1) On the floor of a futures exchange one futures contract is traded where both the long
and short parties are closing out existing positions. What is the resultant change in the
open interest? Circle one.
a. No change
b. Decrease by one
c. Decrease by two
d. Increase by one
2) Form a long butterfly spread using the three call options in the table below.
a. What does it cost to establish the butterfly spread?
b. Calculate each of the Greek measures for this butterfly spread position and explain
how each can be interpreted.
c. How would you make this option portfolio delta neutral? What would be achieved by
doing so?
d. Suppose that tomorrow the price of C1 falls to $12.18 while the prices of C2 and C3
remain the same. Does this create an arbitrage opportunity? Explain.
3) Consider a six month American put option on index futures where the current futures
price is 450, the exercise price is 450, the risk-free rate of interest is 7 percent per
annum, the continuous dividend yield of the index is 3 percent, and the volatility of the
index is 30 percent per annum. The futures contract underlying the option matures in
seven months. Using a three-step binomial tree, calculate
a. the price of the American put option now,
b. the delta of the option with respect to the futures price,
c. the delta of the option with respect to the index level, and
d. the price of the corresponding European put option on index futures.
e. Apply the control variate technique to improve your estimate of the American option
price and of the delta of the option with respect to the futures price.
Note that the Black-Scholes price of the European put option is $36.704 and the delta
with respect to the futures price given by Black-Scholes is 0.442.
4) When there are dividends (circle one)
a. It is never optimal to exercise a call option early
b. It can be optimal to exercise a call option at any time
c. It is only ever optimal to exercise a call option immediately after an ex-dividend date
d. None of the above
5) A stock price is $100. Volatility is estimated to be 20% per year. What is the an
estimate of the standard deviation of the change in the stock price in one week (circle
one)
a. $0.38
b. $2.77
c. $3.02
d. $0.76
6) To be investment grade a company has to have a credit rating of (Circle one)
a. AA or better
b. A or better
c. BBB or better
d. BB or better
7) Gamma measures (circle one)
a. The rate of change of delta with the asset price
b. The rate of change of the portfolio value with the passage of time
c. The sensitivity of the portfolio value to interest rate changes
d. None of the above
8) The equity tranche of iTraxx is responsible for losses in the following range (Circle
one)
a. 0 to 3%
b. 0 to 4%
c. 0 to 5%
d. 0 to 6%
9) Theta measures (circle one)
a. The rate of change of delta with the asset price
b. The rate of change of the portfolio value with the passage of time
c. The sensitivity of a portfolio value to interest rate changes
d. None of the above
10) When the zero curve is upward sloping, which two of the following is true? (circle
two)
a. The one-year zero rate is always greater than the forward rate for the period between
1 year and 1.5 years
b. The one-year zero rate is always less than the forward rate for the period between 1
year and 1.5 years
c. The one-year par yield is always greater than the one-year zero rate
d. The one-year par yield is always less than the one-year zero rate
11) The short term risk-free rate usually used by derivatives traders in the
over-the-counter market is (circle one)
a. The Treasury rate
b. The LIBOR rate
c. The repo rate
d. The commercial paper rate
12) An investor shorts 100 shares when the share price is $50 and closes out the
position six months later when the share price is $43. The shares pay a dividend of $3
per share during the six months. How much does the investor gain?
13) An interest rate swap with a principal of $100 million involves the exchange of 5%
per annum (semiannually compounded) for 6-month LIBOR. The remaining life is 14
months. Interest is exchanged every six months. The 2 month, 8 month and 14 month
rates are 4.5%, 5%, and 5.4% with continuous compounding. Six-month LIBOR was
5.5% four months ago. What is the value of the swap?
14) A variable, x, starts at 10 and follows a generalized Wiener process
dx =a dt+b dz
where a = 2, b = 3, and dz is a Wiener process.
(i) What is the mean value of the variable after three years?
(ii) What is the standard deviation of the value of the variable after three years?
(iii) What is the mean value of the variable after six months?
(iv) What is the standard deviation of the value of the variable after six months?
15) Which of the following is
16) The estimate of the volatility of an asset made at the end of Tuesday is 1% per day.
On Wednesday the return realized on the asset is 5%. Update the volatility estimate
i. Use the EWMA model with = 0.9
ii. Use GARCH (1,1) with = 0.000003, =0.05, and =0.94
17) The Deutschemark-Canadian dollar exchange rate is currently 1.0000. At the end of
6 months it will be either 1.1000 or 0.9000. What is the value of a 6 month option to
sell one million Canadian dollars for 1.05 million deutschemarks. Verify that the answer
given by risk neutral valuation is the same as that given by no-arbitrage arguments. Is
the option the same as one to buy 1.05 million deutschemarks for 1 million Canadian
dollars? Assume that risk-free interest rates in Canada and Germany are 8% and 6% per
annum respectively.