Introduction

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

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

a butcher, a succesful ¬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 ¬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 “joy of simulation” is that you do not

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

to acquire a considerable experience valuing ¬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 ¬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 ¬nancial mar-

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

to ¬nancial derivatives such as options and futures. These are ¬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 ¬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 “real options”,

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 ¬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 ¬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 ¬rst hurdles faced before adopting stochastic or random models in

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

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

pendence on ¬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 e¬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 in¬‚uencing the process and their e¬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 insu¬cient information on these forces

to make a more accurate prediction of the outcome. Most ¬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 ¬‚ow of information. In the aggregate there is an extraordinary

number of forces and information that in¬‚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 ¬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 su¬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 “numeraire” in deference to the French in¬‚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 ¬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 ¬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 e¬ect of exogenous variables

or identify which of several schemes is more e¬cient or more pro¬table. The

variables that have the greatest e¬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 ¬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 e¬ect on the results. Is this lack of ¬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 “what if”. 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 e¬ect of an aircraft of ¬‚ying under

extreme conditions and is used to analyse the ¬‚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 sacri¬ce. Two individuals may

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

have quite di¬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 speci¬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 ¬ll in. Nevertheless, the importance of model

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

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

without any evidence that they ¬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 company™s stock is presently valued at around $11.50 per share,

a careful analysis has determined that it ¬ts su¬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 o¬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 di¬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-competitor™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 payo¬ to our ¬rm depends on

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

the following table.

Competitor™s Strategy

Bid I II III

Your 12 3 2 -2

Bid 13 1 -4 4

14 0 -5 5

The payo¬s to the competitor are somewhat di¬erent and given below

Competitor™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-

petitor™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 competitor™s gain. A game with

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

analytically. De¬ne the 3 — 3 matrix of payo¬s to your company by A and the

9

payo¬ matrix to your competitor by B,

⎛ ⎞ ⎛ ⎞

32 -2 -1 -2 3

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

A = ⎜ 1 -4 4 ⎟ , B = ⎜ 0 -6 ⎟ .

4

⎝ ⎠ ⎝ ⎠

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 competitor™s strategies

are q1 , q2 , q3 . Then if we denote

⎛ ⎞ ⎛ ⎞

p q

⎜1 ⎜1

⎟ ⎟

⎜ ⎟ ⎜ ⎟

p = ⎜ p2 ⎟ , and q = ⎜ q2 ⎟,

⎝ ⎠ ⎝ ⎠

p3 q3

P3 P3

we can write the expected payo¬ 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 ¬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 payo¬. 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 opponent™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 opponent™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 opponent™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 payo¬ to us of 0 and to the competitor 1/3. This seems to indicate that

the strategies should be “mixed” 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 doesn™t matter what your frequencies are, your

2]

2

average is 0). If you were to believe a single ¬xed strategy is always your “best”

then your competitor could presumably determine what your “best” 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 e¬ort. Evidently, in a game such as this, there is no clear de¬nition of

what an optimal strategy would be, since one might plan one™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

“random number” 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 · 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.