Problem
A student would like to receive $S_1^3$ euros at time $T = 2$, where $S_t$
is the share price. Assume the Black-Scholes model is valid, the risk-free rate is $r = 0.1$, and current share price is $S_0 = 100$.
How much the student should pay now at $t = 0$?
My attempt
$$S_1 = e^{-r(T+1)}E^*( S_t^3 | F_T),$$
where $E^*$ is expectation for probability where $W^* = W_t + \frac{\mu - r}{\sigma} t$ is Brownian motion, $F_T$ is a filtration, and
$$S_t = S_0 \exp(\mu t) \exp(\sigma W_t -\frac{\sigma^2 t}{2})$$
Therefore, $$S_1 = e^{-r(T+1)}E^*( S_0^3 \exp(3r) \exp(3\sigma W_t -\frac{3\sigma^2 t}{2}) | F_T) = e^{-r(T+1)} S_0^3 \exp(3r T) \exp(-\frac{3\sigma^2 t}{2}) E^*(\exp(3\sigma W_t | F_T)$$ I think that the latter expectation is Martingale, so we may substitute $T = 2$ instead of $t$. Then the result will be
$$ = e^{-0.1*3} \cdot 100^3 \cdot e^{3*0.1*2} \cdot e^{-3\sigma^2} \cdot e^{3 \sigma W^*_2}$$ where $W^*_2 = W_2 +\frac{\mu - r}{\sigma}\cdot 2$.
I am pretty sure that this solution is not correct, plus I think the result should not contain a random variable $\sigma W^*_2$. What am I doing wrong?
You should wonder how that expectation yields a random variable if the current price is known. Consider $t<T$. The random variable $S_t^3$ is $\mathcal{F}_t$-measurable and $\mathcal{F}_t\subset \mathcal{F}_T$. Thus the sum to be received is decided at $t$ and we have the payoff structure $$h(S_T,T)=h(S_t,t)=S_t^3 \ \ \ \ \forall T >t$$ So the fair price is given by $$F(S_0,0)=e^{-rT}\mathbb{E}^\mathbb{Q}[h(S_T,T)|S_0]=e^{-rT}\mathbb{E}^\mathbb{Q}[S_t^3|S_0]=e^{-rT}e^{3(\ln(S_0)+(r-0.5\sigma^2)t)}\mathbb{E}^\mathbb{Q}[e^{3\sigma W_t^\mathbb{Q}}]$$ Since $3\sigma W_t^\mathbb{Q}\sim N(0,3^2\sigma^2t)$ then $e^{3\sigma W_t^\mathbb{Q}} \sim \textrm{LogNorm}(0,3^2\sigma^2t)$ so $\mathbb{E}^\mathbb{Q}[e^{3\sigma W_t^\mathbb{Q}}]=e^{\frac{3^2\sigma^2 t}{2}}$. Notice that everything is under the risk neutral dynamics.