Hi there,So the price of a European call isApparently, one can interpret it as discounted expected* value of benefit minus discounted expected* value of cost of the call.I can prove that KN(d-) is the expected* cost of this contract, but I really don't know where to start to prove the interpretation of the first term.Also, the info I see in two different books (Hull and Crack) sounds contradictory (and no, I cannot prove either)1. S(0) e^[(r-d)T] N(d1)= expected benefit at time T for holding a call under risk-neutral measure = expected value of a variable that equals S(T) if S(T) > K and is zero otherwise in a risk-neutral world2. S(0) e^[(r-d)T] N(d1) = expected terminal stock price conditional on the option finishing in-the-money, i.e. E*[S(T) | S(T) > K], under risk-neutral measureHow can 2 be the same as 1? Aren't we working in a completely different probability space?Can I have a hint to help me connect the dots please? I'm sure I'm being super-thick, sorry.Many thanks*under risk-neutral measure

I find the best thing to do when looking at these sorts of questions is to think extremes. Imagine we've got a really out of the money option, so N(d1) is really low, as will be S(0) e^(-dT) N(d1) e(rT) .This is consistent with definition 1. Yet, definition 2 can't possibly be less than K, so it can't be right.

Thanks gjlipman. But I would really like to prove the result.For KN(d_2), I see how that is my expected cost (K multiplied by the probability of being in the money at maturity=N(d_2)), but for the first part I'm at a loss.Thanks in advance...

Hi pizza,the definition 1 is correctthe definition 2 should be changed in E*[S(T)/(S(0)*exp(r-d)T)|S(T)>K]see this threadRegardsP.

If you're trying to prove that S(0) e^[(r-d)T] N(d1) is the expected value of the what we get, the most intuitive proof to me is the one given in the appendix to the chapter "The B-S-M Model" of Hull. It isn't trivial, but straight forward enough when you read it.