2. Deduce that r must belong to (d − 1, u − 1) for the market to be arbitrage-free.

If the market is viable, there exists a probability P¤ equivalent to P, under which ( ˜ Sn) is a martingale. Thus, according to Question 1, E¤(Tn+1|Fn)=1 + r, and therefore E¤(Tn+1)=1 + r. Since Tn+1 is equal to either of d and u with non-zero probability, we necessarily have (1 + r) 2

(d, u).