a Free Lunch ?

526 NOTICES OF THE AMS VOLUME 51, NUMBER 5 The notion of arbitrage is crucial in the modern theory of finance. It is the cornerstone of the option pricing theory due to F. Black and M. Scholes (published in 1973, Nobel Prize in Economics 1997). The underlying idea is best explained by telling a little joke. A finance professor and a normal person go on a walk, and the normal person sees a €100 bill lying on the street. When the normal person wants to pick it up, the finance professor says: “Don’t try to do that. It is absolutely impossible that there is a €100 bill lying on the street. Indeed, if it were lying on the street, somebody else would already have picked it up.” How about financial markets? There it is already much more reasonable to assume that there are no arbitrage possibilities, i.e., that there are no €100 bills lying around waiting to be picked up. Let us illustrate this with an easy example. Consider the trading of dollars versus euros which takes place simultaneously at two exchanges, say in New York and Frankfurt. Assume for simplicity that in New York the $/€ rate is 1:1. Then it is quite obvious that in Frankfurt the exchange rate (at the same moment of time) also is 1:1. Let us have a closer look why this is indeed the case. Suppose to the contrary that you can buy in Frankfurt a dollar for €0.999. Then, indeed, the so-called “arbitrageurs” (these are people with two telephones in their hands and three screens in front of them) would quickly act to buy dollars in Frankfurt and simultaneously sell the same amount of dollars in New York, keeping the margin in their (or their bank’s) pocket. Note that there is no normalizing factor in front of the exchanged amount and the arbitrageur would try to do this on a scale as large as possible. It is rather obvious that in the above-described situation the market cannot be in equilibrium. A moment’s reflection reveals that the market forces triggered by the arbitrageurs will make the dollar rise in Frankfurt and fall in New York. The arbitrage possibility will disappear when the two prices become equal. Of course “equality” here is to be understood as an approximate identity where, even for arbitrageurs with very low transaction costs, the above scheme is not profitable any more. This brings us to a first, informal and intuitive, definition of arbitrage: an arbitrage opportunity is the possibility to make a profit in a financial market without risk and without net investment of capital. The principle of no arbitrage states that a mathematical model of a financial market should not allow for arbitrage possibilities. To apply this principle to less trivial cases, we consider a—still extremely simple—mathematical model of a financial market: there are two assets, called the bond and the stock. The bond is riskless; hence by definition we know what it is worth tomorrow. For (mainly notational) simplicity we neglect interest rates and assume that the price of a bond equals €1 today as well as tomorrow, i.e., B0 = B1 = 1. The more interesting feature of the model is the stock, which is risky: we know its value today, ?W H A T I S . . .