(Change in volatility caused by change of numéraire). Let $S(t)$ and $N(t)$ be the prices of two assets, denominated in a common currency, and let $\sigma$ and $\nu$ denote their volatilities, which we assume are constant. We assume also that the interest rate $r$ is constant. Then
$$
\begin{aligned}
d S(t) & =r S(t) d t+\sigma S(t) d \widetilde{W}_1(t), \\
d N(t) & =r N(t) d t+\nu N(t) d \widetilde{W}_3(t),
\end{aligned}
$$
where $\widetilde{W}_1(t)$ and $\widetilde{W}_3(t)$ are Brownian motions under the risk-neutral measure $\widetilde{\mathbb{P}}$. We assume these Brownian motions are correlated, with $d \widetilde{W}_1(t) d \widetilde{W}_3(t)=$ $\rho d t$ for some constant $\rho$.
(i) Show that $S^{(N)}(t)=\frac{S(t)}{N(t)}$ has volatility $\gamma=\sqrt{\sigma^2-2 \rho \sigma \nu+\nu^2}$. In other words, show that there exists a Brownian motion $\widetilde{W}_4$ under $\tilde{\mathbb{P}}$ such that
$$
\frac{d S^{(N)}(t)}{S^{(N)}(t)}=\text { (Something) } d t+\gamma d \widetilde{W}_4(t) .
$$
(ii) Show how to construct a Brownian motion $\widetilde{W}_2(t)$ under $\widetilde{\mathbb{P}}$ that is independent of $\widetilde{W}_1(t)$ such that $d N(t)$ may be written as
$$
d N(t)=r N(t) d t+\nu N(t)\left[\rho d \widetilde{W}_1(t)+\sqrt{1-\rho^2} d \widetilde{W}_2(t)\right] .
$$
(iii) Using Theorem 9.2.2, determine the volatility vector of $S^{(N)}(t)$. In other words, find a vector $\left(v_1, v_2\right)$ such that
$$
d S^{(N)}(t)=S^{(N)}(t)\left[v_1 d \widetilde{W}_1^{(N)}(t)+v_2 d \widetilde{W}_2^{(N)}(t)\right],
$$
where $\widetilde{W}_1(t)$ and $\widetilde{W}_2(t)$ are independent Brownian motions under $\widetilde{\mathbb{P}}^{(N)}$. Show that
$$
\sqrt{v_1^2+v_2^2}=\sqrt{\sigma^2-2 \rho \sigma \nu+\nu^2} .
$$