IEOR E4706: Financial Engineering: Discrete-Time Models c ? 2010 by Martin Haugh Martingale Pricing Theory These notes develop the modern theory of martingale pricing in a discrete-time, discrete-space framework. This theory is also important for the modern theory of portfolio optimization as the problems of pricing and portfolio optimization are now recognized as being intimately related. We choose to work in a discrete-time and discrete-space environment as this will allow us to quickly develop results using a minimal amount of mathematics: we will use only the basics of linear programming duality and martingale theory. Despite this restriction, the results we obtain hold more generally for continuous-time and continuous-space models once various technical conditions are satisfi ed. This is not too surprising as one can imagine approximating these latter models using our discrete-time, discrete-space models by simply keeping the time horizon fi xed and letting the number of periods and states go to infi nity in an appropriate manner. 1 Notation and Defi nitions for Single-Period Models We fi rst consider a one-period model and introduce the necessary defi nitions and concepts in this context. We will then extend these defi nitions to multi-period models. t = 0t = 1 c c 1 c 2 QQ QQ QQ QQ QQc m c PP PP PP PPc Let t = 0 and t = 1 denote the beginning and end, respectively, of the period. At t = 0 we assume that there are N + 1 securities available for trading, and at t = 1 one of m possible states will have occurred. Let S(i) 0 denote the time t = 0 value of the ithsecurity for 0 i N, and let S(i) 1 (j ) denote its payoff at date t = 1 in the event that joccurs. Let P = (p1,.,pm) be the true probability distribution describing the likelihood of each state occurring. We assume that pk 0 for each k. Arbitrage A type A arbitrage is an investment that produces immediate positive reward at t = 0 and has no future cost at t = 1. An example of a type A arbitrage would be somebody walking up to you on the street, giving you a Martingale Pricing Theory2 positive amount of cash, and asking for nothing in return, either then or in the future. A type B arbitrage is an investment that has a non-positive cost at t = 0 but has a positive probability of yielding a positive payoff at t = 1 and zero probability of producing a negative payoff then. An example of a type B arbitrage would be a stock that costs nothing, but that will possibly generate dividend income in the future. In fi nance we always assume that arbitrage opportunities do not exist1since if they did, market forces would quickly act to dispel them. Linear Pricing Defi nition 1 Let S(1) 0 and S(2) 0 be the date t = 0 prices of two securities whose payoff s at date t = 1 are d1 and d2, respectively2. We say that linear pricing holds if for all 1and 2, 1S(1) 0 + 2S(2) 0 is the value of the security that pays 1d1+ 2d2at date t = 1. It is easy to see that absence of type A arbitrage implies that linear pricing holds. As we always assume that arbitrage opportunities do not exist, we also assume that linear pricing always holds. Elementary Securities, Attainability and State Prices Defi nition 2 An elementary security is a security that has date t = 1 payoff of the form ej = (0,.,0,1,0,.,0), where the payoff of 1 occurs in state j. As there are m possible states at t = 1, there are m elementary securities. Defi nition 3 A security or contingent claim, X, is said to be attainable if there exists a trading strategy, = 01. N+1T, such that X(1) . . . X(m) = S(0) 1 (1).S(N) 1 (1) . . . . . . . . . S(0) 1 (m).S(N) 1 (m) 0 . . . N . (1) In shorthand we write X = S1 where S1 is the m (N + 1) matrix of date 1 security payoff s. Note that j represents the number of units of the jthsecurity purchased at date 0. We call the replicating portfolio. Example 1 (An Attainable Claim) Consider the one-period model below where there are 4 possible states of nature and 3 securities, i.e. m = 4 and N = 2. At t = 1 and state 3, for example, the values of the 3 securities are 1.03 , 2 and 4, respectively. t = 0t = 1 ( ( ( c 1.0194, 3.4045, 2.4917 hhhhhh hh h HH HH HH HH H c1 1.03, 3, 2 c2 1.03, 4, 1 c3 1.03, 2, 4 c4 1.03, 5, 2 The claim X = 7.47 6.97 9.97 10.47Tis an attainable claim since X = S1 where = 1 1.5 2Tis a replicating portfolio for X. 1This is often stated as assuming that “there is no free lunch”. 2d1 and d2are therefore m 1 vectors. Martingale Pricing Theory3 Note that the date t = 0 cost of the three securities has nothing to do with whether or not a claim is attainable. We can now give a more formal defi nition of arbitrage in our one-period models. Defi nition 4 A type A arbitrage is a trading strategy, , such that S0 0 for 1 i 4. Defl ating by the Numeraire Security Let us recall that there are N + 1 securities and that S(i) 1 (