ńņš. 1 |

AND FINANCE

Don L. McLeish

September, 2004

ii

Contents

1 Introduction 1

2 Some Basic Theory of Finance 13

Introduction to Pricing: Single Period Models . . . . . . . . . . . . . . 13

Multiperiod Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Determining the Process Bt . . . . . . . . . . . . . . . . . . . . . . . . . 30

Minimum Variance Portfolios and the Capital Asset Pricing Model. . . 35

Entropy: choosing a Q measure . . . . . . . . . . . . . . . . . . . . . 56

Models in Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . 67

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3 Basic Monte Carlo Methods 97

....................................... 97

Uniform Random Number Generation . . . . . . . . . . . . . . . . . . 98

Apparent Randomness of Pseudo-Random Number Generators . . . . 109

Generating Random Numbers from Non-Uniform Continuous Distri-

butions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Generating Random Numbers from Discrete Distributions . . . . . . . 166

Random Samples Associated with Markov Chains . . . . . . . . . . . 176

Simulating Stochastic Partial Diļ¬erential Equations. . . . . . . . . . . 186

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

iii

iv CONTENTS

4 Variance Reduction Techniques 203

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Variance reduction for one-dimensional Monte-Carlo Integration. . . . 207

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

5 Simulating the Value of Options 255

Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Pricing a Call option under stochastic interest rates. . . . . . . . . . . 266

Simulating Barrier and lookback options . . . . . . . . . . . . . . . . . 269

Survivorship Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

6 Quasi- Monte Carlo Multiple Integration 301

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Theory of Low discrepancy sequences . . . . . . . . . . . . . . . . . . 307

Examples of low discrepancy sequences . . . . . . . . . . . . . . . . . . 310

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

CONTENTS v

Dedication: to be added

Acknowledgement 1 I am grateful to all of the past students of Statistics 906

and the Masterā™s of Finance program at the University of Waterloo for their pa-

tient reading and suggestions to improve this material, especially Keldon Drudge

and Hristo Sendov. I am also indebted to my colleagues, Adam Kolkiewicz and

Phelim Boyle for their contributions to my understanding of this material.

Chapter 1

Introduction

Experience, how much and of what, is a valuable commodity. It is a major

diļ¬erence between an airline pilot and a New York Cab driver, a surgeon and

a butcher, a succesful ļ¬nanceer and a cashier at your local grocers. Experience

with data, with its analysis, experience constructing portfolios, trading, and

even experience losing money (one experience we all think we could do without)

are all part of the education of the ļ¬nancially literate. Of course, few of us

have the courage to approach the manager of our local bank and ask for a few

million so we can acquire this experience, and fewer still managers have the

courage to acceed to our request. The ājoy of simulationā is that you do not

need to have a Boeing 767 to ļ¬‚y one, and that you donā™t need millions of dollars

to acquire a considerable experience valuing ļ¬nancial products, constructing

portfolios and testing trading rules. Of course if your trading rule is to buy

condos in Florida because you expect boomers to all wish to retire there, a

computer simulation will do little to help you since the ingredients to your

decision are largely psychological (yours and theirs), but if it is that you should

hedge your current investment in condos using ļ¬nancial derivatives real estate

companies, then the methods of computer simulation become relevant.

1

2 CHAPTER 1. INTRODUCTION

This book concerns the simulation and analysis of models for ļ¬nancial mar-

kets, particularly traded assets like stocks, bonds. We pay particular attention

to ļ¬nancial derivatives such as options and futures. These are ļ¬nancial instru-

ments which derive their value from some associated asset. For example a call

option is written on a particular stock, and its value depends on the price of

the stock at expiry. But there are many other types of ļ¬nancial derivatives,

traded on assets such as bonds, currency markets or foreign exchange markets,

and commodities. Indeed there is a growing interest in so-called āreal optionsā,

those written on some real-world physical process such as the temperature or

the amount of rainfall.

In general, an option gives the holder a right, not an obligation, to sell or

buy a prescribed asset (the underlying asset) at a price determined by the

contract (the exercise or strike price). For example if you own a call option on

shares of IBM with expiry date Oct. 20, 2000 and exercise price $120, then

on October 20, 2000 you have the right to purchase a ļ¬xed number , say 100

shares of IBM at the price $120. If IBM is selling for $130 on that date, then

your option is worth $10 per share on expiry. If IBM is selling for $120 or less,

then your option is worthless. We need to know what a fair value would be

for this option when it is sold, say on February 1, 2000. Determining this fair

value relies on sophisticated models both for the movements in the underlying

asset and the relationship of this asset with the derivative, and is the subject of

a large part of this book. You may have bought an IBM option for two possible

reasons, either because you are speculating on an increase in the stock price,

or to hedge a promise that you have made to deliver IBM stocks to someone

in the future against possible increases in the stock price. The second use of

derivatives is similar to the use of an insurance policy against movements in

an asset price that could damage or bankrupt the holder of a portfolio. It is

this second use of derivatives that has fueled most of the phenomenal growth

in their trading. With the globalization of economies, industries are subject to

3

more and more economic forces that they are unable to control but nevertheless

wish some form of insurance against. This requires hedges against a whole

litany of disadvantageous moves of the market such as increases in the cost of

borrowing, decreases in the value of assets held, changes in a foreign currency

exchange rates, etc.

The advanced theory of ļ¬nance, like many areas where advanced mathemat-

ics plays an important part, is undergoing a revolution aided and abetted by

the computer and the proliferation of powerful simulation and symbolic math-

ematical tools. This is the mathematical equivalent of the invention of the

printing press. The numerical and computational power once reserved for the

most highly trained mathematicians, scientists or engineers is now available to

any competent programmer.

One of the ļ¬rst hurdles faced before adopting stochastic or random models in

ļ¬nance is the recognition that for all practical purposes, the prices of equities in

an eļ¬cient market are random variables, that is while they may show some de-

pendence on ļ¬scal and economic processes and policies, they have a component

of randomness that makes them unpredictable. This appears on the surface to

be contrary to the training we all receive that every eļ¬ect has a cause, and every

change in the price of a stock must be driven by some factor in the company or

the economy. But we should remember that random models are often applied

to systems that are essentially causal when measuring and analyzing the vari-

ous factors inļ¬‚uencing the process and their eļ¬ects is too monumental a task.

Even in the simple toss of a fair coin, the result is predetermined by the forces

applied to the coin during and after it is tossed. In spite of this, we model it

as a random variable because we have insuļ¬cient information on these forces

to make a more accurate prediction of the outcome. Most ļ¬nancial processes

in an advanced economy are of a similar nature. Exchange rates, interest rates

and equity prices are subject to the pressures of a large number of traders,

government agencies, speculators, as well as the forces applied by international

4 CHAPTER 1. INTRODUCTION

trade and the ļ¬‚ow of information. In the aggregate there is an extraordinary

number of forces and information that inļ¬‚uence the process. While we might

hope to predict some features of the process such as the average change in price

or the volatility, a precise estimate of the price of an asset one year from to-

day is clearly impossible. This is the basic argument necessitating stochastic

models in ļ¬nance. Adoption of a stochastic model does neither implies that

the process is pure noise nor that we are unable to forecast. Such a model is

adopted whenever we acknowledge that a process is not perfectly predictable

and the non-predictable component of the process is of suļ¬cient importance to

warrant modeling.

Now if we accept that the price of a stock is a random variable, what are

the constants in our model? Is a dollar of constant value, and if so, the dollar

of which nation? Or should we accept one unit of a index what in some sense

represents a share of the global economy as the constant? This question concerns

our choice of what is called the ānumeraireā in deference to the French inļ¬‚uence

on the theory of probability, or the process against which the value of our assets

will be measured. We will see that there is not a unique answer to this question,

nor does that matter for most purposes. We can use a bond denominated in

Canadian dollars as the numeraire or one in US dollars. Provided we account

for the variability in the exchange rate, the price of an asset will be the same.

So to some extent our choice of numeraire is arbitrary- we may pick whatever

is most convenient for the problem at hand.

One of the most important modern tools for analyzing a stochastic system

is simulation. Simulation is the imitation of a real-world process or system. It

is essentially a model, often a mathematical model of a process. In ļ¬nance,

a basic model for the evolution of stock prices, interest rates, exchange rates

etc. would be necessary to determine a fair price of a derivative security.

Simulations, like purely mathematical models, usually make assumptions about

the behaviour of the system being modelled. This model requires inputs, often

5

called the parameters of the model and outputs a result which might measure the

performance of a system, the price of a given ļ¬nancial instrument, or the weights

on a portfolio chosen to have some desirable property. We usually construct the

model in such a way that inputs are easily changed over a given set of values,

as this allows for a more complete picture of the possible outcomes.

Why use simulation? The simple answer is that is that it transfers work

to the computer. Models can be handled which have greater complexity, and

fewer assumptions, and a more faithful representation of the real-world than

those that can be handled tractable by pure mathematical analysis are possible.

By changing parameters we can examine interactions, and sensitivities of the

system to various factors. Experimenters may either use a simulation to provide

a numerical answer to a question, assign a price to a given asset, identify optimal

settings for controllable parameters, examine the eļ¬ect of exogenous variables

or identify which of several schemes is more eļ¬cient or more proļ¬table. The

variables that have the greatest eļ¬ect on a system can be isolated. We can

also use simulation to verify the results obtained from an analytic solution.

For example many of the tractable models used in ļ¬nance to select portfolios

and price derivatives are wrong. They put too little weight on the extreme

observations, the large positive and negative movements (crashes), which have

the most dramatic eļ¬ect on the results. Is this lack of ļ¬t of major concern when

we use a standard model such as the Black-Scholes model to price a derivative?

Questions such as this one can be answered in part by examining simulations

which accord more closely with the real world, but which are intractable to

mathematical analysis.

Simulation is also used to answer questions starting with āwhat ifā. For

example, What would be the result if interest rates rose 3 percentage points

over the next 12 months? In engineering, determining what would happen under

more extreme circumstances is often referred to as stress testing and simulation

is a particularly valuable tool here since the scenarios we are concerned about are

6 CHAPTER 1. INTRODUCTION

those that we observe too rarely to have a substantial experience of. Simulations

are used, for example, to determine the eļ¬ect of an aircraft of ļ¬‚ying under

extreme conditions and is used to analyse the ļ¬‚ight data information in the

event of an accident. Simulation often provides experience at a lower cost than

the alternatives.

But these advantages are not without some sacriļ¬ce. Two individuals may

choose to model the same phenomenon in diļ¬erent ways, and as a result, may

have quite diļ¬erent simulation results. Because the output from a simulation

is random, it is sometimes harder to analyze- some statistical experience and

tools are a valuable asset. Building models and writing simulation code is not

always easy. Time is required both to construct the simulation, validate it, and

to analyze the results. And simulation does not render mathematical analysis

unnecessary. If a reasonably simple analytic expression for a solution exists,

it is always preferable to a simulation. While a simulation may provide an

approximate numerical answer at one or more possible parameter values, only

an expression for the solution provides insight to the way in which it responds

to the individual parameters, the sensitivities of the solution.

In constructing a simulation, you should be conscious of a number of distinct

steps;

1. Formulate the problem at hand. Why do we need to use simulation?

2. Set the objectives as speciļ¬cally as possible. This should include what

measures on the process are of most interest.

3. Suggest candidate models. Which of these are closest to the real-world?

Which are fairly easy to write computer code for? What parameter values

are of interest?

4. If possible, collect real data and identify which of the above models is

most appropriate. Which does the best job of generating the general

7

characteristics of the real data?

5. Implement the model. Write computer code to run simulations.

6. Verify (debug) the model. Using simple special cases, insure that the code

is doing what you think it is doing.

7. Validate the model. Ensure that it generates data with the characteristics

of the real data.

8. Determine simulation design parameters. How many simulations are to

be run and what alternatives are to be simulated?

9. Run the simulation. Collect and analyse the output.

10. Are there surprises? Do we need to change the model or the parameters?

Do we need more runs?

11. Finally we document the results and conclusions in the light of the simula-

tion results. Tables of numbers are to be avoided. Well-chosen graphs are

often better ways of gleaning qualitative information from a simulation.

In this book, we will not always follow our own advice, leaving some of

the above steps for the reader to ļ¬ll in. Nevertheless, the importance of model

validation, for example, cannot be overstated. Particularly in ļ¬nance where data

is often plentiful, highly complex mathematical models are too often applied

without any evidence that they ļ¬t the observed data adequately. The reader is

advised to consult and address the points in each of the steps above with each

new simulation (and many of the examples in this text).

Example

Let us consider the following example illustrating a simple use for a simu-

lation model. We are considering a buy-out bid for the shares of a company.

Although the companyā™s stock is presently valued at around $11.50 per share,

a careful analysis has determined that it ļ¬ts suļ¬ciently well with our current

8 CHAPTER 1. INTRODUCTION

assets that if the buy-out were successful, it would be worth approximately

$14.00 per share in our hands. We are considering only three alternatives,

an immediate cash oļ¬er of $12.00, $13.00 or $14.00 per share for outstanding

shares of the company. Naturally we would like to bid as little as possible, but

we expect a competitor to virtually simultaneously make a bid for the company

and the competitor values the shares diļ¬erently. The competitor has three bid-

ding strategies that we will simply identify as I, II, and III. There are costs

associated with any pair of strategies (our bid-competitorā™s bidding strategy)

including costs associated with losing a given bid to the competitor or paying

too much for the company. In other words, the payoļ¬ to our ļ¬rm depends on

the amount bid by the competitor and the possible scenarios are as given in

the following table.

Competitorā™s Strategy

Bid I II III

Your 12 3 2 -2

Bid 13 1 -4 4

14 0 -5 5

The payoļ¬s to the competitor are somewhat diļ¬erent and given below

Competitorā™s Strategy

I II III

Your 12 -1 -2 3

Bid 13 0 4 -6

14 0 5 -5

For example, the combination of your bid=$13 per share and your com-

petitorā™s strategy II results in a loss of 4 units (for example four dollars per

share) to you and a gain of 4 units to your competitor. However it is not always

the case that the your loss is the same as your competitorā™s gain. A game with

this property is called a zero-sum game and these are much easier to analyze

analytically. Deļ¬ne the 3 Ć— 3 matrix of payoļ¬s to your company by A and the

9

payoļ¬ matrix to your competitor by B,

ā ā ā ā

32 -2 -1 -2 3

ā ā ā ā

ā ā ā ā

A = ā 1 -4 4 ā , B = ā 0 -6 ā .

4

ā ā ā ā

0 -5 5 0 5 -5

Provided that you play strategy i = 1, 2, 3 (i.e. bid $12,$13,$14 with proba-

bilities p1 , p2 , p3 respectively and the probabilities of the competitorā™s strategies

are q1 , q2 , q3 . Then if we denote

ā ā ā ā

p q

ā1 ā1

ā ā

ā ā ā ā

p = ā p2 ā , and q = ā q2 ā,

ā ā ā ā

p3 q3

P3 P3

we can write the expected payoļ¬ to you in the form pi Aij qj . When

i=1 j=1

written as a vector-matrix product, this takes the form pT Aq. This might be

thought of as the average return to your ļ¬rm in the long run if this game were

repeated many times, although in the real world, the game is played only once.

If the vector q were known to you, you would clearly choose pi = 1 for the

row i corresponding to the maximum component of Aq since this maximizes

your payoļ¬. Similarly if your competitor knew p, they would choose qj = 1

for the column j corresponding to the maximum component of pT B. Over

the long haul, if this game were indeed repeated may times, you would likely

keep track of your opponentā™s frequencies and replace the unknown probabilities

by the frequencies. However, we assume that both the actual move made by

your opponent and the probabilities that they use in selecting their move are

unknown to you at the time you commit to your strategy. However, if the game

is repeated many times, each player obtains information about their opponentā™s

taste in moves, and this would seem to be a reasonable approach to building a

simulation model for this game. Suppose the game is played repeatedly, with

each of the two players updating their estimated probabilities using information

gathered about their opponentā™s historical use of their available strategies. We

10 CHAPTER 1. INTRODUCTION

may record number of times each strategy is used by each player and hope that

the relative frequencies approach a sensible limit. This is carried out by the

following Matlab function;

function [p,q]=nonzerosum(A,B,nsim)

% A and B are payoff matrices to the two participants in a game.

Outputs

%mixed strategies p and q determined by simulation conducted nsim

times

n=size(A); % A and B have the same size

p=ones(1,n(1)); q=ones(n(2),1); % initialize with positive weights

on all strategies

for i=1:nsim % runs the simulation nsim times

[m,s]=max(A*q); % s=index of optimal strategy for

us

[m,t]=max(p*B); % =index of optimal strategy for

competitor

p(s)=p(s)+1; % augment counts for us

q(t)=q(t)+1; % augment counts for competitor

end

p=p-ones(1,n(1)); p=p/sum(p); %remove initial weights from counts

and then

q=q-ones(n(2),1); q=q/sum(q); % convert counts to relative frequencies

The following output results from running this function for 50,000 simula-

tions.

[p,q]=nonzerosum(A,B,50000)

This results in approximately p0 = [ 2 0 1 ] 1 1

and q 0 = [0 with an

2]

3 3 2

average payoļ¬ to us of 0 and to the competitor 1/3. This seems to indicate that

the strategies should be āmixedā or random. You should choose a bid of $12.00

with probability around 2/3, and $14.00 with probability 1/3. It appears that

11

the competitor need only toss a fair coin and select between B and C based on

its outcome. Why randomize your choice? The average value of the game to

you is 0 if you use the probabilities above (in fact if your competitor chooses

1 1

probabilities q 0 = [0 it doesnā™t matter what your frequencies are, your

2]

2

average is 0). If you were to believe a single ļ¬xed strategy is always your ābestā

then your competitor could presumably determine what your ābestā strategy is

and act to reduce your return (i.e. substantially less than 0) while increasing

theirs. Only randomization provides the necessary insurance that neither player

can guess the strategy to be employed by the other. This is a rather simple

example of a two-person game with non-constant sum (in the sense that A+B

is not a constant matrix). Mathematical analysis of such games can be quite

complex. In such case, provided we can ensure cooperation, participants may

cooperate for a greater total return.

There is no assurance that the solution above is optimal. In fact the above

solution is worth an average of 0 per game to us and 1/3 to our competitor.

If we revise our strategy to p0 = [ 2 21

for example, our average return is

9 9 ],

3

still 0 but we have succeeded in reducing that of our opponent to 1/9. The

solution we arrived at in this case seems to be sensible solution, achieved with

little eļ¬ort. Evidently, in a game such as this, there is no clear deļ¬nition of

what an optimal strategy would be, since one might plan oneā™s play based on

the worst case, or the best case scenario, or something in between such as an

average? Do you attempt to collaborate with your competitor for greater total

return and then subsequently divide this in some fashion? This simulation has

emulated a simple form of competitor behaviour and arrived at a reasonable

solution, the best we can hope for without further assumptions.

There remains the question of how we actually select a bid with probabilities

2/3, 0 and 1/3 respectively. First let us assume that we are able to choose a

ārandom numberā U in the interval [0,1] so that the probability that it falls

in any given subinterval is proportional to the length of that subinterval. This

12 CHAPTER 1. INTRODUCTION

means that the random number has a uniform distribution on the interval [0,1].

Then we could determine our bid based on the value of this random number

from the following table;

2/3 Ā· U < 1

If U < 2/3

Bid 12 13 14

The way in which U is generated on a computer will be discussed in more

detail in chapter 2, but for the present note that each of the three alternative

bids have the correct probabilities.

Chapter 2

Some Basic Theory of

Finance

Introduction to Pricing: Single Period Models

Let us begin with a very simple example designed to illustrate the no-arbitrage

approach to pricing derivatives. Consider a stock whose price at present is $s.

Over a given period, the stock may move either up or down, up to a value su

where u > 1 with probability p or down to the value sd where d < 1 with

probability 1 ā’ p. In this model, these are the only moves possible for the stock

in a single period. Over a longer period, of course, many other values are

possible. In this market, we also assume that there is a so-called risk-free bond

available returning a guaranteed rate of r% per period. Such a bond cannot

default; there is no random mechanism governing its return which is known

upon purchase. An investment of $1 at the beginning of the period returns a

guaranteed $(1 + r) at the end. Then a portfolio purchased at the beginning

of a period consisting of y stocks and x bonds will return at the end of the

period an amount $x(1 + r) + ysZ where Z is a random variable taking

13

14 CHAPTER 2. SOME BASIC THEORY OF FINANCE

values u or d with probabilities p and 1 ā’ p respectively. We permit owning

a negative amount of a stock or bond, corresponding to shorting or borrowing

the correspond asset for immediate sale.

An ambitious investor might seek a portfolio whose initial cost is zero (i.e.

x + ys = 0) such that the return is greater than or equal to zero with positive

probability. Such a strategy is called an arbitrage. This means that the investor

is able to achieve a positive probability of future proļ¬ts with no down-side risk

with a net investment of $0. In mathematical terms, the investor seeks a point

(x, y) such that x + ys = 0 (net cost of the portfolio is zero) and

x(1 + r) + ysu ā„ 0,

x(1 + r) + ysd ā„ 0

with at least one of the two inequalities strict (so there is never a loss and a

non-zero chance of a positive return). Alternatively, is there a point on the line

y = ā’ 1 x which lies above both of the two lines

s

1+r

y=ā’ x

su

1+r

y=ā’ x

sd

and strictly above one of them? Since all three lines pass through the origin,

we need only compare the slopes; an arbitrage will NOT be possible if

1+r 1 1+r

ā’ Ā·ā’ Ā·ā’ (2.1)

sd s su

and otherwise there is a point (x, y) permitting an arbitrage. The condition for

no arbitrage (2.1) reduces to

d u

(2.2)

<1<

1+r 1+r

So the condition for no arbitrage demands that (1 + r ā’ u) and (1 + r ā’ d)

have opposite sign or d Ā· (1 + r) Ā· u. Unless this occurs, the stock always

has either better or worse returns than the bond, which makes no sense in a

INTRODUCTION TO PRICING: SINGLE PERIOD MODELS 15

free market where both are traded without compulsion. Under a no arbitrage

assumption since d Ā· (1 + r) Ā· u, the bond payoļ¬ is a convex combination or

a weighted average of the two possible stock payoļ¬s; i.e. there are probabilities

0 Ā· q Ā· 1 and (1 ā’ q) such that (1 + r) = qu + (1 ā’ q)d. In fact it is easy to

solve this equation to determine the values of q and 1 ā’ q.

(1 + r) ā’ d u ā’ (1 + r)

and 1 ā’ q =

q= , .

uā’d uā’d

Denote by Q the probability distribution which puts probabilities q and 1 ā’ q

on these points su, sd. Then if S1 is the value of the stock at the end of the

period, note that

1 1 1

(qsu + (1 ā’ q)sd) =

EQ (S1 ) = s(1 + r) = s

1+r 1+r 1+r

where EQ denotes the expectation assuming that Q describes the probabilities

of the two outcomes.

In other words, if there is to be no arbitrage, there exists a probability mea-

sure Q such that the expected price of future value of the stock S1 discounted

to the present using the return from a risk-free bond is exactly the present value

of the stock. The measure Q is called the risk-neutral measure and the prob-

abilities that it assigns to the possible outcomes of S are not necessarily those

that determine the future behaviour of the stock. The risk neutral measure

embodies both the current consensus beliefs in the future value of the stock and

the consensus investorsā™ attitude to risk avoidance. It is not usually true that

1

= s with P denoting the actual probability distribution describing

1+r EP (S1 )

the future probabilities of the stock. Indeed it is highly unlikely that an investor

would wish to purchase a risky stock if he or she could achieve exactly the same

expected return with no risk at all using a bond. We generally expect that

to make a risky investment attractive, its expected return should be greater

than that of a risk-free investment. Notice in this example that the risk-neutral

measure Q did not use the probabilities p, and 1 ā’ p that the stock would go

16 CHAPTER 2. SOME BASIC THEORY OF FINANCE

up or down and this seems contrary to intuition. Surely if a stock is more likely

to go up, then a call option on the stock should be valued higher!

Let us suppose for example that we have a friend willing, in a private trans-

action with me, to buy or sell a stock at a price determined from his subjectively

assigned distribution P , diļ¬erent from Q. The friend believes that the stock

is presently worth

psu + (1 ā’ p)sd

1

6= s since p 6= q.

EP S1 =

1+r 1+r

Such a friend oļ¬ers their assets as a sacriļ¬ce to the gods of arbitrage. If the

friendā™s assessed price is greater than the current market price, we can buy on

the open market and sell to the friend. Otherwise, one can do the reverse.

Either way one is enriched monetarily (and perhaps impoverished socially)!

So why should we use the Q measure to determine the price of a given asset

in a market (assuming, of course, there is a risk-neutral Q measure and we are

able to determine it)? Not because it precisely describes the future behaviour

of the stock, but because if we use any other distribution, we oļ¬er an intelligent

investor (there are many!) an arbitrage opportunity, or an opportunity to make

money at no risk and at our expense.

Derivatives are investments which derive their value from that of a corre-

sponding asset, such as a stock. A European call option is an option which

permits you (but does not compel you) to purchase the stock at a ļ¬xed future

date ( the maturity date) or for a given predetermined price, the exercise price

of the option). For example a call option with exercise price $10 on a stock

whose future value is denoted S1 , is worth on expiry S1 ā’ 10 if S1 > 10 but

nothing at all if S1 < 10. The diļ¬erence S1 ā’ 10 between the value of the stock

on expiry and the exercise price of the option is your proļ¬t if you exercises the

option, purchasing the stock for $10 and sell it on the open market at $S1 .

However, if S1 < 10, there is no point in exercising your option as you are

not compelled to do so and your return is $0. In general, your payoļ¬ from pur-

INTRODUCTION TO PRICING: SINGLE PERIOD MODELS 17

chasing the option is a simple function of the future price of the stock, such as

V (S1 ) = max(S1 ā’ 10, 0). We denote this by (S1 ā’ 10)+ . The future value of

the option is a random variable but it derives its value from that of the stock,

hence it is called a derivative and the stock is the underlying.

A function of the stock price V (S1 ) which may represent the return from a

portfolio of stocks and derivatives is called a contingent claim. V (S1 ) repre-

sents the payoļ¬ to an investor from a certain ļ¬nancial instrument or derivative

when the stock price at the end of the period is S1 . In our simple binomial

example above, the random variable takes only two possible values V (su) and

V (sd). We will show that there is a portfolio, called a replicating portfolio, con-

sisting of an investment solely in the above stock and bond which reproduces

these values V (su) and V (sd) exactly. We can determine the corresponding

weights on the bond and stocks (x, y) simply by solving the two equations in

two unknowns

x(1 + r) + ysu = V (su)

x(1 + r) + ysd = V (sd)

V (su)ā’yā— su

V (su)ā’V (sd)

Solving: y ā— = and xā— = By buying y ā— units of

.

suā’sd 1+r

stock and xā— units of bond, we are able to replicate the contingent claim V (S1 )

exactly- i.e. produce a portfolio of stocks and bonds with exactly the same

return as the contingent claim. So in this case at least, there can be only one

possible present value for the contingent claim and that is the present value

of the replicating portfolio xā— + y ā— s. If the market placed any other value

on the contingent claim, then a trader could guarantee a positive return by a

simple trade, shorting the contingent claim and buying the equivalent portfolio

or buying the contingent claim and shorting the replicating portfolio. Thus this

is the only price that precludes an arbitrage opportunity. There is a simpler

18 CHAPTER 2. SOME BASIC THEORY OF FINANCE

expression for the current price of the contingent claim in this case: Note that

1 1

(qV (su) + (1 ā’ q)V (sd))

EQ V (S1 ) =

1+r 1+r

1 1+rā’d u ā’ (1 + r)

= ( V (su) + V (sd))

1+r uā’d uā’d

= xā— + y ā— s.

In words, the discounted expected value of the contingent claim is equal to

the no-arbitrage price of the derivative where the expectation is taken using the

Q-measure. Indeed any contingent claim that is attainable must have its price

determined in this way. While we have developed this only in an extremely

simple case, it extends much more generally.

Suppose we have a total of N risky assets whose prices at times t = 0, 1,

j j

are given by (S0 , S1 ), j = 1, 2, ..., N. We denote by S0 , S1 the column vector of

initial and ļ¬nal prices

ā ā ā ā

1 1

S0 S1

ā ā ā ā

ā2 ā ā2 ā

ā S0 ā ā S1 ā

ā ā ā ā

ā ā ā ā

ā. ā ā. ā

ā ā ā ā

S0 = ā ā , S1 = ā ā

ā. ā ā. ā

ā ā ā ā

ā ā ā ā

ā ā ā ā

ā. ā ā. ā

ā ā ā ā

N N

S0 S1

where at time 0, S0 is known and S1 is random. Assume also there is a riskless

asset (a bond) paying interest rate r over one unit of time. Suppose we borrow

money (this is the same as shorting bonds) at the risk-free rate to buy wj units

P j

of stock j at time 0 for a total cost of wj S0 . The value of this portfolio at

P j j

wj (S1 ā’ (1 + r)S0 ). If there are weights wj so that

time t = 1 is T (w) =

this sum is always non-negative, and P (T (w) > 0) > 0, then this is an arbitrage

opportunity. Similarly, by replacing the weights wj by their negative ā’wj ,

there is an arbitrage opportunity if for some weights the sum is non-positive

and negative with positive probability. In summary, there are no arbitrage op-

INTRODUCTION TO PRICING: SINGLE PERIOD MODELS 19

portunities if for all weights wj P (T (w) > 0) > 0 and P (T (w) < 0) > 0 so

T (w) takes both positive and negative values. We assume that the moment

P j j

generating function M (w) = E[exp( wj (S1 ā’ (1 + r)S0 ))] exists and is an an-

alytic function of w.Roughly the condition that the moment generating function

is analytic assures that we can expand the function in a series expansion in w.

This is the case, for example, if the values of S1 , S0 are bounded. The following

theorem provides a general proof, due to Chris Rogers, of the equivalence of the

no-arbitrage condition and the existence of an equivalent measure Q. Refer to

the appendix for the technical deļ¬nitions of an equivalent probability measure

and the existence and properties of a moment generating function M (w).

Theorem 2 A necessary and suļ¬cient condition that there be no arbitrage op-

j

portunities is that there exists a measure Q equivalent to P such that EQ (S1 ) =

j

1

for all j = 1, ..., N.

1+r S0

P j j

wj (S1 ā’ (1 + r)S0 ))] and

Proof. Deļ¬ne M (w) = E exp(T (w)) = E[exp(

consider the problem

min ln(M (w)).

w

The no-arbitrage condition implies that for each j there exists Īµ > 0,

j j

P [S1 ā’ (1 + r)S0 > Īµ] > 0

and therefore as wj ā’ ā while the other weights wk , k 6= j remain ļ¬xed,

X j j j j

wj (S1 ā’(1+r)S0 ))] > C exp(wj Īµ)P [S1 ā’(1+r)S0 > Īµ] ā’ ā as wj ā’ ā.

M (w) = E[exp(

Similarly, M (w) ā’ ā as wj ā’ ā’ā. From the properties of a moment gen-

erating function (see the appendix) M (w) is convex, continuous, analytic and

ā‚M

M (0) = 1. Therefore the function M (w) has a minimum wā— satisfying =0

ā‚wj

or

ā‚M (w)

= 0 or (2.3)

ā‚wj

j j

E[S1 exp(T (w))] = (1 + r)S0 E[exp(T (w))]

20 CHAPTER 2. SOME BASIC THEORY OF FINANCE

or

j

E[exp(T (w))S1 ]

j

S0 = .

(1 + r)E[exp(T (w))]

Deļ¬ne a distribution or probability measure Q as follows; for any event A,

EP [IA exp(w0 S1 )]

Q(A) = .

EP [exp(w0 S1 )]

The Radon-Nikodym derivative (see the appendix) is

exp(w0 S1 )]

dQ

= .

EP [exp(w0 S1 )]

dP

dQ

Since ā > > 0, the measure Q is equivalent to the original probability mea-

dP

sure P (in the intuitive sense that it has the same support). When we calculate

expected values under this new measure, note that for each j,

dQ j

j

EQ (S1 ) = EP [ S]

dP 1

j

EP [S1 exp(w0 S1 )]

=

EP [exp(w0 S1 )]

j

= (1 + r)S0 .

or

1

j j

S0 = EQ (S1 ).

1+r

Therefore, the current price of each stock is the discounted expected value of the

future price under this ārisk-neutralā measure Q.

Conversely if

1

j j

S0 , for all j (2.4)

EQ (S1 ) =

1+r

holds for some measure Q then EQ [T (w)] = 0 for all w and this implies that the

random variable T (w) is either identically 0 or admits both positive and negative

values. Therefore the existence of the measure Q satisfying (2.4) implies that

there are no arbitrage opportunities.

The so-called risk-neutral measure Q is constructed to minimize the cross-

entropy between Q and P subject to the constraints E(S1 ā’ (1 + r)S0 ) = 0

MULTIPERIOD MODELS. 21

where cross-entropy is deļ¬ned in Section 1.5. If there N possible values of the

random variables S1 and S0 then (2.3) consists of N equations in N unknowns

and so it is reasonable to expect a unique solution. In this case, the Q measure

is unique and we call the market complete.

The theory of pricing derivatives in a complete market is rooted in a rather

trivial observation because in a complete market, the derivative can be replicated

with a portfolio of other marketable securities. If we can reproduce exactly the

same (random) returns as the derivative provides using a linear combination of

other marketable securities (which have prices assigned by the market) then the

derivative must have the same price as the linear combination of other securities.

Any other price would provide arbitrage opportunities.

Of course in the real world, there are costs associated with trading, these

costs usually related to a bid-ask spread. There is essentially a diļ¬erent price for

buying a security and for selling it. The argument above assumes a frictionless

market with no trading costs, with borrowing any amount at the risk-free bond

rate possible, and a completely liquid market- any amount of any security can be

bought or sold. Moreover it is usually assumed that the market is complete and

it is questionable whether complete markets exist. For example if a derivative

security can be perfectly replicated using other marketable instruments, then

what is the purpose of the derivative security in the market? All models,

excepting those on Fashion File, have deļ¬ciencies and critics. The merit of the

frictionless trading assumption is that it provides an accurate approximation

to increasingly liquid real-world markets. Like all useful models, this permits

tentative conclusions that should be subject to constant study and improvement.

Multiperiod Models.

When an asset price evolves over time, the investor normally makes decisions

about the investment at various periods during its life. Such decisions are made

22 CHAPTER 2. SOME BASIC THEORY OF FINANCE

with the beneļ¬t of current information, and this information, whether used

or not, includes the price of the asset and any related assets at all previous

time periods, beginning at some time t = 0 when we began observation of the

process. We denote this information available for use at time t as Ht . Formally,

Ht is what is called a sigma-ļ¬eld (see the appendix) generated by the past, and

there are two fundamental properties of this sigma-ļ¬eld that will use. The ļ¬rst

is that the sigma-ļ¬elds increase over time. In other words, our information

about this and related processes increases over time because we have observed

more of the relevant history. In the mathematical model, we do not āforgetā

relevant information: this model ļ¬ts better the behaviour of youthful traders

than aging professors. The second property of Ht is that it includes the value

of the asset price SĻ„ , Ļ„ Ā· t at all times Ļ„ Ā· t. In measure-theoretic language, St

is adapted to or measurable with respect to Ht . Now the analysis above shows

that when our investment life began at time t = 0 and we were planning for the

next period of time, absence of arbitrage implies a risk-neutral measure Q such

1

that EQ ( 1+r S1 ) = S0 . Imagine now that we are in a similar position at time

t, planning our investment for the next unit time. All expected values should

be taken in the light of our current knowledge, i.e. given the information Ht .

An identical analysis to that above shows that under the risk neutral measure

Q, if St represents the price of the stock after t periods, and rt the risk-free

one-period interest rate oļ¬ered that time, then

1

St+1 |Ht ) = St . (2.5)

EQ (

1 + rt

Suppose we let Bt be the value of $1 invested at time t = 0 after a total

of t periods. Then B1 = (1 + r0 ), B2 = (1 + r0 )(1 + r1 ), and in general

Bt = (1 + r0 )(1 + r1 )...(1 + rtā’1 ). Since the interest rate per period is announced

at the beginning of this period, the value Bt is known at time t ā’ 1. If you

owe exactly $1.00 payable at time t, then to cover this debt you should have an

MULTIPERIOD MODELS. 23

investment at time t = 0 of $E(1/Bt ), which we might call the present value

of the promise. In general, at time t, the present value of a certain amount

$VT promised at time T (i.e. the present value or the value discounted to the

present of this payment) is

Bt

|Ht ).

E(VT

BT

Now suppose we divide (2.5) above by Bt. We obtain

St+1 1 1 1 St

|Ht ) = EQ ( St+1 |Ht ) = St+1 |Ht ) =

EQ ( EQ ( .

Bt+1 Bt (1 + rt ) Bt 1 + rt Bt

(2.6)

Notice that we are able to take the divisor Bt outside the expectation since Bt

is known at time t (in the language of Appendix 1, Bt is measurable with re-

spect to Ht+1 ). This equation (2.6) describes an elegant mathematical property

shared by all marketable securities in a complete market. Under the risk-neutral

measure, the discounted price Yt = St /Bt forms a martingale. A martingale

is a process Yt for which the expectation of a future value given the present is

equal to the present i.e.

E(Yt+1 |Ht ) = Yt .for all t. (2.7)

Properties of a martingale are given in the appendix and it is easy to show that

for such a process, when T > t,

E(YT |Ht ) = E[...E[E(YT |HT ā’1 )|HT ā’2 ]...|Ht ] = Yt . (2.8)

A martingale is a fair game in a world with no inļ¬‚ation, no need to consume

and no mortality. Your future fortune if you play the game is a random vari-

able whose expectation, given everything you know at present, is your present

fortune.

Thus, under a risk-neutral measure Q in a complete market, all marketable

securities discounted to the present form martingales. For this reason, we often

refer to the risk-neutral measure as a martingale measure. The fact that prices of

24 CHAPTER 2. SOME BASIC THEORY OF FINANCE

marketable commodities must be martingales under the risk neutral measure has

many consequences for the canny investor. Suppose, for example, you believe

that you are able to model the history of the price process nearly perfectly, and

it tells you that the price of a share of XXX computer systems increases on

average 20% per year. Should you use this P ā’measure in valuing a derivative,

even if you are conļ¬dent it is absolutely correct, in pricing a call option on

XXX computer systems with maturity one year from now? If you do so, you are

oļ¬ering some arbitrager another free lunch at your expense. The measure Q,

not the measure P , determines derivative prices in a no-arbitrage market. This

also means that there is no advantage, when pricing derivatives, in using some

elaborate statistical method to estimate the expected rate of return because this

is a property of P not Q.

What have we discovered? In general, prices in a market are determined as

expected values, but expected values with respect to the measure Q. This is true

in any complete market, regardless of the number of assets traded in the market.

For any future time T > t, and for any derivative deļ¬ned on the traded assets

Bt

in a market whose value at time t is given by Vt , EQ ( BT VT |Ht ] = Vt = the

market price of the derivative at time t. So in theory, determining a reasonable

price of a derivative should be a simple task, one that could be easily handled

by simulation. Suppose we wish to determine a suitable price for a derivative

whose value is determined by some stock price process St . Suppose that at

time T > t, the value of the derivative is a simple function of the stock price at

that time VT = V (ST ). We may simply generate many simulations of the future

value of the stock and corresponding value of the derivative ST , V (ST ) given the

current store of information Ht . These simulations must be conducted under the

measure Q. In order to determine a fair price for the derivative, we then average

the discounted values of the derivatives, discounted to the present, over all the

simulations. The catch is that the Q measure is often neither obvious from

the present market prices nor statistically estimable from its past. It is given

MULTIPERIOD MODELS. 25

implicitly by the fact that the expected value of the discounted future value of

traded assets must produce the present market price. In other words, a ļ¬rst

step in valuing any asset is to determine a measure Q for which this holds. Now

in some simple models involving a single stock, this is fairly simple, and there

is a unique such measure Q. This is the case, for example, for the stock model

above in which the stock moves in simple steps, either increasing or decreasing

at each step. But as the number of traded assets increases, and as the number

of possible jumps per period changes, a measure Q which completely describes

the stock dynamics and which has the necessary properties for a risk neutral

measure becomes potentially much more complicated as the following example

shows.

Solving for the Q Measure.

Let us consider the following simple example. Over each period, a stock price

provides a return greater than, less than, or the same as that of a risk free

investment like a bond. Assume for simplicity that the stock changes by the

factor u(1 + r) (greater) or (1 + r) (the same) d(1 + r)(less) where u > 1 > d =

1/u. The Q probability of increases and decreases is unknown, and may vary

from one period to the next. Over two periods, the possible paths executed by

this stock price process are displayed below assuming that the stock begins at

time t = 0 with price S0 = 1.

[FIGURE 2.1 ABOUT HERE]

In general in such a tree there are three branches from each of the nodes

at times t = 0, 1 and there are a total of 1 + 3 = 4 such nodes. Thus, even

if we assume that probabilities of up and down movements do not depend on

how the process arrived at a given node, there is a total of 3 Ć— 4 = 12 unknown

parameters. Of course there are constraints; for example the sum of the three

probabilities on branches exiting a given node must add to one and the price

26 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Figure 2.1: A Trinomial Tree for Stock Prices

process must form a martingale. For each of the four nodes, this provides two

constraints for a total of 8 constraints, leaving 4 parameters to be estimated.

We would need the market price of 4 diļ¬erent derivatives or other contingent

claims to be able to generate 4 equations in these 4 unknowns and solve for

them. Provided we are able to obtain prices of four such derivatives, then we

can solve these equations. If we denote the risk-neutral probability of ā™upā™ at

each of the four nodes by p1 , p2 , p3 , p4 then the conditional distribution of St+1

given St = s is:

Stock value su(1 + r) s(1 + r) sd(1 + r)

uā’d uā’1

1ā’ = 1 ā’ kpi

Probability pi 1ā’d pi 1ā’d pi = cpi

Consider the following special case, with the risk-free interest rate per period

r, u = 1.089, S0 = $1.00. We also assume that we are given the price of four

call options expiring at time T = 2. The possible values of the price at time

T = 2 corresponding to two steps up, one step up and one constant, one up

one down, etc. are the values of S(T ) in the set

{1.1859, 1.0890, 1.0000, 0.9183, 0.8432}.

Now consider a ācall optionā on this stock expiring at time T = 2 with strike

MULTIPERIOD MODELS. 27

price K. Such an option has value at time t = 2 equal to (S2 ā’ K) if this is

For brevity we denote this by (S2 ā’ K)+ . The

positive, or zero otherwise.

present value of the option is EQ (S2 ā’ K)+ discounted to the present, where

K is the exercise price of the option and S2 is the price of the stock at time 2.

Thus the price of the call option at time 0 is given by

V0 = EQ (S2 ā’ K)+ /(1 + r)2

Assuming interest rate r = 1% per period, suppose we have market prices of four

call options with the same expiry and diļ¬erent exercise prices in the following

table;

K =Exercise Price V0 =Call Option Price

T =Maturity

0.867 2 0.154

0.969 2 .0675

1.071 2 .0155

1.173 2 .0016

If we can observe the prices of these options only, then the equations to be

solved for the probabilities associated with the measure Q equate the observed

price of the options to their theoretical price V0 = E(S2 ā’ K)+ /(1 + r)2 .

1

(1.186 ā’ 1.173)p1 p2

0.0016 =

(1.01)2

1

[(1.186 ā’ 1.071)p1 p2 + (1.089 ā’ 1.071){p1 (1 ā’ kp2 ) + (1 ā’ kp1 )p2 }]

0.0155 =

(1.01)2

1

[0.217p1 p2 + 0.12{p1 (1 ā’ kp2 ) + (1 ā’ kp1 )p2 }

0.0675 =

(1.01)2

+ 0.031{(1 ā’ kp1 )(1 ā’ kp2 ) + cp1 p2 + cp1 p4 )}

1

[0.319p1 p2 + 0.222{p1 (1 ā’ kp2 ) + (1 ā’ kp1 )p2 }

0.154 =

(1.01)2

+ 0.133{(1 ā’ kp1 )(1 ā’ kp2 ) + cp1 p2 + cp1 p4 )}

+ 0.051{{cp1 (1 ā’ kp4 ) + (1 ā’ kp1 )cp3 }].

28 CHAPTER 2. SOME BASIC THEORY OF FINANCE

While it is not too diļ¬cult to solve this system in this case one can see that

with more branches and more derivatives, this non-linear system of equations

becomes diļ¬cult very quickly. What do we do if we observe market prices for

only two derivatives deļ¬ned on this stock, and only two parameters can be

obtained from the market information? This is an example of what is called

an incomplete market, a market in which the risk neutral distribution is not

uniquely speciļ¬ed by market information. In general when we have fewer

equations than parameters in a model, there are really only two choices

(a) Simplify the model so that the number of unknown parameters and the

number of equations match.

(b) Determine additional natural criteria or constraints that the parameters

must satisfy.

In this case, for example, one might prefer a model in which the probability

of a step up or down depends on the time, but not on the current price of the

stock. This assumption would force equal all of p2 = p3 = p4 and simplify the

system of equations above. For example using only the prices of the ļ¬rst two

derivatives, we obtain equations, which, when solved, determine the probabilities

on the other branches as well.

1

(1.186 ā’ 1.173)p1 p2

0.0016 =

(1.01)2

1

[(1.186 ā’ 1.071)p1 p2 + (1.089 ā’ 1.071){p1 (1 ā’ kp2 ) + (1 ā’ kp1 )p2 }]

0.0155 =

(1.01)2

This example reļ¬‚ects a basic problem which occurs often when we build a

reasonable and ļ¬‚exible model in ļ¬nance. Frequently there are more parameters

than there are marketable securities from which we can estimate these parame-

ters. It is quite common to react by simplifying the model. For example, it

is for this reason that binomial trees (with only two branches emanating from

each node) are often preferred to the trinomial tree example we use above, even

though they provide a worse approximation to the actual distribution of stock

MULTIPERIOD MODELS. 29

returns.

In general if there are n diļ¬erent securities (excluding derivatives whose value

is a function of one or more of these) and if each security can take any one of m

diļ¬erent values, then there are a total of mn possible states of nature at time

t = 1. The Q measure must assign a probability to each of them. This results in

a total of mn unknown probability values, which, of course must add to one, and

result in the right expectation for each of n marketable securities. To uniquely

determine Q we would require a total of mn ā’ n ā’ 1 equations or mn ā’ n ā’ 1

diļ¬erent derivatives. For example for m = 10, n = 100, approximately one with

a hundred zeros, a prohibitive number, are required to uniquely determine Q.

In a complete market, Q is uniquely determined by marketed securities, but

in eļ¬ect no real market can be complete. In real markets, one asset is not

perfectly replicated by a combination of other assets because there is no value

in duplication. Whether an asset is a derivative whose value is determined by

another marketed security, together with interest rates and volatilities, markets

rarely permit exact replication. The most we can probably hope for in practice

is to ļ¬nd a model or measure Q in a subclass of measures with desirable features

under which

Bt

V (ST )|Ht ] ā Vt for all marketable V. (2.9)

EQ [

BT

Even if we had equalities in (2.9), this would represent typically fewer equa-

tions than the number of unknown Q probabilities so some simpliļ¬cation of the

model is required before settling on a measure Q. One could, at oneā™s peril,

ignore the fact that certain factors in the market depend on others. Similar

stocks behave similarly, and none may be actually independent. Can we, with

any reasonable level of conļ¬dence, accurately predict the eļ¬ect that a lowering

of interest rates will have on a given bank stock? Perhaps the best model

for the future behaviour of most processes is the past, except that as we have

seen the historical distribution of stocks do not generally produce a risk-neutral

30 CHAPTER 2. SOME BASIC THEORY OF FINANCE

measure. Even if historical information provided a ļ¬‚awless guide to the future,

there is too little of it to accurately estimate the large number of parameters

required for a simulation of a market of reasonable size. Some simpliļ¬cation of

the model is clearly necessary. Are some baskets of stocks independent of other

combinations? What independence can we reasonably assume over time?

As a ļ¬rst step in simplifying a model, consider some of the common measures

of behaviour. Stocks can go up, or down. The drift of a stock is a tendency in

one or other of these two directions. But it can also go up and down- by a lot

or a little. The measure of this, the variance or variability in the stock returns

is called the volatility of the stock. Our model should have as ingredients these

two quantities. It should also have as much dependence over time and among

diļ¬erent asset prices as we have evidence to support.

Determining the Process Bt .

We have seen in the last section that given the Q or risk-neutral measure, we can,

at least in theory, determine the price of a derivative if we are given the price Bt

of a risk-free investment at time t (in ļ¬nance such a yardstick for measuring and

discounting prices is often called a ānumeraireā). Unfortunately no completely

liquid risk-free investment is traded on the open market. There are government

treasury bills which, depending on the government, one might wish to assume

are almost risk-free, and there are government bonds, usually with longer terms,

which complicate matters by paying dividends periodically. The question dealt

with in this section is whether we can estimate or approximate an approximate

risk-free process Bt given information on the prices of these bonds. There are

typically too few genuinely risk-free bonds to get a detailed picture of the process

Bs , s > 0. We might use government bonds for this purpose, but are these

genuinely risk-free? Might not the additional use of bonds issued by other large

corporations provide a more detailed picture of the bank account process Bs ?

DETERMINING THE PROCESS BT . 31

Can we incorporate information on bond prices from lower grade debt? To

do so, we need a simple model linking the debt rating of a given bond and the

probability of default and payoļ¬ to the bond-holders in the event of default. To

begin with, let us assume that a given basket of companies, say those with a

common debt rating from one of the major bond rating organisations, have a

common distribution of default time. The thesis of this section is that even if

no totally risk-free investment existed, we might still be able to use bond prices

to estimate what interest rate such an investment would oļ¬er.

We begin with what we know. Presumably we know the current prices of

marketable securities. This may include prices of certain low-risk bonds with

face value F , the value of the bond on maturity at time T. Typically such a bond

pays certain payments of value dt at certain times t < T and then the face value

of the bond F at maturity time T, unless the bond-holder defaults. Let us assume

for simplicity that the current time is 0. The current bond prices P0 provide some

information on Bt as well as the possibility of default. Suppose we let Ļ„ denote

the random time at which default or bankruptcy would occur. Assume that the

eļ¬ect of possible default is to render the payments at various times random so

for example dt is paid provided that default has not yet occurred, i.e. if Ļ„ > t,

and similarly the payment on maturity is the face value of the Bond F if default

has not yet occurred and if it has, some fraction of the face value pF is paid.

When a real bond defaults, the payout to bondholders is a complicated function

of the hierarchy of the bond and may occur before maturity, but we choose this

model with payout at maturity in any case for simplicity. Then the current

price of the bond is the expected discounted value of all future payments, so

X 1 pF F

I(Ļ„ Ā· T ) +

P0 = EQ ( ds I(Ļ„ > s) + I(Ļ„ > T ))

Bs BT BT

{s;0<s<T }

X

ā’1

ā’1

ds EQ [Bs I(Ļ„ > s)] + F EQ [BT (p + (1 ā’ p)I(Ļ„ > T ))]

=

{s;0<s<T }

32 CHAPTER 2. SOME BASIC THEORY OF FINANCE

The bank account process Bt that we considered is the compounded value at

time of an investment of $1 deposited at time 0. This value might be random

but the interest rate is declared at the beginning of each period so, for example,

Bt is completely determined at time t ā’ 1. In measure-theoretical language, Bt

is Htā’1 measurable for each t. With Q is the risk-neutral distribution

X

ā’1

ā’1

P0 = EQ { ds Bs Q(Ļ„ > s|Hsā’1 ) + F BT (p + (1 ā’ p)Q(Ļ„ > T |HT ā’1 ))}.

{s;0<s<T }

This takes a form very similar to the price of a bond which does not default but

with a diļ¬erent bank account process. Suppose we deļ¬ne a new bank account

f

process Bs , equivalent in expectation to the risk-free account, but that only

pays if default does not occur in the interval. Such a process must satisfy

f

EQ (Bs I(Ļ„ > s)|Hsā’1 ) = Bs .

f

From this we see that the process Bs is deļ¬ned by

Bs

f on the set Q[Ļ„ > s|Hsā’1 ] > 0.

Bs =

Q[Ļ„ > s|Hsā’1 ]

In terms of this new bank account process, the price of the bond can be rewritten

as

X

g g

ā’1 ā’1 ā’1

P0 = EQ { ds Bs + (1 ā’ p)F BT + pF BT }.

{s;0<s<T }

If we subtract from the current bond price the present value of the guaranteed

payment of pF, the result is

X

g g

ā’1 ā’1 ā’1

P0 ā’ = EQ { ds Bs + (1 ā’ p)F BT }.

pF EQ (BT )

{s;0<s<T }

This equation has a simple interpretation. The left side is the price of the

bond reduced by the present value of the guaranteed payment on maturity F p.

The right hand side is the current value of a risk-free bond paying the same

f

dividends, with interest rates increased by replacing Bs by Bs and with face

value F (1 ā’ p) all discounted to the present using the bank account process

DETERMINING THE PROCESS BT . 33

f

Bs . In words, to value a defaultable bond, augment the interest rate using the

probability of default in intervals, change the face value to the potential loss of

face value on default and then add the present value of the guaranteed payment

on maturity.

Typically we might expect to be able to obtain prices of a variety of bonds

issued on one ļ¬rm, or ļ¬rms with similar credit ratings. If we are willing to

assume that such ļ¬rms share the same conditional distribution of default time

f

Q[Ļ„ > s|Hsā’1 ] then they must all share the same process Bs and so each

observed bond price P0 leads to an equation of the form

X gā’1

ds vs + (1 ā’ p)F BT + pF vT .

e

P0 =

{s;0<s<T }

gā’1 ā’1

in the unknowns vs = EQ (Bs ), ...s Ā· T. and vT = EQ (BT ). If we assume

e

that the coupon dates of the bonds match, then k bonds of a given maturity

e

T and credit rating will allow us to estimate the k unknown values of vs . Since

the term vT is included in all bonds, it can be estimated from all of the bond

prices, but most accurately from bonds with very low risk.

Unfortunately, this model still has too many unknown parameters to be

generally useful. We now consider a particular case that is considerably simpler.

While it seems unreasonable to assume that default of a bond or bankruptcy

of a ļ¬rm is unrelated to interest rates, one might suppose some simple model

which allows a form of dependence. For most ļ¬rms, one might expect that

the probability of survival another unit time is negatively associated with the

interest rate. For example we might suppose that the probability of default in

the next time interval conditional on surviving to the present is a function of

the current interest rate, for example

a + (b ā’ 1)rt

ht = Q(Ļ„ = t|Ļ„ ā„ t, rt ) = .

1 + a + brt

The quantity ht is a more natural measure of the risk at time t than are

other measures of the distribution of Ļ„ and the function ht is called the hazard

34 CHAPTER 2. SOME BASIC THEORY OF FINANCE

function. If the constant b > 1+a, then theāhazardā ht increases with increasing

interest rates, otherwise it decreases. In case the default is independent of the

interest rates, we may put b = 1 + a in which case the hazard is a/(1 + a). Then

on the set [Ļ„ ā„ s]

1 + rs e

f e

Bs = Bsā’1 = (1 + a + brs )Bsā’1

1 ā’ hs

which means that the bond is priced using a similar bank account process but

one for which the eļ¬ective interest rate is not rs but a + brs . The diļ¬erence

a + (b ā’ 1)rs between the eļ¬ective interest rate and rs is usually referred to as

the spread and this model justiļ¬es using a linear function to model this spread.

Now suppose that default is assumed independent of the past history of interest

rates under the risk-neutral measure Q. In this case, b = 1 + a and the spread

is a(1 + rs ) ' a ' a/(1 + a) provided both a and rs is small. So in this case

the spread gives an approximate risk-neutral probability of default in a given

time interval, conditional on survival to that time.

We might hope that the probabilities of default are very small and follow a

relatively simple pattern. If the pattern is not perfect, then little harm results

provided that indeed the default probabilities are small. Suppose for example

that the time of default follows a geometric distribution so that the hazard is

constant ht = h = a/(1 + a). Then

f

Bs = (1 + a)s Bs for s > 0.

f

Bs grows faster than Bs and it grows even faster as the probability of default h

increases. The eļ¬ective interest rate on this account is approximately a units

per period higher.

Given only three bond prices with the same default characteristics, for ex-

ample, and assuming constant interest rates so that Bs = (1 + r)s , we may solve

for the values of the three unknown parameters (r, a, p) equations of the form

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.35

X

P0 ā’ pF (1 + r)ā’T = (1 + a + r + ar)ā’s ds + (1 ā’ p)F (1 + a + r + ar)ā’T .

0<s<T

Market prices for a minimum of three diļ¬erent bonds would allow us to solve

for the unknowns (r, a, p) and these are obtainable from three diļ¬erent bonds.

Minimum Variance Portfolios and the Capital As-

set Pricing Model.

Let us begin by building a model for portfolios of securities that captures many

of the features of market movements. We assume that by using the methods of

the previous section and the prices of low-risk bonds, we are able to determine

the value Bt of a risk-free investment at time t in the future. Normally these

values might be used to discount future stock prices to the present. However

for much of this section we will consider only a single period and the analysis

will be essentially the same with our without this discounting.

Suppose we have a number n of potential investments or securities, each

risky in the sense that prices at future dates are random. Suppose we denote

the price of these securities at time t by Si (t), i = 1, 2, ..., n. There is a better

measure of the value of an investment than the price of a security or even the

change in the price of a security Si (t) ā’ Si (t ā’ 1) over a period because this does

not reļ¬‚ect the cost of our initial investment. A common measure on investments

that allows to obtain prices, but is more stable over time and between securities

is the return. For a security that has prices Si (t) and Si (t + 1) at times t and

t + 1, we deļ¬ne the return Ri (t + 1) on the security over this time interval by

Si (t + 1) ā’ Si (t)

Ri (t + 1) = .

Si (t)

For example a stock that moved in price from $10 per share to $11 per share

over a period of time corresponds to a return of 10%. Returns can be measured

36 CHAPTER 2. SOME BASIC THEORY OF FINANCE

in units that are easily understood (for example 5% or 10% per unit time) and

are independent of the amount invested. Obviously the $1 proļ¬t obtained on

the above stock could has easily been obtained by purchasing 10 shares of a

stock whose value per share changed from $1.00 to $1.10 in the same period

of time, and the return in both cases is 10%. Given a sequence of returns and

the initial value of a stock Si (0), it is easy to obtain the stock price at time t

from the initial price at time 0 and the sequence of returns.

Si (t) = Si (0)(1 + Ri (1))(1 + Ri (2))...(1 + Ri (t))

= Si (0)Ī t (1 + Ri (s)).

s=1

Returns are not added over time they are multiplied as above. A 10% return

followed by a 20% return is not a 30% return but a return equal to (1 + .1)(1 +

.2) ā’ 1 or 32%. When we buy a portfolio of stocks, the individual stock returns

combine in a simple fashion to give the return on the whole portfolio. For

example suppose that we wish to invest a total amount $I(t) at time t. The

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