by a foreign sub-contractor may change as a

44

Some �nancial instruments

result of adverse movements in exchange rates �a factor which isoutside the direct control of the main contractor. At di¤erent points intime the risks may be viewed di¤erently and call for protective action.It is true that in the end this is a musical chairs kind of game andsomeone is eventually caught out and left to pay for the loss if anydoes indeed occur. But so long as the payments levied each time acontract transfers the risk to some particular party are enough to coverthe balance between gains and losses of the insuring party, this is abusiness proposition. Despite obvious di¤erences there is a familyresemblance to insurance against �re: for the right premium and alarge enough number of policies the insurer does make money �and ifby a rare accident he does not, then he recovers by raising the contractpremiums later.

The contracts at stock exchanges are drawn up to standards decided by Standardized contracts.the stock exchange, for example, specifying the unit size of a singleorder (e.g. one contract for corn will be for 5000 bushels), the date ofdelivery and in the case of commodities the quality and location ofdelivery. Those issued by �nance houses need not follow the samestandards. Contracts entered into at exchanges contain requirements asto methods of payments which are designed to protect both partiesagainst default. For contracts that require payments to be made atfuture dates rather than solely at the time of issue, the contractingparties must maintain so-called �margin accounts�into which paymentsmust be made at regular times up to the expiry of the contract tore�ect changes in the current value of the contract. This is both toensure that the level of possible default loss is maintained nearlyconstant and low, and also so that if a default occurs to one partybefore the termination date of the contract the damage may be limitedby closing out. We do not discuss the details (which vary betweeninstruments), but strongly advise you to read about this in e.g. Hull(Section 2.2 pp. 23�25 and Section 8.8 pp. 194�195). It is also a goodidea to read up how trades through a stock exchange are organized(instructions from your broker relayed to his representative at theexchange, from there transmitted to his trader on the �oor of theexchange, who will buy/sell a contract at prevailing prices, or wait forthe right price). See Hull Section 2.1 pp. 18�19 for an example.

In case you do not know anything about �nancial instruments let usreview some basic facts about them.

3.8 Some �nancial instruments

First of all there are money loans or deposits which earn interest eitherat �xed-rates or at �oating rates. Fixed rates are quoted for varioustime intervals: short term, long terms. See Hull Section 4.1 page 76 onthe London Interbank O¤er Rate (LIBOR). In regard to many central�state banks�the deposit rate for a speci�c time interval is consideredas risk-free. The idea that there is a risk-free rate is key to themathematical assessment of risk �whenever we aim to protect a riskby holding a collection (or �portfolio�to give the o¢ cial term) ofcontracts and the portfolio is seen to be risk-free �albeit for only ashort period � its growth in value over the period will be evaluated asthough its initial value was �money�deposited at the short-term ratethen in force. This begs the question: which of the various competinginterest rates on the market is regarded as the risk-free rate of ourtheory? One answer is the rate which banks use between themselves:

45


LIBOR. Since we may be calculating the e¤ects of contractingactivities to be undertaken in the future and we do not know the futureshort-term interest rate, one way out is to use as a substitute thecurrently available long-term rate for that time interval. Alternativeprocedures call for a mathematical model of the future interest rates.Here conceptually one needs to refer to the market for �loans�, that isthe �bond�market. A bond sells a legally-binding promise to pay astated amount of money at a speci�ed future date in exchange for anamount of money to be o¤ered immediately by the agreed buyer of thebond. This is how one may obtain a loan for the speci�ed �term ofyears�, and can in principle be done competitively by reference tocandidate buyers, for instance by auction. The ratio of the amount tobe received later to the necessarily smaller amount being o¤eredimmediately by the buyer of the bond can be interpreted (in variousways) as arising from an interest rate for the term of the loan. Thusone talks about the term structure of the interest rates (i.e. how theinterest rate depends on the term of the bond).

Of course these interest rates are tied to the currency of the centralnational bank. The central bank sets its own currency�s short-terminterest rate guided by the performance of the economy as a whole andits relation to other economies (crudely speaking!). Evidently the ratesbeing o¤ered on the bond market are one of the indications of aneconomy�s performance; indeed the government itself issues and sellsbonds. The central bank rate is re�ected in the rates quoted by theprincipal banks of the country. In turn the central bank�s ratein�uences the amount bond buyers are willing to pay for bonds.

An exception to this story is the Eurodollar, which name is applied tothe stock of US dollars held outside the US (see Hull, Section 6.4 pp.137�142).

The exchange rates between di¤erent currencies are set by the marketfor supply and demand of currency between countries (some of itarising from a consumption demand and some speculatively).

Deposit accounts pay speci�ed sums at speci�ed dates to the accountholder. In this respect the bond is super�cially a similar instrument.The bond originates predominantly either from governments or from a Bonds.large company (corporation) and is issued to raise funds for theoriginator�s economic activity. The bond is characterized by its facevalue, namely the stated amount in the originator�s currency which willbe paid at term (at the maturity date bond) when it is said to beredeemed, and coupons stating amounts of money to be paid to theholder of the bond at speci�ed dates (the coupon is speci�ed as anannual amount but is paid in two installments semi-annually). Thesemay be zero amounts in which case one speaks of a zero-coupon bond,or discount bond. Treasury bills and Treasury notes are the namesgiven to government bonds of one�s country. (We note that T-billshave a zero coupon and maturity up to two years, T-notes have acoupon and maturity of between 2 and 10 years, T-bonds have amaturity in excess of 10 years �all this at time of issue.) The buyer ofthe bond may choose to value the bond by reference to its presentvalue using appropriate bank interest rates if it is his belief that thesewill be the interest rates in force in the future when determiningwhether or not to buy a bond at the o¤ered price. Bonds once issuedand sold may be sold on by their initial buyers. We may thus observethe prices at which bonds of various maturities are being traded in themarket and ask what interest rates would give a bond its current price.

46


Such implied interest rates are called yields. A plot of the yield Bond yields as impliedinterest rates.against the maturity date gives rise to a yield curve � see Hull, Section

4.5 pp. 82�84.

Note that we use only continuous compounding rates. Suppose azero-coupon bond costs p when the principle sum/face value is f .Then the gain is g = f � p and the yield is y over the remaining timeto maturity t is given by

pert = f:

Hence

r =1

tln(1 +

g

p):

For instance a zero coupon bond with face value $100 which iscurrently priced at $97.5 and which matures in 3 months (=1/4 year)has a yield of 4 ln(1 + 2:5=97:5) = 0:1012 = 10:12%:

The following table identi�es the observed yields on bonds of maturitiesof 1, 2, 3, 4, 5 years. The bonds all have a face value of 100 dollarsand zero coupons.

Maturity (years) Yield to Maturity Implied rate in each year1 10% 10%2 10.5% 11%3 10.8% 11.4%4 11% 11.6%5 11.1% 11.5%

Thus the price of the �rst bond in the table (expiring in one year andwith yield 10%) is

100e0:1;

while the second bond (with two years�maturity and yield of 10.5%over the two years) is

100e2�0:105:

In the absence of any further information we conclude that if theconstant interest rate of 10% is applied over the �rst year and aconstant interest rate of r is applied for the second year, then

100e2�0:105 = 100e0:1100er;

or

2 � 0:105 = :01 + r:

One says that the implied forward rate for the second year is r = 11%.The yield 10.5% is thus the average of the two interest rates of 10%and 11%. This simple averaging that we have just observed is theresult of employing �continuous compounding�.

We can plot the yield rates against maturity to obtain the following

47


plot.

2 3 4 5

10.2

10.4

10.6

10.8

11

Fig 3.1: A plot of yield rates against maturity in years.

Thus the �yield curve�obtained here is a rising one. The plot of theimplied forward rates, however, is not rising.

2 3 4 5

10.25

10.5

10.75

11

11.25

11.5

Fig 3.2: A plot of forward rates.


Here is a table similar to Hull, Section 4.5 page 42:

No. Face value Time to expiry Annual coupon Price of bond1 100 1/4 0 97.52 100 1/2 0 94.93 100 1 0 90.04 100 1 1/2 8 96.05 100 2 12 101.66 100 2 3/4 10 99.8

Verify that the implied 3-month rate is 10.12% (p.a.),the 6-month rateis 10.47%, the 12-month rate is 10.54%. Use these rates to deduce the18-month rate from No. 4 bond. Go on to �nd the 2-year rate using allthe previous rates. For the �nal bond � recall that payments are madeafter 3, 9,... months up to maturity. Use a linear interpolation betweenr = the 2 3/4 rate and the 2-year rate already established and derive aformula for the present value of the cash �ow. Set this equal to theprice and solve for r using Mathematica. Hint: The command

FindRoot[�[x]; fx; 0:1g]

48


instructs Mathematica to solve �(x) = 0 for x starting with a guessedsolution of x = 10%: Note that throughout the question half theannual coupon is paid out semi-annual installments.

We now consider the central notion of a �share�and try to explain a Stocks and shares.variety of words used almost interchangeably. A share, or moreproperly, a �share of the stock�(i.e. share of the �equity�capital, asde�ned below) of a limited company might be described jokingly as aperpetual bond with unspeci�ed redemption value, unspeci�ed couponvalues, together with the right to participate proportionately to one�sstake in the excess value of the company�s assets should the companybe wound up. This residual claim (i.e. to the excess of assets overliabilities, if there be any) comes with a legally-enshrined limitation asto the liabilities of the shareholders (for which see below). Seriouslyspeaking and in a nutshell: shares give partial ownership of thecompany�s assets and future income in return for money. Indeed wheninvestors give money to a company they are described as having an�equitable�interest in the company. Evidently they demand some formof �guarantee�(�surety�, or �security�) of having something for theirmoney. The underlying notion of equitability or �fairness�(�somethingin return for your money�) gives rise to a number of di¤erent wordswhich describe the paper certi�cates behind the contractualarrangements that exist between the investor and the �rm. The wordsused include: share, stock, equity and security.

Asset is a very broad word indeed for all the means of value creation.You should consider this point in detail as the intricacy of the detailhides the extent to which the value of assets is subject to uncertainties.It includes not only the more obvious �physical�assets which generateincome, such as the means of production, both �physical capital�(plant,inputs and outputs) and �human capital�, but also the associated�architecture�of resources such as legal powers or rights (for example:patents, monopolies, ownership of �brand names�, contracts for supply).There are other so-called �intangible�assets such as �reputation�and�goodwill�(a �loyalty base�on the customer, or demand side, and on thesupply side). Their creation and maintenance has very tangible costs,and these too play their part in income generation. See for exampleJohn Kay�s inspirational book, Foundations of Corporate Success.

A share entitles the shareholder to vote at shareholders�meetings andso gives further rights, albeit in limited forms, over the choice ofcompany strategy (for instance agreement to a merger) which include,in the extreme, the right to sack the directors. The assets owe theirexistence to the investment of cash in the company. Whilst this cashmay be in exchange for shares, some of it can have its origin througheither loans or through bond issuance. These actions create liabilities.However, a shareholder�s liability is limited only to his �stake�� that ishe may lose only his investment and no more. This is the essence ofthe �limited liability company�, a foundation stone of �capitalism�. Thusin the event of the company�s liquidation its assets are used �rst tosettle debts (such as bank loans) and the obligations of its bonds(prioritised by covenants or contracts) and then the excess value ofassets, if there be any at all, is divided among the shareholders. (Theremay be prioritisation between di¤erent share issues.) In the worst casescenario the shareholders receive zero and need not concern themselveswith outstanding debts. The liquidation pay-o¤ to the shareholders

49


graphed against asset value, as shown in Figure 3.3, thus exhibits a�hockey-stick�shape, with the kink representing the debt level. This isthe same as the graph for the pay-o¤ from a call-options. This is thekey feature behind any limitation on liabilities.

00 . 5 1 1 . 5 2

1

0 . 8

0 . 6

0 . 4

0 . 2

0

Fig. 3.3: Pay-o¤ to shareholders against value of the �rmwhen the debt is 1.

The liquidation procedure for dividing assets thus distinguishes between Debt and equity.two kinds of liability: the one to the creditors (loan givers) andbond-holders which is called debt, the other to the shareholders whichis called equity. The total of shares multiplied by the market price ofshares is known as the market capitalization of the shares and soassesses the value of the equity. The ratio between the two forms ofassets, known as leverage, may be assessed at any moment by referenceto the current price on the market of the shares and bonds.

Turning to the brighter side of the matter: the shareholder receivespayments called dividends which are made at speci�ed (quarterly) Dividends.dates, the amounts being determined on each occasion by the board ofdirectors. The value of the share as assessed by its market priceinvolves in part an estimate of the future dividend stream and also aview on the company�s performance. It is therefore not surprising thatthe share price drops immediately after the share has �gone ex-dividend�(the dividend has become payable). As to the view on the company�sperformance, this is based on various kinds of information such asmarket demand for its product, market supply for its inputs (includinglabour), as well as its management strengths. The share price istherefore in�uenced by the arrival of new information includinggovernment statistics and the fortunes of competitors including tradingstatements. Some of this information has a known arrival date and theo¢ cial statistics may often be guessed from independent sources. Ofteninformation impacts on more than one company. Stock exchangestherefore compute weighted averages of the prices across various groupsof companies which therefore act as indicators of the response of themarket, or of relevant sectors of the market to new information. Theseare known as indices. See Hull, Section 3.5 page 60 for a discussionboth of the weights and the composition of the best known indices.

3.9 A �rst look at futures and forwards

The older method of insurance against exposure to risk is to enter intoa binding agreement for delivery or receipt of assets at a �xed price ata �xed future date. This takes out any uncertainty as to price. It is notnecessarily an optimal procedure for risk management since on the

50

A �rst look at futures and forwards

delivery date the then current price for the asset may be either greateror less. One might say that 50% of the time one is worse o¤. Of course50% of the time one is better o¤. Either way one has o¤-loaded therisk of being worse o¤ onto another party whose job it is to managerisk. This is done at the cost of foregoing the 50% chance of beingbetter o¤. There are in essence two versions of a contract for deliveryin the future: a forward contract and a futures contract. Easier tounderstand is the forward contract since the date of delivery is speci�edexactly. The futures contract has a delivery date speci�ed less narrowly� say during the course of a speci�ed month: see Hull, Section 1.2 p.4and Section 2.2, p.21 for details. The di¤erence between the twocontracts may appear small, and under certain assumptions theirvalue/price is identical (Hull, Appendix 3A p.78), but one mustcompare them with care. It is of course the futures prices that arequoted on the exchange and in the newspapers. See Section 2.4 p.31for an account of the published information which includes openingprices (just after the opening bell) and average closing prices � justbefore the closing bell (known as settlement prices � since marginaccounts are settled at these prices) as well as daily highs and lows andlifetime highs and lows (prices traded so far since the initial appearanceof the contract), together with the number of outstanding contracts(open interest) and estimated volume of trading.

The futures contract is said to be written on an underlying asset andis itself a derivative security of the underlying asset.

But could the risk be better managed? It will be helpful to introducesome technical terms. One talks about taking a position in an assetwhich is long �when contracting to buy �or short �when contractingto sell.(As a mnemonic note that the party in the short position in acontract will become short of whatever asset it sells o¤.) The pay-o¤to each of the two positions against the prevailing current price (spotprice) is shown in the diagram given a delivery price K:

A long forward hedge.

0 . 5

1

1

0 . 5

00 . 5 1 1 . 5 2

Fig. 3.4: Pay-o¤ from a long position in a forwardcontract with delivery price of K = 1 dollar. Pro�table

when spot prices exceed 1 dollar.

51


0 . 5

1

1

0 . 5

00 . 5 1 1 . 5 2

Fig. 3.5: Pay-o¤ from a short position in a forward contractwith delivery price of K = 1 dollar. Pro�table when spot

prices are below 1 dollar.

A short forward hedge.

Thus the futures contract for an asset exposed to uncertainty willevidently be in taken in the same position to the contemplated positionof the company in the future.

For example, if the company is to sell an asset it will be in the shortposition (e.g. it may wish to sell assets which will come into itspossession, or may need to sell o¤ holdings to acquire funds). Ittherefore takes a short futures position (selling position) to safeguardfunds to be received using a delivery price K: If the price at thedelivery date is S and S < K then a gain is made. See Hull, Example3.1, Section 3.3, p.55.

Note in this example it is necessary to close out the short futuresposition somewhat before being in the anticipated short position, bypicking a prior maturity date as close as possible to the originalanticipated selling date. The acquisition of the short futures contract iscalled a short hedge.

Use of the word hedge indicates in the �rst instance an intention to actso as to protect against loss of value. The �hedges�of ordinary day liferefer to bushes; but these were planted around the perimeters (or edge)of �elds to protect the crops in the �eld from wind. We will often �ndthat holders of a portfolio of assets will want to supplement theirportfolio with additional assets; often these additional assets are�derivatives�� that is, contracts with pay-o¤s that are derived from thevalue of some simple asset such as a stock or bond.

3.10 Optimal hedges

The futures hedges considered above are only �perfect hedges�if thefuture course of history turns out just as predicted. But, one can dobetter than hedge with a futures contract equal in size and position tothe hedger�s own liability. In the �rst place we can look for �optimalhedges�: measure the level of risk by variance and then minimize.

Suppose we know, or can estimate, the correlation coe¢ cient betweenthe change in the spot price of the asset and the change in the futuresprice. If we hold a portfolio � comprising a quantity h of short futurescontracts each quoting a price F and an asset of value S, then change

52

The �duration�-based hedge: a �rst-order hedge

in value of the portfolio is

�� = �S � h�F;

where for instance �S = S2 � S1 is the gain in value of the assetbetween time 1 (now) and time 2 (the future). Note the negative signagainst the futures contract signifying the short position. Theunderlying asset is thought of as currently in the possession of thehedger (as though bought) and therefore in a long position. Of courseif the positions on the asset and the futures contract are reversed, thesigns are also reversed on both quantities. In either case, however, thevariance is

var(��) = cov(��;��) = h��;��i= var(�S) + h2var(�F )� 2hcov(�S;�F )

and minimizing this U-shaped quadratic polynomial in h over h yields(from the �rst-order condition of calculus) Optimal hedge ratio.

h =cov(�S;�F )

var(�F )=��S�F�2F

=��S�F

:

Indeed, by de�nition

var(�S) = �2S ; var(�F ) = �2F ; cov(�S;�F ) = ��S�F :

The formula gives an optimal hedge ratio.


A company intends to hedge a forthcoming acquisition of 1 milliongallons of jet fuel in 3 months�time. Over this period it believes that�jet = 0:032: The company considers using a futures contract onheating oil, for which in the same period it believes that �oil = 0:04: If�jet;oil = 0:8; how many futures contracts should be bought if thestock-exchange unit of futures contract is 42,000 gallons? What sort(short/long) hedge position is required? [Compare with Hull, Example3.3, Section 3.4, page 59].

3.11 The �duration�-based hedge: a �rst-orderhedge

Let us see another hedge calculation (Hull, Section 4.8 p.89) in regardto bonds. If the yield �calculated on a constant continuouscompounding rate basis - is y; then by de�nition the bond price isrelated to the coupon payments and the face value by the formula

B =nX1

cie�yti : (3.3)

Here the coupon payments are ci and are made at the dates ti (andthis includes as cn the face vale of the bond). Let us calculate theamount by which the bond price will change if the yield were to change

53


by a small amount �y. We begin by computing that

dB

dy= �

nX1

citie�yti

= �BnX1

ticie

�yti

B:

Since the coe¢ cients of the terms ti in the second line are positive andby (3.3) sum to unity, they may be interpreted as weights applied tothe times of coupon payments, thereby giving rise to an average timeof payment de�ned by

DB =nX1

ticie

�yti

B:

This quantity is known as the duration of the bond. Thus we haveapproximately:

�B =dB

dy�y = �BDB�y:

See Hull, Section 4.6 p.101 for a calculation of duration.

Suppose a portfolio � consists of bond currently priced at S and aquantity h units of a futures contract for delivery of bonds (othersimilar interest rate dependent asset) with a current (�spot�) deliveryprice F of the bond. Then the change in the portfolio value consequenton the yield y changing by �Y may be computed approximately byreference to duration to be

�� = �S � h�F= SDS�y � hFDF�y

= (SDS � hFDF )�y:

So taking Duration-based hedge ratio.

h =SDS

FDF;

where this ratio is known as the �price sensitivity ratio�or�duration-based ratio�, we obtain a hedge which has, to �rst-orderapproximation, a zero change in value. One says that the hedge has a�delta of zero�or is �delta neutral�. Evidently the validity of thisapproach is dependent on the behaviour of �B=B: The error in theduration approximation, given by Taylor�s Theorem is governed by Second order hedging.

d2B

d2y;

so a better hedge may be obtained by constructing a wider portfoliowhich will enable o¤-setting control over this further quantity. Portfoliomanagers talk about the convexity measure of the portfolio; see e.g.Hull, Section 4.9 p.92 (a similar second-order idea will be discussedlater under the name gamma, Hull, Section 15.6 p.355). The remarkto be made here is that the delta neutrality has been obtained bybalancing the e¤ects of the same unknown change �y through itse¤ects on two distinct though related quantities.

For example see Hull, Section 4.8 p.89.

Before we give our next hedging example (the beta hedge) we need tostudy the forward contract.

54

Arbitrage arguments

3.12 Arbitrage arguments

We can identify the forward price in a forward contract by reference towhat is known as an arbitrage argument. Originally the term arbitragewas used to describe the activity of buying a commodity at one price inone market and selling at a higher price at another market.

Setting the forward price by arbitrage argument

Our argument will be based on the current/spot price of the assetassuming a constant risk-free rate in the period between current time tand the delivery date T: We claim the forward price is

F = Ser(T�t);

where S is the spot price. Thus

S = Fe�r(T�t):

In other words the �present-value�of F agrees with the spot price (i.e.the price as at present). The forward price is the current pricecompounded up to what is called (by analogy with present value) its�future value�at maturity.

Indeed, assume a price is o¤ered of F > Ser(T�t): A risk-freeopportunity to make money �known as an arbitrage opportunity �occurs as follows. Borrow S dollars, buy the asset for S dollars withthe intention of selling the asset at the delivery date T for F dollars(this is called taking a short future position); the loan can be repaidfrom the proceeds of the sale since the debt amounts to only Ser(T�t).The positive excess F � Ser(T�t) is a sure pro�t.

If the inequality is reversed (i.e. F < Ser(T�t)), we try to make apro�t equal to the formula Ser(T�t) � F: We can do this, by borrowinga unit of the asset at time t from someone who does not need it, andsell this asset for S: (This action is called shorting the asset, since wesell something we were short of!) Depositing S dollars until date Tthen yields a cash balance of Ser(T�t) in the deposit account, and thisequals the �rst term in our hoped-for pro�t formula. We can also takea long position in a forward contract (i.e. o¤er to buy the asset)enabling us to buy back the asset for F dollars at time T: At maturityaccept delivery of the asset paying F out of the invested monies. Sopaying F dollars out of the Ser(T�t) allows us to obtain a unit of theasset in order to settle the asset loan. (This action is called �closing outthe short position��by returning the assets.) We are left with apositive excess pro�t of Ser(T�t) � F:

For examples see Hull, Section 3.1 p.47.


Suppose that the underlying asset in the forward contract pays aknown dividend of D dollars at time t = 1. Suppose that D isexpressed relative to the initial time t = 0 asset price as D = qS, thenq is called the �dividend rate�. Verify by arbitrage argument that theforward price F agreed at time t = 0 is S(1 + r � q).

55



Suppose that the underlying asset gives its holder a dividend paid inthe asset (e.g. more shares if the asset is a share, or more foreigncurrency, if the asset represents a foreign currency deposit) and that ateach moment of time t the dividend rate is a (known) constant q: Thisis taken to mean that over the time interval [t; t+�t] the dividendexpressed relative to the current (spot) asset price St of the asset isworth Steq�t: Verify by arbitrage argument that the forward price F isSe(r�q)T : How does this generalize the result of the preceding activity?

3.13 Forward contracts written on an index

Consider a futures contract on an index I which re�ects the market �thought of as a grand portfolio. Since the index is a hypotheticalportfolio, let us write I =

Pi wiSi; where wi are weights and Si are

the asset prices for the assets comprising the index. Thus the forwardprice FI for delivery of the hypothetical portfolio at time T is

FI =Xi

wiSier(T�t) = Ier(T�t):

(This formula assumes that none of the assets yields a dividend in theinterim.) We need to clarify here that the index itself is a valuation of ahypothetical portfolio of assets by reference to their spot prices.Consequently, the future delivery envisaged in the contract ought intheory to be that of the very same hypothetical portfolio of assets thatde�nes the index; however, the practice is that the future is settled dayby day in the money equivalent of the value of the hypotheticalportfolio.

Now observe that to �rst order

�F = F (t+�t)� F (t)

=@F

@I�I +

@F

@t�t

= er(T�t)�I + Ier(T�t)(�r)�t;

and so�F

F=�I

I� r�t:

Thus for small time changes it is approximately true that that thepercentage growth in forward price is equal to the percentage growth inthe market:

�F

F=�I

I:

But this is slightly misleading, in that the instantaneous indicators areconnected as follows:

1

F� dFdt=1

I� dIdt� r:

Note that �I =Pwi�Si only if we assumes no changes have been

made in the relative weights in the composition of the index. These

56

Beta hedging: a perfect hedge

weights are of course selected so as to re�ect the total marketcapitalization of the share �along the lines of

wi =NiSiPj NiSj

:

Thus the weights may be subject to alteration in situations when anasset has a dramatic change in value. See Hull, Section 3.5, p.60.

3.14 Beta hedging: a perfect hedge

The next example of hedging is motivated by a theoretical result. Thetheory seeks to explain what is called the �excess return�of an asset, ormore generally a portfolio of assets, that is, the di¤erence between thereturn of the asset and the riskless rate. The theory relates this excessto the �market portfolio�, that is, the quantities of all assets held by thetotality of market agents. In an appropriately de�ned one-period model,known as the Capital Asset Pricing Model, or just �CAPM�(pronounced�cap-em�), it may be shown that if the market is in equilibrium, thenthe excess-return of any asset (or portfolio) is proportional to theexcess return of the �market portfolio�. The relevant coe¢ cient ofproportionality is called the � of the asset. This coe¢ cient may beinterpreted as measuring risks associated with the asset.

Thus, the beta � of a portfolio measures the ratio of the excess returnon the portfolio over the risk-free rate on the one hand, to the excessreturn of the market as a whole over the risk-free rate on the other. Itmay be assessed using regression techniques from statistics.

Now suppose we want to protect a portfolio currently worth � againstexcessive movements in stock prices over some not unreasonably longperiod of time. Let �t denote the length of the time period. Weregard the behaviour of the markets over this one period as beingmodelled by the CAPM.

Thus we have for any portfolio that

return� r �t=��

�� r�t = �(

�M

M� r�t): (3.4)

Here �� is the change in the value of the portfolio at the end of theperiod and M represents the market regarded as a grand portfolio. Therisk-free rate per unit time is r; so the return on risklessly-depositeddollars over the period is r�t: Evidently �M

M is the change in value of1 dollar invested in the market for the period of time �t (since Mdollars changes in value by �M). In practice we replace M in thisrelation by a suitable index, so we come to regard M as though it werethe hypothetical portfolio de�ning the index. Now we may rewrite theequation (3.4) de�ning beta, thus:

��

�= �+ �

�M

M;

where, for convenience, � is introduced to denote

� = r�t(1� �):

Consider hedging the original portfolio of assets worth � with a beta of�; by taking a short position of an amount h units of a future contractwritten on a su¢ ciently wide stock-index currently o¤ering the future

57


delivery at the end of the period �t for a price F . We think of thefuture contract on the index as a proxy for the market, so we musthave approximately

�F

F=�M

M� r�t:

Hence the change in the value of the �hedged portfolio�comprising theoriginal portfolio and the future (or to use the jargon: its delta) is

�� h�F

= (�+ ��M

M)�� hF (�M

M� r�t)

= ��+ (�� hF )�MM

+ rhF�t:

So taking the hedge ratio h to be The beta hedge ratio.

h =��

F;

we have a portfolio which is approximately independent of theuncertain change �M in the stock index. It should thus enjoy a returnapproximating to the risk-free rate, and indeed our calculations agreewith this, since

��+ rhF�t = r�t(1� �) + r��t = r�t:


A portfolio with � = 1:5 is currently worth $2.1M. The holder is fearfulof market movements in the next quarter but does not want to sell o¤the portfolio in order to avoid transaction costs. It is proposed tohedge with futures of maturity 4 months. The index future stands at300 and the unit contract is 500 times the index; in this case a singlecontract costs $300�500. (The index is to be imagined as the price wehave to pay to buy a unit of asset, and the contract is for 500 suchunits.) It is of course proposed that the futures contract be closed outafter 3 months. What position should be taken and how manycontracts should be bought?

3.15 E¢ cient market hypothesis

A famous assumption states that the operation of the markets ise¢ cient, which is supposed to mean that the current share pricere�ects all the publicly available information. See Campbell, Lo andMacKinlay p. 22 for a careful explanation of several ways that thisassumption may be formalized (which includes consideration also ofprivate information). As a consequence, if the market is �in Various senses of e¢ ciency.equilibrium�, market participants cannot know in advance (i.e. withcertainty) that the price of a share is to rise (gain in value) by someamount tomorrow; for otherwise current demand (to take possession ofthe gain) would drive the price today immediately upwards by thatsame amount: the universal anticipation of change would lead toone-directional trading. Similarly, a known and sure tendency for aprice to decrease would trigger a strong tendency to sell at the current

58

E¢ cient market hypothesis

better price; a continued selling volume would immediately bring theprice down, perhaps dramatically, with the bulk of the sale playingpresumably into the hands of players speculating on a comeback. Thuswe see a self-ful�lling prophecy unfolding. (Such dramatic events canindeed take place, as they did on Black Monday October 19, 1987 �see Hull, Section 5.9 page 112 and see also Hull Chapter 32 inparticular Section 32.2, page 736. However, that was an exceptionalday, whereas our concern is �with the rule�rather than the exception.But exceptions, though they might be held rare, do occur.) This lack ofcertainty in price direction signi�es that we are only one step away fromthe language of probability: the expected price in the next instant is thesame as the current price (after a due discounting adjustment to makemoney of di¤erent time-moments comparable). Gamblers viewing thegame of speculating on the price of the share will say the game is a fairone. The fairness assumption is of course open to empirical veri�cation. Investment as a �fair�game.See �Capital Ideas�for a riveting account of early empirical observationson this by Alfred Cowles in the 1920�s (see Bernstein�s Capital ideasp.30). To sum up: the �ndings were that all �self-avowed expertopinion�on future price movements were in the long run no better thanthe toss of a coin. In consequence, the view that one cannot beat themarket led to the growth of portfolio management with the intention ofat least following the market by reference to measurements of pricebehaviour in the market as a whole, that is by way of indices. This isreminiscent of the establishment of the principle of conservation ofenergy in physics after periods of searching for the �perpetuum mobile�- a perpetually working machine requiring no power.

The economist Paul Samuelson, writing in 1965 (see �Proof thatproperly anticipated prices �uctuate randomly�, Industrial ManagementReview, 6(2) 1965, pp.41�49, sought to explain the �fair game�phenomenon by reference to the behaviour of the �futures price�FT;tviewing it as a random variable (more precisely random process). Thefutures price, as explained in Section 3.9, is the price contracted atdate t for the delivery of a unit of a speci�ed asset at date T . Supposethat the price of the asset at time t is modelled as having a probabilityof the following form.

Pr[St � xjS0; : : : ; St�1] = P (x; t;S0; : : : ; St�1)

and that the futures price is arrived by way of expectation as

FT;t = EP [ST jS0; : : : ; St] for t = 0; 1; : : : ; T:

(Note that this includes the case t = T; for which FT;T = ST :) Byassumption, the next period�s future price will depend on past assetprices and on the next period�s asset price, that is

FT;t+1 = EP [ST jS0; : : : ; St; St+1]:

Today�s expectation of tomorrow�s forward price, denoted

EP [FT;t+1jS0; :::; St]

is thus equal to

EP [EP [ST jS0; : : : ; St+1]jS0; :::; St]:

By the Law of Iterated Expectation (see Activity 6.7) this equals

EP [ST jS0; : : : ; St];

59


and, by assumption, this is just FT;t: Thus we have

FT;t = EP [FT;t+1]:

Technicalities apart, the process fFT;t : t = 1; 2; :::; Tg is said to be amartingale �meaning that any time in the ongoing process theexpected value at the next instant is the currently observed value. Wewill discover, that the �discounted�share is similarly a martingale �although under a di¤erent probability law, but derived from P:

This brings us to the question of how to model the behaviour of share Bachelier�s model.prices. The �rst mathematical attempt at the turn of the twentiethcentury (Theory of Speculation, 1900) by Louis Bachelier wasunappreciated by contemporaries and went into apparent oblivion formore than half a century, partly due to lack of empirical research andpartly due to the absence of appropriate computing power (see CapitalIdeas, p.18). Bachelier�s ideas thus waited for their rediscovery. SeeLouis Bachelier�s Theory of Speculation: the origin of modern �nance.



compute the terminal value in a deposit account of an initialdeposit and of a deposit stream

compute the present value of a single future payment and of anincome stream

compute the expected value of a defaulting income stream.

60

Chapter 4

One period, one risky asset,and two states



replicate and value a contract in a one-period single asset model,

de�ne what is meant by a forward contract, a call option and a putoption,

compute the forward price,

replicate the call and put options in the current context,

compute the risk-neutral (�synthetic�) probability and use it tovalue a forward contract, a call option and a put option.

4.2 Essential reading

There is no speci�c essential reading for this chapter. It is essentialthat you do some reading, but the topics discussed in this chapter areadequately covered in so many texts on the �applications of calculus�that it would be arti�cial and unnecessarily limiting to specify precisepassages from precise texts. The list below gives examples of relevantreading. (For full publication details, see Chapter 1.)

Pliska, S.R. Introduction to Mathematical Finance. Chapter 1.

Luenberger, D. Investment Science. Chapter 10.

4.3 Introduction

This chapter is a gentle introduction to most of the guiding principlesthat will be used in future chapters. The market model studied herehas one riskless asset and just one risky asset. We are concerned withonly two dates, an initial date which we call t = 0 and a terminal datewhich we will call either t = T or t = 1: There are only two possiblestates of nature at the terminal date; these two states, which we callhigh and low, are determined by whether the risky asset price is higherat time t = T than it was previously at time t = 0 and represented bythe number SH , or lower than previously and represented by thenumber SL: It is assumed that SL < SH . This can be represented

61


pictorially thus.

SH%

S0&

SL

Furthermore, it is assumed that the asset pays no dividends during theperiod under discussion. We routinely relegate the matter of dividendsto a separate consideration � see activity 5.10. This representation bytwo states, though at �rst sight apparently unrealistic, is a basicbuilding block for a richer, later development. Its two saving graces are:(i) the insight it o¤ers on what the mathematical issues are, and (ii) theopportunity that it gives for quick, �back of the envelope�calculationsof option values. Of course the �ball-park��gures that the simplemodel generates depend on how representative the two prices are. Inthe normal course of discussion textbooks do not consider how to picktwo representative prices. So as most textbooks ignore this point, wedevote a section to this question in order to give a persuasive approach.

It is precisely because SH 6= SL that the asset is referred to as �risky�.The price of this asset at time t = 0 is denoted by S0: The price attime t = T , such as it may be, is denoted by ST : Thus, formally, theprice ST is modelled as a random variable. There is also a risklessasset, whose price is initially 1=(1 + r) and whose value at terminaldate is 1 = (1; 1)t. We assume that r > 0, and identify this with aconstant interest rate in force for the time interval between the initialand terminal dates. This riskless asset is most easily interpreted asmodelling a safe bank deposit account, since a bank may be construedas receiving a payment of 1=(1 + r) dollars at time t = 0; in exchangefor which it promises to o¤er 1 dollar at time t = T: Other institutions,such as governments, also o¤er contracts of a similar nature �promising a �xed payment at time t = T in exchange for moneyreceived by them at time t = 0: Such contracts are called bonds, but itis usual for the contract to o¤er in addition �coupons�which may beexchanged for money at speci�ed dates prior to t = T: A bond whicho¤ers no coupons (and so no interim payments) is called a�zero-coupon�bond. Thus the riskless asset may be regarded asmodelling a very special kind of bond. The riskless asset may also bereferred to as a �zero-coupon bond�. Similar contracts are o¤ered byother institutions. Our aim is to identify a way for valuing two-personcontracts which specify payments to be made at the terminal dateaccording to whether the state is high or low, by considering whatpayments should be made at the initial date to induce the two partiesto agree the contract. For clarity, in a general context, the two partiesto the contract will be referred to as He and She, a practice borrowedfrom two-person game theory. The distinction between the two is thata positive payment may then be interpreted as He paying Her, ordelivering an asset to Her, while a negative payment signi�es thereverse transaction. In a speci�c context such as that of a forward or acall option, conventionally He is referred to as the writer of the forwardor call, and She the holder of the forward or call. Alternatively He isreferred to as going �short of the asset�(or �short the asset�, or�shorting the asset�), since after delivering the asset to Her, He willindeed be short of it. She for her part is then �long in the asset�or just�long the asset�. The joke is that She is never said to long the asset.

At the end of this chapter you will be able to answer questions like thisone (see Activity 4.2): A leitmotif for this chapter.

62

Forward and future contracts

�A company whose non-dividend-paying shares currently trade at 50p isnegotiating a takeover. If the takeover takes place the price will rise to75p, but will otherwise fall to 35p. Ignoring the rate of interest,estimate a fair price for the right (without obligation) to buy a shareafter the event for 50p. On your estimate, is the price worth paying ifyour belief in the probability of the take-over is 20%?�

4.4 Forward and future contracts

4.4.1 Forward contracts

For its simplicity we begin by considering the forward contract on a�tradeable asset�such as shares in a context where both sides to thecontract may be assumed not to default on the agreement, i.e. theycarry out their contractual obligations. The forward contract will helpus understand futures contracts which are a practical implementationof the same idea, but managed by a third party such as an Exchange insuch a way as to make the e¤ects of default negligible.

A forward contract (abbreviated to �forward�), discussed in Chapter 3,is an agreement made at some date known as the initial date toexchange at a speci�ed future date (known as the terminal date) oneunit of a speci�ed asset for a �xed amount of money (the �forwardprice�- which is not abbreviated) which is speci�ed in the agreement atits inception (i.e. when the agreement is signed at the initial date). Nofurther payment is required of either party, neither at the initial date,nor at any later date.

It is worth re�ecting on the principal di¤erences between anon-defaultable dollar bond and what would have to be done to removethese di¤erences.

Firstly, the tradeable asset of the bond is money, that is money isexchanged for money. In some ways this is a minor detail as it concernsa numeraire, indeed the parties to the bond could agree to use someassets other than dollars. It could have been pounds sterling, or euros.It need not even be money: one party could have agreed to o¤er afraction of a unit of some tradeable asset that is to be replaced with afull unit of the same asset.

Secondly, and more importantly, the timing of the exchange of assets iscompletely di¤erent: there is a time delay between the movement ofassets. In the bond one side to the contract makes a payment at theinitial date, and at the later terminal date it is the other party thatmakes a payment. In the forward the movement of both assets is at theterminal date.

The only way to remove the second distinction is to permit into thecontract a limiting situation: allow the terminal date to move to theinitial date. Time shrinkage! This kind of trick is conceptually veryuseful in understanding the valuation of the forward.

4.4.2 Choosing the forward price

Suppose you have entered into an obligation to supply, in one year�stime, one share in some company, at a monetary price �xed today. How

63


should you �x the price F , if no other payments are scheduled and youcannot even hope to know the spot market price in 12 months�time?

Since the contract refers to money and to the shares, and to yourpresent and future actions including the delivery of a share, begin byasking yourself what information you have today about all the factorsin the contract.

Firstly, where could you obtain the share which you promise to o¤er tothe other party, secondly what do you know:

(i) about the value of the shares, and

(ii) the value of money?

Let�s take these questions in reverse.

We already know what the value of next year�s money is by reference tothe price today of a bond maturing in 12 months�time.

We also know the value of the shares by reference to their price today.

Where can you get the shares? You can buy them at today�s pricetoday. You can buy them next year at next year�s price! Of course youmay buy them at all the intermediate business days in the year atwhatever price they will then be. You may be able to buy them todayfrom other parties willing to undertake your obligation for a quotedprice, or at any intermediate date. Which should you choose?

Of course you can try to guess the future dates, but then you openyourself to the risk of a bad guess �and could su¤er a loss fromsettling the contract. There is only one certainty: today�s prices for thedelivery of the share and today�s one-year rate r.

So you can borrow some money now, buy the share now at its spotprice S0. Since your liability to the moneylender is S0(1 + r) if your�xed price of delivery in 12 months�time was F = S0(1 + r) or greater,you will discharge your debt with the proceeds of the sale no matterwhat the spot price at time t = 1 is. We can remember the result byrestating it in the form

F � = S0;

that is, the present value of the forward price is today�s price.

To con�rm the �nding, let�s shrink time so that r = 0: Then it isobvious that the forward price should be S0:

The strategy of borrowing and buying immediately on entering thecontract leaves your own wealth at zero after the transaction has takene¤ect. So we say that S0(1 + r) is the forward price which isrisk-neutral. You remain neutral to risk. The strategy is called aperfect hedge, since it guarantees that no loss or pro�t occurs as aresult of entering into the contract.

4.4.3 Valuing the contract

What is the value of the contract to the two parties? This is a seriousquestion and evidently only makes sense if you specify the time. Let�sinvestigate.

64

Forward and future contracts

At terminal date the value to the recipient of the asset is S1 � F: Thatis because that party can immediately sell the asset for S1 havingbought it for F .

To the party delivering the asset the contract is worth F � S1: That isbecause F is received upon giving up an asset currently worth S1:Alternatively, we can rephrase the valuation as saying that to settle theliability this party needs to buy the asset for S1 and sell for F:

At any earlier time t (including the initial time) we can repeat theargument, taking care to value the money payment correctly. The assetto be delivered may be bought at the spot price St say. The moneypayment F to be received at time t = 1 has current value Fe�r(1�t),so the contract value for the delivering side is Fe�r(1�t) � St: Inparticular, at time t = 0 we have Fe�r � S0 = 0 since F = S0e

r:

In summary: at inception the value of the forward contract is zero.This makes sense, because neither side to the contract pays to enterthe contract when the forward price is F: The contract value thereafterchanges according to the level of the spot price St:

4.4.4 Future contracts

We just saw that entering a forward contract is costless to both parties,but that after a time t the value of the contract is in general positive toone party and negative to the other. A party who sees the contract asbeing negative at the terminal date would surely gain from nothonouring its side of the contract details.

How can one prevent such default occurring?

One obvious way is for the parties to o¤er each other guarantees byway of a safe third party, for instance by placing a large enough depositwith the third party. The drawback in this idea is that in the �rst placeit is unclear what large enough means, and secondly a large deposit islikely to be very burdensome.

A better way to reduce the burden and to prevent default is to use athird party who receives a more reasonably-sized deposit and alsocollects at the end of each business day an appropriate incremental�balancing�payment (known therefore as a �margin call�) from Margin call.whichever party faces a negatively-valued contract, in order to pay thisto the other party. The size of the daily payment is such as to make thevaluation of the remaining contractual obligation equal to zero. Thisprocedure is referred to as �marking to market�. Evidently, this has the Marking to market.same e¤ect as though the parties had entered a rolling series of one-dayforwards running all the way to the agreed terminal date. In particular,this means that parties entering a futures contract at date t withterminal date T face a new futures price, Ft;T that changes every day.

So far the e¤ect of the margin call is that the default risk is reduced tothe size of the risk in a single day�s value swing. The risk in a one-daydefault can in turn be reduced by requiring the parties to maintain adeposit adequate to meet a one-day value swing; for example thedeposit can be in the form of unrequired assets, which will only be soldto meet an unful�lled margin call. The size of the deposit can thus bedecided given the asset�s currently observed behaviour and included inthe calculation of the margin call. We note a further e¤ect of marking

65


to market. Inability to meet margin calls acts as an early warning signalof �nancial �ill-health�. In recent history several famous defaults wereindeed triggered by the inability to settle margin calls and these serveto illustrate the point. Perhaps the most famous is the failure of Long Long Term Capital

Management.Term Capital Management. See for instance, Nicholas Dunbar�s book,Inventing Money: The Story of Long-Term Capital Management andthe Legends Behind It or Roger Lowenstein�s When Genius Failed: TheRise and Fall of Long-Term Capital Management.

In summary, the original contract is regarded as embedded in a contractof each of the two parties with a third party known as a �clearinghouse�. Assuming a constant interest rate for the life of the contract,the daily increments with the clearing house (up to the terminal date)yield in total the terminal value of the initial contract to each party.

4.5 A call option struck at the low SH

We have seen that it is costless to enter into a forward contract. Nowwe consider valuing at time t = 0 a call option contract with exerciseprice (also known as the �strike price�) K and exercise date t = 1. Wewill recall its de�nition from Chapter 2 and we will see that to hold acall option one must pay a positive price. The option confers on itsholder the choice of whether or not to buy one unit of asset in exchangefor K dollars. Evidently, if S1 > K; the contract is worth S1 �K tothe holder of the call at time t = 1. But is otherwise worthless. Indeedif S1 � K one is better o¤ buying at the spot price than for K:

To make the argument simple we assume as follows.

i) there are only two possible spot prices S1 that the asset may take attime t = 1 : a high one SH and a low one SL with SL < SH , asdepicted in the �gure below.

SH%

S0&

SL

ii) K = SL:

The call option thus enables the holder to buy at the price SL whateverthe spot price is.

See Activity 4.1 below to deduce a price for the general case, whereSL < K < SH ; from the particular case K = SL.

To begin with we suppose that the interest rate is zero. We reconsiderthis point at the end of the section.

The writer of the call option thus needs to have a fund of SH � SLdollars to ful�ll their side of the bargain in the event that S1 = SH , butnot otherwise. This observation however does not help us much, beingmerely a restatement.

What does help, is to notice that the writer of the option needs to holdone unit of asset at time t = T to satisfy the contract, since then theneeds of the counterparty to receive the asset are met whatever the

66

A call option struck at the low SH

spot price may be.

One obvious action that the writer can follow is to buy the asset attime t = 0: He can do this by charging a payment c for the call andborrowing S0 � c to buy the asset.

If the asset turns out to be worth SL the writer can o¤er the asset tothe market or the counterparty at SL. If the asset price turns out to beSH , then the writer must o¤er it for sale at SL to the call optionholder. In either case the writer has to settle the loan S0 � c with SL,and this will be the case if

S0 � c = SL; i.e. c = S0 � SL:

Thus on agreeing the contract for a price S0 � SL the writer borrowsSL. This means that he has in total S0 dollars in hand and for this sumhe can buy the asset. He can then pay back SL after meeting hiscontractual obligations.

The activity just described is referred to as hedging.

Comment. We implicitly assumed that c > 0; that is, that S0 > SL: Itis also implicit in this simple model that there are parties interested inentering this contract. This means that SL < S0 < SH : There wouldbe no point in entering a call contract if S0 < SL < SH : One might aswell buy at the price S0 and wait for a larger or smaller riskless pro�t.

Example. Value the call with strike 35 when prices (in pence) areassumed to behave as shown. Ignore interest rate considerations.

75%

50&

35

Solution. The call is worth c = 50� 35 = 15 pence.


In the two-state model just described, show that if SL < K < SH thenthe fair price of the call option with exercise price K is

SH �KSH � SL

� c; i.e. SH �KSH � SL

� (S0 � SL):

Hint:

(ST �K)+ =SH �KSH � SL

(ST � SL)+:

Example (continued). Value the call with strike (exercise price) 50when prices (in pence) are assumed to behave as shown earlier. Againignore interest rate considerations.

Solution. Using the last activity, the price is just under 10 pence � infact it is

75� 5075� 35 � c =

5

815:

67


If there is a constant positive interest rate in force during the life of thecontract, we can re-tell the story in present value terms. Recall ourconvention that the present value of a payment S made at time T isdenoted S�:Thus

S� = Se�rT :

So, writingS�L = SLe

�rT ;

which denotes the present value of SL, the premium for the call is thenS0 � S�L. Borrowing S�L enables the writer to buy the asset for S0:Settling the contract at time T brings in SL dollars which is the valueof the loan by time T since S�Le

rT = SL:

Here again we implicitly assume that S�L < S0 < S�H :

4.6 Valuation using replication

In both of the last two examples, we determined the value of the claimby examining what assets the writer of the claim should have at theterminal time to meet his liabilities exactly. We now check that in ourbasic model any claim can be valued by this procedure.

Suppose the general claim pays XH in the high state and XL in thelow state. We check whether there is a portfolio comprising k units ofcash and h units of the risky asset such that

hSH + k(1 + r) = XH ;

hSL + k(1 + r) = XL:

This amounts to solving two equations in two unknowns. The solutionexists and is unique, because we have

h =XH �XL

SH � SL;

k =1

1 + r

�XL �

XH �XL

SH � SLSL

�=

1

1 + r

�SHXL � SLXL �XHSL +XLSL

SH � SL

�=

1

1 + r

�SHXL �XHSL

SH � SL

�:

The equations were soluble because the matrix

S =

�SH 1SL 1

�whose columns describe the prices of the risky asset and the risklessasset according to states of nature has rank two. Of course the reasonfor this is that SH 6= SL: (Otherwise the formulas for h and k would bemeaningless.)

4.7 Valuation using expected value

4.7.1 Synthetic probabilities for the call and the forward

To satisfy the intuition that the value of a contract is an expectation,let us use the simple two-state model of future asset prices. What

68

Valuation using expected value

�synthetic probabilities�should we allocate to the two states to ensurethat the price of a call with exercise price K = SL, which we know tobe S0 � S�L is equal to the expected value of the contractual pay-o¤?

To solve this problem, let pH denote the probability we allocate to theprice SH and let pL be the probability we allocate to the price SL:Then we want

pH(S�H � S�L) + pL � 0 = S0 � S�L;

pH + pL = 1;

pH � 0; pL � 0:

Notice that in the �rst equation we are careful to state valuations onboth sides of the equation in present value form. The �rst twoequations imply that

pH =S0 � S�LS�H � S�L

;

pL =S�H � S0S�H � S�L

;

so the non-negativity condition is met if and only if

S�L � S0 � S�H :


A company whose non-dividend-paying shares currently trade at 50p isnegotiating a takeover. If the takeover takes place the price will rise to75p, but will otherwise fall to 35p. Ignoring the rate of interest howwould you hedge a one-month call with strike price 50p on the stockwith a single purchase of the stock? How does this lead to an estimateof the fair value of the call? On this estimate, is the call worth buyingin the belief that the probability of the takeover is 20%?


Show that the call option premium may be rewritten in the form

S0qL � qHK�;

where

qL = qLS�LS0;

so that the two terms in (ST �K)+ are discounted by factors betweenzero and one.

What about the forward contract? There, we would wish the forwardprice F , which we recall satis�es F � = S0, to be selected so that atentry the value of the contract is zero, namely

pH(S�H � F �) + pL(S�L � F �) = 0;

pH + pL = 1;

pH � 0; pL � 0:

69


Notice that we have written out the requirements in the �rst equationusing present value terms.

Equivalently, we thus require that

qHS�H + qLS

�L = F �;

qH + qL = 1;

qH � 0; qL � 0:

Setting F � = S0 and solving for qH and qL, we obtain

qH =S0 � S�LS�H � S�L

;

qL =S�H � S0S�H � S�L

:

We again obtain exactly the same synthetic probabilities; in the nextsubsection we will see why. In summary, in both cases we have thesynthetic probability characterized by the equation

qHS�H + qLS

�L = S0;

that is,EQ[S

�T ] = S0:

One way to read this equation, is as a consistency requirement. Theclaim which pays ST at time T is valued under this expectationaccording to the initial cost of purchasing the risky asset in the model,namely S0:

De�nition: A probability Q = (qH ; qL) which allocates a probability, Risk neutral measure.respectively to the high and low state of the model is said to be arisk-neutral measure if

EQ[S�T ] = S0

andqH > 0; qL > 0:

We recall that S�T denotes the present value of the payment ST :

In the model of this chapter a risk-neutral measure exists and is unique.In more general models this need not be the case.


Show that the synthetic probability which prices the put optioncorrectly is again the risk-neutral probability.


Initially the exchange rate from a foreign dollar to a pound is at parity.Suppose in a year the exchange rate rises to u or falls to d: so that onedollar may be bought either for u pounds or d pounds according tocircumstance.

price in pounds: now !1 !2dollar 1 u dpound 1 1 1

70

Valuation using expected value

The domestic rate is r and the foreign rate is s: Show that there is noarbitrage opportunity if

d <1 + r

1 + s< u:

Assuming absence of arbitrage, �nd the risk-neutral measure Q treatingpounds as numeraire. What is the price �now�(in pounds) of a claimpaying (i) 1 dollar in state !2 and zero otherwise; (ii) 1 pound in state!2 and zero otherwise. Verify your result by �nding the portfolio whichreplicates the �rst of these claims.

What is the relationship between the risk-neutral measure Q and therisk-neutral measure P with dollars as numeraire? [Hint: What are theclaim prices now in dollars?]

4.7.2 General claims

Recall that the claim may be replicated by a portfolio of k units of cashand h units of the risky asset, when

hS�H + k = X�H ;

hS�L + k = X�L:

Suppose we compute the expected value of the claim X using aprobability Q = (qH ; qL); when

EQ[S�T ] = S0:

Then

EQ[X�] = qHX

�H + qLX

�L

= qH (hS�H + k) + qL(hS

�L + k)

= hEQ[S�T ] + k

= hS0 + k:

So the initial cost of a portfolio replicating the claim agrees with theexpected value of the claim under the risk-neutral measure.


Check that the claim (1; 0) costs qH and the claim (0; 1) costs qL:The condition that qH and qL are positive thus ensures that both theseclaims have a positive valuation.


Let 1A denote the indicator function de�ned on to be 1 if ! 2 A, Put-call parity.and 0 otherwise. If A is the event that ST � K check that

(K � ST )+ = (K � ST )1A;(ST �K)+ = (ST �K)1nA:

71


Let PT = (K � ST )+ and CT = (ST �K)+ denote the European putand call pay-o¤s at exercise (i.e. at maturity). Deduce that

PT � CT = K � ST :

[Hint: 1nA = 1� 1A:]

Let p0 and c0 denote the time 0 price of a European put and callrespectively.

Deduce thatp0 � c0 = K� � S0: (4.1)

[Hint: Discount the put-call parity relation and take risk-neutralexpectations.]

Explain why the identity (4.1) may be interpreted in words as saying�put �call = bond �asset�.

The identity is referred to as the put-call parity relation.


Notice that the identity does not depend on any assumptions regardingthe structure of : That is, it does not matter whether time ismodelled as continuous or discrete, nor, in the latter case, on thenumber of time periods.

4.8 Ball park �gures: on picking representativeprices

* The material of this section is not examinable. It is includedfor interest only.

Suppose it is commonly agreed that end of year pro�t X per share willlie in the interval [0; 12] and that every value in the range is equallylikely. Convert the continuum of possible prices into two representativeprices either side of the mean of 6. The average price below the meanis 3 and the average price above the mean is 9. The situation may thusbe modelled by:

9%

6&

3

Note that the risk-neutral measure for this model assigns to the up anddown outcomes the respective probabilities: ( 12 ;

12 ), in agreement with

the market�s belief that 6 is the mean of 9 and 3:

The equation which identi�es the risk-neutral measure given the pricingmodel S0 ! fSU ; SDg, namely

qU =S0 � SDSU � SD

;

72

An equilibrium interpretation

may be read another way. If qU is given as the proportion of themarket betting on the �up�, the market maker selects the price ratioaccording to the equation in order to clear the market. However thesingle equation

qUSU + qDSD = S0

cannot identify (uniquely) the two values SU ; SD: Further informationis required. Let�s suppose that S0 = 1 and let�s write Uand D forSU ; SD . We could attempt to solve the equation

qU + (1� q)D = 1

simultaneously with something like

pU + (1� p)D = 1

Alas the solution is U = D = 1:

4.9 An equilibrium interpretation

* The material of this section is not examinable. It is includedfor interest only.

To justify the very simple numbers in our �nal interpretation of therisk-neutral measure as measuring market beliefs, we need the followingobservations.


Observations on the risk-neutral measure. Suppose that theinterest rate is zero. Show that if all prices in the standard two-statemodel are shifted by the same amount then the risk-neutral measureremains unchanged. Thus, a change of origin in the numeraire leavesthe measure unchanged (invariant). Verify that the risk-neutralmeasure is also preserved also a rescaling of the asset price.

In the following very simple model of asset price evolution, the assetshould in fact be interpreted as a portfolio of assets with a zero netcost of purchase zero (�set-up cost�of zero). With clarity in mind, wewill at �rst suppose that the portfolio leads either to a gain or to a lossof one dollar. (We consider the more general situation later.)

1%

0&

�1

In this model the risk-neutral measure assigns equal probability for thehigh and low prices.

Now suppose we consider those market participants who take longpositions in the asset. That is, these are agents who have bought theasset and they have bought the asset because they believe it will rise invalue. If the predicted gain occurs, so that the asset is worth +1, thatgain should be interpreted as occurring by way of these agents selling

73


the asset. We may regard this sale as being to those agents whoinitially took short positions in the asset and who are therefore nowrequired to �close out�their positions. The net proceeds of thesetransactions will be zero provided there are equal numbers on the longand short sides. Indeed if the price goes up those in the long positionreceive their reward from those in the short position. A similar storymay be told if the price goes down, except that the direction ofpayments is reversed. The model does not predict which situation is tooccur, and the market in its role as a clearing-house for claims betweenthe two populations of traders is a �risk-neutral�party at the conclusionof these transactions, since it is left holding no assets.

Said another way, as long as initial positions on the market of thismodel are arranged to balance exactly, the proportion of participants onthe long and short sales are both 1

2 of the totality of participants. Thusthe risk-neutral measure in this case conveys the information that thenumber of bets is equal on both sides.

Now suppose that the market participants may choose their positionsarbitrarily on the long side (buying initially and selling at the �nal dateside) and short side (selling initially and buying later). Here we assumethey may not place further orders conditional on the prices in the nextperiod. Suppose that there are twice as many long positions as shortpositions.

The exchange now sets prices to balance the two sides at thetermination date. Suppose that it sets the terminal low side atSL = �1: Then it must set SH so that the revenues from both sidesagree:

2SH + SL = 0;

orSHSL

= �12:

The risk-neutral measure for the model

12

%0&

�1

has q = qH =132

= 23 . Thus the value of qH con�rms that 2=3 of the

market are betting on the up.

More generally, the equation

0 = qSH + (1� q)SL

with SL = �1 and SH=(�SL) = h, rewritten in the form

0 = qh+ (1� q)(�1);

may be read as saying that the market maker sets the ratio of thebuying and selling prices to clear the market. Thus q represents theproportion of participants betting on a price rise and (1� q) theproportion betting on a fall.

74

Solutions to selected activities

4.10 Solutions to selected activities


Clearly the two answers di¤er by a factor of d: Consider a portfolio(H;K) of H dollars and K pounds. After one period, if the state is !1one dollar is worth u pounds. But one dollar earns interest, so thedollar deposit is worth u(1 + s) pounds. To replicate the claim we need

H(1 + s)u+K(1 + r) = 0;

H(1 + s)d+K(1 + r) = d:

Then H(1 + s)(u� d) = �d, i.e.

H =�d

(u� d)(1 + s) ; K =ud

(u� d)(1 + r)

so the fair price is

H +K =ud(1 + s)� d(1 + r)(u� d)(1 + r)(1 + s)

=ud� d(1 + r)=(1 + s)

(u� d)(1 + r)

=u� (1 + r)=(1 + s)

(u� d)d

(1 + r):

Note that there is an sure arbitrage if the fair price m is negative;otherwise (H;K �m) is of zero price initially, yet pays �m pounds inone state and �m+ d pounds in the other, both of which are positive.

Working in terms of terminal values we have prices as follows:

price: now !1 !2dollar 1=(1 + s) u dpound 1=(1 + r) 1 1

and using the pound as numeraire, the risk-neutral probability is

Q(!1) =(1 + r)=(1 + s)� d

(u� d) ; Q(!2) =u� (1 + r)=(1 + s)

(u� d) :

The expected value of the claim X(!1) = 0; X(!2) = d is

u� (1 + r)=(1 + s)(u� d)

d

1 + r:

The claims have the same price in dollars because of current parity andsince one dollar is worth d pounds in the state !2 we must have

P (!2)1

1 + s= Q(!2)

d

1 + r;

i.e.

P (!2) = Q(!2)d

(1 + r)=(1 + s):

Similarly for !1 with u for d: Thus

P (!) = Q(!)Z(!);

75


where

Z =1

(1 + r)=(1 + s)(u; d):

Alternatively, from �rst principles and using the dollar as numeraire,we have the table of exchange rates:

price in dollars: now !1 !2dollar 1 1 1pound 1 1=u 1=d

and working with terminal values:

price: now !1 !2dollar 1=(1 + s) 1 1pound 1=(1 + r) 1=u 1=d

Hence

P (!1) =1=d� (1 + s)=(1 + r)

1=d� 1=u ; P (!2) =(1 + s)=(1 + r)� 1=u

1=d� 1=u ;

P (!1) =u

(1 + r)=(1 + s)

(1 + r)=(1 + s)� du� d ;

P (!2) =d

(1 + r)=(1 + s)

(1 + r)=(1 + s)� uu� d :

The claim (i) is Y (!2) = 1, Y (!1) = 0:

(1 + s)=(1 + r)� 1=u1=d� 1=u � 1

1 + s

=ud(1 + s)=(1 + r)� d

u� d � 1

1 + s

=u� (1 + r)=(1 + s)

u� d � d

(1 + r):



replicate and value a contract in a one-period single asset model

de�ne what is meant by a forward contract, a call option and a putoption

compute the forward price

replicate the call and put options in the current context

compute the risk-neutral (�synthetic�) probability and use it tovalue a forward contract, a call option and a put option.

4.11 Sample examination questions

The following are sample questions or parts of questions.

76

Sample examination questions

1. You and a friend jointly agree to book a week�s skiing holiday atChristmas 2007 and the friend bets 100 dollars with you that theweather will be so bad that you will not be able to ski all week (i.e.you lose 100 dollars if the weather is bad and you gain 100 dollarsif the weather is good). You subsequently discover that if you buythe same holiday from a di¤erent agent for 40 dollars more (each)then both of you will be paid 100 dollars each by way ofcompensation if the weather turns out so bad that you are not ableto ski all week. Explain why the alternative travel agent is in factselling you a call option contingent on the weather. Use this optionto determine the value of the bet assuming that you pay the twoextra booking fees and collect both compensations. Ignore anyinterest rate considerations.

2. Let V be the value of a derivative written on an asset whose timet = 0 value is S and whose value in the next period at time t = 1will be Su or Sd; where d < 1 = r < u and r is the interest ratefor the period. Suppose the derivative value in the next period iscorrespondingly Vu and Vd: Show by constructing an appropriateportfolio that

(1 + r)V =(Vu � Vd)(1 + r) + Vdd� Vud

u� d :

77


78

ch1-4

Documents

Transcript of ch1-4