Charles Edward Perugini (1839-1918) was an Italian-born English painter.
The topics are
- Currency Markets
- Cross-Currency Market Model
- Currency Forward Contracts and Options
- Foreign Equity Forward Contracts
- Foreign Market Futures Contracts
- Foreign Equity Options
An arbitrage-free model of the domestic security market is extended by assuming that trading in foreign assets, such as foreign risk-free bonds and foreign stocks (and their derivatives), is allowed. We will work within the classic Black-Scholes framework. More specifically, both domestic and foreign risk-free interest rates are assumed throughout to be nonnegative constants, and the foreign stock price and the exchange rate are modelled by means of geometric Brownian motions. This implies that the foreign stock price, as well as the price in domestic currency of one unit of foreign currency (i.e. the exchange rate) will have lognormal probability distributions at future times. Note that in order to avoid perfect correlation between these two processes, the underlying noise process should be modeled by means of a multidimensional, rather than a one-dimensional Brownian motion. The main goal is to establish explicit valuation formulas for various kinds of currency and foreign equity options. Also, we will provide some indications concerning the form of the corresponding hegdeing strategies. It is clear that foreign market contracts of certain kinds should be hedged both against exchange rate moments and against the fluctuations of relevant foreign equities.
\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}
Currency Markets.
Frequently, we will be interested in assets, risk-free bonds and risky stocks, in several countries simultaneously. For simplicity, we restrict
attention to just two countries, the domestic country with interest rate \(r_{d}\) and the foreign country with interest rate \(r_{f}\). We can write
\[\mu_{t}^{(d)}=e^{r_{d}t}\mbox{ and }B_{f}(t)=e^{r_{f}t}\]
for the domestic and foreign “bank accounts”, respectively. The link between the domestic and foreign economies is the exchange-rate process \({\bf Q}\) say, by which one passes from denomination in foreign to domestic currency. The fluctuations in exchange rates depend on multiplicity of factors: the interest rates \(r_{d}\), \(r_{f}\), the performance of the two economies, the policies of the two governments and central banks, the corresponding things in countries significantly linked to them, the markets’ perceptions of all of these, etc. Thus a multidimensional noise process is appropriate for modeling purposes; the simplest plausible model is geometric Brownian motion with \(d\)-dimensional noise process
\[d{\bf Q}(t)={\bf Q}(t)[\boldsymbol{\mu}_{{\bf Q}}dt+\boldsymbol{\sigma}_{{\bf Q}}d{\bf W}_{t}],{\bf Q}(0)>{\bf 0},\]
where \(\boldsymbol{\mu}_{{\bf Q}}\) is a constant drift (which may have either sign), \(\boldsymbol{\sigma}_{{\bf Q}}\) is a positive vector of volatilites, \({\bf W}_{t}\) is a standard \(d\)-dimensional Brownian motion. Solving, one obtains
\[{\bf Q}(t)={\bf Q}(0)\exp\left [\boldsymbol{\sigma}_{{\bf Q}}{\bf W}_{t}+\left (\boldsymbol{\mu}_{{\bf Q}}-\frac{1}{2}
\parallel \boldsymbol{\sigma}_{{\bf Q}}\parallel^{2}\right )^{2}t\right ],\]
where \(\parallel\cdot\parallel\) denotes the Euclidean norm in \(\mathbb{R}^{d}\).
The value of the foreign savings account \(B_{f}(t)\) is \(B_{f}(t){\bf Q}(t)\) in domestic currency, and \(B_{f}(t){\bf Q}(t)/\mu_{t}^{(d)}\) when discounted by the domestic interest rate. We write this process as \({\bf Q}^{*}(t)\)
\[{\bf Q}^{*}(t)=B_{f}(t){\bf Q}(t)/\mu_{t}^{(d)}=e^{(r_{f}-r_{d})t}{\bf Q}(t)\mbox{ for }t\in [0,T].\]
Its dynamics are given by
\[d{\bf Q}^{*}(t)={\bf Q}^{*}(t)[\boldsymbol{\mu}_{{\bf Q}}+r_{f}-r_{d})dt+\boldsymbol{\sigma}_{{\bf Q}}d{\bf W}_{t}]\]
with solution
\[{\bf Q}^{*}(t)={\bf Q}(0)=\exp\left [\boldsymbol{\sigma}_{{\bf Q}}
{\bf W}_{t}+\left (\boldsymbol{\mu}_{{\bf Q}}+r_{f}-r_{d}-\frac{1}{2}
\parallel\boldsymbol{\sigma}_{{\bf Q}}\parallel^{2}\right )t\right ].\]
To avoid arbitrage between the domestic and foreign bond markets, we need to pass to an equivalent martingale measure eliminating the drift term in the dynamics above. We have a \(d\)-dimensional noise process, and cannot expect uniqueness of equivalent martingale measure unless there are \(d\) independent traded assets available. When this is the case, there exists a unique equivalent martingale measure \(\bar{\mathbb{P}}\), called the domestic martingale measure, giving the dynamics as
\[d{\bf Q}(t)={\bf Q}(t)[(r_{d}-r_{f})dt+\boldsymbol{\sigma}_{{\bf Q}}d\tilde{{\bf W}}(t)],{\bf Q}(0)>{\bf 0}\]
with \(\tilde{{\bf W}}\) a \(\bar{\mathbb{P}}\)-Brownian motion. This is the risk-neutral probability measure for an investor reckoning everything in terms of the domestic currency. Risk-neutral valuation gives the price process of a contingent claim \(X\) as
\[\Pi_{t}(X)=e^{-r_{d}(T-t)}\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}[X|{\cal F}_{t}].\]
Consider now an agent involved in international trade, wishing to limit his exposure to adverse movements in the exchange-rate process \({\bf Q}\). He will seek to purchase an option protecting him against this, in the same way that an agent dealing in a stock \(S\) will purchase an option in the stock. Of course, an exchange rate is not a tangible asset in the sense that a stock is; nevertheless, it is possible and helpful to treat options on currency in a way closely analogous to how we treat options on stock. For this purpose, consider the forward price at time of one unit of the foreign currency to be delivered at the settlement date \(T\) in terms of the domestic currency. It is natural to call this the forward exchange rate, \(F_{{\bf Q}}(t,T)\) say. In terms of the bond-price processes, one has
\[F_{{\bf Q}}(t,T)=e^{(r_{d}-r_{f})(T-t)}{\bf Q}(t)=B_{f}(t,T){\bf Q}(t)/B_{d}(t,T)\mbox{ for }t\in [0,T],\]
a relationship known as interest-rate parity. The absence of arbitrage requires that the forward exchange premium must be the difference \(r_{d}-r_{f}\) between the two exchange rates.
Currency options may now be constructed, and priced, analogously to options on stock. For example, a standard currency European call option may be constructed with payoff
\[C_{{\bf Q}}(T)=({\bf Q}(T)-K)^{+},\]
where \({\bf Q}(T)\) is the spot exchange rate at the option’s delivery date and \(K\) is the strike price. Finding the arbitrage price of this option is formally the same as finding the price of a futures option with the forward price of the stock replaced by the forward exchange rate.
Theorem. The arbitrage price of the currency European futures call option above is
\[C_{{\bf Q}}(t)=e^{-r_{d}(T-t)}\cdot [F(t)N(d_{1}(F(t),T-t))-KN(d_{2}(F(t),T-t))],\]
where \(F(t)=F_{{\bf Q}}(t,T)\) is the forward exchange rate and
\[d_{1,2}(F,t)=\frac{\log (F/K)\pm\frac{1}{2}\sigma_{{\bf Q}}^{2}t}{\sigma_{{\bf Q}}\sqrt{t}}.\]
\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}
Cross-Currency Market Model.
All processes considered in what follows are defined on a common filtered probability space \((\Omega ,{\bf F},P)\), where the filtration \({\bf F}\) is assumed to be the \(P\)-augmentation of the natural filtration generated by a \(d\)-dimensional Brownian motion \({\bf W}=(W_{1},\cdots ,W_{d})\). The domestic and foreign interest rates, \(r_{d}\) and \(r_{f}\), are assumed to be given real numbers. Consequently, the domestic and foreign savings accounts satisfy
\[B_{t}^{d}=\exp (r_{d}\cdot t)\mbox{ and }B_{t}^{f}=\exp (r_{f}\cdot t)\mbox{ for all }t\in [0,T],\]
where \(B_{t}^{d}\) and \(B_{t}^{f}\) are denominated in units of domestic and foreign currency, respectively. The exchange rate process \(Q\), which is used to convert foreign payoffs into domestic currency, is modeled by the following stochastic differential equation
\begin{equation}{\label{museq71}}\tag{1}
dQ_{t}=Q_{t}\cdot (\mu_{Q}dt+\boldsymbol{\sigma}_{Q}\cdot d{\bf W}_{t})\mbox{ for }Q_{0}>0,
\end{equation}
where \(\mu_{Q}\in \mathbb{R}\) is a constant drift coefficient and \(\boldsymbol{\sigma}_{Q}\in \mathbb{R}^{d}\) denotes a constant volatility vector. As usual, we have
\[\boldsymbol{\sigma}_{Q}\cdot d{\bf W}_{t}=\sum_{i=1}^{d} \sigma_{Q}^{i}dW_{t}^{i}.\]
Let us make clear that the exchange rate process \(Q\) is denominated in units of domestic currency per unit of foreign currency; that is, \(Q_{t}\) represents the domestic price at time \(t\) of one unit of the foreign currency. Note that the exchange rate process \(Q\) cannot be treated on an equal basis with the price processes of domestic assets; put another way, the foreign currency cannot be seen as just an additional traded security in the domestic market model, unless the impact of the foreign interest rate is taken into account. The process \(Q\) plays an important role as a tool which allows the conversion of foreign market cash flows into units of domestic currency. Moreover, it can also play the role of an option’s underlying “asset”.
Domestic Martingale Measure.
In view of (\ref{museq71}), the exchange rate at time \(t\) equals
\[Q_{t}=Q_{0}\cdot\exp\left (\boldsymbol{\sigma}_{Q}\cdot {\bf W}_{t}
+\left (\mu_{Q}-\frac{1}{2}\cdot\parallel\boldsymbol{\sigma}_{Q}\parallel^{2}\right )\cdot t\right ).\]
Let us introduce an auxiliary process \(Q^{*}\) given by the equality
\[Q_{t}^{*}\equiv\frac{B_{t}^{f}\cdot Q_{t}}{B_{t}^{d}}=e^{(r_{f}-r_{d})t}\cdot Q_{t}\mbox{ for all }t\in [0,T].\]
It is clear that \(Q_{t}^{*}\) represents the value at time \(t\) of the foreign savings account, when converted into the domestic currency, and discounted by the current value of the domestic savings account. Moreover, it is useful to observe that \(Q^{*}\) satisfies
\[Q_{t}^{*}=Q_{0}\exp\left (\boldsymbol{\sigma}_{Q}\cdot {\bf W}_{t}+\left (\mu_{Q}+r_{f}-r_{d}-\frac{1}{2}\cdot\parallel
\boldsymbol{\sigma}_{Q}\parallel^{2}\right )\cdot t\right ),\]
or equivalently, that the dynamics of \(Q^{*}\) are
\[dQ^{*}_{t}=Q_{t}^{*}\cdot\left ((\mu_{Q}+r_{f}-r_{d})dt+\boldsymbol{\sigma}_{Q}\cdot d{\bf W}_{t}\right ).\]
It is clear that the process \(Q^{*}\) follows a martingale under the original probability measure if and only if the drift coefficient
\(\mu_{Q}\) satisfies \(\mu_{Q}=r_{d}-r_{f}\). We shall frequently make use of the process \(\tilde{B}_{t}^{f}\), which equals
\[\tilde{B}_{t}^{f}\equiv B_{t}^{f}\cdot Q_{t}=e^{r_{f}t}\cdot Q_{t}\mbox{ for all }t\in [0,T].\]
Note that \(\tilde{B}_{t}^{f}\) represents the value at time \(t\) of a unit investment in a foreign savings account, expressed in units of the domestic currency. In order to exclude arbitrage between investments in domestic and foreign bonds, we have to assume that the drift coefficient of the exchange rate process equals \(r_{d}-r_{f}\) under an equivalent probability measure \(\mathbb{P}^{*}\), hereafter referred to as the martingale measure of the dometic market, or briefly, the domestic martingale measure. It is worthwhile to observe that a martingale measure \(\mathbb{P}^{*}\) is not unique in general. Indeed, in our framework, the martingale measure \(\mathbb{P}^{*}\) is associated with a solution \(\hat{\boldsymbol{\eta}}\in\mathbb{R}^{d}\) of the following equation
\[\mu_{Q}+r_{f}-r_{d}+\boldsymbol{\sigma}_{Q}\cdot\hat{\boldsymbol{\eta}}=0,\]
for which the uniqueness of a solution need not hold in general. Still, for any solution \(\hat{\boldsymbol{\eta}}\) of this equation, the probability measure \(\mathbb{P}^{*}\) given by the usual formula (ref. Theorem \ref{mustb21} with \(\hat{\boldsymbol{\eta}}=\boldsymbol{\gamma}_{s}\))
\[\frac{d\mathbb{P}^{*}}{d\mathbb{P}}=\exp\left (\hat{\boldsymbol{\eta}}\cdot
{\bf W}_{T}-\frac{1}{2}\cdot\parallel\hat{\boldsymbol{\eta}}\parallel^{2}\cdot T\right )\mbox{ \(P\)-a.s.}\]
can play the role of a martingale measure associated with the domestic market. In addition, the process \({\bf W}^{*}\), which equals
\[{\bf W}_{t}^{*}={\bf W}_{t}-\hat{\boldsymbol{\eta}}t\mbox{ for all }t\in [0,T]\]
follows a \(d\)-dimensional Brownian motion under \(\mathbb{P}^{*}\) (with respect to the underlying filtration).
The uniqueness of the martingale measure can be gained by introducing the possibility of trading in additional foreign or domestic assets, say, foreign or domestic stocks. In other words, the uniqueness of a martingale measure holds if the number of (non-redundant) traded assets, including the domestic savings account, equals \(d+1\), where \(d\) stands for the dimensionality of the underlying Brownian motion. For instance, if no domestic stocks are traded and only one foreign stock is considered, to guarantee the uniqueness of a martingale measure \(\mathbb{P}^{*}\), and thus the completeness of the market, it is enough to assume that \({\bf W}\), and thus also \({\bf W}^{*}\), is a two-dimensional Brownian motion. In such a case, the market model involves three primary securities; the domestic and foreign savings account (or equivalently, domestic and foreign bonds) and a foreign stock. In any case, the dynamics of the exchange rate \(Q\) under the domestic martingale measure \(\mathbb{P}^{*}\) are easily seen to be
\begin{equation}{\label{museq73}}\tag{2}
dQ_{t}=Q_{t}\cdot\left ((r_{d}-r_{f})dt+\boldsymbol{\sigma}_{Q}\cdot d{\bf W}_{t}^{*}\right )\mbox{ for }Q_{0}>0,
\end{equation}
where \({\bf W}^{*}\) follows a \(d\)-dimensional Brownian motion under \(\mathbb{P}^{*}\). Intuitively, the domestic martingale measure \(\mathbb{P}^{*}\) is a risk-neutral probability as seen from the perspective of a domestic investor; that is, an investor who constantly denominates the prices of all assets in units of domestic currency. It is clear that the arbitrage price \(\Pi_{t}(X)\), in units of domestic currency, of any contingent claim \(X\), which settles at time \(T\) and is also denominated in the domestic currency, equals
\begin{equation}{\label{museq74}}\tag{3}
\Pi_{t}(X)=e^{-r_{d}(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}[X|{\cal F}_{t}].
\end{equation}
If a time \(T\) claim \(Y\) is denominated in units of foreign currency, its arbitrage price at time \(t\), expressed in units of domestic currency, is given by the formula
\[\Pi_{t}(Y)=e^{-r_{d}(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}[Q_{T}\cdot Y|{\cal F}_{t}].\]
Notice that the arbitrage price of such a claim can be alternatively evaluated using the martingale measure associated with the foreign market, and ultimately converted into domestic currency using the current exchange rate \(Q_{t}\). For this purpose, we need to introduce an arbitrage-free probability measure associated with the foreign market, referred to as the {\bf foreign martingale measure}.
Foreign Martingale Measure.
We now take the perspective of a foreign-based investor; that is, an investor who consistently denominates his or her profits and losses in units of foreign currency. Since \(Q_{t}\) is the price at time \(t\) of one unit of foreign currency in the domestic currency, it is evident that the price at time \(t\) of one unit of the domestic currency, expressed in units of foreign currency, equals \(R_{t}=1/Q_{t}\). From Ito’s formula, we have
\[dQ_{t}^{-1}=Q_{t}^{-2}dQ_{t}+Q_{t}^{-3}d\langle Q,Q\rangle\mathbb{E}_{t},\]
or more explicitly
\[dR_{t}=-R_{t}\cdot\left ((r_{d}-r_{f})dt+\boldsymbol{\sigma}_{Q}
\cdot d{\bf W}_{t}^{*}\right )+R_{t}\cdot\parallel\boldsymbol{\sigma}_{Q}\parallel^{2}dt.\]
Therefore, the dynamics of \(R\) under \(\mathbb{P}^{*}\) are given by the expression
\[dR_{t}=R_{t}\cdot\left ((r_{f}-r_{d})dt-\boldsymbol{\sigma}_{Q}\cdot (d{\bf W}_{t}^{*}-\boldsymbol{\sigma}_{Q}dt)\right ).\]
Equivalently,
\begin{equation}{\label{museq75}}\tag{4}
dR_{t}^{*}=-R_{t}^{*}\cdot\boldsymbol{\sigma}_{Q}\cdot (d{\bf W}_{t}^{*}-\boldsymbol{\sigma}_{q}dt),
\end{equation}
where we denote by \(R^{*}\) the following process
\[R_{t}^{*}\equiv R_{t}\cdot e^{(r_{d}-r_{f})t}=e^{-r_{f}t}\cdot R_{t}\cdot B_{t}^{d}\mbox{ for all }t\in [0,T].\]
Observe that \(R^{*}\) represents the price process of the domestic savings account, expressed in units of foreign currency, and discounted using the foreign risk-free interest rate. By virtue of (\ref{museq75}), it is easily seen that the process \(R^{*}\) follows a martingale under a probability measure \(\tilde{\mathbb{P}}\), equivalent to \(\mathbb{P}^{*}\), which satisfies
\begin{equation}{\label{museq76}}\tag{5}
\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}^{*}}=\eta_{T}\mbox{ \(\mathbb{P}^{*}\)-a.s.}
\end{equation}
on \((\Omega ,{\cal F}_{T})\), where \(\eta\) equals
\begin{equation}{\label{museq77}}\tag{6}
\eta_{t}=\exp\left (\boldsymbol{\sigma}_{Q}\cdot {\bf W}_{t}^{*}-
\frac{1}{2}\cdot\parallel\boldsymbol{\sigma}_{Q}\parallel^{2}\cdot t\right )\mbox{ for all }t\in [0,T].
\end{equation}
Any probability measure \(\tilde{\mathbb{P}}\) defined in this way is referred to as the martingale measure of the foreign market. If the uniqueness of a domestic martingale measure \(\mathbb{P}^{*}\) is not valid, the uniqueness of a foreign market martingale measure \(\tilde{\mathbb{P}}\) does not hold either. However, under any foreign martingale measure \(\tilde{\mathbb{P}}\), we have
\[dR_{t}^{*}=-R_{t}^{*}\cdot\boldsymbol{\sigma}_{Q}\cdot d\tilde{{\bf W}}_{t}^{*},\]
where the process \(\tilde{{\bf W}}_{t}={\bf W}_{t}-\boldsymbol{\sigma}_{Q}t\) follows a \(d\)-dimensional Brownian motion under \(\tilde{\mathbb{P}}\). It is useful to observe that the dynamics of \(R\) under the martingale measure \(\tilde{\mathbb{P}}\) are given by the following counterpart of (\ref{museq73})
\begin{equation}{\label{museq79}}\tag{7}
dR_{t}=R_{t}\left ((r_{f}-r_{d})dt-\boldsymbol{\sigma}_{Q}\cdot d\tilde{{\bf W}}_{t}\right ).
\end{equation}
In financial interpretation, a foreign market martingale measure \(\tilde{\mathbb{P}}\) is any probability measure on \((\Omega ,{\cal F}_{T})\) equivalent to \(P\) that excludes arbitrage opportunities between risk-free and risky investments in both economies, as seen from the perspective of a foreign-based investor. For any attainable contingent claim \(X\), which settles at time \(T\) and is denominated in units of domestic currency, the arbitrage price at time \(t\) in units of foreign currency is given by the equality
\[\tilde{\Pi}_{t}(X)=e^{-r_{f}(T-t)}\cdot \mathbb{E}_{\tilde{\mathbb{P}}}[R_{T}\cdot X|{\cal F}_{t}]\mbox{ for all }t\in [0,T].\]
We assume here that \(\tilde{\mathbb{P}}\) is associated with \(\mathbb{P}^{*}\) through (\ref{museq76}).
Proposition. The following formula is valid for any \({\cal F}_{T}\)-measurable random variable \(X\) provided that the conditional expectation is well-defined.
\begin{equation}{\label{museq710}}\tag{8}
\mathbb{E}_{\tilde{\mathbb{P}}}[X|{\cal F}_{t}]=\mathbb{E}_{\mathbb{P}^{*}}\left [\left .X\cdot\exp\left (
\boldsymbol{\sigma}_{Q}\cdot ({\bf W}_{T}^{*}-{\bf W}_{t}^{*})-
\frac{1}{2}\cdot\parallel\boldsymbol{\sigma}_{Q}\parallel^{2}\cdot (T-t)\right )\right |{\cal F}_{t}\right ].\end{equation}
Proof. By virtue of the Bayes formula in Proposition \ref{musa04}, we get
\[\mathbb{E}_{\tilde{\mathbb{P}}}[X|{\cal F}_{t}]=\frac{\mathbb{E}_{\mathbb{P}^{*}}[\eta_{T}\cdot X|
{\cal F}_{t}]}{\mathbb{E}_{\mathbb{P}^{*}}[\eta_{T}|{\cal F}_{t}]},\]
and thus (observe that \(\eta\) follows a martingale under \(\mathbb{P}^{*}\))
\[\mathbb{E}_{\tilde{\mathbb{P}}}[X|{\cal F}_{t}]=\frac{\mathbb{E}_{\mathbb{P}^{*}}[\eta_{T}\cdot X|
{\cal F}_{t}]}{\eta_{t}}=\mathbb{E}_{\mathbb{P}^{*}}\left [\left .\frac{\eta_{T}\cdot X}
{\eta_{t}}\right |{\cal F}_{t}\right ]\]
as expected from (\ref{museq77}). \(\blacksquare\)
Foreign Stock Price Dynamics.
Let \(S_{t}^{f}\) be the foreign currency price at time \(t\) of a foreign traded stock whch pays no dividends. In order to exclude arbitrage, we assume that the dynamics of the price process \(S^{f}\) under the foreign martingale measure \(\tilde{\mathbb{P}}\) are
\begin{equation}{\label{museq711}}\tag{9}
dS_{t}^{f}=S_{t}^{f}\cdot (r_{f}dt+\boldsymbol{\sigma}_{S^{f}}\cdot\tilde{{\bf W}}_{t})\mbox{ for }S_{0}^{f}>0
\end{equation}
with a constant volatility \(\boldsymbol{\sigma}_{S^{f}}\in \mathbb{R}^{d}\). This means that the stock price process \(S^{f}\) follows
\begin{equation}{\label{museq712}}\tag{10}
dS_{t}^{f}=S_{t}^{f}\cdot\left ((r_{f}-\boldsymbol{\sigma}_{Q}\cdot
\boldsymbol{\sigma}_{S^{f}})dt+\boldsymbol{\sigma}_{S^{f}}\cdot d{\bf W}_{t}^{*}\right )
\end{equation}
under the domestic martingale measure \(\mathbb{P}^{*}\) associated with \(\tilde{\mathbb{P}}\). For the purpose of pricing foreign equity options, we will sometimes find it useful to convert the price of the underlying foreign stock into the domestic currency. We write \(\tilde{S}_{t}^{f}=Q_{t}\cdot S_{t}^{f}\) to denote the price of a foreign stock \(S^{f}\) expressed in units of domestic currency. Using Ito’s formula, and the dynamics under \(\mathbb{P}^{*}\) of the exchange rate \(Q\), which are
\begin{equation}{\label{museq713}}\tag{11}
dQ_{t}=Q_{t}\cdot\left ((r_{d}-r_{f})dt+\boldsymbol{\sigma}_{Q}\cdot d{\bf W}_{t}^{*}\right ),
\end{equation}
one finds that under the domestic martingale measure \(\mathbb{P}^{*}\), the process \(\tilde{S}_{t}^{f}\) satisfies
\begin{equation}{\label{museq714}}\tag{12}
d\tilde{S}_{t}^{f}=\tilde{S}_{t}^{f}\cdot\left (r_{d}dt+
(\boldsymbol{\sigma}_{Q}+\boldsymbol{\sigma}_{S^{f}})\cdot d{\bf W}_{t}^{*}\right ).
\end{equation}
The last equality shows that the price process \(\tilde{S}^{f}\) behaves as the price process of a domestic stock in the classic Black-Scholes framework; however, the corresponding volatility coefficient is equal to the superposition \(\boldsymbol{\sigma}_{Q}+ \boldsymbol{\sigma}_{S^{f}}\) of two volatilities; the foreign stock price volatility and the exchange rate volatility. By defining in the usual
way the class of admissible trading strategies, one may now easily construct a market model in which there is no arbitrage between investments in foreign and domestic bonds and stocks.
\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}
Currency Forward Contracts and Options.
We consider derivative securities whose value depends exclusively on the fluctuations of exchange rate \(Q\), as opposed to those securities which depend also on some foreign equities.
Forward Exchange Rate.
Let us consider a foreign exchange forward contract, written at time \(t\), which settles at the future date \(T\). The asset to be delivered by the party assuming a short position in the contract is a pre-specified amount of foreign currency, say \(1\) unit. The party who assumes a long position in a currency forward contract is obliged to pay a certain number of units of a domestic currency, the {\bf delivery price}. As usual, the delivery price that makes the forward contract worthless at time \(t\leq T\) is called the forward price at time \(t\) of one unit of the foreign currency to be delivered at the settlement date \(T\). In the present context, it is natural to refer to this forward price as the forward exchange rate. We will write \(F_{Q}(t,T)\) to denote the forward exchange rate.
Proposition. The forward exchange rate \(F_{Q}(t,T)\) at time \(t\) for the settlement date \(T\) is given by the following formula
\begin{equation}{\label{museq715}}\tag{13}
F_{Q}(t,T)=e^{(r_{d}-r_{f})(T-t)}\cdot Q_{t}\mbox{ for all }t\in [0,T].
\end{equation}
Proof. It is easily seen that if (\ref{museq715}) does not hold, risk-free profitable opportunities arise between the domestic and foreign market. \(\blacksquare\)
Relationship (\ref{museq715}), commonly known as the interest rate parity, asserts that the forward exchange premium must equal, in the market equilibrium, the interest rate differential \(r_{d}-r_{f}\). A relatively simple version of the interest rate parity still holds even when the domestic and foreign interest rates are no longer deterministic constants, but follow stochastic processes. Under uncertain interest rates, we need to introduce the price processes \(p_{d}(t,T)\) and \(p_{f}(t,T)\) of the domestic and foreign zero-coupon bonds with maturity \(T\). Suppose that zero-coupon bonds with maturity \(T\) are traded in both domestic and foreign markets. Then equality (\ref{museq715}) may be extended to cover the case of stochastic interest rates. Indeed, it is not hard to show, by means of no-arbitrage arguments, that
\begin{equation}{\label{museq716}}\tag{14}
F_{Q}(t,T)=\frac{p_{f}(t,T)}{p_{d}(t,T)}\cdot Q_{t}\mbox{ for all }t\in [0,T],
\end{equation}
where \(p_{d}(t,T)\) and \(p_{f}(t,T)\) stand for the respective time \(t\) prices of the domestic and foreign zero-coupon bonds with maturity \(T\). Notice that in (\ref{museq716}), both \(p_{d}(t,T)\) and \(p_{f}(t,T)\) are to be seen as the domestic and foreign discount factors rather than the prices. Indeed, it is natural to express prices in units of the corresponding currencies, while discount factors are simply the corresponding real numbers. Finally, it follows immediately from (\ref{museq713}) that for any fixed settlement date \(T\), the forward price dymanics under the martingale measure (of the domestic economy) \(\mathbb{P}^{*}\) are
\begin{equation}{\label{museq717}}\tag{15}
dF_{Q}(t,T)=F_{Q}(t,T)\cdot\boldsymbol{\sigma}_{Q}\cdot d{\bf W}_{t}^{*},
\end{equation}
and \(F_{Q}(T,T)=Q_{T}\). In what follows, we will use the last formula as a convenient argument to show that the forward and futures exchange rates agree (at least as long as the interest rates are deterministic).
Currency Option Valuation Formula.
As a first example of a currency option, we consider a standard European call option, whose payoff at the expiry date \(T\) equals
\[C_{T}^{Q}\equiv N\cdot (Q_{T}-K)^{+},\]
where \(Q_{T}\) is the spot price of the deliverable currency (i.e. the spot exchange rate at the option’s expiry date), \(K\) is the strike price in units of domestic currency per foreign unit, and \(N>0\) is the nominal value of the option, expressed in units of the underlying foreign currency. It is clear that payoff from the option is expressed in the domestic currency; also, there is no loss of generality if we assume that \(N=1\). Summarizing, we consider an option to buy one unit if a foreign currency at a pre-specified price \(K\), which may be exercised at the date \(T\) only.
\begin{equation}{\label{musp722}}\tag{16}\mbox{}\end{equation}
Proposition \ref{musp722}. The arbitrage price, in units of domestic currency, of a currency European call option is given by the risk-neutral valuation formula
\begin{equation}{\label{museq718}}\tag{17}
C_{t}^{Q}=e^{-r_{d}(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .
(Q_{T}-K)^{+}\right |{\cal F}_{t}\right ]\mbox{ for all }t\in [0,T].
\end{equation}
Moreover, the price \(C_{t}^{Q}\) is given by the following expression
\[C_{t}^{Q}=Q_{t}\cdot e^{-r_{f}(T-t)}\cdot N(h_{1}(Q_{t},T-t))-K\cdot e^{-r_{d}(T-t)}\cdot N(h_{2}(Q_{t},T-t)),\]
where \(N\) is the standard Gaussian cumulative distribution function, and
\[h_{1,2}(q,t)=\frac{\ln q-\ln K+(r_{d}-r_{f}\pm\frac{1}{2}\cdot \boldsymbol{\sigma}_{Q}^{2})\cdot t}
{\boldsymbol{\sigma}_{Q}\cdot\sqrt{t}}.\]
Proof. Let us first examine a trading strategy in risk-free domestic and foreign bonds, which we call a {\bf currency trading strategy} in what follows. Formally, by a currency trading strategy we mean an adapted stochastic process \(\boldsymbol{\phi}=(\phi^{1},\phi^{2})\). In financial interpretation, \(\phi^{1}\cdot\tilde{B}_{t}^{f}\) and \(\phi^{2}\cdot B_{t}^{d}\) represent the amounts of money invested at time \(t\) in foreign and domestic bonds. It is important to note that both amounts are expressed in units of domestic currency. A currency trading strategy \(\boldsymbol{\phi}\) is said to be self-financing if its wealth process \(V(\boldsymbol{\phi})\), which equals
\[V_{t}(\boldsymbol{\phi})=\phi_{t}^{1}\cdot\tilde{B}_{t}^{f}+\phi^{2}_{t}\cdot B_{t}^{d}\mbox{ for all }t\in [0,T],\]
where \(\tilde{B}_{t}^{f}=B_{t}^{f}\cdot Q_{t}\) and \(B_{t}^{d}=e^{r_{d}t}\), satisfies the following relationship
\[dV_{t}(\boldsymbol{\phi})=\phi^{1}_{t}d\tilde{B}_{t}^{f}+\phi^{2}_{t}dB_{t}^{d}.\]
For the discounted wealth process \(V_{t}^{*}(\boldsymbol{\phi})=e^{-r_{d}t}\cdot V_{t}(\boldsymbol{\phi})\) of a self-financing currency trading strategy, we easily get
\[dV_{t}^{*}(\boldsymbol{\phi})=\phi_{t}^{1}d(e^{-r_{d}t}\cdot\tilde{B}_{t}^{f})=\phi^{1}_{t}dQ_{t}^{*}.\]
On the other hand, by virtue of (\ref{museq713}), the dynamics of the process \(Q^{*}\), under the domestic martingale measure \(\mathbb{P}^{*}\), are given by the expression
\[dQ_{t}^{*}=\sigma_{Q}\cdot Q_{t}^{*}dW_{t}^{*}.\]
Therefore, the discounted wealth \(V^{*}(\boldsymbol{\phi})\) of any self-financing currency trading strategy \(\boldsymbol{\phi}\) follows a martingale under \(\mathbb{P}^{*}\). This justifies the risk-neutral valuation formula (\ref{museq718}). Taking into account the equality \(Q_{T}=\tilde{B}_{T}^{f}\cdot e^{-r_{f}t}\), one gets also
\begin{align*}
C_{t}^{Q} & =e^{-r_{d}(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left . (Q_{T}-K)^{+}\right |{\cal F}_{t}\right ]\\
& =e^{-r_{f}T}\cdot e^{-r_{d}(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .
(\tilde{B}_{T}^{f}-K\cdot e^{r_{f}T})^{+}\right |{\cal F}_{t}\right ]\\
& =e^{-r_{f}T}\cdot C(\tilde{B}_{t}^{f},T-t,K\cdot e^{r_{f}T},r_{d},\sigma_{Q}),
\end{align*}
where \(C\) stands for the standard Black-Scholes call option price. More explicitly, we have
\begin{align*}
C_{t}^{Q} & =e^{-r_{f}T}\cdot\left (\tilde{B}_{t}^{f}\cdot N(d_{1}(
\tilde{B}_{t}^{f},T-t))-K\cdot e^{r_{f}T}\cdot e^{-r_{d}(T-t)}\cdot N(d_{2}(\tilde{B}_{t}^{f},T-t))\right )\\
& =Q_{t}\cdot e^{-r_{f}(T-t)}\cdot N(d_{1}(\tilde{B}_{t}^{f},T-t))-K\cdot e^{-r_{d}(T-t)}\cdot N(d_{2}(\tilde{B}_{t}^{f},T-t)).
\end{align*}
This proves the formula since
\[d_{i}(\tilde{B}_{t}^{f},T-t,K\cdot e^{r_{f}T},r_{d},\sigma_{Q})=h_{i}(Q_{t},T-t)\]
for \(i=1,2\). Finally, one finds immediately that the first component of the self-financing currency strategy that replicates the options equals
\[\phi_{t}^{1}=e^{-r_{f}T}\cdot N(d_{1}(\tilde{B}_{t}^{f},T-t))=e^{-r_{f}T}\cdot N(h_{1}(Q_{t},T-t)).\]
Therefore, to hedge a short position, the writer of the currency call should invest at time \(t\leq T\) the amount (expressed in units of foreign currency)
\[\phi_{t}^{1}\cdot B_{t}^{f}=e^{-r_{f}(T-t)}\cdot N(h_{1}(Q_{t},T-t))\]
in foreign market risk-free bonds (or equivalently, in the foreign savings account). In addition, he or she should also invest the amount (denominated in domestic currency)
\[C_{t}^{Q}-Q_{t}\cdot e^{-r_{f}(T-t)}\cdot N(h_{1}(Q_{t},T-t))\]
in the domestic savings account. \(\blacksquare\)
A comparison of the currency option valuation formula established in Proposition \ref{musp722} with expression (\ref{museq620}) shows that the exchange rate \(Q\) can be formally seen as the price of a fictitious domestic “stock”. Under such a convention, the foreign interest rate \(r_{f}\) can be interpreted as a dividend yield that is continuously paid by this fictitious stock.
It is easy to derive the put-call relationship for currency options. Indeed, the payoff in domestic currency of a portfolio composed of one long call option and one short put option is
\[C_{T}^{Q}-P_{T}^{Q}=(Q_{T}-K)^{+}-(K-Q_{T})^{+}=Q_{T}-K,\]
where we assume that the options are written on one unit if foreign currency. Consequently, for any \(t\in [0,T]\), we have
\begin{equation}{\label{museq719}}\tag{18}
C_{t}^{Q}-P_{t}^{Q}=e^{-r_{f}(T-t)}\cdot Q_{t}-e^{-r_{d}(T-t)}\cdot K.
\end{equation}
Suppose that the strike level \(K\) is equal to the current value of the forward exchange rate \(F_{Q}(t,T)\). By substituting (\ref{museq715}) into (\ref{museq719}), we get
\[C_{t}^{Q}-P_{t}^{Q}=0\mbox{ for all }t\in [0,T]\]
so that the arbitrage price of the call option with exercise price \(K\) equal to the forward exchange rate \(F_{Q}(t,T)\) coincides with the value of the corresponding put option.
We may also rewrite the currency option valuation formula of Proposition \ref{musp722} in the following way
\[C_{t}^{Q}=e^{-r_{d}(T-t)}\cdot\left (F_{t}\cdot N(\tilde{d}_{1}(F_{t},T-t))-K\cdot N(\tilde{d}_{2}(F_{t},T-t))\right ),\]
where \(F_{t}=F_{Q}(t,T)\) and
\[\tilde{d}_{1,2}(x,t)=\frac{\ln x-\ln K\pm\frac{\sigma_{Q}^{2}}{2}\cdot t}
{\sigma_{Q}\cdot\sqrt{t}}\mbox{ for all }(x,t)\in \mathbb{R}_{+}\times (0,T].\]
This shows that the currency option valuation formula can be seen as a variant of the Black futures formula (\ref{museq68}). Furthermore, it is possible to reexpress the replicating strategy of the option in terms of domestic bonds and currency forward contracts. Let us mention that under the present assumptions of deterministic domestic and foreign interest rates, the distinction between the currency futures price and forward exchange rate is not essential. In market practice, currency options are frequently hedged by taking positions in forward and futures contracts, rather than by investing in foreign risk-free bonds. Similar strategies are used to hedge the risk associated with options written on a foreign stock. Hence there is a need to analyze investments in foreign market futures contracts in some detail.
\begin{equation}{\label{d}}\tag{D}\mbox{}\end{equation}
Foreign Equity Forward Contracts.
In a global equity market, an investor may link his foreign stock and currency exposures in a large varieties of ways. More specifically, he or she may choose to combine his or her investments in foreign equities with different degrees of protection against adverse moves in exchange rates and stock prices, using forward and futures contracts as well as a variety of options.
Forward Price of a Foreign Stock.
We will first consider an ordinary forward contract with a foreign stock being the underlying asset to be delivered; that is, an agreements to buy a stock on a certain date at a certain delivery price in a specified currency. It is natural to distinguish between the two following cases
(i) when the delivery price \(K^{f}\) is denominated in the foreign currency,
(ii) when it is expressed in the domestic currency, in this case, the delivery price will be denoted by \(K^{d}\).
Let us clarify that in both situations, the value of the forward contract at the settlement date \(T\) is equal to the spread between the stock price at this date and the delivery price expressed in foreign currency. The terminal payoff is then converted into units of domestic currency at the exchange rate that prevails at the settlement date \(T\). Summarizing, in units of domestic currency, the terminal payoffs from the long positions are
\[V_{T}^{d}(K^{f})=Q_{T}\cdot (S_{T}^{f}-K^{f})\]
in the first case, and
\[V_{T}^{d}(K^{d})=Q_{T}\cdot (S_{T}^{f}-K_{d}/R_{T})=(Q_{T}\cdot S_{T}^{f}-K^{d})=(\tilde{S}_{T}^{f}-K^{d})\]
if the second case is considered.
Case (i). Observe that the foreign-currency payoff at settlement of the forward contract equals \(X_{T}=S_{T}^{f}-K^{f}\). Therefore its value at time \(t\), denominated in the foreign currency, is
\[V_{t}^{f}(K^{f})=e^{-r_{f}(T-t)}\cdot \mathbb{E}_{\tilde{\mathbb{P}}}[S_{T}^{f}-K^{f}|
{\cal F}_{t}]=S_{t}^{f}-e^{-r_{f}(T-t)}\cdot K^{f}.\]
Consequently, when expressed in the domestic currency, the value of the contract at time \(t\) equals
\[V_{t}^{d}(K^{f})=Q_{t}\cdot (S_{t}^{f}-e^{-r_{f}(T-t)}\cdot K^{f}).\]
We conclude that the forward price of the stock \(S^{f}\), expressed in units of foreign currency, equals
\[F_{S^{f}}^{f}(t,T)=e^{r_{f}(T-t)}\cdot S_{t}^{f}\mbox{ for all }t\in [0,T].\]
Case (ii). In this case, equality (\ref{museq74}) yields immediately
\[V_{t}^{d}(K^{d})=e^{-r_{d}(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}[\tilde{S}_{T}^{f}-K^{d}|{\cal F}_{t}];\]
hence, by virtue of (\ref{museq714}), the domestic-currency value of the forward contract with the delivery price \(K^{d}\) denominated in domestic currency equals
\[V_{t}^{d}(K^{d})=Q_{t}\cdot S_{t}^{f}-e^{-r_{d}(T-t)}\cdot K^{d}.\]
This implies that the forward price of a foreign stock in domestic currency
equals
\[F_{S^{f}}^{d}=e^{r_{d}(T-t)}\cdot\tilde{S}_{t}^{f}\mbox{ for all }t\in [0,T],\]
so that, somewhat surprisingly, it is independent of the foreign risk-free interest rate \(r_{f}\), To explain this apparent parodox, observe that in order to determine the forward price \(F_{S^{f}}^{d}(t,T)\), one can find first the delivery price from the perspective of the foreign-based investor, \(F_{S^{f}}^{f}(t,T)\), and then convert its value into domestic currency using the appropriate forward exchange rate. Indeed, such calculations yield
\[F_{S^{f}}^{f}(t,T)\cdot F_{Q}(t,T)=e^{r_{f}(T-t)}\cdot S_{t}^{f}\cdot
e^{(r_{d}-r_{f})(T-t)}\cdot Q_{t}=e^{r_{d}(T-t)}\cdot Q_{t}\cdot S_{t}=F_{S^{f}}^{d}(t,T),\]
so that the interest rate \(r_{f}\) drops out from the final result.
Quanto Forward Contracts.
The aim here is to examine a quanto forward contract on a foreign stock (Generally speaking, a financial asset is termed to be a quanto product if it is denominated in a currency other than that in which it is usually traded.). Such a contract is also known as a guaranteed exchange rate forward contract (a GER forward contract for short). To describe the intuition that underpins the concept of a quanto forward contract, let us consider an investor who expects a certain foreign stock to appreciate significantly over the next period, and who wishes to capture this appreciation in his or her portfolio. Buying the stock, or taking a long position in it through a forward contract or call option, leaves the investor exposed to exchange rate risk. To avoid having his return depend on the performance of the foreign currency, he needs a guarantee that he can close his foreign stock position at an exchange rate close to the one that prevails at present. This can be done by entering a quanto forward or option contract in a foreign stock. We start by defining precisely what is meant by a quanto forward contract in a foreign stock \(S^{f}\). As before, the payoff of a guaranteed exchange rate forward contract on a foreign stock at settlement date \(T\) is the difference between the stock price st time \(T\) and the delivery price denominated in the foreign currency, say \(K^{f}\). However, this payoff is converted into domestic currency at a predetermined exchange rate, denoted by \(\bar{Q}\) in what follows. More fomally, denoting by \(V_{t}^{d}(K^{f},\bar{Q})\) the time \(t\) value in domestic currency of the quanto fprward contract, we have
\[V_{T}^{d}(K^{f},\bar{Q})=\bar{Q}\cdot (S_{T}^{f}-K^{f}).\]
We wish to determine the right value of such a cpntract at time \(t\) before the settlement. Notice that the terminal payoff of a quanto forward contract does not account for the future exchange rate fluctuations during the life of a contract. Nevertheles, as we shall see in what follows, its value \(V_{t}^{d}(K^{f},\bar{Q})\) depends on the volatlity coefficient \(\boldsymbol{\sigma}_{Q}\) of the exchange rate process; more precisely, on the inner product \(\boldsymbol{\sigma}_{Q}\cdot\boldsymbol{\sigma}_{S^{f}}\) that determines the instantaneous covariance between the logarithmic returns of the stock price and the exchange rate. By virtue of the risk-neutral valuation formula, the value at time \(t\) of the quanto forward contract equals (in domestic currency)
\[V_{t}^{d}(K^{f},\bar{Q})=\bar{Q}\cdot e^{-r_{d}(T-t)}\cdot\left (
\mathbb{E}_{\mathbb{P}^{*}}[S_{T}^{f}|{\cal F}_{t}]-K^{f}\right ).\]
To find the conditional expectation \(\mathbb{E}_{\mathbb{P}^{*}}[S_{T}^{f}|{\cal F}_{t}]\), observe that by virtue of (\ref{museq712}), the process \(\hat{S}_{t}=e^{-\delta t}\cdot S_{t}^{f}\) follows a martingale under \(\mathbb{P}^{*}\), provided that we take \(\delta =r_{f}-\boldsymbol{\sigma}_{Q}\cdot\boldsymbol{\sigma}_{S^{f}}\). Consequently, we find easily that
\[\mathbb{E}_{\mathbb{P}^{*}}[S_{T}^{f}|{\cal F}_{t}]=e^{\delta T}\cdot \mathbb{E}_{\mathbb{P}^{*}}[
\hat{S}_{T}|{\cal F}_{t}]=e^{\delta T}\cdot\hat{S}_{t}=e^{\delta (T-t)}\cdot S_{t}^{f},\]
and thus
\begin{equation}{\label{museq721}}\tag{19}
V_{t}^{d}(K^{f},\bar{Q})=\bar{Q}\cdot e^{-r_{d}(T-t)}\cdot\left (e^{(r_{f}-\boldsymbol{\sigma}_{Q}\cdot
\boldsymbol{\sigma}_{S^{f}})(T-t)}\cdot S_{t}^{f}-K^{f}\right ).
\end{equation}
This in turn implies that the forward price at time \(t\) associated with the quanto forward contract that settles at time \(T\) equals (in units of foreign qurrency)
\[\hat{F}_{S^{f}}^{f}(t,T)=e^{(r_{f}-\boldsymbol{\sigma}_{Q}\cdot
\boldsymbol{\sigma}_{S^{f}})(T-t)}\cdot S_{t}^{f}=\mathbb{E}_{\mathbb{P}^{*}}[S_{T}^{f}|{\cal F}_{t}].\]
It is interesting to note that \(\hat{F}_{S^{f}}^{f}(t,T)\) is simply the conditional expectation of the stock price at the settlement date \(T\), as seen at time \(t\) from the perspective of a domestic-based investor. Furthermore, at least when \(\lambda =\boldsymbol{\sigma}_{Q}\cdot\boldsymbol{\sigma}_{S^{f}})\geq 0\), it can also be interpreted as the forward price of a fictitious dividend-paying stock with \(\lambda =\boldsymbol{\sigma}_{Q}\cdot\boldsymbol{\sigma}_{S^{f}})\) playing the role of the dividend yield.
\begin{equation}{\label{e}}\tag{E}\mbox{}\end{equation}
Foreign Market Futures Contracts.
Let us consider an investor who assumes positions in foreign market futures contracts. We need to translate, in an appropriate way, the marking to market feature of futures contracts. To this end, we assume that the profits or losses from futures positions are immediately (i.e. continuously) converted into domestic currency. Let us start by examing investments in foreign market futures contracts in a discrete-time framework. Suppose that \(f_{t}=f_{S^{f}}(t,T)\) represents the foreign market futures price of some asset \(S^{f}\). We consider a finite collection of dates, \(t_{0}=0<t_{1}< \cdots <t_{N}=T\), that is assumed to represent the set of dates when the futures contracts are marked to market. If at any date \(t_{i}\) an investor assumes \(\alpha_{t_{i}}\) positions in foreign market futures contracts, and then holds the portfolio unchanged up to the date \(t_{i+1}>t_{i}\), the cumulative profits or losses incurred by the investor up to the terminal date \(T\) are given by the expression
\begin{equation}{\label{museq2061}}\tag{20}
\sum_{i=0}^{N-1}\alpha_{t_{i}}\cdot Q_{t_{i+1}}\cdot (f_{t_{i+1}}-f_{t_{i}}),
\end{equation}
where we assume that all cash flows resulting from the marking to market procedure are immediately converted into domestic currency. The sum (\ref{museq2061}) may be given the following equivalent form
\[\sum_{i=0}^{N-1}\alpha_{t_{i}}\cdot Q_{t_{i}}\cdot (f_{t_{i+1}}-
f_{t_{i}})+\sum_{i=0}^{N-1}\alpha_{t_{i}}\cdot (Q_{t_{i+1}}-Q_{t_{i}})\cdot (f_{t_{i+1}}-f_{t_{i}}).\]
Assume now that the number of resettlement dates tends to infinity, the time interval \([0,T]\) being fixed. In view of properties of the cross-variation of continuous semimartingales, the equality above leads to the following self-financing condition for continuously re-balanced portfolios.
\begin{equation}{\label{musd741}}\tag{21}\mbox{}\end{equation}
Definition \ref{musd741}. An adapted process \(\tilde{\boldsymbol{\phi}}=(\tilde{\phi}^{1},\tilde{\phi}^{2})\) is a self-financing trading strategy in foreign market futures contracts with the price process \(f_{t}=f_{S^{f}}(t,T)\), and in a domestic savings account with the price process \(B_{t}^{d}\), if
\begin{equation}{\label{museq722}}\tag{22}
dV_{t}(\tilde{\boldsymbol{\phi}})=\tilde{\phi}^{1}_{t}\cdot (
Q_{t}df_{t}+d\langle Q,f\rangle\mathbb{E}_{t})+\tilde{\phi}_{t}^{2}\cdot dB_{t}^{d},
\end{equation}
where the wealth process equals
\[V_{t}(\tilde{\boldsymbol{\phi}})=\tilde{\phi}_{t}^{2}\cdot B_{t}^{d}\]
for \(t\in [0,T]\). \(\sharp\)
In the sequel, we will assume that the futures price \(f=f_{S^{f}}\) follows a martingale under the foreign market martingale measure \(\tilde{\mathbb{P}}\). More specifically, the dynamics of the foreign market futures price \(f_{S^{f}}\) are given by the following expression
\begin{equation}{\label{museq723}}\tag{23}
df_{t}=f_{t}\cdot\boldsymbol{\sigma}_{f}\cdot d\tilde{{\bf W}}_{t},
\end{equation}
where \(\boldsymbol{\sigma}_{f}=\boldsymbol{\sigma}_{S^{f}}\) is a constant volatility vector. Combining (\ref{museq713}) with (\ref{museq723}), we get the following equivalent form of (\ref{museq722})
\[dV_{t}(\tilde{\boldsymbol{\phi}})=\tilde{\phi}_{t}^{1}\cdot Q_{t}
\cdot f_{t}\cdot\left (\boldsymbol{\sigma}_{f}\cdot d\tilde{\bf W}_{t}+
\boldsymbol{\sigma}_{f}\cdot\boldsymbol{\sigma}{Q}dt\right )+\tilde{\phi}_{t}^{2}\cdot r_{d}\cdot B_{t}^{d}dt,\]
and finally
\[dV_{t}(\tilde{\boldsymbol{\phi}})=\tilde{\phi}_{t}^{1}\cdot Q_{t}
\cdot f_{t}\cdot\boldsymbol{\sigma}_{f}\cdot d{\bf W}_{t}^{*}+\tilde{\phi}_{t}^{2}\cdot r_{d}\cdot B_{t}^{d}dt.\]
Let \(f_{Q}(t,T)\) stand for the futures exchange rate at time \(t\) for the settlement date \(T\); that is, the domestic price at time \(t\) of the \(T\)-maturity futures contract with terminal price \(Q_{T}\). It is natural to postulate that \(f_{Q}(t,T)=\mathbb{E}_{\mathbb{P}^{*}}[Q_{T}|{\cal F}_{t}]\). It follows easily that \(f_{Q}(t,T)=F_{Q}(t,T)\), i.e., the forward and futures exchange rates agree. Equivalently, the dynamics of the futures exchange rate under the domestic martingale measure \(\mathbb{P}^{*}\) are (ref. (\ref{museq717}))
\begin{equation}{\label{museq724}}\tag{24}
df_{Q}(t,T)=f_{Q}(t,T)\cdot\boldsymbol{\sigma}_{Q}\cdot d{\bf W}_{t}^{*}
\end{equation}
with the terminal condition \(f_{Q}(T,T)=Q_{T}\).
Definition. By the domestic futures price of the foreign asset \(S^{f}\) we mean the process \(f_{S^{f}}^{d}(t,T)\), denominated in units of domestic currency, which satisfies the terminal condition
\begin{equation}{\label{museq725}}\tag{25}
f_{S^{f}}^{d}(T,T)=Q_{T}\cdot f_{S^{f}}(T,T)=Q_{T}\cdot S_{T}^{f},
\end{equation}
and such that for an arbitrary self-financing futures trading strategy \(\tilde{\boldsymbol{\phi}}\) in foreign market futures contracts, there exists a self-financing (in the usual sense) futures strategy \(\boldsymbol{\phi}\) such that $latex dV_{t}(\boldsymbol{\phi})=
dV_{t}(\tilde{\boldsymbol{\phi}})$. \(\sharp\)
More explicitly, we postulate that for any strategy \(\tilde{\boldsymbol{\phi}}\) satisfying the conditions of Definition \ref{musd741}, there exists a trading strategy \(\boldsymbol{\phi}=(\phi^{1},\phi^{2},\phi^{3})\) such that the corresponding wealth process
\[V_{t}(\boldsymbol{\phi})=\phi_{t}^{3}\cdot B_{t}^{d}\mbox{ for }t\in [0,T],\]
satisfies the standard self-financing condition for the futures market, namely
\[dV_{t}(\boldsymbol{\phi})=\phi_{t}^{1}df_{S^{f}}^{d}(t,T)+\phi_{t}^{2}df_{Q}(t,T)+\phi_{t}^{3}dB_{t}^{d},\]
and the instantaneous gains or losses from both strategies are identical. Intuitively, using the domestic futures price \(f_{S^{f}}^{d}(t,T)\) and futures exchange rate \(f_{Q}(t,T)\), we are able to mimic positions in foreign market futures contracts on foreign assets by entering contracts on the corresponding domestic futures market. This in turn implies that the valuation and hedging of foeign market futures options can be reduced to pricing of domestic futures options. The aim is now to show that it is possible to express \(f_{S^{f}}^{d}(t,T)\) in terms of the futures echange rate \(f_{Q}(t,T)\) and the foreign market futures price \(f_{S^{f}}(t,T)\). Indeed, we have the following result.
Proposition. The domestic futures price \(f_{S^{f}}^{d}(t,T)\) of the foreign market asset \(S^{f}\) for the settlement date \(T\) satisfies
\begin{equation}{\label{museq726}}\tag{26}
f_{S^{f}}^{d}(t,T)=f_{Q}(t,T)\cdot f_{S^{f}}(t,T)
\end{equation}
for all \(t\in [0,T]\).
Proof. It is clear that (\ref{museq726}) implies the terminal equality (\ref{museq725}). Furthermore, using (\ref{museq723}), (\ref{museq724}) and Ito’s formula, we get
\begin{equation}{\label{museq727}}\tag{27}
dZ_{t}=Z_{t}\cdot (\boldsymbol{\sigma}_{f}+\boldsymbol{\sigma}_{Q})\cdot d{\bf W}_{t}^{*},
\end{equation}
where we write \(Z_{t}=f_{Q}(t,T)\cdot f_{S^{f}}(t,T)\). We conclude that \(Z\) follows a \(\mathbb{P}^{*}\)-martingale, and satisfies the terminal condition \(Z_{T}=Q_{T}\cdot S_{T}^{f}\). It is now not difficult to check that \(Z\) is the domestic futures price of the foreign asset \(S^{f}\). We skip the details. \(\blacksquare\)
In order to justify equality (\ref{museq726}) in a more intuitive way, let us consider a specific trading strategy. We combine here a self-financing trading strategy in foreign market futures contracts with a dynamic portfolio of exchange rate futures contracts. Let \(f_{S^{f}}^{d}(t,T)\) be given by (\ref{museq726}). Suppose that at any time \(t\leq T\), the strategy \(\boldsymbol{\eta}=(\eta^{1},\eta^{2},\eta^{3})\) is long in \(\eta_{t}^{1}\) foreign market futures contracts and long in \(\eta_{t}^{2}\) exchange rate futures contracts, where (this choice of trading strategy was discussed in Jamshidian \cite{jam94b})
\[\eta_{t}^{1}=\frac{\psi_{t}\cdot f_{S^{f}}^{d}(t,T)}{Q_{t}\cdot f_{S^{f}}(t,T)}\mbox{ and }\eta_{t}^{2}=
\frac{\psi_{t}\cdot f_{S^{f}}^{d}(t,T)}{f_{Q}(t,T)}\mbox{ for all }t\in [0,T],\]
for some process \(\psi\), and instantaneous marked-to-market profits and losses from the foreign positions are continually converted into domestic currency. Furthermore, let \(\eta^{3}\) be the bonds component of the portfolio, so that the portfolio’s wealth in domestic currency is
\[V_{t}(\boldsymbol{\eta})=\eta_{t}^{3}\cdot B_{t}^{d},\]
and we may assume that the portfolio is self-financing. This means that the incremental profits and losses from \(\boldsymbol{\eta}\) satisfy
\[dV_{t}(\boldsymbol{\eta})=\eta_{t}^{1}\cdot (Q_{t}df_{t}+
d\langle Q,f\rangle\mathbb{E}_{t})+\eta_{t}^{2}df_{Q}(t,T)+\eta_{t}^{3}dB_{t}^{d},\]
where we write \(f_{t}=f_{S^{f}}(t,T)\). By applying (\ref{museq723}) and (\ref{museq724}), we get
\[dV_{t}(\boldsymbol{\eta})=\psi_{t}\cdot f_{S^{f}}^{d}(t,T)\cdot (\boldsymbol{\sigma}_{f}\cdot\tilde{\bf W}_{t}+
\boldsymbol{\sigma}_{f}\cdot\boldsymbol{\sigma}_{Q}dt+\boldsymbol{\sigma}_{Q}\cdot d{\bf W}_{t}^{*})+\eta_{t}^{3}dB_{t}^{d}.\]
In view of equality \(\tilde{\bf W}_{t}={\bf W}_{t}^{*}-\boldsymbol{\sigma}_{Q}\cdot t\), the last equality can be simplified as follows
\[dV_{t}(\boldsymbol{\eta})=\psi_{t}\cdot f_{S^{f}}^{d}(t,T)\cdot
(\boldsymbol{\sigma}_{f}+\boldsymbol{\sigma}_{Q})\cdot d{\bf W}_{t}^{*}+\eta_{t}^{3}dB_{t}^{d}.\]
Finally, using (\ref{museq727}), we arrive at the following equality
\[dV_{t}(\boldsymbol{\eta})=\psi_{t}df_{S^{f}}^{d}(t,T)+\eta_{t}^{3}dB_{t}^{d}.\]
This shows that the strategy \(\boldsymbol{\eta}_{t}\) is essentially equivalent to \(\psi_{t}\) positions in the domestic futures contract, whose price process is \(f_{S^{f}}^{d}(t,T)\).
Jamshidian \cite{jam94b} exploits similar arguments to evaluate and hedge various foreign market futures contracts using their domestic (quanto) counterparts. Let us set
\[\tilde{\phi}_{t}^{1}=\frac{g_{t}}{Q_{t}\cdot f_{t}},\]
where
\begin{equation}{\label{museq728}}\tag{28}
g_{t}=f_{t}\cdot e^{-\boldsymbol{\sigma}_{Q}\cdot \boldsymbol{\sigma}_{f}(T-t)}\mbox{ for all }t\in [0,T],
\end{equation}
and \(f_{t}\) represents the foreign market futures price of a certain contract. The specific nature of the futures contract in question has no
relevance; we need only to assume that the dynamics of \(f\) under the foreign market martingale measure are
\[df_{t}=f_{t}\cdot\boldsymbol{\sigma}_{f}\cdot d\tilde{\bf W}_{t}\]
with \(f_{T}=X\), where \(\boldsymbol{\sigma}_{f}\) is a constant (or at least deterministic) volatility vector, and \(X\) is an arbitrary \({\cal F}_{T}\)-measurable random variable. If the bond component \(\tilde{\phi}^{2}\) of this strategy is chosen in such a way that the strategy \(\tilde{\boldsymbol{\phi}}=(\tilde{\phi}^{1},\tilde{\phi}^{2})\) is self-financing in the sense of Definition \ref{musd741}, then we have
\begin{align*}
dV_{t}(\tilde{\boldsymbol{\phi}}) &=\frac{g_{t}}{Q_{t}\cdot f_{t}}
\cdot\left (Q_{t}df_{t}+d\langle Q,f\rangle\mathbb{E}_{t}\right )+\tilde{\phi}_{t}^{2}dB_{t}^{d}\\
& =g_{t}\cdot\left (\boldsymbol{\sigma}_{f}\cdot d\tilde{\bf W}_{t}+
\boldsymbol{\sigma}_{f}\cdot\boldsymbol{\sigma}_{Q}dt\right )+\tilde{\phi}_{t}^{2}dB_{t}^{d}\\
& =dg_{t}+\tilde{\phi}_{t}^{2}dB_{t}^{d},
\end{align*}
since an application of Ito’s rule to (\ref{museq728}) yields
\[dg_{t}=g_{t}\cdot\left (\boldsymbol{\sigma}_{f}\cdot d\tilde{\bf W}_{t}+\boldsymbol{\sigma}_{f}\cdot
\boldsymbol{\sigma}_{Q}dt\right )=g_{t}\cdot\boldsymbol{\sigma}_{f}\cdot d{\bf W}_{t}^{*}.\]
Since we have \(g_{T}=f_{T}=X\) (note the distinction with condition (\ref{museq725})), it is clear that the process \(g\) represents the quanto counterpart of the foreign market futures price \(f\). Put another way, in order to hedge a domestic-denominated futures contract whose terminal value in units of the domestic currency equals \(X\), we may take positions either in foreign market futures contracts with price process \(f\), or in their domestic counterparts with price process \(g\).
\begin{equation}{\label{f}}\tag{F}\mbox{}\end{equation}
Foreign Equity Options.
We shall study examples of {\bf foreign equity options}; that is, options whose terminal payoff (in units of domestic currency) depends not only on the future behavior of the exchange rate, but also on the price fluctuations of a certain foreign stock. To value options related to foreign market equities, we shall use either the domestic martingale measure \(\mathbb{P}^{*}\) or the foreign martingale measure \(\tilde{\mathbb{P}}\), whichever will be more convenient.
Options Struck in a Foreign Currency.
Assume that an investor wants to participate in gains in foreign equity, desires protection against losses in that equity, but is unconcerned about the translation risk arising from the potential drop in the exchange rate. We denote by \(T\) the expiry date and by \(K^{f}\) the exercise price of an option. It is essential to note that \(K^{f}\) is expressed in units of foreign currency. The terminal payoff from a {\bf foreign equity call struck in foreign currency} equals
\[C_{T}^{1}\equiv Q_{T}\cdot (S_{T}^{f}-K^{f})^{+}.\]
This means that the terminal payoff is assumed to be converted into domestic currency at the spot exchange rate that prevails at the expiry date. By reasoning in much the same way as in the previous discussions, one can check that the arbitrage price of a European call option at time \(t\) equals
\[C_{t}^{1}=e^{-r_{d}(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .Q_{T}\cdot
(S_{T}^{f}-K^{f})^{+}\right |{\cal F}_{t}\right ].\]
Using (\ref{museq713}), we find that
\[C_{t}^{1}=e^{-r_{d}(T-t)}\cdot Q_{t}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .
(S_{T}^{f}-K^{f})^{+}\cdot\exp\left (\boldsymbol{\sigma}_{Q}\cdot
({\bf W}_{T}^{*}-{\bf W}_{t}^{*})+\lambda (T-t)\right )\right |{\cal F}_{t}\right ],\]
where \(\lambda =r_{d}-r_{f}-\frac{1}{2}\cdot\parallel\boldsymbol{\sigma}_{Q}\parallel^{2}\). Equivalently, using (\ref{museq710}), we get
\[C_{t}^{1}=e^{-r_{f}(T-t)}\cdot Q_{t}\cdot \mathbb{E}_{\tilde{\mathbb{P}}}\left [\left .
(S{T}^{f}-K^{f})^{+}\right |{\cal F}_{t}\right ].\]
Since \(\tilde{\mathbb{P}}\) is the arbitrage-free measure of the foreign economy, it is not hard to establish the following expression
\[C_{t}^{1}=Q_{t}\left (S_{t}^{f}\cdot N(g_{1}(S_{t}^{f},T-t))-K^{f}\cdot e^{-r_{f}(T-t)}\cdot N(g_{2}(S_{t}^{f},T-t))\right ),\]
where
\[g_{1,2}(s,t)=\frac{\ln s-\ln K^{f}+(r_{f}\pm\frac{1}{2}\parallel
\boldsymbol{\sigma}_{S^{f}}\parallel^{2})\cdot t}{\parallel\boldsymbol{\sigma}_{S^{f}}\parallel\cdot\sqrt{t}}.\]
An inspection of the valuation formula above makes clear that a hedging portfolio involves at any instant \(t\) the number \(N(g_{1}(S_{t}^{f},T-t))\) shares of the underlying stock; this stock investment demands the additional borrowing of
\[\beta_{t}^{d}=Q_{t}\cdot K^{f}\cdot e^{-r_{f}(T-t)}\cdot N(g_{2}(S_{t}^{f},T-t))\]
units of the domestic currency, or equivalently, the borrowing of
\[\beta_{t}^{f}=K^{f}\cdot e^{-r_{f}(T-t)}\cdot N(g_{2}(S_{t}^{f},T-t))\]
units of the foreign currency.
As mentioned, when dealing with foreign equity options, one can either do the computations with reference to the domestic economy, or equivalently, one may work within the framework of the foreign economy and then convert the final result into units of domestic currency. The choice of the method depends on the particular form of an option’s payoff; to some extent it is also a matter of taste. In the case considered above, in order to complete the calculations in the domestic economy, one needs to compute directly the following expectation (for the sake of notational simplicity we put \(t=0\))
\[C_{0}^{1}=e^{-r_{d}T}\cdot Q_{0}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left (S_{0}^{f}
\cdot\exp\left (\boldsymbol{\sigma}_{S^{f}}+\boldsymbol{\sigma}_{Q})\cdot {\bf W}_{T}^{*}+(r_{d}-\frac{1}{2}\cdot
\parallel\boldsymbol{\sigma}_{S^{f}}+\boldsymbol{\sigma}_{Q}\parallel^{2})\cdot T\right )
-K^{f}\cdot\exp\left (\boldsymbol{\sigma}_{Q}\cdot {\bf W}_{T}^{*}+\lambda T\right )\right )^{+}\right ],\]
where \(\lambda =r_{d}-r_{f}-\frac{1}{2}\cdot\parallel\boldsymbol{\sigma}_{Q}\parallel^{2}\). This can be done using the following proposition.
\begin{equation}{\label{musl751}}\tag{29}\mbox{}\end{equation}
Proposition \ref{musl751}. Let \((\xi ,\eta )\) be a zero-mean, jointly Gaussian \((\)non-degenerate$)$, two-dimensional random vector on a probability spec \((\Omega ,{\cal F},P)\). Then for arbitrary positive real numbers \(a\) and \(b\), we have
\begin{equation}{\label{museq729}}\tag{30}
\mathbb{E}_{P}\left [\left (a\cdot\exp\left (\xi -\frac{1}{2}\cdot Var(\xi )
\right )-b\cdot\exp\left (\eta -\frac{1}{2}\cdot Var (\eta )\right )\right )^{+}\right ]=a\cdot N(h)-b\cdot N(h-k),
\end{equation}
where
\[h=\frac{1}{k}\cdot\left (\ln a-\ln b\right )+\frac{k}{2}\mbox{ and }k=\sqrt{Var (\xi-\eta )}.\]
Options Struck in Domestic Currency.
Assume now that an investor wishes to receive any positive returns from the foreign market, but wants to be certain that those returns are meaningful when translated back into his own currency. In this case, he might be interested in a foreign equity call struck in domestic
currency with payoff at expiry
\[C_{T}^{2}\equiv (S_{T}^{f}\cdot Q_{T}-K^{d})^{+}=(\tilde{S}_{T}^{f}-K^{d})^{+},\]
where the strike price \(K^{d}\) is expressed in domestic currency. Due to the particular form of the option’s payoff, it is clear that it is now convenient to study the option from the domestic perspective. To find the arbitrage price of the option at time \(t\), it is sufficient to evaluate the following conditional expectation
\[C_{t}^{2}=e^{-r_{d}(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .(\tilde{S}_{T}^{f}-K^{d})^{+}\right |{\cal F}_{t}\right ],\]
and by virtue of (\ref{museq712}) and (\ref{museq713}), the stock price expressed in units of domestic currency \(\tilde{S}_{t}^{f}\) has the following dynamics under \(\mathbb{P}^{*}\)
\[d\tilde{S}_{t}^{f}=\tilde{S}_{t}^{f}\cdot\left (r_{d}dt+
(\boldsymbol{\sigma}_{S^{f}}+\boldsymbol{\sigma}_{Q})\cdot d{\bf W}_{t}^{*}\right ).\]
Therefore, arguing as in the proof of the classic Black-Scholes formula, one finds easily that the option’s price, expressed in units of domestic currency, is given by the formula
\[C_{t}^{2}=\tilde{S}_{t}^{f}\cdot N(l_{1}(\tilde{S}_{t}^{f},T-t))-
e^{-r_{d}(T-t)}\cdot K^{d}\cdot N(l_{2}(\tilde{S}_{t}^{f},T-t)),\]
where
\[l_{1,2}(s,t)=\frac{\ln s-\ln K^{d}+(r_{d}\pm\frac{1}{2}\parallel
\boldsymbol{\sigma}_{S^{f}}+\boldsymbol{\sigma}_{Q}\parallel^{2})
\cdot t}{\parallel\boldsymbol{\sigma}_{S^{f}}+\boldsymbol{\sigma}_{Q}\parallel\cdot\sqrt{t}}.\]
Once again, the value \(N(l_{1}(\tilde{S}_{t}^{f},T-t))\) represents the number of shares of the underlying stock held in the replicating portfolio at time \(t\). To establish the replicating portfolio, an additional borrowing of
\[\beta_{t}^{d}=e^{-r_{d}(T-t)}\cdot K^{d}\cdot N(l_{2}(\tilde{S}_{t}^{f},T-t))\]
units of the domestic currency is required.
Quanto Options.
Assume that an investor wishes to capture positive returns on his or her foreign equity investment, but now desires to eliminate all exchange risk by fixing an advance rate at which the option’s payoff will be converted into domestic currency. At the intuitive level, such a contract can be seen as a combination of a foreign equity option with a currency forward contract. By definition, the payoff of a {\bf quanto call} (i.e., a guaranteed exchange rate foreign equity call option) at expiry is set to be
\[C_{T}^{3}\equiv\bar{Q}\cdot (S_{T}^{f}-K^{f})^{+},\]
where \(\bar{Q}\) is the prespecified exchange rate at which the conversion of the option’s payoff is made. Notice that the quantity \(\bar{Q}\) is denominated in the domestic currency per unit of foreign currency, and the strike price \(K^{f}\) is expressed in units of foreign currency. Since the payoff from the quanto option is expressed in units of domestic currency, its arbitrage price equals
\begin{equation}{\label{museq730}}\tag{31}
C_{t}^{3}=\bar{Q}\cdot e^{-r_{d}(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left . (S_{T}^{f}-K^{f})^{+}\right |{\cal F}_{t}\right ].
\end{equation}
The next result shows that a closed-form solution for the price of a quanto option is easily available.
\begin{equation}{\label{musp751}}\tag{32}\mbox{}\end{equation}
Proposition \ref{musp751}. The arbitrage price at time \(t\) of a European quanto call option with expiry date \(T\) and strike price \(K^{f}\) equals \((\)in units of domestic currency$)$
\[C_{t}^{3}=\bar{Q}\cdot e^{-r_{d}(T-t)}\cdot\left (S_{t}^{f}\cdot
e^{\delta (T-t)}\cdot N(c_{1}(S_{t}^{f},T-t))-K^{f}\cdot N(c_{2}(S_{t}^{f},T-t))\right ),\]
where \(\delta =r_{f}-\boldsymbol{\sigma}_{Q}\cdot\boldsymbol{\sigma}_{S^{f}}\) and
\[c_{1,2}(s,t)=\frac{\ln s-\ln K^{f}+(\delta\pm\frac{1}{2}\cdot\parallel
\boldsymbol{\sigma}_{S^{f}}\parallel^{2})\cdot t}{\parallel\boldsymbol{\sigma}_{S^{f}}\parallel\cdot\sqrt{t}}.\]
Proof. Using (\ref{museq730}), we obtain
\[C_{t}^{3}=\bar{Q}\cdot e^{(\delta -r_{d})(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [
\left .e^{-\delta (T-t)}\cdot (S_{T}^{f}-K^{f})^{+}\right |{\cal F}_{t}\right ].\]
Since the dynamics of \(S^{f}\) under \(\mathbb{P}^{*}\) are (ref. (\ref{museq712}))
\[dS_{t}^{f}=S_{t}^{f}\cdot (\delta dt+\boldsymbol{\sigma}_{S^{f}}\cdot d{\bf W}_{t}^{*}),\]
in order to evaluate the conditional expectation, we can make use of the classic form of the Black-Scholes formula. Namely, we have
\[\mathbb{E}_{\mathbb{P}^{*}}\left [\left .e^{-\delta (T-t)}\cdot (S_{T}^{f}-K^{f})^{+}
\right |{\cal F}_{t}\right ]=C(S_{t}^{f},T-t,K^{f},\delta ,\boldsymbol{\sigma}_{S^{f}}),\]
where the function \(C\) is given by (\ref{museq556}). Upon rearranging, this yields the desired equality. \(\blacksquare\)
The quanto option can also be examined under the foreign market martingale probability; the domestic market method is slightly more convenient. To this end, note that, expressed in units of foreign currency, the price at time \(t\) of the quanto option is given by the expression
\[e^{-r_{f}(T-t)}\cdot\bar{Q}\cdot \mathbb{E}_{\tilde{\mathbb{P}}}\left [\left .R_{T}\cdot
(S_{T}^{f}-K^{f})^{+}\right |{\cal F}_{t}\right ].\]
By virtue of (\ref{museq79}), we have
\[R_{T}=R_{0}\cdot\exp\left (-\boldsymbol{\sigma}_{Q}\cdot
\tilde{\bf W}_{T}+(r_{f}-r_{d}-\frac{1}{2}\cdot\parallel
\boldsymbol{\sigma}_{Q}\parallel^{2})\cdot T\right ).\]
On the other hand, (\ref{museq711}) yields
\[S_{T}^{f}=S_{0}^{f}\cdot\exp\left (\boldsymbol{\sigma}_{S^{f}}\cdot\tilde{\bf W}_{T}+(r_{f}-\frac{1}{2}\cdot\parallel
\boldsymbol{\sigma}_{S^{f}}\parallel^{2})\cdot T\right ).\]
By combining the last two equalities, we arrive at the following expression
\[R_{T}\cdot S_{T}^{f}=R_{0}^{f}\cdot S_{0}^{f}\cdot\exp\left (T\cdot (r_{f}-\gamma )\right )\cdot\exp\left (\boldsymbol{\sigma}\cdot\tilde{\bf W}_{T}-\frac{1}{2}\cdot\parallel\boldsymbol{\sigma}\parallel^{2}\cdot T\right ),\]
where \(\boldsymbol{\sigma}=\boldsymbol{\sigma}_{S^{f}}-\boldsymbol{\sigma}_{Q}\) and \(\gamma =r_{d}-r_{f}+ \boldsymbol{\sigma}_{Q}\cdot\boldsymbol{\sigma}_{S^{f}}\). Let us consider the case \(t=0\). To find the value of the option in units of
domestic currency, it is enough to evaluate
\[C_{0}^{3}=\bar{Q}\cdot Q_{0}\cdot \mathbb{E}_{\tilde{\mathbb{P}}}\left [\left (e^{-r_{f}T}
\cdot R_{T}\cdot S_{T}^{f}-e^{-r_{f}T}\cdot K^{f}\cdot R_{T}\right )^{+}\right ].\]
For this purpose, one can make use of Proposition \ref{musl751} with constants
\[a=S_{0}^{f}\cdot e^{-\gamma T}\mbox{ and }b=K^{f}\cdot e^{-r_{d}T}\]
and with the random variables
\[\xi =(\boldsymbol{\sigma}_{S^{f}}-\boldsymbol{\sigma}_{Q})
\cdot\tilde{\bf W}_{T}\mbox{ and }\eta =-\boldsymbol{\sigma}_{Q}\cdot\tilde{\bf W}_{T}.\]
It is easy to check that \(k=\parallel\boldsymbol{\sigma}_{S^{f}}\parallel\cdot\sqrt{T}\), and
\[h=\frac{\ln S_{0}^{f}-\ln K^{f}+(\delta +\frac{1}{2}\cdot\parallel
\boldsymbol{\sigma}_{S^{f}}\parallel^{2})\cdot T}{\parallel\boldsymbol{\sigma}_{S^{f}}\parallel\cdot\sqrt{T}},\]
where \(\delta =r_{f}-\boldsymbol{\sigma}_{Q}\cdot\boldsymbol{\sigma}_{S^{f}}\). Consequently, by virtue of (\ref{museq729}), we get
\[C_{0}^{3}=\bar{Q}\cdot (a\cdot N(h)-b\cdot N(h-k))=\bar{Q}\cdot
e^{-r_{d}T}\cdot (S_{0}^{f}\cdot e^{\delta T}\cdot N(h)-K^{f}\cdot N(h-k)).\]
The last equality is easily seen to agree with the option valuation formula established in Proposition \ref{musp751}. By proceeding along the same lines, one can also find the price \(P_{t}^{3}\) of a quanto put option with the strike price \(K^{f}\) and the guaranteed exchange rate \(\bar{Q}\). Furthermore, we have the following version of the put-call parity relationship: \(C_{t}^{3}-P_{t}^{3}=V_{t}^{d}(K^{f},\bar{Q})\), where \(V_{t}^{d}(K^{f},\bar{Q})\) represents the value at time \(t\), in domestic currency, of the corresponding guaranteed exchange rate forward contract (see (\ref{museq721})).
Equity-Linked Foreign Exchange Options.
Assume that an investor desires to hold foreign equity regardless of whether the stock price rises or falls (that is, he is indifferent to the
foreign equity exposure), however, wishes to place a floor on the exchange rate risk of his foreign investment. An {\bf equity-linked foreign
exchange call} (an Elf-X call, for short) with payof at expiry (in units of domestic currency)
\[C_{T}^{4}\equiv (Q_{T}-K)^{+}\cdot S_{T}^{f},\]
where \(K\) is a {\bf strike exchange rate} expressed in domestic currency per unit of foreign currency, is thus a combination of a currency option with an equity forward. The arbitrage price (in units of domestic currency) of a European Elf-X call with expiry date \(T\) equals
\[C_{t}^{4}=e^{-r_{d}(T-t)}\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .(Q_{T}-K)^{+}\right |{\cal F}_{t}\right ].\]
We shall first value an equity-linked foreign exchange call option using the domestic martingale measure.
Proposition. The arbitrage price, expressed in domestic currency, of a European equity-linked foreign exchange call option, with strike exchange rate \(K\) and expiry date \(T\), is given by the following formula
\begin{equation}{\label{museq731}}\tag{33}
C_{t}^{4}=S_{t}^{f}\cdot\left (Q_{t}\cdot N(w_{1}(Q_{t},T-t))-K\cdot e^{-\gamma (T-t)}\cdot N(w_{2}(Q_{t},T-t))\right ),
\end{equation}
where \(\gamma =r_{d}-r_{f}+\boldsymbol{\sigma}_{Q}\cdot\boldsymbol{\sigma}_{S^{f}}\) and
\[w_{1,2}(q,t)=\frac{\ln q-\ln K+(\gamma\pm\frac{1}{2}\cdot\parallel
\boldsymbol{\sigma}_{Q}\parallel^{2})\cdot t}{\parallel\boldsymbol{\sigma}_{Q}\parallel\cdot\sqrt{t}}.\]
Proof. As usual, it is sufficient to consider the case \(t=0\). In view of (\ref{museq712}), we have
\[S_{T}^{f}=S_{0}^{f}\cdot\exp\left (\boldsymbol{\sigma}_{S^{f}}\cdot
{\bf W}_{T}^{*}-\frac{1}{2}\cdot\parallel\boldsymbol{\sigma}_{S^{f}}
\parallel^{2}\cdot T+(r_{f}-\boldsymbol{\sigma}_{Q}\cdot\boldsymbol{\sigma}_{S^{f}})\cdot T\right ).\]
We define a probability measure \(O\) on \((\Omega ,{\cal F}_{T})\) by setting
\[\frac{dO}{d\mathbb{P}^{*}}=\exp\left (\boldsymbol{\sigma}_{S^{f}}\cdot
{\bf W}_{T}^{*}-\frac{1}{2}\cdot\parallel\boldsymbol{\sigma}_{S^{f}}\parallel^{2}\cdot T\right )\mbox{ \(P\)-a.s.}\]
Notice that the process \({\bf U}_{t}={\bf W}_{t}^{*}-\boldsymbol{\sigma}_{S^{f}}\cdot t\) follows a Brownian motion under \(O\). Furthermore,
\[C_{0}^{4}=S_{0}^{f}\cdot\exp\left ((r_{f}-r_{d}-\boldsymbol{\sigma}_{Q}\cdot\boldsymbol{\sigma}_{S^{f}})\cdot T
\right )\cdot \mathbb{E}_{O}[(Q_{T}-K)^{+}],\]
and the dynamics of the process \(Q\) under the probability measure \(O\) are
\[dQ_{t}=Q_{t}\cdot\left ((r_{d}-r_{f}+\boldsymbol{\sigma}_{Q}\cdot
\boldsymbol{\sigma}_{S^{f}})dt+\boldsymbol{\sigma}_{Q}\cdot d{\bf U}_{t}\right ).\]
The expectation \(\mathbb{E}_{O}[(Q_{T}-K)^{+}]\) can be evaluated along the same lines as in the standard Black-Scholes model, yielding the following equality
\[C_{0}^{4}=S_{0}^{f}\cdot C(Q_{0},T,K,\gamma ,\boldsymbol{\sigma}_{Q}),\]
which is the asserted formula. \(\blacksquare\)
The Elf-X option can also be valued by taking the perspective of the foreign market-based investor. The option price at time \(0\), when expressed in units of foreign currency, is given by the risk-neutral valuation formula
\[C_{0}^{4}=e^{-r_{f}T}\cdot \mathbb{E}_{\tilde{\mathbb{P}}}\left [(S_{T}^{f}-K\cdot R_{T}\cdot S_{T}^{f})^{+}\right ].\]
To derive equality (\ref{museq731}) from the last formula, one can directly apply Proposition \ref{musl751} with the constants
\[a=S_{0}^{f}\mbox{ and }b=K\cdot R_{0}\cdot S_{0}^{f}\cdot e^{-\gamma T},\]
and the random variables
\[\xi =\boldsymbol{\sigma}_{S^{f}}\cdot\tilde{\bf W}_{T}\mbox{ and }
\eta =(\boldsymbol{\sigma}_{S^{f}}-\boldsymbol{\sigma}_{Q})\cdot\tilde{\bf W}_{T}.\]
Notice that the standard deviation of the Gaussian random variable \(\xi -\eta\) is \(k=\parallel\boldsymbol{\sigma}_{Q}\parallel\cdot \sqrt{T}\), and thus \(h\) equals
\[h=\frac{1}{k}\cdot (\ln a-\ln b)+\frac{k}{2}=\frac{-\ln (R_{0}\cdot K)+
(\gamma +\frac{1}{2}\cdot\parallel\boldsymbol{\sigma}_{Q}\parallel^{2})
\cdot T}{\parallel\boldsymbol{\sigma}_{Q}\parallel\cdot\sqrt{T}}.\]
This implies that the option’s price, expressed in units of domestic currency, equals
\[C_{0}^{4}=S_{0}^{f}\cdot (Q_{0}\cdot N(h)-K\cdot e^{-\gamma T}\cdot N(h-k)).\]
Formula (\ref{museq731}) now follows by standard arguments.
Note that as \(S^{f}\) we may take the price process of any foreign security, not necessarily a stock. For instance, to find the currency option valuation formula, it is enough to take as \(S^{f}\) the price process of a foreign bond which pays one unit of the foreign currency at maturity date \(T\). Under the present assumptions the bond price equals \(p^{f}(t,T)=\exp (-r_{f}(T-t))\) so that the bond price volatility vanishes; that is, \(\boldsymbol{\sigma}_{S^{f}}={\bf 0}\). To derive from (\ref{museq731}) formula (\ref{museq718}) which gives the arbitrage price \(C_{t}^{Q}\) of a European currency call option, it is enough to observe that the relationship \(C_{t}^{4}=\exp (-r_{f}(T-t))\cdot C_{t}^{Q}\) is satisfied for every \(t\in [0,T]\).
