Chapter 9, Change of Numeraire Video Solutions, Stochastic Calculus for Finance II : Continuous-Time Models

Problem 1

This exercise provides an alternate proof of the main assertion of Theorem 9.2.2.
(i) Use Lemma 5.2.2 to prove Remark 9.2.5.
(ii) Let $S(t)$ and $N(t)$ be prices of two assets, denominated in a common currency, and assume $N(t)$ is always strictly positive. Let $\widetilde{\mathbb{P}}$ be the riskneutral measure under which the discounted asset prices $D(t) S(t)$ and $D(t) N(t)$ are martingales. Apply Remark 9.2 .5 to show that $S^{(N)}(t)=$ $\frac{S(t)}{N(t)}$ is a martingale under $\widetilde{\mathbb{P}}^{(N)}$ defined by (9.2.6).

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Problem 2

Portfolios under change of numéraire). Consider two assets with prices $S(t)$ and $N(t)$ given by
$$
\begin{aligned}
& S(t)=S(0) \exp \left\{\sigma \widetilde{W}(t)+\left(r-\frac{1}{2} \sigma^2\right) t\right\}, \\
& N(t)=N(0) \exp \left\{\nu \widetilde{W}(t)+\left(r-\frac{1}{2} \nu^2\right) t\right\},
\end{aligned}
$$
where $\widetilde{W}(t)$ is a one-dimensional Brownian motion under the risk-neutral measure $\widetilde{\mathbb{P}}$ and the volatilities $\sigma>0$ and $\nu>0$ are constant, as is the interest rate $r$. We define a third asset, the money market account, whose price per share at time $t$ is $M(t)=e^{r t}$.

Let us now denominate prices in terms of the numéraire $N$, so that the redenominated first asset price is
$$
\widehat{S}(t)=\frac{S(t)}{N(t)}
$$
and the redenominated money market account price is
$$
\widehat{M}(t)=\frac{M(t)}{N(t)} .
$$

According to Theorem 9.2.2, $d \widehat{S}(t)=(\sigma-\nu) \widehat{S}(t) d \widehat{W}(t)$, where $\widehat{W}(t)=\widetilde{W}(t)-$ $\nu t$.
(i) Compute the differential of $\frac{1}{N(t)}$.
(ii) Compute the differential of $\widehat{M}(t)$, expressing it in terms of $d \widehat{W}(t)$.

Consider a portfolio that at each time $t$ holds $\Delta(t)$ shares of the first asset and finances this by investing in or borrowing from the money market. According to the usual formula, the differential of the value $X(t)$ of this portfolio is
$$
d X(t)=\Delta(t) d S(t)+r(X(t)-\Delta(t) S(t)) d t .
$$

We define
$$
X(t)-\Delta(t) S(t)
$$

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Problem 3

(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
$$
\sqrt{v_1^2+v_2^2}=\sqrt{\sigma^2-2 \rho \sigma \nu+\nu^2} .
$$

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Problem 4

From the differential formulas (9.3.14) and (9.3.15) for the stock and discounted exchange rate in terms of the Brownian motions under the domestic risk-neutral measure, derive the differential formulas (9.3.22) and (9.3.23) for the redenominated money market account and stock discounted at the forcign interest rate and written in terms of the Brownian motions under the foreign risk-neutral measure.

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Problem 5

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
$$
q(t, x)=x e^{-a \tau} N\left(d_{+}(\tau, x)\right)-e^{-r \tau} K N\left(d_{-}(\tau, x)\right),
$$

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Problem 6

Exercise 9.6. Verify equation (9.4.16),
$$
\operatorname{For}_S(t, T) d N\left(d_{+}(t)\right)+d \text { For }_S(t, T) d N\left(d_{+}(t)\right)-K d N\left(d_{-}(t)\right)=0,
$$
in the following steps.
(i) Use (9.4.10) to show that
$$
d_{-}(t)=d_{+}(t)-\sigma \sqrt{T-t} .
$$
(ii) Use (9.4.10) to show that
$$
d_{+}^2(t)-d_{-}^2(t)=2 \log \frac{\text { For }_S(t, T)}{K} .
$$
(iii) Use (ii) to show that
$$
\text { For }_S(t, T) e^{-d_{+}^2(t) / 2}-K e^{-d_{-}^2(t) / 2}=0 .
$$
(iv) Use (9.4.8) and the Itô-Doeblin formula to show that
$$
d d_{+}(t)=\frac{1}{2 \sigma(T-t)^{3 / 2}} \log \frac{\text { For }_S(t, T)}{K} d t-\frac{3 \sigma}{4 \sqrt{T-t}} d t+\frac{1}{\sqrt{T-t}} d \widetilde{W}
$$
(v) Use (i) to show that
$$
d d_{-}(t)=d d_{+}(t)+\frac{\sigma}{2 \sqrt{T-t}} d t .
$$
(vi) Use (iv) and (v) to show that
$$
d d_{+}(t) d d_{+}(t)=d d_{-}(t) d d_{-}(t)=\frac{d t}{T-t} .
$$
(vii) Use the Itô-Doeblin formula to show that
$$
d N\left(d_{+}(t)\right)=\frac{1}{\sqrt{2 \pi}} e^{-d_{+}^2(t) / 2} d d_{+}(t)-\frac{d_{+}(t)}{2(T-t) \sqrt{2 \pi}} e^{-d_{+}^2(t) / 2} d t .
$$
(viii) Use the Itô-Doeblin formula, (v), (i), and (vi) to show that
$$
\begin{gathered}
d N\left(d_{-}(t)\right)=\frac{1}{\sqrt{2 \pi}} e^{-d_{-}^2(t) / 2} d d_{+}(t)+\frac{\sigma}{\sqrt{2 \pi(T-t)}} e^{-d_{-}^2(t) / 2} d t \\
-\frac{d_{+}(t)}{2(T-t) \sqrt{2 \pi}} e^{-d_{-}^2(t) / 2} d t .
\end{gathered}
$$
(ix) Use (9.4.8), (vii), and (iv) to show that
$$
d \text { For }_S(t, T) d N\left(d_{+}(t)\right)=\frac{\sigma \text { For }_S(t, T)}{\sqrt{2 \pi(T-t)}} e^{-d_{+}^2(t) / 2} d t .
$$
(x) Now prove (9.4.16).

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