We will make the assumption that it is demand deposits
at the interbank offering rates in the respective countries that
provide the appropriate definition of the underlying for a currency
option problem. This is consistent with one way an institution
could implement a cash market arbitrage designed to take advantage
of any option mispricing.
The Wall Street Journal provides information regarding
USD LIBOR. We will use this information with currency forward
prices to get an estimate for the foreign offering rate as illustrated
in the previous Tutor break.
Aside: In the following examples the closest LIBOR
rate is applied for example purposes only. No interpolation method
is used.
Problem: What is the
implied volatility and the Greeks for a Deutschmark currency option
trading on the Philadelphia exchange?
Data: Close of trading
October 30, 1996
Reported USD LIBOR rates in the WSW: 1-month 5 3/8%,
3-months 5 ½%
Step 1: Select the subject
Option Calculator in Option Tutor. From the menu item Options
select the sub-menu item calculator. Click on LIBOR and enter
the following LIBOR rate information:
That is, the continuously compounded equivalent rate
for the US is 0.05437. Click on Transfer to transfer this rate
to both the option calculator and the futures calculator.
Step 2: For the close
of trading October 30, the 30-day forward price = 0.6628, and
the spot exchange rate = 0.6616. Select Options from the menu
item in Option Calculator and then select the sub menu item Futures
Calculator. In the regular calculator enter the 30-day life in
the Convert Dates into Maturity and transmit to the future's calculator
by clicking on the button Convert and then Transmit:
Step 3: The futures calculator
now has the time to maturity and the US interest rate. Select
Currencies in the Future's calculator and enter the spot exchange
rate (0.6616) and forward exchange rate (0.6628) as displayed
below. Finally, for this step select Implied Foreign Int. Rate
and then click on Calculate. Your futures calculator is as follows:
The implied rate for Germany is 0.033 from 30-day
forwards on October 30, 1996.
Step 4: The goal of this
next step is to enter this data into Option Tutor's Option Calculator
to solve for implied volatility and the Greeks from the option
price.
Data: Trading terminates 2nd Friday before
3rd Wednesday of the month. That is, they are exercised
at the prices realized on this day. This day is November 15,
1996 which when expressed as proportion of a year equals 0.043836.
66 Nov Put 0.22, Strike price = $0.66 USD for DEM
1, premium 22 units = $6.25 * 22 = $137.50
Underlying = 62,500*0.6616 = 41,350, Strike = 62,500*0.66
= 41,250
European style options
The implied volatility and Greeks are as displayed
below:
Step 5: Consider the
same problem as above but this time for an American, December
maturity option.
Data: 66 Dec Put 0.57, Strike price = $0.66 USD for DEM 1, premium 57 units = $6.25 * 57 = $356.25
Expiration is 3rd Wednesday of December, (December 18, 1996 = 0.13427)
American style
Assume that the same interest rates apply to this
problem i.e., flat from end of November to middle December. This
is a reasonable simplifying assumption if you contrast the 1-month
and 3-month LIBOR rates at this time (5 3/8 for 3-months vrs 5
½ for 3-months)
The trinomial tree numerical procedure is used first
for 100, and then 1000 steps (time partitions). For 100 steps
these estimates are:
and for 1000 steps these estimates are:
You can check that this method is very accurate even
with as few as 100 partitions.
(C) Copyright 1997, OS Financial Trading System