1.1 Arbitrage
The notion of arbitrage is crucial to the modern theory of Finance. It is the corner-stone of the option pricing theory due to F. Black, R. Merton and M. Scholes [BS 73], [M 73] (published in 1973, honoured by the Nobel prize in Economics 1997).
The idea of arbitrage is best explained by telling a little joke: a professor working in Mathematical Finance and a normal person go on a walk and the normal person sees a 100e bill lying on the street. When the normal person wants to pick it up, the professor says: don’t try to do that. It is absolutely
impossible that there is a 100e bill lying on the street. Indeed, if it were lying on the street, somebody else would have picked it up before you. (end of joke) 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 100e bills lying around and waiting to be picked up. Let us illustrate this with an easy example. Consider the trading of $ versus e that takes place simultaneously at two exchanges, say in New York and Frankfurt. Assume for simplicity that in New York the $/e 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 the case. Suppose to the contrary that you can buy in Frankfurt a $ for 0.999e. 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 $ in Frankfurt and simultaneously sell the same amount of $ in New York, keeping the margin in their (or their bank’s) pocket. Note that there is no normalising factor in front of the exchanged amount and the arbitrageur would try to do this on a scale as large as possible.