I have the following problem with its solution, but I keep on getting it wrong. I would be really grateful if someone could please explain to me what I am doing wrong. Thanks! It is part C that I don't understand, but I have included parts A and B to give you an idea of what the problem is about. Thanks a lot in advance!
QUESTION: Let us consider the Black and Scholes model of a financial market where investors can trade in a risky and in a riskless asset. The risky asset follows the dynamics
$$dS_T=\mu S_t dt+\sigma S_t dW_t, t\in\mathbb{R}, S_0=s>0,$$ $\mu, \sigma$ are positive constants while the price of the riskless asset is described by $$dB_t=rB_t dt, B_0=1,$$
where $r\ge 0$ denotes the risk-free rate. Here $W_t, t\in\mathbb{R_+}$ is a Brownian motion. We denote by $F_t, t\in\mathbb{R_+}$ its natural filtration. The trading period is assumed to be the interval $[0,T]$.
PART A Write down the Radon-Nikodym density $dP^*/dP$ of the unique $P^*~P$ such that the discounted price $\tilde{S}_t:=S_t/B_t$ is a $P^*$-martingale, $t\in[0,T]$.
Part A - Answer $$\frac{dP^*}{dP}=\exp\{-\frac{\mu-r}{\sigma}W_T-\frac{1}{2}\left(\frac{\mu-r}{\sigma}\right)^2T\}$$ Part B Write down a formula for $S_T$ using the $P^*$-Brownian motion $$W_t^*=W_t+\frac{\mu-r}{\sigma}t,\quad t\in[0,T].$$ Part B - Answer $$S_T=s\exp\{\left(r-\frac{\sigma^2}{2}\right)T+\sigma W_T^*\}$$ PART C Consider an option with payoff $G=2W_T$ at time $T$. Choose $r=0$, $T=1$, $s=1$, $\mu=2$, $\sigma=2$. Check that in this case $G=\ln(S_T)$.
Part C - Answer $$\ln(S_1)=\left(\mu-\frac{\sigma^2}{2}\right)+2W_1=2W_1$$ MY ATTEMPT We have $$G=\ln(S_1)=\ln\{\exp\{\left(0-\frac{2^2}{2}\right)+2W_1\}\}=-2+2W_1$$
What have I done wrong? It is almost the same as the actual answer, but that $-2$ is not supposed to be there.