1. 1 A Basic One Period Model

One of the key concepts in modern financial theory is "arbitrage'', or rather the lack of it, in financial markets. The no-arbitrage (NA) condition, frequently referred to as the "law of one price'', or by the idiom "no free lunch'', is an essential tool in developing pricing methodologies. The consequences of this condition will be demonstrated by examining a simple model, (see [1]).

The real-world-conscious reader may be dismayed at the simplicity of the model, and may fear for its inability to address concepts of real-world significance, believing it so simple a model that it cannot possibly capture anything of use. However, this model is, in fact, extremely powerful. Though it seems, and indeed is, unrealistic, it provides the foundation for solid intuitive understanding, and can readily be augmented to capture the complexity of the illusive real-world case. Before a formal definition of the no-arbitrage condition is given, an example to illustrate the definition is investigated. The example will be followed throughout this Chapter.

Consider a "simple" market with three securities and only three possible outcomes in the next time period. We refer to these possible outcomes as states of nature. Security 1 has a current price of $60. Its price will decrease to $50 if state 1 occurs, stay at $60 if state 2 occurs, and will increase to $80 if state 3 occurs. Similarly, security 2 has a current price of $105 and its price will be $90, $110, and $130 in states 1, 2, and 3, respectively.

One can think about the three states of nature as collections of exogenous circumstances which will cause the prices of the securities to be as above. At the present time, it is not known which state will be realized in the next period. It is known only that one (and only one) of these states will occur. There is no assumption made about the probability of each state's occurrence, either subjective or objective, except that each state has a positive probability of occurrence. States, in this model, capture the uncertainty about the prices of the securities in the next time period.

The value of the third security in the next period does not depend on which of the three states of nature will occur. To put it differently, its value is not contingent on the state of nature in the next time period. It is a risk-free security, or a bond. In a later Chapter we will see that such a security is a 10% coupon bond. Its value will be $110 in the next time period regardless of which state of nature materializes, and its present value (current price) is $100. Hence, the interest rate in this simple model is 10%. Table 1.1 summarizes the prices and payoffs.

[Maple Math] [Maple Math] [Maple Math] [Maple Math] [Maple Math]
[Maple Math] [Maple Math] [Maple Math] [Maple Math] [Maple Math]
[Maple Math] [Maple Math] [Maple Math] [Maple Math] [Maple Math]
[Maple Math] [Maple Math] [Maple Math] [Maple Math] [Maple Math]

Table 1.1: A One-Period Model

Our simple market is assumed to be a "perfect market'' or frictionless market. It is a a stylized market in which there are no transaction costs, no margin requirements, no taxes, and no limit on short sales. It is a market in which borrowing and lending are done at the same rate of interest, the risk-free rate, which is 10%. All investors in this market agree that each of the three states is a possible outcome in the next time period, and that one (and only one) of the states will actually occur. The investors all agree that each of the states has a positive probability of occurrence; the investors may, however, disagree about the actual value of the probability. We shall keep these assumptions throughout the book.

Consider buying a portfolio in this market of [Maple Math] units of security 1, [Maple Math] of security 2, and [Maple Math] of the bond. We interpret positive values of [Maple Math] and [Maple Math] as buying the securities, or taking a long position, and negative values as short positions. Similarly, a positive value of [Maple Math] means a long position in the bond, lending, and a negative value for [Maple Math] means shorting the bond, which is equivalent to borrowing.

Taking a short position in a security generates positive proceeds. The short seller receives the price of the security being shorted in exchange for the commitment to pay back the value of that security in the next time period, whatever that value may end up being. Thus, taking a short position in security 1, say, for 2 units, ( [Maple Math] ), would cost -2 * $60 =-$120 (i.e., it produces an income of $120). A long position of two units of security 2, [Maple Math] , would cost 2 * $105=$210. The total net cost of such a portfolio is -$120+$210=$90. In general, the net cost of buying a portfolio of [Maple Math] , [Maple Math] , and [Maple Math] units of each of the three securities is

$60S1 + $105S2 + $100SF.

If this last quantity is positive, then establishing this position does indeed cost money; if it is zero, then establishing the position costs nothing; and if it is negative, then establishing the position actually produces income: proceeds of the buy . Portfolios for which $60S1+ $105S2+ $100SF <= $0 (in our example), or more generally, for which the net cost is nonpositive, are referred to as self-financing portfolios. We also use the terminology of a negative-cost portfolio if $60S1 + $105S2 + $100SF is actually negative. Self-financing portfolios require no out-of-pocket cost to establish the position; the buyer, however, may be committed for some payments in the future. These commitments are contingent on the state of nature in the next time period.

We will denote a portfolio by an [Maple Math] , where [Maple Math] is the number of securities in the market. The first component of the [Maple Math] is the number of units of security 1 in the portfolio, the second component is the number of units of security 2 in the portfolio, and so on. Similarly, cash flows will be denoted by an [Maple Math] where [Maple Math] is the number of states. Thus the cash flow of security 1 is ( [Maple Math] ) and is interpreted as $50 in state 1, $60 in state 2, and $80 in state 3. A cash flow should actually be thought of as a random variable taking the values specified in the [Maple Math] . The cash flow ( [Maple Math] ) is a sure amount since it is not contingent on the state of nature. Whenever there is a possibility of confusion (as in the case of the example where [Maple Math] =3), the tuple will be referred to specifically as either a cash flow or as a portfolio. The general form of the cash flow resulting from the transaction of buying the portfolio ( [Maple Math] ) is specified below.

In the current period, the cost of the portfolio is

$60 x S1+ $105 x S2 + $100 x SF,

and in the next time period, the portfolio will pay

$50 x S1+ $90 x S2+ $110 x SF

if state 1 is realized,

$60 x S1+ $110 x S2+ $110 x SF

if state 2 is realized, and

$80 x S1+ $130 x S2+ $110 x SF

if state 3 is realized.

Let us look at a few numerical examples. The array Cash is defined in MAPLE as having four components: Cost , the cost of the portfolio, and IncomeSt1 , IncomeSt2 , IncomeSt3 , the payoffs from the portfolio in states 1, 2, and 3, respectively.

> Cash:=[Cost=60*S1+105*S2+100*SF,IncomeSt1=50*S1+90*S2+110*SF,IncomeSt2=60*S1+110*S2+110*SF,IncomeSt3=80*S1+130*S2+110*SF];

[Maple Math]
[Maple Math]

By substituting different values for ( [Maple Math] ), we seek to investigate the costs and payoffs of different portfolio combinations of these three securities. For example, what would be the cost and payoffs, in the different states of nature, of the portfolio ( [Maple Math] , [Maple Math] , [Maple Math] )?

> subs(S1=1,S2=-3,SF=1,Cash);

[Maple Math]

Establishing this portfolio does not cost money. It actually produces an income of $155. Effectively the buyer of such a portfolio is selling contingent claims . This buyer commits to paying $110 if state 1 occurs, $160 if state 2 occurs, and $200 if state 3 occurs. In return for these commitments, the buyer receives $155 today.

Consider another example, the portfolio [Maple Math] , [Maple Math] , [Maple Math] .

> subs(S1=1/10,S2=-1/10,SF=1/22,Cash);

[Maple Math]

Establishing this position will cost $ [Maple Math] . This position is actually a contingent claim on state 1. If state 1 occurs, the portfolio will pay $1; in any other state, the portfolio will pay nothing. This means that, in this market, the present value of $1 to be obtained contingent on state 1 occurring in the next time period is $ [Maple Math] . It would seem, therefore, that the price of $1, contingent on state 1, is implicit in the structure of the market. We shall return to this important key point in the pricing of derivative securities.

Back to table of contents