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OPTION PRICES AND STATE PRICES

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D. M. Chance, TN97-13 Option Prices and State Prices 1 TEACHING NOTE 97-13: OPTION PRICES AND STATE PRICES1 Version date: July 18, 2008 C:\CLASSES\TEACHING NOTES\TN97-13.DOC A contingent claim is a security that provides a payoff that is dependent (contingent) on something specific happening. An option is one form of a contingent claim in that it provides a positive payoff under the condition that the option expires in-the-money. If the option does not expire in-the-money, the payoff is obviously zero. Another form of a contingent claim is a security that pays $1 in a given outcome and zero otherwise. These outcomes are referred to as states or states of nature and the security is often called a state-contingent claim. Other common names for this type of security are pure security, the term we shall use, and Arrow-Debreu security, the latter arising out of the work of Nobel Laureates Kenneth Arrow and Gerard Debreu. In this Teaching Note, we examine some properties of pure securities and demonstrate how they relate to options. Suppose we are facing a risky situation, which could be something as simple as the next day in the stock market. Let us define the possible outcomes in terms of three states, which might be as simple as the market goes down 2% (state 1), the market is unchanged (state 2), and the market goes up 2% (state 3). Naturally the possible outcomes are infinite and cannot be reduced to such simple statements, but the framework provided by this simplification is, nonetheless, useful and generalizes to the case of a continuous spectrum of states. Consider a stock that will be worth $110 in state 1, $100 in state 2 and $90 in state 3. Another security might be worth $105 in state 1, $101 in state 2 and $98 in state 3. Suppose the risk-free rate is 2%. Then a risk-free security worth $100 today would have a value of $102 in each state. 1These notes have benefitted from helpful conversations with and similar notes of Professor Richard Rendleman of the University of North Carolina at Chapel Hill. The notes also appear in published form in Financial Engineering News, March/April 2003, pp. 8-10. Now consider a state-contingent claim that pays $1 in state 1 and zero in the other states. Another state-contingent claim pays $1 in state 2 and zero in the other states. A third state-contingent claim pays $1 in state 3 and zero in the other states. Our first stock, whose three possible future values are $110, $100 and $90, can be viewed as a portfolio of 110 units of the first-stateD. M. Chance, TN97-13 Option Prices and State Prices 2 contingent claim, 100 units of the second state-contingent claim and 90 units of the third state-contingent claim. Our second security can be viewed as a portfolio of 105 units of the first state-contingent claim, 101 units of the second and 98 units of the third. A risk-free security worth $100 today can be viewed as 102 units of all three state-contingent claims. The price of a state-contingent claim is called a state price. It follows that the price of each security must be the value today of the equivalent portfolio of state-contingent claims. In other words, if we know the state prices, we can determine the security prices. Alternatively, if we know the security prices, we can determine the state prices. The state-contingent claims are the fundamental securities in the market. We cannot literally see them or trade them, but they are there, the financial atoms of the marketplace. Ordinary securities, being combinations of these pure securities, are sometimes called complex securities, though there is nothing particularly complex about them. They are just portfolios of pure securities. Let us now develop a formal framework for understanding these concepts. First let us establish the fact that there must always be at least as many securities as there are states. This is referred to as the spanning property, which means that the pure securities will be sufficient to reproduce any complex securities. Here we shall make the number of securities equal to the number of states. Specifically let there be n states, where each state is identified as state i, i = 1, 2, ...n and n complex securities, with each security defined as security j, j = 1, 2, ...,n with price Sj. Let Xij be the payoff of complex security j in state i. A complex security can be defined in terms of the number of units of each pure security required to replicate the payoffs of the complex security. We can alternatively define each pure security in terms of the number of units of each complex security required to replicate its outcomes. Define pure security i as a security that pays $1 in state i and zero in all other states. Then αij is the number of units of complex security j that should be held to reproduce the payoff of pure security i. Alternatively we can view the payoff Xij as the number of units of pure security i that are implicit in complex security j. Let us now organize this information in a more meaningful way. We shall use both matrix and scalar notation, though the matrix notation is somewhat more useful in facilitating the solution of simultaneous equations. As stated, a pure security is a combination of complex securities. The payoffs of pure security 1 in each of the possible states are as follows: α11X11 + α12X12 + ... + α1nX1n = 1 (outcome in state 1)α11X21 + α12X22 + ... + α1nX2n = 0 (outcome in state 2) . . . α11Xn1 + α12Xn2 + ... + α1nXnn = 0 (outcome in state n). In other words, pure security 1 is a combination of α11 units of complex security 1, α12 units of complex security 2, ..., and α1n units of complex security n. Similarly the payoffs of pure security 2 in each of the possible states are as follows: α21X11 + α22X12 + ... + α2nX1n = 0 (outcome in state 1) α21X21 + α22X22 + ... + α2nX2n = 1 (outcome in state 2) . . . α21Xn1 + α22Xn2 + ... + α2nXnn = 0 (outcome in state n). Pure security 2 is, thus, a combination of α21 units of complex security 1, α22 units of complex security 2 and α2n units of complex security n. The payoffs of pure security n in each of the possible states are as follows: αn1X11 + αn2X12 + ... + αnnX1n = 0 (outcome in state 1) αn1X21 + αn2X22 + ... + αnnX2n = 0 (outcome in state 2) . . . αn1Xn1 + αn2Xn2 + ... + αnnXnn = 1 (outcome in state n). Pure security n is, thus, a combination of αn1 units of complex security 1, αn2 units of complex


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