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 , typically treasury bonds are used as the risk free investment. Now assume we have a stock that has price at time 0. Suppose you enter into a forward agreement to pay at time T for the stock .
Now using the risk free investment we can say that is worth at time 0 (this is because if we invested in the risk free investment at time 0 we would have money at time T).
The assumption of no arbitrage means that since by investing 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
This is one example of using the assumption of no arbitrage. The Wikipedia article on arbitrage shows the other types of arbitrage.