Consider a European option on a stock with price $S$ that at expiration at time $T$ pays $S^\alpha$ where $\alpha$ is some arbitrarily chosen power.
Could someone explain how to use stochastic calculus and Ito’s Lemma to derive the formula for the evolution of the function $f=S^\alpha$ in a Black-Scholes risk-neutral world and evaluate the discounted risk-neutral expected value of the payoff. Thank you very much.
Given the Black-Scholes assumptions, under the risk neutral measure $Q$ the stock price follows:
$$dS(t)=S(t)(rdt+\sigma dW(t))$$
Let $X(t)=S(t)^{\alpha}$. Using Ito's lemma
$$dX(t)=\alpha S(t)^{\alpha-1}dS_t+0.5\alpha(\alpha-1)S(t)^{\alpha-2}dS(t)^2$$
i.e.
$$dX(t)=\alpha X(t)(rdt+\sigma dW_t)+0.5\alpha(\alpha-1)\sigma^2X(t)$$
or
$$dX(t)=X(t)((\alpha r+0.5\alpha(\alpha-1)\sigma^2)dt+\sigma dW(t))$$
This is a GBM with a well-known solution
$$X(t)=X(0)e^{\left(\alpha r+0.5\alpha(\alpha-1)^2\sigma^2-0.5\sigma^2\right)t+\sigma dW(t)}$$
Hence because the strike price is zero the price of this option is
$$e^{-rt}E^Q(X(t))=X(0)e^{t(\alpha-1) r+0.5t\alpha(\alpha-1)^2\sigma^2}$$
