# Arbitrage

I’m going to demonstrate a simple example using forward agreements for stocks of what we mean when we say there is no arbitrage.

First assume that there is a risk free investment that yields continuous interest at rate $r$, typically treasury bonds are used as the risk free investment. Now assume we have a stock $S$ that has price $S_0$ at time 0. Suppose you enter into a forward agreement to pay $F_{0,T}$ at time T for the stock $S$.

Now using the risk free investment we can say that $F_{0,T}$ is worth $F_{0,T}e^{-rT}$ at time 0 (this is because if we invested $F_{0,T}e^{-rT}$ in the risk free investment at time 0 we would have $F_{0,T}$ money at time T).

The assumption of no arbitrage means that $F_{0,T}e^{-rT} = S_0$ since by investing $F_{0,T}e^{-rT}$ in treasury bonds we will own the stock at time T and by buying the stock we also own it at time T.

Thus by assuming no arbitrage the forward price is $F_{0,T} = S_0e^{rT}.$

This is one example of using the assumption of no arbitrage. The Wikipedia article on arbitrage shows the other types of arbitrage.