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) = µS(t)dt + σS(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(k−1)(r+ k 2 σ2)(T −t) .