Hi Mark,

Perhaps I have misunderstood question 8.11 (comparing prices of out-of-the-money American and European digital calls when r =0, asset price follows BM). Please tell me where I am going wrong:

I assume you mean W_t is standard BM, so by risk-neutral pricing, at time t, the European digital has price E(1_{W_T >= K } | F_t ), which by some simple computation and facts about such conditional expectations is just E(1_{W_T >= K} | W_t = x) = P( N(x, sqrt(T-t) ) >= K ) where N(x, sigma) is used to denote a normal random variable with mean x, variance sigma^2. We're assuming that spot is out-of the money, so x < K, in which case this probability is < 0.5.

On the other hand, the price of the digital is E(1_{M_T} >= K | F_t ) and if the spot had previously gone above K, then this value is obviously 1. So I don't make the price of the European being half the price of the American. I get this only when we assume spot is *at-the-money*.

Thanks.