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.

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