金融衍生工具测试题 (26)

1.

A box spread is a combination of a bull spread composed of two call options with strike

prices

X

1

and

X

2

and a bear spread composed of two put options with the same two

strike prices.

a)

Describe the payoff from a box spread on the expiration date of the options.

b)

What would be a fair price for the box spread today?

Define variables as

necessary.

c)

Under what circumstances might an investor choose to construct a box spread?

d)

What sort of investor do you think is most likely to invest in such an option

combination, i.e. a hedger, speculator or arbitrageur?

Explain your answer.

Form a long butterfly spread using the three call options in the table below.

C1

C2

C3

X = $$90

X = $$100

X = $$110

T = 180 days

T = 180 days

T = 180 days

Price

16.3300

10.3000

6.0600

DELTA

0.7860

0.6151

0.4365

GAMMA

0.0138

0.0181

0.0187

THETA

-11.2054

-12.2607

-11.4208

VEGA

20.4619

26.8416

27.6602

RHO

30.7085

25.2515

18.5394

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.

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.

A financial institution trades swaps where 12 month LIBOR is exchanged for a fixed rate of

interest. Payments are made once a year. The one-year swap rate (i.e., the rate that would be

exchanged for 12 month LIBOR in a new one-year swap) is 6 percent. Similarly the two-year

swap rate is 6.5 percent.

2.

3.

4.

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a)

Use this swap data to calculate the one and two year LIBOR zero rates, expressing

the rates with continuous compounding.

b)

What is the value of an existing swap with a notional principal of $$10 million that

has two years to go and is such that financial institution pays 7 percent and receives

12 month LIBOR?

Payments are made once a year.

c)

What is the value of a forward rate agreement where a rate of 8 percent will be

received on a principal of $$1 million for the period between one year and two years?

Note: All rates given in this question are expressed with annual compounding.

5.

The term structure is flat at 5% per annum with continuous compounding. Some

time ago a financial institution entered into a 5-year swap with a principal of $$100

million in which every year it pays 12-month LIBOR and receives 6%. The swap

now has two years eight months to run. Four months ago 12-month LIBOR was

4% (with annual compounding). What is the value of the swap today? What is the

financial institution’s credit exposure on the swap?

An American put option

to sell a Swiss franc for USD has a strike price of

0.80 and a time to maturity of 1 year. The volatility of the Swiss franc is 10%,

the USD interest rate is 6%, and the Swiss franc interest rate is 3% (both interest

rates continuously compounded). The current exchange rate is 0.81. Use a three

time step tree to value the option.

A European call option on a certain stock has a strike price of $$30, a time to

maturity of one year and an implied volatility of 30%. A put option on the same

stock has a strike price of $$30, a time to maturity of one year and an implied

volatility of 33%. What is the arbitrage opportunity open to a trader. Does the

opportunity work only when the lognormal assumption underlying Black-Scholes

holds. Explain the reasons for your answer carefully.

A put option on the S&P 500 has an exercise price of 500 and a time to maturity

of one year. The risk free rate is 7% and the dividend yield on the index is 3%.

The volatility of the index is 20% per annum and the current level of the index is

500. A financial institution has a short position in the option.

a) Calculate the delta, gamma, and vega of the position. Explain how they can be

interpreted.

b) How can the position be made delta neutral?

6.

7.

8.

c) Suppose that one week later the index has increased to 515. How can delta

neutrality be preserved?

9.

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

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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?

10.

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.

An American put futures option has a strike price of

0.55 and a time to maturity

of 1 year. The current futures price is 0.60. The volatility of the futures price is

25% and

the

interest rate(continuously compounded) is 6% per annum.

Use

a four time step tree to value the option.

Is it ever optimal to exercise early an American call option on a) the spot price of

gold, b) the spot price of copper, c) the futures price of gold, and d) the average

price of gold measured between time zero and the current time. Explain your

answers.

The future probability distribution of a stock price has a fatter right tail and

thinner left tail than the lognormal distribution. Describe the effect of this on the

prices of in-the-money and out-of-the- money calls and puts. What is the volatility

smile that would be observed?

A bank has just sold a call option on 500,000 shares of a stock. The strike price is

40; the stock price is 40; the risk- free rate is 5%; the volatility is 30%; and the

time to maturity is 3 months.

a) What position should the company take in the stock for delta neutrality?

b) Suppose that the bank does set up a delta neutral position as soon as the option

has been sold and the stock price jumps to 42 within the first hour of trading.

What trade is necessary to maintain delta neutrality? Explain whether the bank has

gained or lost money in this situation. (You do not need to calculate the exact

amount gained or lost.)

c) Repeat part b) on the assumption that the stock jumps to 38 instead of 42

15.

A bank has sold a product that offers investors the total return (excluding dividends)

on the Toronto 300 index over a one year period.

The return is capped

at 20%.

If the index goes down the original investment of the investor is returned.

11.

12.

13.

14.

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