3. Give examples of arbitrage strategies if the no-arbitrage condition derived in Question 2 is not satised.

Assume for instance that r ·u − 1. By borrowing an amount S0 at time 0, we can purchase one share of the risky asset. At time N, we pay the loan back and sell the risky asset. We realised a prot equal to SN − S0(1 + r)N which is always positive, since SN ¸S0dN. Moreover, it is strictly positive with nonzero probability. This is an arbitrage opportunity. If r ¸u − 1, we can make a riskless prot by short-selling the risky asset.