17. The spot USD-EUR exchange rate is USD1.24/EUR. Consider a one-month (=0.083 years) put option on

17.
The spot USD-EUR exchange rate is USD1.24/EUR. Consider a
one-month (=0.083 years) put option on the EUR with a strike of USD1.25/EUR.
Assume that the volatility of the exchange rate is 12%, the one-month interest
rate on the USD is 3.1%, and the one-month interest rate on the EUR is 3.7%,
both in continuously-compounded terms.

(a)
What is the Black-Scholes price of the put?

(b) If you had written this put on EUR 10 million, what would you do
to delta-hedge your position?

18.
The spot USD-EUR exchange rate is USD1.50/EUR. Consider a
six-month (= 0.5 years) call option on the EUR with a strike of USD1.50/EUR.
Suppose the volatility of the exchange rate is 20%, the six-month interest rate
on the USD is 1.5%, and the six-month interest rate on the EUR is 2.5%, both in
continuously-compounded terms.

Sundaram
& Das: Derivatives – Problems and Solutions . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 201

(a)
What is the Black-Scholes price of the call?

(b) If you had written this call on EUR 100 million, what would you do
to delta-hedge your position?

(a)

19.
The spot USD-EUR exchange rate is USD1.50/EUR. Price a one-month
straddle with an at-the-money-forward (ATMF) strike. The ATMF strike price is
de ned to be that

value of K which equals the forward exchange
rate for that maturity, i.e., for which KerT = SeqT . Assume that the volatility of the exchange rate is 20%, the
six-month interest rate on the USD is 1.5%, and the six-month interest rate on
the EUR is 2.5%, both in continuously-compounded terms.

20.
An option is said to be at-the-money-forward (ATMF) if the strike
price equals the

forward price on the stock for that maturity.
Assume there are no dividends, so the ATMF strike K satis es St = PV (K) = er(T t)K. Show that the value of an ATMF
call in the Black-Scholes world is given by

^

(4)

St [2 N(d1) 1]

^

p

where d1 = [ T
t]=2.

21. Show
that the at-the-money-forward call price (4) is approximately equal to

St p1 pT t (10)
2

Remark: Expression (10) gives us a quick method for calculating
the prices of ATMF calls. Two interesting points about expression (10):

Sundaram
& Das: Derivatives – Problems and Solutions . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 203

(a) It depends on only three parameters (St, , and T t) and the constant ;
in particular, the cumulative normal distribution function N( ) is not
involved.

(b)
It shows that the price of at-the-money-forward calls are
approximately linear in .

These features make the formula above very
easy to use in practice not only to obtain prices of ATMF options, but also to
obtain quick estimates of implied volatility of such options. The next two
questions illustrate these points.
^

22. Using (10), identify the approximate price of an at-the-money
forward call with the following parameters:

(a)
S = 50, T t = 1 month,
and = 0:15.

(b) S = 70, T t = 2 month,
and = 0:25.

23. Suppose an at-the-money forward call with one month to maturity is
trading at a price of C = 0:946 when the stock price is St = 54:77.

(a)
Using the approximation (10), what is the implied volatility on
the call?

(b)
What if the call were trading at C = 1:576 instead?

24. A stock index is currently at 858. A call option with a strike of
850 and 17 days (= 0.047 years) to maturity costs 23.50. Assume an interest
rate of 3%. For simplicity, assume also that the dividend yield on the index is
zero.

(a)
What is the implied volatility?

(b)
If implied volatility went up to 28%, what would happen to the
call’s value?

Sundaram
& Das: Derivatives – Problems and Solutions . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 205

(c) If all the other parameters remained the same, what would the
option value be after one week (i.e., with 10 days or 0.027 years left to
maturity)?

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