I'm not sure if I'm applying Change of Numeraire and Girsanov correctly in part c) and d). Also with the information I got, I don't know how to get a result for e).
Consider a financial market with 2 stocks and a bank account $S_t^{(0)}=1$ for $0\le t\le T$, i.e. the riskless interest rate is $r=0$. Under a risk-neutral martingale measure $Q$, assume the following dynamics of 2 stock prices $\{S_t^{(1)}\}_{0 \le t \le T}$ and $\{S_t^{(2)}\}_{0 \le t \le T}$:
$$dS_t^{(1)}=\sigma_{11}S_t^{(1)}d\widetilde{W}_t^{(1)}$$ with $S_0^{(1)}>0$, and
$$dS_t^{(2)}=\sigma_{21}S_t^{(2)}d\widetilde{W}_t^{(1)}+\sigma_{22}S_t^{(2)}d\widetilde{W}_t^{(2)}$$ with $S_0^{(2)}>0$, with constants $\sigma_{11}$,$\sigma_{21}$,$\sigma_{22}$. Note that $\{\widetilde{W}_t^{(1)}\}_{0 \le t \le T}$ and $\{\widetilde{W}_t^{(2)}\}_{0 \le t \le T}$ are 2 independent standard Brownian motions under the risk-neutral martingale measure $Q$. It is further assumed that the matrix \begin{bmatrix}\sigma_{11}&0\\\sigma_{21}&\sigma_{22}\end{bmatrix} is irreversible.
a) Using Itô's formula, show that the solution of the SDE,
$dS_t^{(2)}=\sigma_{21}S_t^{(2)}d\widetilde{W}_t^{(1)}+\sigma_{22}S_t^{(2)}d\widetilde{W}_t^{(2)}$, with $S_0^{(2)}>0$ is given by
$S_t^{(2)}=S_0^{(2)} \exp(\frac{-1}{2}(\sigma_{21}^2+\sigma_{22}^2)t+\sigma_{21}\widetilde{W}_t^{(1)}+\sigma_{22}\widetilde{W}_t^{(2)} )$.
My solution for a) One can see this by applying, $df=f_tdt+f_xdW_t+\frac{1}{2}f_{xx}d\langle W\rangle_t$ On get then $dS_t^{(2)}=-\frac{1}{2}(\sigma^2_{21}+ \sigma^2_{22})S_t^{(2)}dt+\sigma^2_{21}S_t^{(2)}d\widetilde{W}_t^{(1)}+\sigma^2_{22}S_t^{(2)}d\widetilde{W}_t^{(2)}+\frac{1}{2}(\sigma_{21}^2dt+\sigma_{22}^2dt)$ where here is important to see that since the $\widetilde{W}_t^{(i)}$ are independent, $d\widetilde{W}_t^{(1)}d\widetilde{W}_t^{(2)}=0$. If we simplify we get the desired solution.
b) Explain why we can use the risk-neutral valuation formula to price any contingent claim $X$ in this financial market.
My solution for b) Since the market is complete, due to the fact that the matrix with sigma is invertible and that we are under a risk-neutral martingale measure, we can use risk-neutral valuation to price any contingent claim $X$
c) Consider the following contingent claim or derivative
$$X(T)=S_T^{(2)}1_{\{S_T^{(1)}>K\}}$$.
Using "Change of Numeraire" and $S^{(2)}$ as numeraire, show that the risk-neutral price $\Pi_0^X$ of the contingent claim $X$ at time $t=0$ is given by
$$\Pi_0^X=S_0^{(2)} E_{Q^1}[1_{\{S_T^{(1)}>K\}}]$$ wherethe measure $Q^1$ has to be determined, i.e. determine its Radon-Nikodym derivative with respect to $Q$ $$\frac{dQ^1}{dQ}|F_t$$
my solution to c)
One has $$\frac{dQ^1}{dQ}|F_t=\frac{N_t}{S_t^{(0)}}\frac{S_0^{(0)}}{N_0}$$ since new numeraire is supposed to be $S^{(2)}$ and $S_t^{(0)}=1$ for all $t$ ($r=0$) we get $$\frac{dQ^1}{dQ}|F_t=\frac{S_t^{(2)}}{1}\frac{1}{S_0^{(2)}}$$ and one has $\Pi_0^X=1*E_Q[S_T^{(2)}1_{\{S_T^{(1)}>K\}}]=S_0^{(2)}E_{Q^1}[1_{\{S_T^{(1)}>K\}}]$
d) Consider the probability measure $Q^1$ from (c). Applying Girsanov theorem, determine the stochastic dynamics of $S^{(1)}$ under $Q^1$. Knowing that a general SDE of this form $$dSt=\hat{\mu} S_tdt+\hat{\sigma}S_td\hat{W}_t$$ with, $S_0>0$ has the solution $$St=S_0\exp((\hat{\mu}-\frac{1}{2}\hat{\sigma})t+\hat{\sigma}\hat{W}_t)$$ write down the solution of the derived SDE.
my solution to d) (Updated) Using Girsanov we know that for any equivalent measure we have
$$\frac{dQ^1}{dQ}=\frac{S_T^{(2)}}{1}\frac{1}{S_0^{(2)}}=L_T$$ but at the same time $$L_T=\exp\{-\int_0^T \gamma_s^{(1)}d\widetilde{W}_s^{(1)} -\int_0^T \gamma_s^{(2)}d\widetilde{W}_s^{(2)}-\frac{1}{2}\int_0^T(\gamma_s^{(1)})^2+(\gamma_s^{(2)})^2ds\}$$ since we know that $S_T^{(2)}/S_0^{(2)}=\exp(-\frac{1}{2}(\sigma_{21}^2+\sigma_{22}^2)T+\sigma_{21}\widetilde{W}_T^{(1)}+\sigma_{22}\widetilde{W}_T^{(2)})$, it follows that $\gamma_t^{(1)}=-\sigma_{21}$ and $\gamma_t^{(1)}=-\sigma_{22}$.
Now using the fact that $d\hat{W}_t^{(i)}=d\widetilde{W}_t^{(i)}+\gamma_t^{(i)}dt$ are Brownian motions under $Q^1$
Applying this to $dS_t^{(1)}=\sigma_{11}S_t^{(1)}(d\hat{W}_t^{(1)}+\sigma_{21}dt)$ gives $\hat{\mu}=\sigma_{21}\sigma_{11}$ and $\hat{\sigma}=\sigma_{11}$
e) Combining results from c) and d), determine the risk-neutral price of this derivative $X(T)$ at time $t=0$
Solution to e (according to suggestion) Ok. Since my calculation is correct, I just need to calculate the following now: (I'm taking $S_0^{(2)}$ out of the calculations since it's a constant)
$E_{Q^1}[1_{\{S_T^{(1)}>K\}}]=Q^1(S_T^{1}>K)=Q^1(S_0^{(1)}\exp((\sigma_{11}\sigma_{21}-\frac{1}{2}\sigma_{11}^2)T+\sigma_{11}\hat{W}_T^{(1)})$
and after moving some terms, using the fact that $-W_t=W_t$ and that $W_t\sim N(0,t)$,
$=Q^1(\frac{\ln(\frac{S_0^{(1)}}{K})+(\sigma_{11}\sigma_{12}-\frac{1}{2}\sigma_{11}^2)T}{\sigma_{11}\sqrt T}> Z)$
where Z is standard normal distribution.