Quanto option). A quanto option pays off in one currency the price in another currency of an underlying asset without taking the currency conversion into account. For example, a quanto call on a British asset struck at $\$ 25$ would pay $\$ 5$ if the price of the asset upon expiration of the option is $£ 30$. To compute the payoff of the option, the price 30 is treated as if it were dollars, even though it is pounds sterling.
In this problem we consider a quanto option in the foreign exchange model of Section 9.3. We take the domestic and foreign interest rates to be constants $r$ and $r^f$, respectively, and we assume that $\sigma_1>0, \sigma_2>0$, and $\rho \in(-1,1)$ are likewise constant.
(i) From (9.3.14), show that
$$
S(t)=S(0) \exp \left\{\sigma_1 \widetilde{W}_1(t)+\left(r-\frac{1}{2} \sigma_1^2\right) t\right\} .
$$
(ii) From (9.3.16), show that
$$
Q(t)=Q(0) \exp \left\{\sigma_2 \rho \widetilde{W}_1(t)+\sigma_2 \sqrt{1-\rho^2} \widetilde{W}_2(t)+\left(r-r^f-\frac{1}{2} \sigma_2^2\right) t\right\} .
$$
(iii) Show that
$$
\frac{S(t)}{Q(t)}=\frac{S(0)}{Q(0)} \exp \left\{\sigma_4 \widetilde{W}_4(t)+\left(r-a-\frac{1}{2} \sigma_4^2\right) t\right\},
$$
where
$$
\begin{aligned}
\sigma_4 & =\sqrt{\sigma_1^2-2 \rho \sigma_1 \sigma_2+\sigma_2^2}, \\
a & =r-r^f+\rho \sigma_1 \sigma_2-\sigma_2^2,
\end{aligned}
$$
and
$$
\widetilde{W}_4(t)=\frac{\sigma_1-\sigma_2 \rho}{\sigma_4} \widetilde{W}_1(t)-\frac{\sigma_2 \sqrt{1-\rho^2}}{\sigma_4} \widetilde{W}_2(t)
$$
is a Brownian motion.
(iv) Consider a quanto call that pays off
$$
\left(\frac{S(T)}{Q(T)}-K\right)^{+}
$$
units of domestic currency at time $T$. (Note that $\frac{S(T)}{Q(T)}$ is denominated in units of foreign currency, but in this payoff it is treated as if it is a number of units of domestic currency.) Show that if at time $t \in[0, T]$ we have $\frac{S(t)}{Q(t)}=x$, then the price of the quanto call at this time is
$$
q(t, x)=x e^{-a \tau} N\left(d_{+}(\tau, x)\right)-e^{-r \tau} K N\left(d_{-}(\tau, x)\right),
$$