S follows dS = mu S dt + sigma S dW. Derive the SDEs satisfied by Y = ln S and by Y = S^n, and give E[S_T^n] in closed form.

S follows dS = mu S dt + sigma S dW. Derive the SDEs satisfied by Y = ln S and by Y = S^n, and give E[S_T^n] in closed form.

Approach: Apply Ito with the second order term (1/2) f'' sigma^2 S^2 dt in each case, then integrate the log equation to get the closed form and take the expectation of the power.

d(ln S) = (mu - sigma^2/2) dt + sigma dW, d(S^n) = n S^n (mu + (n-1) sigma^2/2) dt + n sigma S^n dW, and E[S_T^n] = S_0^n exp(n(mu + (n-1) sigma^2/2) T). Ito's lemma applied to the logarithm has f' = 1/S and f'' = -1/S^2, so the second order term is (1/2)(-1/S^2)(sigma^2 S^2) dt = -(sigma^2/2) dt, which is the drift adjustment that separates the arithmetic growth rate mu from the log growth rate. Integrating gives S_T = S_0 exp((mu - sigma^2/2) T + sigma W_T), so S_T is lognormal and the moments follow from the Gaussian moment generating function: E[S_T^n] = S_0^n exp(n(mu - sigma^2/2)T + n^2 sigma^2 T/2), which rearranges to the stated form. The same exponent appears as the drift of the geometric Brownian motion S^n, confirming both computations against each other.

Follow-up: For which n is S^n a martingale when mu = r, and what does that say about pricing a contract paying S_T^2?

Key concepts: Ito's lemma, geometric Brownian motion, lognormal moments, drift adjustment.