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#### Mathematical Techniques in Finance

**Tools for Incomplete Markets**

**By Ales Cerny**

** Princeton University Press **

**Copyright © 2009**

**Princeton University Press**

All right reserved.

All right reserved.

**ISBN: 978-0-691-14121-3**

#### Chapter One

**The Simplest Model of Financial Markets**

The main goal of the first chapter is to introduce the one-period finite state model of financial markets with elementary financial concepts such as *basis assets, focus assets, portfolio, Arrow-Debreu securities, hedging* and *replication*. Alongside the financial topics we will encounter mathematical tools-linear algebra and matrices-essential for formulating and solving basic investment problems. The chapter explains vector and matrix notation and important concepts such as linear independence.

After reading the first two chapters you should understand the meaning of and be able to solve questions of the following type.

**Example 1.1 (replication of securities).** Suppose that there is a risky security (call it stock) with tomorrow's value *S* = 3, 2 or 1 depending on the state of the market tomorrow. The first state (first scenario) happens with probability 1/2, the second with probability 1/6 and the third with probability 1/3. There is also a risk-free security (bond) which pays 1 no matter what happens tomorrow. We are interested in replicating two call options written on the stock, one with strike 1.5 and the second with strike 1.

1. Find a portfolio of the stock, bond and the first call option that replicates the second call option (so-called gamma hedging).

2. If the initial stock price is 2 and the risk-free rate of return is 5%, what is the no-arbitrage price of the second option?

3. Find the portfolio of the bond and stock which is the best hedge to the first option in terms of the expected squared replication error (so-called delta hedging).

This chapter is important for two reasons. Firstly, the one-period model of financial markets is the main building block of a dynamic multi-period model which will be discussed later and which represents the main tool of any financial analyst. Secondly, matrices provide an effective way of describing the relationships among several variables, random or deterministic, and as such they are used with great advantage throughout the book.

**1.1 One-Period Finite State Model**

It is a statement of the obvious that the returns in financial markets are uncertain. The question is how to model this uncertainty. The simplest model assumes that there are only two dates, which we will call today and tomorrow, but which could equally well be called this week and next week, this year and next year, or now and in 10 min. The essential feature of our two-date, one-period model is that no investment decisions are taken between the two dates. One should be thinking of a world which is at a standstill apart from at 12 noon each day when all economic activity (work, consumption, trading, etc.) is carried out in a split second.

It is assumed that we do not know today what the market prices will be tomorrow, in other words the state of tomorrow's world is uncertain. However, we assume that there is only a finite number of scenarios that can take place, each of which is known today down to the smallest detail. One of these scenarios is drawn at random, using a controlled experiment whereby the probability of each scenario being drawn is known today. The result of the draw is made public at noon tomorrow and all events take place as prescribed by the chosen scenario (see Table 1.1 for illustration).

Let us stop for a moment and reflect how realistic the finite state model is. First of all, how many scenarios are necessary? In the above table we have four random variables: the value of the FTSE index, the level of UK base interest rate, UK weather and the result of the Chelsea-Wimbledon football game. Assuming that each of these variables has five different outcomes and that any combination of individual outcomes is possible we would require [5.sup.4] = 625 different scenarios. Given that in finance one usually works with two or three scenarios, 625 seems more than sufficient. And yet if you realize that this only allows five values for each random variable (only five different results of the football match!), then 625 scenarios do not appear overly exigent.

Next, do we know the probability of each of the 625 scenarios? Well, we might have a *subjective* opinion on how much these probabilities are but since the weather, football match or development in financial markets can hardly be thought of as controlled random experiments we do not know what the *objective* probabilities of those scenarios are. There is even a school of thought stating that objective probabilities do not exist; see the notes at the end of the chapter.

Hence the finite state model departs from reality in two ways: firstly, with a small number of scenarios (states of the world) it provides only a patchy coverage of the actual outcomes, and secondly we do not know the objective probabilities of each scenario, we only have our subjective opinion of how much they might be.

**1.2 Securities and Their Payoffs**

Security is a legal entitlement to receive (or an obligation to pay) an amount of money. A security is characterized by its known price today and its generally uncertain payoff tomorrow. What constitutes the payoff depends to some extent on the given security. For example, consider a model with just two scenarios and one security, a share in publicly traded company TRADEWELL Inc. Let us assume that the initial price of the share is 1, and tomorrow it can either increase to 1.2 or drop to 0.9. Assume further that the shareholders will receive a dividend of 0.1 per share tomorrow, no matter what happens to the share price.

The *security payoff* is the amount of money one receives after selling the security tomorrow plus any additional payment such as the dividend, coupon or rebate one is entitled to by virtue of holding the security. In our case the payoff of one TRADEWELL share is 1.3 or 1 depending on the state of the world tomorrow.

Security price plays a dual role. The stock price today is just that-a price. The stock price tomorrow is part of the stock's uncertain payoff.

*Throughout this chapter and for a large part of the next chapter we will ignore today's prices* and will only talk about the security payoffs. We will come back to pricing in Chapter 2, Section 2.5. Throughout this book we assume *frictionless trading*, meaning that one can buy or sell any amount of any security at the market price without transaction costs. This assumption is justified in liquid markets.

**Example 1.2.** Suppose *S* is the stock price at maturity. A *call option* with strike *K* is a derivative security paying

*S* - *K* if *S* > *K*, 0 if *S* [less than or equal to] *K*.

The payoffs of options in Example 1.1 are in Table 1.2.

**1.3 Securities as Vectors**

An *n*-tuple of real numbers is called an *n*-dimensional vector. For

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we write *x, y* [member of] [R.sup.*n*]. Each *n*-dimensional vector refers to a point in *n*-dimensional space. The above is a representation of such a point as a *column vector*, which is nothing other than an *n* x 1 matrix, since it has n rows and 1 column. Of course, the same point can be written as a row vector instead. Whether to use columns or rows is a matter of personal taste, but it is important to be consistent.

**Example 1.3.** Consider the four securities from the introductory example. Let us write the payoffs of each security in the three states (scenarios) as a three-dimensional column vector:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

These securities are depicted graphically in Figure 1.1.

In MATLAB one would write

a1 = [1;1;1]; a2 = [3;2;1]; a3 = [1.5;0.5;0]; a4 = [2;1;0];

**1.4 Operations on Securities**

We can multiply vectors by a scalar. For any [alpha] [member of] R we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This operation represents [alpha] units of security *x*.

**Example 1.4.** Two units of the third security will have the payoff

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we buy two units of the third security today, tomorrow we will collect 3 pounds (dollars, euros) in the first scenario, 1 in the second scenario and nothing in the third scenario. In MATLAB one would type

2*a3;

If we issued (wrote, sold) 1 unit of the fourth security, then our payoff tomorrow would be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In other words, we would have to pay the holder of this security 2 in the first scenario, 1 in the second scenario and nothing in the third scenario. In MATLAB one types

-a4;

Various amounts of securities [*a*.sub.3] and [*a*.sub.4] are represented graphically in Figure 1.2.

One can also add vectors together:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With this operation we can calculate portfolio payoffs. A *portfolio* is a combination of existing securities, which tells us how many units of each security have to be bought or sold to create the portfolio. Naturally, *portfolio payoff* is what the name suggests: the payoff of the combination of securities. The word 'portfolio' is sometimes used as an abbreviation of 'portfolio payoff', creating a degree of ambiguity in the terminology.

**Example 1.5.** A portfolio in which we hold two units of the first option and issue one unit of the second option will have the payoff

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Graphically, this situation is depicted in Figure 1.3. In MATLAB the portfolio payoff is

2*a3-a4;

**1.5 The Matrix as a Collection of Securities**

Often we need to work with a collection of securities (vectors). It is then convenient to stack the column vectors next to each other to form a matrix.

**Example 1.6.** The vectors [*a*.sub.1], [*a*.sub.2], [*a*.sub.3], [*a*.sub.4] from Example 1.3 form a 3 x 4 payoff matrix, which we denote *A*,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The market scenarios (states of the world) are in rows, securities are in columns. In MATLAB

A = [a1 a2 a3 a4];

**1.6 Transposition**

Sometimes we need a *row vector* rather than a column vector. This is achieved by *transposition* of a column vector:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that *x** (transpose of x) is a 1 x *n* matrix. Conversely, transposition of a row vector gives a column vector. Should we perform the transposition twice, we will end up with the original vector:

(x*)* = x.

**Example 1.7.**

[*a*.sup.*.sub.1] = [1 1 1], [*a*.sup.*.sub.2] = [3 2 1], [*a*.sup.*.sub.3] = [1.5 0.5 0], [*a*.sup.*.sub.4] = [2 1 0].

In MATLAB transposition is achieved by attaching a prime to the matrix name. For example, [*a*.sup.*.sub.1] would be written as

a1';

The vectors [*a*.sup.*.sub.1], [*a*.sup.*.sub.2], [*a*.sup.*.sub.3, [*a*.sup.*.sub.4]] stacked under each other form a 4 x 3 matrix *B*

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

in MATLAB

B = [a1'; a2'; a3'; a4'] (1.2)

Matrix *B* from equation (1.1) is in fact the *transpose of matrix A*

*B* = *A**,

thus instead of (1.2) in MATLAB one would simply write

B = A';

In general, we can have an *m* x *n* matrix *M* (denoted *M* [member of] [R.sup.*m* x *n*]), where *m* is the number of rows and *n* is the number of columns. The element in the *i*th row and *j*th column is denoted [*M*.sub.*ij*]. The entire *j*th column is denoted [*M*.sub.*j*] while the entire *i*th row is denoted [*M*.sub.*i*]]. According to our needs we can think of the matrix *M* as if it were composed of *m* row vectors or *n* column vectors:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The transpose of a matrix is obtained by changing the columns of the original matrix into the rows of the transposed matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, for example, [*M*.sup.*.sub.1] = [([*M*.sub.1]).sup.*] and [*M*.sup.*.sub.1] = [([*M*.sub.1]).sup.*], which in words says that the first row of the transposed matrix is the transpose of the first column of the original matrix.

**Example 1.8.** Suppose a 3 x 4 payoff matrix *A* is given. To extract the payoff of the third security in all states, in MATLAB one would simply write

A(:,3);

On the other hand, if one wanted to know the payoff of all four securities in the first market scenario, one would look at the row

A(1,:);

**1.7 Matrix Multiplication and Portfolios**

The basic building block of matrix multiplication is the multiplication of a row vector by a column vector. Let *A* [member of] [R.sup.1x*k*] and *B* [member of] [R.sup.*k*x1]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this simple case the matrix multiplication *AB* is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that *A* is a 1 x *k* matrix, *B* is *k* x 1 matrix and the result is a 1 x 1 matrix. One often thinks of a 1 x 1 matrix as a number.

**Example 1.9.** Suppose that we have a portfolio of the four securities from the introductory example which consists of [*x*.sub.1], [*x*.sub.2], [*x*.sub.3], [*x*.sub.4] units of the first, second, third and fourth security, respectively. In the third state the individual securities pay 1, 1, 0, 0 in turn. The payoff of the portfolio in the third state will be

[*x*.sub.1] x 1 + [*x*.sub.2] x 1 + [*x*.sub.3] x 0 + [*x*.sub.4] x 0.

If we take

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then the portfolio payoff can be written in matrix notation as [*A*.sub.3]*x*.

In general one can multiply a matrix *U* (*m* x *k*) with a matrix *V* (*k* x *n*), regarding the former as *m* row vectors in [R.sup.*k*] and the latter as *n* column vectors in [R.sup.*k*]. One multiplies each of the *m* row vectors in *U* with each of the *n* column vectors in *V* using the simple multiplication rule (1.3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

*(Continues...)*

Excerpted fromMathematical Techniques in FinancebyAles CernyCopyright © 2009 by Princeton University Press. Excerpted by permission.

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