Exercise on arbitrage-free process
Consider the following problem, from Bjork’s Arbitrage Theory in Continuous Time:
Consider the standard Black-Scholes model. Derive the arbitrage free price process for the $ T $ -claim $ \mathcal{X} $ where $ \mathcal{X} $ is given by $ \mathcal{X}={S(T)}^\beta $ . Here $ \beta $ is a known constant.
My approach.
Let $ F(t,s) $ be the price of the claim $ \mathcal{X} $ at time $ t $ , when the underlying spot price is $ s $ .
The Black-Scholes equation for $ F $ is: $$ \begin{align} F_t + rsF_s + \frac12 \sigma^2s^2 F_{ss} - rF &= 0 \ F(T, S(T)) &= S(T)^\beta. \end{align} $$
It is convenient to make a change of variables of the form $ \tilde{F}(t,s) = e^{-rt}F(t,s) $ , so that the associated stochastic process is: $$ \begin{align} dX &= rX dt + \sigma X dW \ X(t) &= s. \end{align} $$
After changing variable to $ Y = \log X $ and integrating, I find $$ X(T) = s \exp\left((r-\frac12 \sigma^2)(T-t) + \sigma(W(T) - W(t))\right). $$ So by the Feynman-Kac formula I have: $$ \begin{align} F(t,s) &= e^{-r(T-t)}\mathbb{E}\left[X(T)^\beta\right] \ &= e^{-r(T-t)}\int_{-\infty}^{\infty}s^\beta e^{\beta z} \exp\left(-\frac12 \frac{(z - (r-\frac12\sigma^2)(T-t))^2}{\sigma^2(T-t)}\right) dz, \end{align} $$ which after some computation gives, if I did not make any mistake: $$ F(t,s)=e^{-r(T-t)}s^\beta \exp\left(\frac12\sigma^2\beta^2(T-t) + (r-\frac12\sigma^2)\beta(T-t)\right). $$
Does it sound right?
Also, regardless of whether the pricing formula is correct, I am not sure if what I found is really the arbitrage free stochastic process for $ \mathcal{X} $ .
If $ X $ is a lognormally distributed variable, $ X = e^{\mu + \nu Z} $ , so $ \ln X $ has mean $ \mu $ and variance $ \nu^2 $ , and $ Z $ is normally distributed, then $$ E\left[ X^n \right] = e^{n\mu + \frac{1}{2} n^2\nu^2} $$ This solves your question with $ X = S_T/S_t $ , $ n = \beta $ , $ \mu = (r -\frac{1}{2} \sigma^2) (T-t) $ and $ \nu = \sigma \sqrt{T-t} $ in the Black Scholes world.
See also the following wiki page for other properties of lognormal distribution: