In
finance, a foreign exchange option (commonly shortened
to just FX option or currency option) is
a derivative financial instrument where the owner has
the right but not the obligation to exchange money denominated
in one currency into another currency at a preagreed
exchange rate on a specified date.
The
FX options market is the deepest, largest and most liquid
market for options of any kind in the world. Most of the
FX option volume is traded OTC and is lightly regulated,
but a fraction is traded on exchanges like the Philadelphia
Stock Exchange, or the Chicago Mercantile Exchange for
options on futures contracts: the global market for exchangetraded
currency options is notionally valued by the Bank for
International Settlements at $158,300 million in 2005.
Example
For
example a GBPUSD FX option might be specified by a contract
allowing the owner to sell Â£1,000,000 and buy $2,000,000
on December 31. In this case the preagreed exchange rate,
or strike price, is 2.0000 GBPUSD or 0.5000 USDGBP and
the notionals are Â£1,000,000 and $2,000,000 (Â£1,000,000
from the eyes of a USD investor, $2,000,000 from the eyes
of a GBP investor).
This
type of contract is both a call on dollars and a put on
sterling, and is often called a GBPUSD put by market participants,
as it is a put on the exchange rate; it could equally
be called a USDGBP call, but isn't, as market convention
is to quote the 2.0000 number (normal quote), not
the 0.5000 number (inverse quote).
If
the rate is lower than 2.0000 GBPUSD come December 31
(say at 1.9000 GBPUSD), meaning that the dollar is stronger
and the pound is weaker, then the option will be exercised,
allowing the owner to sell GBP at 2.0000 and immediately
buy it back in the spot market at 1.9000, making a profit
of (2.0000 USD/GBP  1.9000 USD/GBP)*1,000,000 GBP = 100,000
USD in the process. If they immediately exchanges their
profit into GBP, this amounts to 100,000/1.9000 = 52,631.58
GBP.
Terms
Generally
in thinking about options, one assumes that one is buying
an asset: for instance, you can have a call option on
oil, which allows you to buy oil at a given price. One
can consider this situation more symmetrically in FX,
where one exchanges: a put on GBPUSD allows one
to exchange GBP for USD: it is at once a put on GBP and
a call on USD.
As
a vivid example: people usually consider that in a fast
food restaurant, one buys hamburgers and pays in dollars,
but one can instead say that the restaurant buys dollars
and pays in hamburgers.
There
are a number of subtleties that follow from this symmetry.
 Ratio
of notionals
 The
ratio of the notionals in an FX option is the strike,
not the current spot or forward. Notably,
when constructing an option strategy from FX options,
one must be careful to match the foreign currency
notionals, not the local currency notionals, else the
foreign currencies received and delivered don't offset
and one is left with residual risk.
 Nonlinear
payoff
 The
payoff for a vanilla option is linear in the underlying,
when one denominates the payout in a given numeraire.
In the case of an FX option on a rate, one must
be careful of which currency is the underlying and which
in the numeraire: in the above example, an option on
GBPUSD gives a USD value that is linear in GBPUSD (a
move from 2.0000 to 1.9000 yields a .10 * $2,000,000
/ 2.0000 = $100,000 profit), but has a nonlinear GBP
value in GBPUSD. Conversely, the GBP value is linear
in the USDGBP rate, while the USD value is nonlinear
in the USDGBP rate. This is because inverting a rate
has the effect of , which is nonlinear.
 Change
of numeraire
 the
implied volatility of an FX option depends on the numÃ©raire
of the purchaser, again because of the nonlinearity
of .
Hedging
with FX options
Corporations
primarily use FX options to hedge uncertain future
cash flows in a foreign currency. The general rule is
to hedge certain foreign currency cash flows with
forwards, and uncertain foreign cash flows
with options.
Suppose
a United Kingdom manufacturing firm is expecting to be
paid US$100,000 for a piece of engineering equipment to
be delivered in 90 days. If the GBP strengthen against
the US$ over the next 90 days the UK firm will lose money,
as it will receive less GBP when the US$100,000 is converted
into GBP. However, if the GBP weaken against the US$,
then the UK firm will gain additional money: the firm
is exposed to FX risk. Assuming that the cash flow is
certain, the firm can enter into a forward contract to
deliver the US$100,000 in 90 days time, in exchange for
GBP at the current forward rate. This forward contract
is free, and, presuming the expected cash arrives, exactly
matches the firm's exposure, perfectly hedging their FX
risk.
If
the cash flow is uncertain, the firm will likely want
to use options: if the firm enters a forward FX contract
and the expected USD cash is not received, then
the forward, instead of hedging, exposes the firm to FX
risk in the opposite direction.
Using
options, the UK firm can purchase a GBP call/USD put option
(the right to sell part or all of their expected income
for pounds sterling at a predetermined rate), which will:
 protect
the GBP value that the firm will receive in 90 day's
time (presuming the cash is received)
 cost
at most the option premium (unlike a forward, which
can have unlimited losses)
 yield
a profit if the expected cash is not received but FX
rates move in its favor
Valuing
FX options: The GarmanKohlhagen model
As
in the BlackScholes model for stock options and
the Black model for certain interest rate options, the
value of an european option on a FX rate is typically
calculated by assuming that the rate follows a lognormal
process.
In
1983 Garman and Kohlhagen extended the BlackScholes model
to cope with the presence of two interest rates (one for
each currency). Suppose that r_{d} is the riskfree
interest rate to expiry of the domestic currency and r_{f}
is the foreign currency riskfree interest rate (where
domestic currency is the currency in which we obtain the
value of the option; the formula also requires that FX
rates  both strike and current spot be quoted in terms
of "units of domestic currency per unit of foreign currency").
Then the domestic currency value of a call option into
the foreign currency is

The
value of a put option has value

where :


 S_{0}
is the current spot rate
 K
is the strike price
 N
is the cumulative normal distribution function
 r_{d}
is domestic risk free rate
 r_{f}
is foreign risk free rate
 and
Ïƒ is the volatility of the
FX rate.