**10.3 Currency Option Pricing Model
**

A |

currency option
gives its owner the right to buy/sell one currency for another at a fixed
exchange rate for a specified period of time. We
will adopt the convention that an
option specifies delivery of one* *unit
of one currency in return for *k* units
of the other.

In practice, currency
options come in both the European and American variety, although much of the
volume is taken up by the American options.
Further, the actual quantities delivered depend on the currencies; we
will explore this later when we look at examples of currency options.

In this topic, we derive
the option pricing model for currency options. This model, developed by Garman and Kolhagen (1983), is
applied widely in practice by currency traders.

In topic 10.2, __Interest
Rate Parity Relationship__,
we referred to different
currencies in terms of "strike" or "deliverable" currencies.
For example, if $U.S. is the strike currency and the Deutschmark, dm, is
the deliverable currency, a currency call option gives you the right to buy 1dm
for $*k*U.S.

If dm is the strike
currency, and $U.S. is the deliverable currency, a put option gives you the
right to sell $1U.S. for *k'*
dm.

If *k'=1/k*, you can immediately see that a call option (with deliverable
dm) is the same as a put option (with deliverable $U.S.).

For example, options
traded at the Philadelphia Stock Exchange (PHLX) have strikes in
$U.S. per one unit of deliverable currency (e.g., Australian *$*,
British pound, Canadian *$* etc.).
In each of these countries, currency options are traded with $U.S. as the
deliverable currency, with the home currency being the strike currency.
If the strikes are the "same" ( i.e., *k'=1/k*),
the call option in the U.S. should be priced exactly the same as the
(equivalent) put option in the other country.

**Notation
**

In order to develop an
option pricing model for currency options, we again have to be careful with
notation. One change from before is
that we use *K* for the strike price
instead of *X*.

*C*
= price of a currency call option that is defined in units of its strike
currency;

*P*
= price of a currency put option that is defined in units of its strike
currency;

*S*
= rate of exchange for one unit of deliverable currency in exchange for *x*
units of the strike currency;

*K*
= the strike price of the option;

*T*
= the time left to maturity for the currency option;

s
= the volatility of *S;*

= the risk-free interest rate for a pure discount (zero-coupon) bond with the
same maturity as the currency option in the strike country;

*r*_{d }= the risk-free interest rate for a pure discount bond with the same maturity as
the currency option in the deliverable country (in practice, this is the
Eurorate for the deliverable currency).

**The Exchange Rate Process**

The exchange
rate follows the process *dS* = m*Sdt*
+ s*Sdz*.
Note that *S *is quoted as units
of deliverable currency per unit of strike currency. If you find it more intuitive, you may want to think of your
domestic currency as the numerator and the foreign currency as the denominator.
For example, if your home currency is the dollar, and the foreign
currency is the yen, then the exchange rate is the number of dollars per yen.
This means that if *S* goes up,
you get more dollars per yen, and the dollar has thus depreciated.

**What
Makes up the Riskless Hedge
**

As you have seen
repeatedly, the first step in valuing an option is to find a risk-free portfolio
involving the option. With a stock
option, you saw that holding delta units of the stock and (plus/minus) one unit
of the option gave such a portfolio.

What do we do with an
exchange rate? The problem is that you cannot “hold” the exchange rate, but
only assets in the different currencies. Therefore,
the portfolio is constructed by holding risk-free bonds in the two currencies.
However, we have to be careful that when we construct this, we measure
everything in the same currency.

Let *B**d*
be the value of the bond in the deliverable currency, and *B**k* the value in the strike currency. Then, the value of *B**d* in the strike currency is *SB**d*,
and depends on the exchange rate.

If you find strike and
deliverable currencies confusing, think of the case of an American investor
interested in yen options traded on the Philadelphia Exchange.
For such an investor, the strike currency is the U.S. dollar and the
deliverable currency is the yen. *B**d* is a Japanese bond, and *S*
is the number of dollars per yen.

The riskless portfolio
is constructed by holding *B**d* and the option, and measuring everything in the strike
currency. In units of the strike
currency, the two assets are *SB**d* and *C*.

**The Option Pricing Formula
**

It turns out that the
formula for the currency call option is the same as that for an option on a
stock paying a __continuous dividend yield__
(see Chapter 9, topic 9.2) where the yield is *r**d*.
This is quite intuitive if you think of the asset *SB*d.
This asset has a drift rate equal to m + *r**d*, which is the depreciation/appreciation of the exchange rate
plus the return on the bond. The
formal equivalence of this is derived in Chapter 12, topic 12.4, __Application:
Currency Options__.

The Garman-Kolhagen
European call option price is:

and since log*(S'/K)
= *log*(S/K) - r**dT*

The European put option
price is:

where *d*1 and *d*2
are as before.

**Example
**

Consider the following information for options on the US
Dollar and British Pound. The spot
exchange rate is 1.73 (dollars per pound).
The 3-month risk-free interest rate in the US is 5%, while that in
Britain is 6.45%. The volatility of
the exchange rate is estimated to be 0.15.
Then, the value of a call option with strike price 1.7 is just over 6
cents (0.0629). The value of a put
option with the same strike is 4 cents.

As before, a major
feature of the currency option pricing model is that it provides traders with an
analytical description of the hedge parameters. These parameters are discussed in topic 10.4, __Currency
Options: Hedge Parameters__.