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Questions and Answers of
Corporate Finance
Section 10.1 gives an example of a situation where the value of a European call option decreases with the time to maturity. Give an example of a situation where the value of a European put option
Explain why the arguments leading to put–call parity for European options cannot be used to give a similar result for American options.
What is a lower bound for the price of a six-month call option on a non-dividend-paying stock when the stock price is $80, the strike price is $75, and the risk-free interest rate is 10% per annum?
A diagonal spread is created by buying a call with strike price K2 and exercise date T2 and selling a call with strike price K1 and exercise date T1(T2 > T1). Draw a diagram showing the value of
Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that
Use put–call parity to show that the cost of a butterfly spread created from European puts is identical to the cost of a butterfly spread created from European calls.
A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices
Construct a table showing the payoff from a bull spread when puts with strike prices K1 and K2 are used (K2 > K1).
An investor believes that there will be a big jump in a stock price, but is uncertain as to the direction. Identify six different strategies the investor can follow and explain the differences among
How can a forward contract on a stock with a particular delivery price and delivery date be created from options?
A box spread comprises four options. Two can be combined to create a long forward position and two can be combined to create a short forward position. Explain this statement.
What is the result if the strike price of the put is higher than the strike price of the call in a strangle?
One Australian dollar is currently worth $0.64. A one-year butterfly spread is set up using European call options with strike prices of $0.60, $0.65, and $0.70. The risk-free interest rates in the
An index provides a dividend yield of 1% and has a volatility of 20%. The risk-free interest rate is 4%. How long does a principal-protected note, created as in Example 11.1, have to last for it to
Investor’s Profit/Loss in Problem 11.20b when short maturity call is worth more than long maturity call
Three put options on a stock have the same expiration date and strike prices of $55, $60, and $65. The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can be created.
Draw a diagram showing the variation of an investor’s profit and loss with the terminal stock price for a portfolio consisting of a. One share and a short position in one call option b. Two
Describe the trading position created in which a call option is bought with strike price K1 and a put option is sold with strike price K2 when both have the same time to maturity and K2 > K1. What
A bank decides to create a five-year principal-protected note on a non-dividend-paying stock by offering investors a zero-coupon bond plus a bull spread created from calls. The risk-free rate is 4%
Suppose that the price of a non-dividend-paying stock is $32, its volatility is 30%, and the risk-free rate for all maturities is 5% per annum. Use DerivaGem to calculate the cost of setting up the
Use put–call parity to relate the initial investment for a bull spread created using calls to the initial investment for a bull spread created using puts.
Explain how an aggressive bear spread can be created using put options.
A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding.
A stock price is currently $80. It is known that at the end of four months it will be either $75 or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of
A stock price is currently $40. It is known that at the end of three months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the
A stock price is currently $50. Over each of the next two three-month periods it is expected to go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with continuous compounding.
For the situation considered in Problem 12.12, what is the value of a six-month European put option with a strike price of $51? Verify that the European call and European put prices satisfy
A stock price is currently $25. It is known that at the end of two months it will be either $23 or $27. The risk-free interest rate is 10% per annum with continuous compounding.
Calculate u, d, and p when a binomial tree is constructed to value an option on a foreign currency. The tree step size is one month, the domestic interest rate is 5% per annum, the foreign interest
The volatility of a non-dividend-paying stock whose price is $78, is 30%. The risk-free rate is 3% per annum (continuously compounded) for all maturities. Calculate values for u, d, and p when a
A stock index is currently 1,500. Its volatility is 18%. The risk-free rate is 4% per annum (continuously compounded) for all maturities and the dividend yield on the index is 2.5%. Calculate values
The futures price of a commodity is $90. Use a three-step tree to value (a) A nine-month American call option with strike price $93 (b) A nine-month American put option with strike price $93. The
The current price of a non-dividend-paying biotech stock is $140 with a volatility of 25%. The risk-free rate is 4%. For a three-month time step:(a) What is the percentage up movement?(b) What is the
Using a “trial-and-error” approach, estimate how high the strike price has to be in Problem 12.17 for it to be optimal to exercise the option immediately.
In Problem 12.19, suppose that a trader sells 10,000 European call options. How many shares of the stock are needed to hedge the position for the first and second three-month period? For the second
A stock price is currently $50. It is known that at the end of six months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculate the
A stock price is currently $30. During each two-month period for the next four months it is expected to increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a two-step tree to
Repeat Problem 12.25 for an American put option on a futures contract. The strike price and the futures price are $50, the risk-free rate is 10%, the time to maturity is six months, and the
A stock index is currently 990, the risk-free rate is 5%, and the dividend yield on the index is 2%. Use a three-step tree to value an 18-month American put option with a strike price of 1,000 when
Calculate the value of nine-month American call option on a foreign currency using a three-step binomial tree. The current exchange rate is 0.79 and the strike price is 0.80 (both expressed as
Consider a European call option on a non-dividend-paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the
A stock price is currently $50. It is known that at the end of two months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of
Prove that, with the notation in the chapter, a 95% confidence interval for ST is between
A portfolio manager announces that the average of the returns realized in each of the last 10 years is 20% per annum. In what respect is this statement misleading?
Assume that a non-dividend-paying stock has an expected return of μ and a volatility of σ. An innovative financial institution has just announced that it will trade a derivative that pays off
What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30%
What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per
A call option on a non-dividend-paying stock has a market price of $2.50. The stock price is $15, the exercise price is $13, the time to maturity is three months, and the risk-free interest rate is
Show that the Black–Scholes–Merton formula for a call option gives a price that tends to
Explain carefully why Black’s approach to evaluating an American call option on a dividend-paying stock may give an approximate answer even when only one dividend is anticipated. Does the answer
Consider an American call option on a stock. The stock price is $70, the time to maturity is eight months, the risk-free rate of interest is 10% per annum, the exercise price is $65, and the
A stock price is currently $50 and the risk-free interest rate is 5%. Use the DerivaGem software to translate the following table of European call options on the stock into a table of implied
A stock price is currently $50. Assume that the expected return from the stock is 18% per annum and its volatility is 30% per annum. What is the probability distribution for the stock price in two
Show that the Black–Scholes–Merton formulas for call and put options satisfy put–call parity.
Show that the probability that a European call option will be exercised in a risk-neutral world is, with the notation introduced in this chapter, N(d2). What is an expression for the value of a
Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows:30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 33.0,32.9, 33.0, 33.5, 33.5, 33.7, 33.5,
A financial institution plans to offer a derivative that pays off a dollar amount equal to S2T at time T where ST is the stock price at time T. Assume no dividends. Defining other variables as
Assume that the stock in Problem 13.26 is due to go ex-dividend in 1.5 months. The expected dividend is 50 cents. a. What is the price of the option if it is a European call? b. What is the price of
Consider an American call option when the stock price is $18, the exercise price is $20, the time to maturity is six months, the volatility is 30% per annum, and the risk-free interest rate is 10%
Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to
A stock price is currently $40. Assume that the expected return from the stock is 15% and its volatility is 25%. What is the probability distribution for the rate of return (with continuous
A stock price has an expected return of 16% and a volatility of 35%. The current price is $38. a) What is the probability that a European call option on the stock with an exercise price of $40 and a
Explain how you would do the analysis to produce a chart.
A company’s CFO says: “The accounting treatment of stock options is crazy. We granted 10,000,000 at-the-money stock options to our employees last year when the stock price was $30. We estimated
A company has granted 2,000,000 options to its employees. The stock price and strike price are both $60. The options last for 8 years and vest after two years. The company decides to value the
(a) Hedge funds earn a management fee plus an incentive fee that is a percentage of the profits (after fees and expenses), if any that they generate. How is a fund manager motivated to behave with
A company has granted 500,000 options to its executives. The stock price and strike price are both $40. The options last for 12 years and vest after four years. The company decides to value the
The Dow Jones Industrial Average on January 12, 2007 was 12,556 and the price of the March 126 call was $2.25. Use the DerivaGem software to calculate the implied volatility of this option. Assume
Consider a stock index currently standing at 250. The dividend yield on the index is 4% per annum, and the risk-free rate is 6% per annum. A three-month European call option on the index with a
An index currently stands at 696 and has a volatility of 30% per annum. The risk-free rate of interest is 7% per annum and the index provides a dividend yield of 4% per annum. Calculate the value of
Show that if C is the price of an American call with exercise price K and maturity T on a stock paying a dividend yield of q, and P is the price of an American put on the same stock with the same
Show that a European call option on a currency has the same price as the corresponding European put option on the currency when the forward price equals the strike price.
Would you expect the volatility of a stock index to be greater or less than the volatility of a typical stock? Explain your answer.
Does the cost of portfolio insurance increase or decrease as the beta of a portfolio increases? Explain your answer.
Suppose that a portfolio is worth $60 million and the S&P 500 is at 1200. If the value of the portfolio mirrors the value of the index, what options should be purchased to provide protection against
Consider again the situation in Problem 15.16. Suppose that the portfolio has a beta of 2.0, the risk-free interest rate is 5% per annum, and the dividend yield on both the portfolio and the index is
An index currently stands at 1,500. European call and put options with a strike price of 1,400 and time to maturity of six months have market prices of 154.00 and 34.25, respectively. The six-month
A total return index tracks the return, including dividends, on a certain portfolio. Explain how you would value (a) forward contracts and (b) European options on the index.
A stock index currently stands at 300 and has a volatility of 20%. The risk-free interest rate is 8% and the dividend yield on the index is 3%. Use a three-step binomial tree to value a six-month put
What is the put–call parity relationship for European currency options
Can an option on the yen-euro exchange rate be created from two options, one on the dollar-euro exchange rate, and the other on the dollar-yen exchange rate? Explain your answer.
Prove the results in equation (15.1), (15.2), and (15.3) using the portfolios indicated.
The spot price of an index is 1,000 and the risk-free rate is 4%. The prices of three month European call and put options when the strike price is 950 are 78 and 26. Estimate (a) The dividend yield
The USD/euro exchange rate is 1.3000. The exchange rate volatility is 15%. A US company will have to pay 1 million euros in three months. The euro and USD risk-free rates are 5% and 4%,
In Business Snapshot 15.1 what is the cost of a guarantee that the return on the fund will not be negative over the next 10 years?
The one-year forward price of the Mexican peso is $0.0750 per MXN. The U.S. risk-free rate is 1.25%. The exchange rate volatility is 13%. What is the value of one-year European call and put options
Suppose that the spot price of the Canadian dollar is U.S. $0.95 and that the Canadian dollar/U.S. dollar exchange rate has a volatility of 8% per annum. The risk-free rates of interest in Canada
Show that the formula in equation (15.9) for a put option to sell one unit of currency A for currency B at strike price K gives the same value as equation (15.8) for a call option to buy K units of
A foreign currency is currently worth $1.50. The domestic and foreign risk-free interest rates are 5% and 9%, respectively. Calculate a lower bound for the value of a six-month call option on the
A futures price is currently 40. It is known that at the end of three months the price will be either 35 or 45. What is the value of a three-month European call option on the futures with a strike
Consider a two-month call futures option with a strike price of 40 when the risk-free interest rate is 10% per annum. The current futures price is 47. What is a lower bound for the value of the
Consider a four-month put futures option with a strike price of 50 when the risk-free interest rate is 10% per annum. The current futures price is 47. What is a lower bound for the value of the
A futures price is currently 60 and its volatility is 30%. The risk-free interest rate is 8% per annum. Use a two-step binomial tree to calculate the value of a six-month European call option on the
In Problem 16.12 what value does the binomial tree give for a six-month European put option on futures with a strike price of 60? If the put were American, would it ever be worth exercising it
A futures price is currently 25, its volatility is 30% per annum, and the risk-free interest rate is 10% per annum. What is the value of a nine-month European call on the futures with a strike price
A futures price is currently 70, its volatility is 20% per annum, and the risk-free interest rate is 6% per annum. What is the value of a five-month European put on the futures with a strike price
Suppose that a one-year futures price is currently 35. A one-year European call option and a one-year European put option on the futures with a strike price of 34 are both priced at 2 in the market.
“The price of an at-the-money European call futures option always equals the price of a similar at-the-money European put futures option.” Explain why this statement is true.
Suppose that a futures price is currently 30. The risk-free interest rate is 5% per annum. A three-month American call futures option with a strike price of 28 is worth 4. Calculate bounds for the
Show that if C is the price of an American call option on a futures contract when the strike price is K and the maturity is T, and P is the price of an American put on the same futures contract with
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