if you know that the a security pays out an amount of max{ln(S/X),0} at T, how can we use risk neutral valuation to calculate the price of the security at time t?

anyone?

What is S? what is X?But generally yes... if you can find the dynamics of S/X under the risk neutral measure then you can price the security. Depending on the dynamics it may have to be done numerically, but I would say that in this case a closed fom is likely.If for instance S/X is lognormal, (say exp(Y) = S/X) then you are evaluating E[Y; Y>0] where Y is normally distributedwhich is something like sigma*exp(-mu^2/2*sigma^2) +mu * P(Y>0)

S is the stock price and X is the strike price. shouldn't the expected value have something with N(d1) and N(d2) in it?

Ok, so under the risk neutral measuredS_t = r S_t dt + sigma S_t dWt (if you are doing this in the normal BS framework)Solving this SDE gives usS_t= S_0exp((r-0.5sigma^2)t + sigmaW_t)so So Ln(S_t/X) is normally distributed with mean = ln(S_0/X) +(r-0.5*sigma^2)t and variance sigma^2 tThe value V in the RN world isV = exp(-rt)E[Phi(S)] where Phi is the payoff functionSo, in this case V = exp(-rt)E[Ln(S_t/X) ; Ln(S_t/X>0]let Z_t = ln(S_t/X)so V = exp(-rt)E[Z_t; Z_t>0]= exp(-rt) *1/sqrt(2*Pi)*S * int_{0}^{infty}z * exp(-(z-M)^2/2*S^2 dzwhere M is the mean and S is the stdev of Zdo the integral, and substitute back in, and that is your answer.