4. From now on, we assume that r 2 (d − 1, u − 1) and we write p=

(u − 1 − r)/(u − d). Show that ( ˜ Sn) is a P-martingale if and only if the random variables T1, T2, . . . , TN are independent, identically distributed

(IID) and their distribution is given by P(T1 =d)=p=1 − P(T1 =u).

Conclude that the market is arbitrage-free and complete.