hree popular types of currency options are:  European options traded on the exchange rate (Philadelphia Stock Exchange), American options traded on the exchange rate (Philadelphia Stock Exchange), and American options traded on currency futures (Chicago Board Options Exchange).  For currency options, the last trading day is the Friday before the third Wednesday of the settlement month. For options on the futures, the last trading day is the second Friday before the third Wednesday.  Settlement is on the Saturday.

Online, you can apply the Option Calculator to value some currency option examples.  We will assume that the strike currency is the U.S.$ and the delivery currencies are pounds, Deutschmarks, and the yen.  For the former two (pounds and Deutschmarks) option values have been driven by speculation as to whether Europe's Exchange Rate Mechanism (ERM) (The ERM was first formed on March 13, 1979 to manage target ranges for the exchange rates of member countries.) will hold together or not.

For example, the British pound joined the ERM on October 8, 1990, but pulled out (along with the Italian lira) on September 16, 1992.  This precipitated a currency crisis in which the implied volatilities for pounds and Deutschmarks rose from the 11%-12% range to over 20% by the end of October.  Implied volatilities then fell back sharply during 1993.  Pressure again built up for a realignment of target ranges when the German Bundesbank failed to deliver interest rate cuts on Thursday, July 29, 1993.  The foreign exchange market response was again hectic, leading to heavy intervention by the European central banks to support ERM targets as required by ERM regulations.  Ultimately, the mark/franc targets were adjusted to resolve this issue.  This was followed by implied volatilities settling down to just above historic levels.

In this situation, the Black-Scholes assumption of constant volatility was clearly violated.  Furthermore, it is unlikely that projected volatilities will remain constant as we move across different option maturities because of potential currency crises.

To apply the model, we need to make an assumption about volatilities. During the period we are examining, there was no currency crisis.  For the purposes of this exercise, we will use the historical level of 11% for the Deutschmark.

Assume the values in Table 13.11 as reported in The Wall Street Journal on March 31, 1994 (closing prices for March 30, 1994, Philadelphia Exchange).  Using the information in Tables 13.11 and 13.12, combined with the LIBOR information, let's calculate currency option values.

First, we'll provide some examples to step you through the details.

Table 13.11

Currency Option Values 

Table 13.12

Settlement Dates

Spot exchange rates at the close on March 30, 1994 (U.S.$ equivalent), were

Pounds:   1.4800

Marks:      0.5968

Yen:         0.009713

Consider first the European put option, strike 59, June maturity, defined on the German mark.  The deliverable currency is 62,500 marks, and the strike currency is U.S.$.  Prices are quoted in units of U.S.$ 0.0001 ("points") per mark.  As a result, the last traded price equal to 0.94 cents implies a contract price of 0.0094*(62,500) = $587.50.  At the spot exchange rate, the total strike currency for 62,500 marks is 0.59*(62,500) = U.S.$ 36,875 and the current underlying asset value is the U.S.$ value of 62,500 marks, which equals 0.5968*(62,500) = $37,300.

Gathering this together implies that:

S = $37,300

X = $36,875

 = 0.039375  (3-month LIBOR, Table 13.8)

= 0.0570536 (3-month LIBOR, Table 13.8)

T  = 0.2

s  = 0.11 (assumed)

Observed market price p = $562.5

Online, you can compute the price using the Garman-Kolhagen model.  This price is $584.60.

In terms of quoted cents, this equals $584.60/62,500 = 0.009356 which in terms of quoted price is $0.935 in units of $0.0001, very close to the actual $0.94 price.

You may be asking  whether the volatility estimate holds up across other contracts and times.

To answer this question, consider the April European option contract on the Deutschmark with a 59 strike, settled April 16.

By working through the same steps, you can verify that:

April contract:

Maturity T = 0.0466

S = $37,300

X = 0.59*(62,500) = $36,875

Using the one-month LIBOR as an approximation yields:

 = 0.036875

= 0.0578572

s = 0.11 (assumed)

Predicted Market Price C = $578.10

To convert to quoted cents, we divide the total contract price by 62,500 and scale into units of 0.0001:

Price in cents:   $578.10/62,500 = 0.92.

Market price = $0.90.

Similarly, we can check whether the value of the European put option with the same characteristics (April, strike 0.59) is similarly overestimated.

Maturity T = 0.0466

S = $37,300

X = 0.59*(62,500) = $36,875

Using the one-month LIBOR as an approximation yields:

 = 0.036875

= 0.0578572

s = 0.11 (assumed)

Online, applying the calculator you can verify the price to equal $190.20.  Converting this back to a price in cents is:

Price in cents:   190.20/62,500 = $.30

Market price in cents:  $0.28.

From these put and call prices, it is apparent that the implied volatility is less than our assumed 11% for short-dated options.  You can use the calculator to find the implied volatilities for the April European put and call options.

First convert the cents per unit to a total contract price:

Put price p = 62,500*(0.28) = $175.00

Call price c = 62,500*(0.90) = $562.50

Plugging each value into the calculator, and switching on the implied volatility calculator, you will get:

Implied put volatility = 0.105

Implied call volatility = 0.105

If you assume that the model provides a reasonable estimate of implied volatility, then,  at least over the short run, it appears that the market is not anticipating any crisis, since the implied volatility for the German mark is quite low.