6
Foreign Exchange
We will see in this chapter how to apply the Black-Scholes formulas to value
currency options and options on foreign assets. We will also discuss currency
forwards and futures, quanto forwards, and return swaps.
For concreteness, we will call one currency the “domestic” currency and
the other the “foreign” currency. Let X(t) denote the exchange rate at time t
measured in units of the domestic currency per unit of the foreign currency.
Exchange rates can be confusing, because we can look at them from the per-
spective of either currency, so it may help to keep in mind that X(t) here
means the price of a unit of the foreign currency, just as we might consider
the price of a stock. When we speak of the “cost” or “value” of something
without specifying the currency, it should be understood to be the domestic
currency that we mean. If S is the price of a foreign asset, denominated in
units of the foreign currency, we can convert it into a domestic asset price
simply by multiplying by the exchange rate: X(t)S(t) is the price of the asset,
denominated in the domestic currency. For example, if the domestic currency
is dollars and the foreign currency is yen, then S is in units of yen and X is
units of dollars per unit of yen, so XS is in units of dollars.
Throughout the chapter, we will maintain assumptions similar to the
Black-Scholes assumptions. There is a foreign asset with price S in the foreign
currency. It has a constant dividend yield q and a constant volatility σs. The
exchange rate has a constant volatility σx and a constant correlation ρ with
the foreign asset. There is a domestic risk-free asset with constant interest
rate r and a foreign risk-free asset with constant interest rate rf . The term
“risk free” means of course that they are risk-free in their respective curren-
cies. For example, an investment in the foreign risk-free asset is not risk free
to a domestic investor, because of exchange rate risk.
112 6 Foreign Exchange
Currency Options
A European call option on the exchange rate X pays max(0, X(T )−K) at its
maturity T , where K is the strike price (in domestic currency). The underlying
asset should be regarded as the foreign risk-free asset, the domestic price of
which fluctuates with the exchange rate. An investment in the foreign risk-
free asset grows via reinvestment of interest at rate rf , just as the number of
shares held of a stock grows via reinvestment of dividends at rate q, if q is its
constant dividend yield. In particular, the cost at date 0 of obtaining one unit
of foreign currency at date T is the cost at date 0 of e−rfT units of foreign
currency, which is e−rfTX(0). Thus, the exchange rate is analogous to a stock
price, with the foreign risk-free rate being its dividend yield. This means we
can apply the Black-Scholes formulas to value currency calls and puts:
Calls and puts on foreign currency can be valued by the Black-Scholes formu-
las with inputs X(0) = initial asset price, r = risk-free rate, σx = volatility,
and rf = dividend yield.
Options on Foreign Assets Struck in Foreign
Currency
An option on a foreign asset, with the strike price defined in the foreign
currency, can be priced with the Black-Scholes formula, assuming the volatility
and dividend yields of the asset are constant and that the (foreign) interest
rate is constant. This must be true, because we did not need to specify the
currency (dollars, yen, etc.) when deriving the Black-Scholes formula. The
value given by the Black-Scholes formula is in the same currency as the asset.
To obtain a value in domestic currency for an option on a foreign asset, we
simply multiply the Black-Scholes formula by the current exchange rate.
Options on Foreign Assets Struck in Domestic
Currency
A call option with domestic strike price K on the foreign asset with price S
pays
max(X(T )S(T )−K, 0)
at its maturity T . The underlying price X(T )S(T ) is the value in domestic cur-
rency of the portfolio that with starts with e−qT units of the asset and reinvests
dividends until date T . Thus, we can use the Black-Scholes formula to value
Currency Forwards and Futures 113
it, taking the initial asset price to be e−qTX(0)S(0) and the dividend rate
to be zero (or taking the initial asset price to be X(0)S(0) and the dividend
rate to be q). The volatility that should be input into the Black-Scholes for-
mula is the volatility of the domestic currency price e−q(T−t)X(t)S(t), which
is the same as the volatility of X(t)S(t). According to the formula (), the
volatility of the domestic currency price XS is
σ =
√
σ2x + σ2s + 2ρσxσs . ()
We conclude:
Calls and puts on foreign assets struck in domestic currency can be valued
by the Black-Scholes formulas with inputs X(0)S(0) = initial price, r =
risk-free rate, () = volatility, q = dividend yield.
Currency Forwards and Futures
Consider a forward contract maturing at some date T on one unit of foreign
currency. In keeping with our convention for options, we will always assume
(without loss of generality) that a forward contract is written on a single unit
of currency. Let F (t) denote the forward price (in domestic currency) at date
t ≤ T . This means that someone who purchases (goes long) the contract at
date t will receive a unit of foreign currency, worth X(T ), at date T and must
pay F (t) at date T . The value of the long contract at date T is therefore
X(T )−F (t). The value at date T of a short contract initiated at date t is the
opposite: F (t)−X(T ). Naturally, the forward price F (t) is called the “forward
exchange rate.”
The deepest market for currency is the inter-bank forward market, but
futures contracts are also traded on exchanges. The difference between for-
wards and futures is that futures are “marked to market” daily. Thus, there
are daily cash flows with a futures contract, whereas the only cash flows on
a forward contract occur at the maturity of the forward. In both cases, there
is no cash flow at the time the contract is bought/sold, so its market value is
zero. In Sect. we will discuss futures contracts further. In particular, we
will show, assuming continuous marking to market, that if there is a constant
(domestic) risk-free rate—or, more generally, if there is an instantaneous risk-
free rate that changes over time in a non-random way—then futures prices
must equal forward prices in the absence of arbitrage opportunities. Thus, our
assumptions in this chapter imply that currency futures prices should equal
currency forward prices. We will consider currency forwards in the remainder
of this section.
A forward contract on a traded asset can always be created synthetically
simply by buying the asset and holding it, using borrowed money to finance
114 6 Foreign Exchange
the purchase and to finance any storage costs, assuming the storage costs can
be estimated in advance. If the asset pays dividends or generates other pos-
itive cash flows, then we do not need to purchase the entire amount covered
by the forward contract, because we can accumulate additional amounts of
the asset by reinvesting the dividends. There are no storage costs on cur-
rency and its dividend yield is equal to the foreign risk-free rate. A forward
contract on one unit of foreign currency maturing at date T can be created
synthetically at date 0 by buying e−rfT units of foreign currency and bor-
rowing the cost e−rfTX(0) at the domestic risk-free rate. This will lead to
ownership of one unit of foreign currency at date T and a liability, including
interest, of e(r−rf )TX(0) at date T . Thus, the forward price at date 0 must be
F (0) = e(r−rf )TX(0); otherwise, one could arbitrage by buying the forward
and “selling” the synthetic forward, or vice versa. More generally,
The forward exchange rate at date t, for a contract maturing at T , must be
F (t) = e(r−rf )(T−t)X(t) . ()
The relation () is called “covered interest parity.” The name stems from
the fact that an investment in one of the risk-free assets (foreign or domestic)
financed by borrowing in the other, with the currency risk hedged (“covered”)
by a forward contract, is certain to generate zero value (otherwise, it would
be an arbitrage opportunity).1
Suppose that one has made a commitment to pay a certain amount of
foreign currency (perhaps to a foreign manufacturer) at some date in the
future. The exchange rate risk that this commitment entails can obviously
be hedged by buying the currency forward. However, one can also create a
synthetic forward, by buying currency today and investing it in the foreign
risk-free asset. The cash outflow can be incurred today, or it can be deferred
by borrowing the cost of the currency at the domestic risk-free rate. In the
latter case, we have created a true synthetic forward. In either case, we would
call this a “money market hedge” because we have utilized the foreign money
market (risk-free asset) to create the hedge.
Later in this chapter we will construct replicating strategies for various
contracts using the foreign risk-free asset and the domestic risk-free asset.
1 A relation analogous to covered interest parity holds for any forward contract if
the underlying asset has a constant dividend yield and storage costs that are a
constant proportion of the value of the units stored. For commodities, the term
“dividend yield” must be interpreted in a broad sense, and is usually called “con-
venience yield,” because ownership of the physical asset may produce abnormal
profits during temporary shortages, an advantage that is not obtained by owning
a forward contract on the asset, just as dividends are not received by the owner
of a forward contract. Thus, one must consider the “convenience” of owning the
physical asset as an advantage analogous to dividends.
Currency Forwards and Futures 115
One can interpret these replicating strategies as money market hedges or
synthetic currency forwards. In practice, it will often be more convenient to
use actual forwards rather than using the foreign risk-free asset. Using actual
currency forwards produces an equivalent (given that we are not considering
transaction costs) replicating strategy. Here is, in abstract, the way we convert
from money market hedges to hedges using forwards. As we have discussed,
Long Currency Forward = Long Synthetic Currency Forward
= Long Foreign Risk-Free Asset
+ Short Domestic Risk-Free Asset .
Subtracting a short position is the same as adding a long position, so we can
rearrange this as
Long Currency Forward + Long Domestic Risk-Free Asset
= Long Foreign Risk-Free Asset .
Thus, an investment in the foreign risk-free asset can be replaced in any repli-
cating strategy by long currency forwards and an investment in the domestic
risk-free asset.
To be more precise about the sizes of the investments, consider replacing
a money market hedge with a forward hedge at some date t prior to the
maturity of the forward and analyze the replacement per unit of the money
market hedge (per unit of foreign currency invested in the foreign risk-free
asset). One unit of foreign currency invested in the foreign risk-free asset at
date t will grow to erf (T−t) units by date T . Thus, the corresponding forward
contract should be on erf (T−t) units of currency. The value at date t of both
sides of the above equation should be the same, and the value of a forward
contract at the date of initiation is zero, so the investment in the domestic
risk-free asset should be the domestic currency equivalent of one unit of foreign
currency, which is the exchange rate X(t). Thus, we have
erf (T−t) Long Currency Forwards
+ X(t) Long in the Domestic Risk-Free Asset
= 1 Unit of Foreign Currency Long in the Foreign Risk-Free Asset .
()
To check this, consider holding the portfolios until date T . As explained
in the first paragraph of this section, the currency forwards will have value
erf (T−t)[X(T ) − F (t)], which by covered interest parity is erf (T−t)X(T ) −
er(T−t)X(t). When we include the long position in the domestic risk-free asset
116 6 Foreign Exchange
with accumulated interest, the value at date T of the portfolio on the left-
hand side of () is erf (T−t)X(T ). On the other, the right-hand side of ()
with accumulated interest will consist of erf (T−t) units of foreign currency,
also worth erf (T−t)X(T ).
Quantos
A “quanto” is a derivative written on a foreign asset the value of which is
converted to domestic currency at a fixed exchange rate. In other words,
the contract pays in the domestic currency and the exchange rate is part of
the contract. Such contracts are very useful for investors who want to bet
on foreign assets but do not want exposure to exchange rate risk. Such an
investor could simply buy the foreign asset and hedge the currency risk by
selling currency futures or forwards, but doing so is a bit tricky because the
amount of currency that needs to be hedged depends on how well the foreign
asset does. Thus, quantos can be desirable contracts. Of course, when an
investor purchases a quanto, the problem of hedging the exchange rate risk
has simply been transferred to the seller. In this and the following section,
we will see how to value and how to replicate a contract that pays the price
of a foreign asset at some future date T with the price translated into the
domestic currency at a fixed exchange rate. The replicating strategy is the
strategy that would be followed by the seller (or by an investor who wants to
create a synthetic on his own). Specifically, in this section we will determine
the value at date 0 (in domestic currency) of a contract that pays X¯S(T )
at date T , where X¯ is a fixed exchange rate. Later in the chapter, we will
consider quanto forwards and quanto options.
In addition to being practically useful, this contract is an excellent example
for demonstrating the methodology of pricing and hedging. The best way to
proceed in problems of this general type is to first value the contract and
then calculate the replicating As discussed in Sect. , valuation
is simplified by choosing a numeraire that will cancel the randomness in the
contract payoff. Our numeraire must be a non-dividend-paying (domestic)
asset price, so we can choose Z(t) = X(t)eqtS(t) to be the numeraire asset
price. This is the value in the domestic currency of a strategy that is long
one unit of the foreign asset at date 0 and which reinvests the dividends of
the asset into new shares. As we will see immediately, using it as numeraire
introduces randomness into the payoff through the exchange rate, and that
poses some complications. Applying our fundamental pricing formula (),
the value of the contract is
2 We did the same thing in Chap. 3: we first derived the Black-Scholes formula and
then found the replicating strategy (delta hedge) by differentiating the formula.
Quantos 117
Z(0)EZ
[
X¯S(T )
Z(T )
]
= e−qTX(0)S(0)EZ
[
X¯S(T )
X(T )S(T )
]
= e−qT X¯S(0)EZ
[
X(0)
X(T )
]
. ()
Now we need to evaluate EZ [X(0)/X(T )], which is the expected growth
of 1/X when Z is used as the numeraire. We will show that
EZ
[
X(0)
X(T )
]
= exp {(rf − r − ρσxσs)T} . ()
This implies:
The value at date 0 of a contract that pays X¯S(T ) at date T , where X¯ is
a fixed exchange rate and S is the foreign price of an asset with a constant
dividend yield q, is
exp {(rf − r − q − ρσxσs)T} X¯S(0) . ()
We will now prove (). The assumption that S and X have constant volatilities
and correlation means that
dX
X
= µx dt + σx dBx ,
dS
S
= µs dt + σs dBs ,
for some (possibly random) µx and µs, where Bs and Bx are Brownian motions with
correlation equal to ρ. From Itoˆ’s formula, we have
dZ
Z
= q dt +
d(XS)
XS
= (q + µx + µs + ρσxσs) dt + σx dBx + σs dBs
= (q + µx + µs + ρσxσs) dt + σ
(σx
σ
dBx +
σs
σ
dBs
)
= (q + µx + µs + ρσxσs) dt + σ dB ,
where we define σ in () and B by B(0) = 0 and
dB =
σx
σ
dBx +
σs
σ
dBs .
As discussed in Sect. , B is a Brownian motion and σ is the volatility of Z.
Notice that the correlation of X and Z is
(dB)(dBx) =
(σx
σ
dBx +
σs
σ
dBs
)
(dBx)
=
σx + ρσs
σ
dt .
118 6 Foreign Exchange
Now we use () in Sect. which gives the drift of an asset when another risky
asset is used as the numeraire. We use Z as the numeraire and X as the other as-
set, regarding X as the domestic price of an asset with dividend yield rf as before.
Therefore, we substitute rf for q in (), substitute σ for the volatility of the nu-
meraire asset price, substitute σx for the volatility of the other asset, and substitute
(σx + ρσs)/σ for their correlation. This yields
dX
X
=
(
r − rf + σ2x + ρσxσs
)
dt + σx dB
∗
x ,
where B∗x is a Brownian motion when Z is the numeraire. Now we apply Itoˆ’s formula
for ratios to obtain
d(1/X)
1/X
= −dX
X
+
(
dX
X
)2
= (rf − r − ρσxσs) dt + σx dB∗x .
This implies that 1/X is a geometric Brownian motion with growth rate rf − r −
ρσxσs, from which () follows.
Replicating Quantos
The assets we will use to replicate the payoff X¯S(T ) are the foreign asset
with price S, the foreign risk-free asset, and the domestic risk-free asset. At
the end of this section, we will explain how to replace the foreign risk-free
asset with currency forwards, as discussed in Sect. . Before beginning the
calculations, we can make the following intuitive observations:
• The payoff X¯S(T ) has exposure to the foreign asset price S, so the repli-
cating portfolio must be long the foreign asset.
• The payoff X¯S(T ) has no exposure to the exchange rate, so the replicating
portfolio cannot have any exposure to the exchange rate either. Thus, the
long position in the foreign risky asset must be offset by an equal short
position in the foreign risk-free asset.
• As a result of the previous observation, the value of the replicating port-
folio, displayed in (), will equal the investment in the domestic risk-free
asset.
Consequently, our real task is to compute the number of shares of the foreign
asset that should be held, the remainder of the replicating portfolio being
thereby determined.
The value of the replicating portfolio at any date t ≤ T must be the value
at date t of receiving the payoff X¯S(T ) at date T . We have calculated this
value at date 0, and, clearly, the formula () applies to general dates t, when
we replace the time T to maturity by T − t and the asset price S(0) at the
date of valuation by S(t). That is, the value of the portfolio at any date t ≤ T
must be V (t) defined as
Replicating Quantos 119
V (t) = exp {(rf − r − q − ρσxσs)(T − t)} X¯S(t) . ()
As just noted, we will need to invest this amount in the domestic risk-free
asset at date t. What remains to be done is to calculate the size of the long
position in the foreign risky asset and the offsetting short position in the
foreign risk-free asset.
From Itoˆ’s formula, we have
dV
V
= −(rf − r − q − ρσxσs) dt + dS
S
. ()
Equivalently,
dV = (r + q − rf + ρσxσs)V dt + V dS
S
. ()
On the other hand, consider a strategy that invests a(t) units of the domestic
currency in the foreign asset, b(t) units of the domestic currency in the foreign
risk-free asset, and c(t) units of the domestic currency in the domestic risk-
free asset. Let W = a+ b+ c denote the value of this portfolio. The return on
the foreign asset, per unit of domestic currency invested, is
d(XeqtS)
XeqtS
= q dt +
dX
X
+
dS
S
+
(
dX
X
)(
dS
S
)
= (q + ρσxσs) dt +
dX
X
+
dS
S
. ()
Similarly, the rate of return on the foreign risk-free asset is
d(erf tX)
erf tX
= rf dt +
dX
X
, ()
and of course the rate of return on the domestic risk-free asset is r dt. There-
fore, the change in the value of the portfolio will be
dW = a
[
(q + ρσxσs) dt +
dX
X
+
dS
S
]
+ b
[
rf dt +
dX
X
]
+ cr dt
= (aq + aρσxσs + brf + cr) dt + (a + b)
dX
X
+ a
dS
S
. ()
The change () of the portfolio value will match the change () of V if
and only if
a = V, b = −V, c = V . ()
This implies:
120 6 Foreign Exchange
The strategy that replicates the payoff X¯S(T ) at date T is to invest V (t)
units of domestic currency in the foreign asset, where V (t) is defined in ().
This will purchase
V (t)
X(t)S(t)
=
X¯
X(t)
exp {(rf − r − q − ρσxσs)(T − t)} ()
shares of the foreign asset. This position is financed entirely by borrowing
at the foreign risk-free rate. On the other hand, the same amount V (t) of
the domestic currency is invested in the domestic risk-free asset.
From our analysis at the end of Sect. , we know that the foreign risk-free
asset in this replicating strategy can be replaced by currency forwards. The
strategy here involves borrowing at the foreign risk-free rate, so we should
replace “long” by “short” in (). Borrowing V (t) units of domestic currency
means borrowing V (t)/X(t) units of the foreign currency. Therefore, ()
gives us:
An equivalent strategy for replicating the payoff X¯S(T ) at date T is to
invest V (t) units of domestic currency in the foreign asset and to be short
erf (T−t)V (t)/X(t) currency forwards at date t.
At the beginning of the previous section, we noted that an investor who
wants to bet on a foreign asset but does not want the exchange rate exposure
could simply buy the asset and sell the currency forward. This shows how much
of the asset he should buy and how much currency he should sell forward.
It is important to note that this strategy involves continuously buying
and selling forwards, just as it involves continuously trading the foreign asset.
Buying at date t a forward contract sold at date s < t cancels the delivery
obligation on the contract sold at s and leaves a cash flow of F (s) − F (t) to
be paid/received at the maturity date T . Therefore, the strategy accumulates
a liability or asset, depending on the direction the forward price moves, to be
received at T . On the other hand, maintaining an investment of V (t) in the
foreign asset will generate cash flows as the asset is sold or purchased over
time. As () shows, whether it is sold or purchased depends on the direction
the exchange rate moves. These cash flows should be invested or borrowed at
the domestic risk-free rate. Thus, there is a liability or asset to be received
at date T that is not shown in the boxed statement immediately above, and
there is an investment or liability in the domestic risk-free asset that is not
shown. It can be demonstrated that these cancel each other: if profits are made
from trading forwards, then they (more precisely, their present value) will be
consumed by the cost of buying the foreign asset, and vice versa. Hedging
with forwards (and with futures) is considered in more detail in Sect. .
Quanto Options 121
Quanto Forwards
In this section, we consider a contract similar to that of the previous section,
except that it is a pure forward, meaning that the payment for the contract
occurs at date T . We maintain all of the assumptions of the previous section.
The payment at date T is in domestic currency, and we define the quanto
forward price in units of domestic currency. Specifically, a long quanto forward
contract, initiated at date t and maturing at date T and initiated at the
forward price F ∗(t) will pay
X¯S(T )− F ∗(t)
at date T . The forward price F ∗(t) should be the price that makes this contract
have a value of 0 at date t.
We already know how to replicate the underlying payoff X¯S(T ) of the
forward contract at the cost V (t) defined in (). Thus, the synthetic quanto
forward is to purchase the replicating strategy and to borrow the cost V (t) at
the domestic risk-free rate. This leads to the liability er(T−t)V (t) at date T .
Therefore, we have:
The quanto forward price is
F ∗(t) = er(T−t)V (t) = exp {(rf − q − ρσxσs)(T − t)} X¯S(t) . ()
Notice that borrowing V in domestic currency to finance the replicating
strategy of the previous section — ., the domestic currency investments
described in () — means eliminating the domestic risk-free investment
c = V required in the previous section. The replicating strategy for the quanto
forward is simply to invest V in the foreign asset and to finance the investment
entirely by borrowing at the foreign risk-free rate. As in the previous section,
borrowing at the foreign risk-free rate can be replaced by borrowing at the
domestic risk-free rate and selling currency forwards.
Quanto Options
Consider now a European call option on a foreign asset, with strike K set in
the domestic currency and the value of the foreign asset being converted to
domestic currency at a fixed exchange rate X¯. This is called a “quanto call.”
We maintain all of the assumptions of the previous two sections.
The value of the quanto call at maturity is max(0, X¯S(T )−K). To value
this, we make use of what we learned in Sect. . Namely, the portfolio with
value V defined in () replicates the payoff X¯S(T ): in each state of the
world, V (T ) = X¯S(T ). Therefore, the quanto call is equivalent to a standard
122 6 Foreign Exchange
European call on the portfolio with domestic currency price V . The value
is therefore given by the Black-Scholes formula. From the formula () for
the dynamics of V , we see that the volatility of V is the same as that of S;
therefore, we should input σs as the volatility in the Black-Scholes formula.
Furthermore, the portfolio V is non-dividend-paying (it is the value of a claim
to X¯S(T ) at date T with no interim cash flows), so the dividend rate in the
Black-Scholes formula should be zero. Thus, we have:
The value of a quanto call is
V (0)N(d1)− e−rTK N(d2)
= exp {(rf − r − q − ρσxσs)T} X¯S(0)N(d1)− e−rTK N(d2) , ()
where
d1 =
log
(
V (0)
K
)
+
(
r + 12σ
2
s
)
T
σs
√
T
=
log
(
X¯S(0)
K
)
+
(
rf − q − ρσxσs + 12σ2s
)
T
σs
√
T
, ()
d2 = d1 − σs
√
T . ()
Likewise, the value of a quanto put is given by the Black-Scholes formula:
e−rTK N(−d2)− V (0)N(−d1) .
Notice that this is simply the Black-Scholes option formula with inputs V (0) =
initial asset price, K = exercise price, r = interest rate, σs = volatility, 0 =
dividend yield, and T = time to maturity.
We can hedge a written quanto call the same way we hedge a written or-
dinary call: we buy delta shares of the underlying and borrow the difference
between the cost of the delta shares and the option value. However, for the
quanto call, the underlying should be regarded as the portfolio with value V
described in Sect. . This portfolio consists of investing V (0) units of domes-
tic currency in the foreign asset, borrowing the same amount at the foreign
risk-free rate, and investing V (0) units of domestic currency in the domestic
risk-free asset. The delta of the call is N(d1), so the hedge consists of investing
N(d1)V (0) units of domestic currency in the foreign asset, borrowing the same
amount at the foreign risk-free rate, and investing N(d1)V (0) in the domestic
risk-free asset. The difference between the cost of this portfolio and the value
of the option is
N(d1)V (0)− [V (0)N(d1)− e−rTK N(d2)] = e−rTK N(d2) .
Return Swaps 123
This amount is to be borrowed at the domestic risk-free rate. Thus, the net
investment in the domestic risk-free asset is
N(d1)V (0)− e−rTK N(d2) ,
which is just the value of the option. To summarize:
To delta-hedge a written quanto call, one should invest N(d1)V (0) units
of domestic currency in the foreign asset, borrow the same amount at the
foreign risk-free rate, and invest the value of the option in the domestic
risk-free asset.
As in Sect. , borrowing N(d1)V (0) units of domestic currency at the
foreign risk-free rate can be replaced by borrowing the same amount at the
domestic risk-free rate and selling erfT N(d1)V (0)/X(0) currency forwards.
This results in:
An equivalent delta hedge for a written quanto call is to invest N(d1)V (0)
units of domestic currency in the foreign asset, sell erfT N(d1)V (0)/X(0)
currency forward contracts at the market forward price F (0), and borrow
e−rTK N(d2) at the domestic risk-free rate.
Return Swaps
There are many types and applications of return swaps, but here is one im-
portant example that involves the concepts discussed in this chapter. Suppose
an investor wants to receive at date T the difference in the rates of return
of two assets that are denominated in different currencies. The return will
be calculated on a given “notional principal.” For example, an investor may
want to receive at the end of a year the Nikkei rate of return minus the rate
of return on the S&P 500, calculated on a $1 million notional principal. If the
Nikkei earns 15% over the year and the S&P earns 10%, then the payment to
the investor is 5% of $1 million. If the reverse happens—the Nikkei earns 10%
and the S&P earns 15%—then the investor must pay 5% of $1 million to the
counterparty.
To model this, let Sf denote the price of a foreign asset and Sd the price
of a domestic asset. Assume they have constant dividend yields qf and qd. If
the returns are calculated excluding dividends, as is likely to be the case, then
the payment to the investor is(
Sf (T )− Sf (0)
Sf (0)
− Sd(T )− Sd(0)
Sd(0)
)
A =
(
Sf (T )
Sf (0)
− Sd(T )
Sd(0)
)
A ,
124 6 Foreign Exchange
where A denotes the notional principal. Of course, the investor may want the
reverse swap, and we consider this particular case only for concreteness.
The swap may have nonzero market value at date 0, which means that
some payment will have to be made upfront. To eliminate this, we can add
a “swap spread” into the contract, affecting the cash flow at date T . This is
a constant number a (which may be positive or negative), and including it
changes the payment to the investor to(
a +
Sf (T )
Sf (0)
− Sd(T )
Sd(0)
)
A . ()
The question we will address here is: what is the “fair” swap spread; ., for
what number a does the cash flow () have zero market value at date 0?
If the value is zero, then it is zero for any notional principal A, so we
can conveniently take A = 1. The cash flow consists of three pieces, all of
which are to be received/paid at date T : the constant a, the gross return on
the foreign asset, and the gross return on the domestic asset. The value at
date 0 of receiving a units of domestic currency is obviously e−rTa. As we
have observed several times before, the value at date 0 of receiving Sd(T )
units of domestic currency at date T is e−qdTSd(0), because this is the cost
of enough shares to accumulate to one share at date T via reinvestment of
dividends. Therefore, the value at date 0 of receiving Sd(T )/Sd(0) at date T
is e−qdTSd(0)/Sd(0) = e−qdT .
What remains is to calculate the value of receiving Sf (T )/Sf (0) units of
domestic currency at date T . We can do this by interpreting 1/Sf (0) as the
fixed exchange rate X¯ in the definition of a We need to assume as
before that the foreign asset price Sf and the exchange rate have constant
volatilities and a constant correlation. Denoting the volatilities by σs and σx
and the correlation by ρ as before, equation () shows that the value of
receiving X¯Sf (T ) = Sf (T )/Sf (0) units of domestic currency at date T is
exp {(rf − r − qf − ρσxσs)T} .
Adding up the pieces, the value at date 0 of the cash flow () (with A = 1)
is
e−rTa + exp {(rf − r − qf − ρσxσs)T} − e−qdT ,
so we conclude:
The fair swap spread, which equates the value at date 0 of receiving the cash
flow () at date T to zero, is
a = exp {(r − qd)T} − exp {(rf − qf − ρσxσs)T} . ()
3 To make sense of the units, note that the cash flow of Sf (T )/Sf (0) units of
domestic currency can be calculated as Sf (T ) units of foreign currency times
1/Sf (0) units of domestic currency per unit of foreign currency. Therefore, the
units of 1/Sf (0) can be taken to be the units of an exchange rate.
Uncovered Interest Parity 125
Uncovered Interest Parity in the Risk-Neutral
Probabilities
When we use numerical methods to value American and path-dependent op-
tions, as in Chap. 5, we typically focus on the dynamics of asset prices under
the risk-neutral measure. To apply these results to currency options or op-
tions on foreign assets, we need to know the dynamics of the exchange rate
under the risk-neutral measure. Because we can view the exchange rate as
the domestic price of an asset with the foreign risk-free rate rf being its div-
idend yield, we have already calculated these dynamics in equation () of
Sect. . The result is:
The exchange rate X must satisfy
dX
X
= (r − rf ) dt + σx dB∗x , ()
where B∗x is a Brownian motion under the risk-neutral measure.
This equation has an interesting interpretation in terms of “uncovered
interest parity,” which is the theory that differences in interest rates across
currencies will be offset on average by appreciation/depreciation of the cur-
rencies. In other words, it is the theory that the strategy of borrowing in
low-interest-rate currencies to invest in high-interest-rate currencies will not
earn money on average because of depreciation of the high-interest-rate cur-
rency relative to the low-interest-rate currency. It is well known that this
theory is not always true in reality. However, equation () shows that it is
true when we calculate expectations using the risk-neutral measure.
To see the interpretation of equation () as uncovered interest parity,
suppose that the foreign interest rate rf is lower than the domestic rate r.
Then one may be tempted to borrow at the foreign rate and invest at the
domestic rate. This would create a short position in the foreign currency.
Equation () states that the exchange rate is expected (under the risk-
neutral measure) to appreciate at the rate r−rf ; thus, repayment of the foreign
currency will be more expensive in terms of domestic currency, offsetting the
interest rate differential.
Problems
. Create an Excel worksheet to compare the values of call options on foreign
assets that are (i) struck in foreign currency or (ii) struck in domestic currency.
Prompt the user to input X(0), S(0), K, r, rf , σx, σs, ρ, q and T . Take the
strike price of the option struck in foreign currency to be K and take the
strike price of the option struck in domestic currency to be X(0)K (so K is
126 6 Foreign Exchange
interpreted as an amount in foreign currency). You should be able to confirm,
for example, that if r = rf and ρ ≥ 0 then the option struck in domestic
currency is more valuable.
. Repeat the preceding problem comparing (i) call options struck in foreign
currency, versus (ii) quanto call options. Use the same inputs as in the preced-
ing problem and take the fixed exchange rate in the quanto to be X¯ = X(0).
You should be able to confirm, for example, that if r = rf and ρ ≥ 0 then the
option struck in foreign currency is more valuable.
. Create an Excel worksheet in which the user inputs r and rf and
the exchange rate. Compute the forward exchange rate at maturities T =
, , . . . , and plot the forward rate against the maturity in a scatter
plot. A market is said to be in “contango” if this curve is upward sloping and
to be in “backwardation” if this curve is downward sloping. For currencies,
what determines whether the market is in contango or in backwardation?
. Create a VBA subroutine to simulate a path of the exchange rate and
the forward exchange rate under the risk-neutral measure, prompting the user
to input X(0), r, rf , σx, and the maturity T of the forward contract.
. Create a VBA subroutine to simulate a path of the exchange rate under
the actual probability measure, prompting the user to input X(0), σx, and the
expected rate of growth µ of the exchange rate under the actual probability
measure. Prompt the user also to input S(0), r, rf , σs, q, ρ, a fixed exchange
rate X¯, a maturity T , and a number of periods N . Calculate the gain/loss
from the portfolio that promises to pay X¯S(T ) at date T and uses a discretely
rebalanced hedge, rebalancing at dates t1, . . . tN = T , where ti− ti−1 = T/N ,
similar to the calculation in the function Simulated_Delta_Hedge_Profit.
Use the money-market hedge, which means investing V (0) at date 0, holding
the number of shares of the foreign asset shown in () at each date ti,
and having a short position in the foreign risk-free asset of the same value
at each date ti. Cash flows generated at each date from buying/selling the
foreign asset and lending/borrowing at the foreign risk-free rate should be
withdrawn/deposited in the domestic risk-free asset. Note: Because of discrete
rebalancing, this is not a perfect hedge, and the investment in the domestic
risk-free asset will not always equal V (t).
. Repeat the previous exercise using the forward contract hedge discussed
in Sect. . The cash flows generated from trading forwards cannot be with-
drawn/deposited in the domestic risk-free asset, because they do not materi-
alize until the maturity of the forward. You will have to create a variable to
keep track of the net asset/liability and include it in the valuation at date T .
Uncovered Interest Parity 127
. Derive the money-market hedge and the forward contract hedge for a
written quanto put.
. Suppose a customer has contracted with you for a return swap in which
the customer will receive the cash flow () for some number a, where A = 1.
How can you hedge this?
