Consider a security that pays

S(T)k at time T (k 1)

where the price S(t) is governed by the standard model

dS(t) = S(t)dt + S(t)dW(t).

Using Black-Scholes-Merton equation, show that the price of this security at time t < T is given by

c(t, S(t)) = S(0) ke(k1)(r+ k 2 2)(T t) .