Chapter 2 No-Arbitrage Principle

In this section we are going to state the most fundamental assumption about the market. In brief, we shall assume that the market does not allow for risk-free profits with no initial investment.

For example, a possibility of risk-free profits with no initial investment can emerge when market participants make a mistake. Suppose that dealer \(A\) in New York offers to buy British pounds at a rate \(d_A = 1.62\) dollars to a pound, while dealer \(B\) in London sells them at a rate \(d_B = 1.60\) dollars to a pound. If this were the case, the dealers would, in effect, be handing out free money. An investor with no initial capital could realise a profit of \(d_A − d_B = 0.02\) dollars per each pound traded by taking simultaneously a short position with dealer \(B\) and a long position with dealer \(A\). The demand for their generous services would quickly compel the dealers to adjust the exchange rates so that this profitable opportunity would disappear.

2.1 Assumption 1.6 No-Arbitrage Principle

There is no admissible portfolio with initial value \(V(0) = 0\) such that \(V (1) > 0\) with non-zero probability.

In other words, if the initial value of an admissible portfolio is zero, \(V (0) =0\), then \(V (1) = 0\) with probability 1. This means that no investor can lock in a profit without risk and with no initial endowment. If a portfolio violating this principle did exist, we would say that an arbitrage opportunity was available.

Arbitrage opportunities rarely exist in practice. If and when they do, the gains are typically extremely small as compared to the volume of transactions, making them beyond the reach of small investors. In addition, they can be more subtle than the examples above. Situations when the No-Arbitrage Principle is violated are typically short-lived and difficult to spot. The activities of investors (called arbitrageurs) pursuing arbitrage profits effectively make the market free of arbitrage opportunities.

The exclusion of arbitrage in the mathematical model is close enough to reality and turns out to be the most important and fruitful assumption. Arguments based on the No-arbitrage Principle are the main tools of financial mathematics.