Question 1

Leland,O'Brien and Rubinstein (who invented portfolio insurance)came up with a product called "supershares." The product works as follows.It starts with two conventional funds: an index fund owning stocks in the S&P 500 and a money-market fund.
The index fund is divided into two securities.One of those securities is called a "dividend share" and gives the holder the right to all dividends paid during three years and all price appreciation up to 25% during those three years.The other security is called an "appreciation share" and gives the holder the right to all price appreciation above 25% during those three years.
The money-market fund is also divided into two securities.One of those securities is called a "money market income supershare" and the other security is called a "protection supershare".The money market supershare gives the holder the right to all interest income during three years.At the end of those three years,the holder may also get back some or all of the principal value,depending on how well the stock market performs.For every 1% that the S&P 500 has fallen below its current level,the principal value payable to the holder of a money market supershare is reduced by 1% and is instead paid to the holder of a protection supershare (which also has a three-year lifetime).
Assume that the current level of the S&P 500 index is 277,that the standard deviation of the index is 25%,and that the average dividend yield on the index is 4%.Also assume that the current six-month T-bill rate is 8% (which is also the rate on the money-market account)and that money-market fund securities are in units of $100.
(Hints: Remember that the value of the dividend share plus the appreciation share equals the current level of the S&P 500 index and that the value of the money market income supershare plus the protection supershare equals the value of the money market fund.Also,you can adjust for dividends in the Black-Scholes formula by subtracting the dividend yield from the riskless rate and then using this adjusted rate instead of the riskless rate in the formula.)
a. Estimate the value of the appreciation share.
b. Estimate the value of the dividend share.
c. Estimate the value of the protection supershare.
d. Estimate the value of the money market income supershare.

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Question 2

Consider the following data for a stock and a call option on that stock: S₀ = $50,S₁ = $75 or $100,E = $50,and r = 1.10.Derive the hedge ratio ( )and the price of the call option.

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Question 3

Assume that it is now December 2002.You are given the following information: December 2003 gold futures contract price = 515.60/troy oz.;
spot gold price = 481.40/ troy oz.;
annualized interest rate = 6%;
annualized carrying cost of gold = 2%.
a. The above information presents an arbitrage opportunity. Describe what you would have to do now to set up the arbitrage.
b. What would you have to do in December to unwind the position in part a? What is the arbitrage profit?

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Question 4

You have just learned through the grapevine that Pfizer (currently at $50.625 per share)may be a takeover candidate at $65 per share.You would like to speculate on the rumor,but you are worried that the stock will drop significantly if the rumor is false.Therefore,you have decided to use options to exploit this information.You are given the following option data for today,May 15th:
calls puts

a. Set up an option position that will best exploit the information you have, assuming that the takeover will happen by September 16 (the expiration day of the September options).
b. Assume now that the annualized standard deviation of Pfizer's stock price is 0.40 and that the six-month T-bill rate is 6%. Furthermore, assume that Pfizer pays a quarterly dividend of 50 cents and that the dividends are paid in April, July, October and January. What would your Black-Scholes estimates be for the options in the position that you have described in part a?
c. If the beta of Pfizer stock is 1.0, what is the beta of the position that you have set up in part a?
d. What are the deltas of the options that you have chosen for your position?

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Question 5

Assume that one can purchase gold bars or 6-month futures on gold bars.
a. Using the law of one price, derive the relationship between the spot price of gold and the futures contract price.
b. Assume that the futures are underpriced relative to the contract price just derived. What action should an investor take?

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Question 6

You are a portfolio manager who has just discovered the possibilities of stock-index futures.Assume that today is January 12.
a. Assume that you have the resources to buy and hold the stocks in the S&P 500. The current level of S&P 500 index is 258.90, the June S&P 500 futures contract is selling for 260.15, the annualized rate on the T-bill expiring on the expiration date of the futures contract is 6%, and the annualized dividend yield on S&P 500 stocks is 3%. Assume that dividends are paid out continuously over the year. Is there a potential for arbitrage, and, if so, how would you go about setting up the arbitrage?
b. Assume now that you are known for your stock selection skills. You have 10,000 shares of Texacola (now selling for 38) in your portfolio, and you are worried about the direction of the market until June. You would like to protect yourself against market risk by using the December S&P 500 futures contract (which is currently at 260.15). If Texacola's beta is 0.8, how would you go about creating this protection?

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Question 7

You are convinced that the next three months are going to be boom or bust months for IBM.Foreseeing a major increase or decrease in the stock price,you set up a position in options where you buy July 120 calls at $8.75 and you buy July 120 puts at $8.25.IBM is currently selling for $119.
a. Draw the payoff diagram of cash flows on this position.
b. What are the break-even points on the upside and downside of this position?
c. Now assume that IBM has a variance of 0.08. Using the Black-Scholes model to value these two options, do you still think that you should take the above position? Why or why not? (Today is December 21, 2002; the options expire on July 18, 2003; the annualized riskless rate is 6%; ignore dividends.)
d. What is the implied variance in the July 120 call?

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Question 8

Consider a portfolio consisting of long positions in both 6-month Treasury bills and call options.What is the payoff pattern (potential cash flows)and what is this portfolio equivalent to? (Hint: Use put-call parity.)

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Question 9

Suppose you are holding the following portfolio: 100 shares of XYZ stock plus a call option contract for 100 shares of XYZ with exercise price of 50 and a June expiration date plus a put option contract for 100 shares of XYZ with exercise price of 40 and a June expiration date.
a. What will this portfolio be worth on expiration date if XYZ is at 35? At 45? At 55?
b. Can this portfolio ever be worth less than $3500? Can it be worth more than $10,000? Explain.
c. Answer the questions in part a and part b if your portfolio involves a short position in the call option contract instead of a long position. (I.e., you are long 100 shares of stock and the put contract, and short the call option contract.)

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Question 10

By example,diagram,and/or verbal description,demonstrate how put options can be used to construct a "floor" under the return on a portfolio of common stocks.

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Question 11

The trade deficit numbers are expected out on February 19,and you think that market will move a lot,either up or down.You want to take advantage of this using options.You are given the following stock-index option data for today,January 14 (the current level of the index is 236.99,and the annualized T-bill rate is 6%):

a. Find at least two options in the above listing that violate arbitrage
conditions.
b. How would you set up a position using February options to take advantage
of the volatility from the trade deficit numbers? (The February options expire
on the evening of February 19.)
c. What are the breakeven points for the position in part b? (You can draw a
payoff diagram if you want to.)
d. Assume no dividends are paid and that the variance in the stock index is
0.09, and use the Black-Scholes model to value the February 235 call and
the February 235 put.

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Question 12

You have been asked to determine the theoretical bounds on a futures contract on gold.You are supplied with the following information: spot price of gold = $400/troy oz.; time to expiration on futures contract = 6 months; riskless rate = 10%; borrowing rate for marginal investor = 12%; lending rate for marginal investor = 8%; storage costs for gold = $20/troy oz.per year.Short sellers can hope to recover only half of the storage costs that they save by short selling.
a. What is the upper bound on the theoretical futures price?
b. (Recalculate the new bounds.)
b. What is the lower bound on the theoretical futures price?
c. Assume that investors are required to put up a 10% margin on all futures transactions and that only cash (no T-bills) can be used to meet margin requirements. Evaluate the effect this would have on the upper and lower bounds you estimated in part a and part

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Question 13

Consider the purchase of a put option with an exercise price of $40 and a cost of $5 and the purchase of a call option with the same expiration date and on the same stock with an exercise price of $45 and a cost of $6.Graph the profit of this combination.Be sure to label all points.

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Question 14

Consider the following table of partial cash flows:

Assume that the risk-free rate over the period is 10% and price the put option.

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Question 15

You are interested in putting together an option position on Flaming Yon,Inc.The current stock price is 58.25,and The Wall Street Journal of December 11,2002,reports the following prices for the various listed options on the stock:

a. Assume that you buy a January 50 call and a January 60 put. Draw the payoff diagram for this position at maturity.
b. What are the upside and downside breakeven points for this position?

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Question 16

The following is a listing of option prices on Perdida Enterprises on December 12,2002:

The current stock price is 45.75,and the riskless rate is 7%.
a. Consider the following position: sell one January 40 call; buy two January 45 calls; sell one January 50 call. Evaluate the net cash flows on this position at expiration for different stock prices, and draw a payoff diagram.
b. Are the three January put options correctly priced relative to the corresponding call options? (Assume that there are 42 days on the January option.)

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Question 17

Tribbles are soft,furry creatures that reproduce themselves by dividing (like amoebas do)into two tribbles every 60 days.If a tribble now costs $1,what should be the price of a forward contract that expires in 60 days (immediately after the reproduction takes place)for one tribble?

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Question 18

An investment bank has come up with a new product on the S&P 500 index.It offers an appreciation share,the owner of which is entitled to all price appreciation over 10% in the first two years and over 20% in the next three years after that.If the current index value is 250,the riskless rate is 8%,the dividend yield on the index is 3%,and the annualized standard deviation of the index is 20%,what is the value of this share? (Hint: There might be more than one option in this share.)

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Question 19

Consider a futures contract on Japanese yen.Derive the relationship between the spot $/yen exchange rate and the futures contract price.(Hint: Consider an investment in the riskless asset of each country.)

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Question 20

Assume that you are a mutual fund manager with a total portfolio value of $100 million.You estimate the beta of the fund to be 1.25.You would like to hedge against market movements by using stock index futures.You observe that the S&P 500 June futures are selling for 260.15 and that the index is at 256.90.
a. How many stock index futures would you have to sell to protect yourself against market risk?
b. If the riskless rate is 6% and the market risk premium is 8%, what return would you expect to make on the mutual fund (assuming you don't hedge)?
c. How much would you expect to make if you hedge away all market risk?

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Quiz 4: Security Analysis and Portfolio Theory

Consider the following data for a stock and a call option on that stock: S0 = $50,S1 = $75 or $100,E = $50,and r = 1.10.Derive the hedge ratio ( )and the price of the call option.