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Questions and Answers of
Corporate Finance
The futures price of an asset is currently 78 and the risk-free rate is 3%. A six-month put on the futures with a strike price of 80 is currently worth 6.5. What is the value of a six-month call on
Calculate the price of a three-month European call option on the spot price of silver. The three-month futures price is $12, the strike price is $13, the risk-free rate is 4%, and the volatility of
Use a three-step tree to value an American put futures option when the futures price is 50, the life of the option is 9 months, the strike price is 50, the risk-free rate is 3%, and the volatility
It is February 4. July call options on corn futures with strike prices of 260, 270, 280, 290, and 300 cost 26.75, 21.25, 17.25, 14.00, and 11.375, respectively. July put options with these strike
Calculate the price of a six-month European put option on the spot value of the S&P 500. The six-month forward price of the index is 1,400, the strike price is 1,450, the risk-free rate is 5%,
The strike price of a futures option is 550 cents, the risk-free rate of interest is 3%, the volatility of the futures price is 20%, and the time to maturity of the option is 9 months. The futures
Calculate the implied volatility of soybean futures prices from the following information concerning a European put on soybean futures:Current futures price ……………………..525Exercise
Suppose you buy a put option contract on October gold futures with a strike price of $1,800 per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the October
Suppose you sell a call option contract on April live cattle futures with a strike price of 90 cents per pound. Each contract is for the delivery of 40,000 pounds. What happens if the contract is
A bank’s position in options on the dollar–euro exchange rate has a delta of 30,000 and a gamma of – 80,000. Explain how these numbers can be interpreted. The exchange rate (dollars per euro)
Suppose that a stock price is currently $20 and that a call option with an exercise price of $25 is created synthetically using a continually changing position in the stock. Consider the following
What is the delta of a short position in 1,000 European call options on silver futures? The options mature in eight months, and the futures contract underlying the option matures in nine months. The
In Problem 17.11, what initial position in nine-month silver futures is necessary for delta hedging? If silver itself is used, what is the initial position? If one-year silver futures are used, what
A financial institution has just sold 1,000 seven-month European call options on the Japanese yen. Suppose that the spot exchange rate is 0.80 cent per yen, the exercise price is 0.81 cent per yen,
Under what circumstances is it possible to make a European option on a stock index both gamma neutral and vega neutral by adding a position in one other European option?
A fund manager has a well-diversified portfolio that mirrors the performance of the S&P 500 and is worth $360 million. The value of the S&P 500 is 1,200, and the portfolio manager would like
Repeat Problem 17.17 on the assumption that the portfolio has a beta of 1.5. Assume that the dividend yield on the portfolio is 4% per annum.a) If the fund manager buys traded European put options,
Show by substituting for the various terms in equation (17.4) that the equation is true for: a) A single European call option on a non-dividend-paying stock b) A single European put option on a
Consider a one-year European call option on a stock when the stock price is $30, the strike price is $30, the risk-free rate is 5%, and the volatility is 25% per annum. Use the DerivaGem software to
Suppose that $70 billion of equity assets are the subject of portfolio insurance schemes. Assume that the schemes are designed to provide insurance against the value of the assets declining by more
Does a forward contract on a stock index have the same delta as the corresponding futures contract? Explain your answer.
A financial institution has the following portfolio of over-the-counter options on sterling:A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8. a. What position in
Consider again the situation in Problem 17.24. Suppose that a second traded option with a delta of 0.1, a gamma of 0.5, and a vega of 0.6 is available. How could the portfolio be made delta, gamma,
Use DerivaGem to check that equation (17.4) is satisfied for the option considered in Section 17.1. (DerivaGem produces a value of theta “per calendar day.” The theta in equation (17.4) is “per
Use the DerivaGem Application Builder functions to reproduce Table 17.2. (Note that in Table 17.2 the stock position is rounded to the nearest 100 shares.) Calculate the gamma and theta of the
A deposit instrument offered by a bank guarantees that investors will receive a return during a six-month period that is the greater of (a) zero and (b) 40% of the return provided by a market index.
What does it mean to assert that the theta of an option position is –0.1 when time is measured in years? If a trader feels that neither a stock price nor its implied volatility will change, what
The Black–Scholes–Merton price of an out-of-the-money call option with an exercise price of $40 is $4. A trader who has written the option plans to use a stop-loss strategy. The trader’s plan
A four-step Cox–Ross–Rubinstein binomial tree is used to price a one-year American put option on an index when the index level is 500, the strike price is 500, the dividend yield is 2%, the
Use a three-time-step tree to value a nine-month American call option on wheat futures. The current futures price is 400 cents, the strike price is 420 cents, the risk-free rate is 6%, and the
A three-month American call option on a stock has a strike price of $20. The stock price is $20, the risk-free rate is 3% per annum, and the volatility is 25% per annum. A dividend of $2 is expected
A one-year American put option on a non-dividend-paying stock has an exercise price of $18. The current stock price is $20, the risk-free interest rate is 15% per annum, and the volatility of the
A two-month American put option on a stock index has an exercise price of 480. The current level of the index is 484, the risk-free interest rate is 10% per annum, the dividend yield on the index is
How would you use the control variate approach to improve the estimate of the delta of an American option when the binomial tree approach is used?
How would you use the binomial tree approach to value an American option on a stock index when the dividend yield on the index is a function of time?
Estimate delta, gamma, and theta from the tree in Example 18.1. Explain how each can be interpreted.
The DerivaGem Application Builder functions enable you to investigate how the prices of options calculated from a binomial tree converge to the correct value as the number of time steps increases.
A six-month American call option on a stock is expected to pay dividends of $1 per share at the end of the second month and the fifth month. The current stock price is $30, the exercise price is $34,
A one-year American call option on silver futures has an exercise price of $9.00. The current futures price is $8.50, the risk-free rate of interest is 12% per annum, and the volatility of the
An American put option to sell a Swiss franc for dollars has a strike price of $0.80 and a time to maturity of one year. The volatility of the Swiss franc is 10%, the dollar interest rate is 6%, the
Consider an option that pays off the amount by which the final stock price exceeds the average stock price achieved during the life of the option. Can this be valued from a binomial tree using
A nine-month American put option on a non-dividend-paying stock has a strike price of $49. The stock price is $50, the risk-free rate is 5% per annum, and the volatility is 30% per annum. Use a
Using Table 19.2 calculate the implied volatility a trader would use for an 11-month option with a strike price of 0.98
What problems do you think would be encountered in testing a stock option pricing model empirically?
Suppose that a central bank’s policy is to allow an exchange rate to fluctuate between 0.97 and 1.03. What pattern of implied volatilities for options on the exchange rate would you expect to see?
Option traders sometimes refer to deep-out-of-the-money options as being options on volatility. Why do you think they do this?
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 European put option on the same stock has a strike price of
Suppose that the result of a major lawsuit affecting a company is due to be announced tomorrow. The company’s stock price is currently $60. If the ruling is favorable to the company, the stock
An exchange rate is currently 0.8000. The volatility of the exchange rate is quoted as 12% and interest rates in the two countries are the same. Using the lognormal assumption, estimate the
The price of a stock is $40. A six-month European call option on the stock with a strike price of $30 has an implied volatility of 35%. A six month European call option on the stock with a strike
“The Black–Scholes–Merton model is used by traders as an interpolation tool.” Discuss this view.
Using Table 19.2 calculate the implied volatility a trader would use for an 8-month option with a strike price of 1.04.
Consider a European call and a European put with the same strike price and time to maturity. Show that they change in value by the same amount when the volatility increases from a level, , to a new
Data for a number of stock indices are provided on the author’s Web site: www.rotman.utoronto.ca/ hull/dataChoose an index and test whether a three standard deviation down movement happens more
Data for a number of foreign currencies are provided on the author’s Web site:www.rotman.utoronto.ca/ hull/dataChoose a currency and use the data to produce a table similar to Table 19.1.
A futures price is currently $40. The risk-free interest rate is 5%. Some news is expected tomorrow that will cause the volatility over the next three months to be either 10% or 30%. There is a 60%
A company is currently awaiting the outcome of a major lawsuit. This is expected to be known within one month. The stock price is currently $20. If the outcome is positive, the stock price is
A company’s stock is selling for $4. The company has no outstanding debt. Analysts consider the liquidation value of the company to be at least $300,000 and there are 100,000 shares outstanding.
A stock price is currently $20. Tomorrow, news is expected to be announced that will either increase the price by $5 or decrease the price by $5. What are the problems in using Black–Scholes to
What volatility smile is likely to be observed for six-month options when the volatility is uncertain and positively correlated to the stock price?
The calculations for the four-index example at the end of Section 20.6 assume that the investments in the DJIA, FTSE 100, CAC40, and Nikkei 225 are $4 million, $3 million, \$1 million, and $2
Consider a position consisting of a $100,000 investment in asset A and a $100,000 investment in asset B. Assume that the daily volatilities of both assets are 1% and that the coefficient of
The volatility of a certain market variable is 30% per annum. Calculate a 99% confidence interval for the size of the percentage daily change in the variable.
Explain how an interest rate swap is mapped into a portfolio of zero-coupon bonds with standard maturities for the purposes of a VaR calculation.
Explain why the linear model can provide only approximate estimates of VaR for a portfolio containing options.
Some time ago a company entered into a forward contract to buy £1 million for $1.5 million. The contract now has six months to maturity. The daily volatility of a six-month zero-coupon sterling bond
The most recent estimate of the daily volatility of the U.S. dollar–sterling exchange rate is 0.6%, and the exchange rate at 4 p.m. yesterday was 1.5000. The parameter λ in the EWMA model is 0.9.
(a) Calculate the current estimate of the covariance between the assets.(b) On the assumption that the prices of the assets at close of trading today are $20.5 and $40.5, update the correlation
Suppose that the daily volatility of the FT-SE 100 stock index (measured in pounds sterling) is 1.8% and the daily volatility of the dollar/sterling exchange rate is 0.9%. Suppose further that the
Suppose that in Problem 20.17 the correlation between the S&P 500 Index (measured in dollars) and the FT-SE 100 Index (measured in sterling) is 0.7, the correlation between the S&P 500 index
Suppose that the portfolio considered in Section 20.2 has (in $000s) 3,000 in DJIA, 3,000 in FTSE, 1,000 in CAC40, and 3,000 in Nikkei 225. Use the spreadsheet on the author’s web site to calculate
Use the spreadsheets on the author’s web site to calculate the one-day 99% VaR, using the basic methodology in Section 20.2 if the four-index portfolio considered in Section 20.2 is equally divided
At the end of Section 20.6, the VaR for the four-index example was calculated using the model-building approach. How does the VaR calculated change if the investment is $2.5 million in each index?
What is the effect of changing λ from 0.94 to 0.97 in the EWMA calculations in the four-index example at the end of Section 20.6? Use the spreadsheets on the author’s web site.
Consider a position consisting of a $300,000 investment in gold and a $500,000 investment in silver. Suppose that the daily volatilities of these two assets are 1.8% and 1.2%, respectively, and that
Consider a portfolio of options on a single asset. Suppose that the delta of the portfolio is 12, the value of the asset is $10, and the daily volatility of the asset is 2%. Estimate the 1-day 95%
A common complaint of risk managers is that the model building approach (either linear or quadratic) does not work well when delta is close to zero. Test what happens when delta is close to zero in
An Excel spreadsheet containing daily data on a number of different exchange rates and stock indices can be downloaded from the author’s Web site:http://www.rotman.utoronto.ca/hull/dataChoose one
Suppose that in Problem 20.28 the price of silver at the close of trading yesterday was $16, its volatility was estimated as 1.5% per day, and its correlation with gold was estimated as 0.8. The
Suppose that the price of gold at close of trading yesterday was $600, and its volatility was estimated as 1.3% per day. The price at the close of trading today is $596. Update the volatility
A bank has written a call option on one stock and a put option on another stock. For the first option the stock price is 50, the strike price is 51, the volatility is 28% per annum, and the time to
A company uses an EWMA model for forecasting volatility. It decides to change the parameter λ from 0.95 to 0.85. Explain the likely impact on the forecasts.
Explain the difference between value at risk and expected shortfall.
Use the DerivaGem software to value a European swaption that gives you the right in two years to enter into a 5-year swap in which you pay a fixed rate of 6% and receive floating. Cash flows are
If the yield volatility for a five-year put option on a bond maturing in 10 years time is specified as 22%, how should the option be valued? Assume that, based on today’s interest rates the
A corporation knows that in three months it will have $5 million to invest for 90 days at LIBOR minus 50 basis points and wishes to ensure that the rate obtained will be at least 6.5%. What position
Explain carefully how you would use (a) spot volatilities and (b) flat volatilities to value a five-year cap.
What other instrument is the same as a five-year zero-cost collar in which the strike price of the cap equals the strike price of the floor? What does the common strike price equal?
Suppose that the 1-year, 2-year, 3-year, 4-year and 5-year LIBOR/swap zero rates are 6%, 6.4%, 6.7%, 6.9%, and 7%. The price of a 5-year semiannual cap with a principal of $100 at a cap rate of 8% is
Show that V1 + f = V2 where V1 is the value of a swaption to pay a fixed rate of Rk and receive LIBOR between times T1 and T2, f is the value of a forward swap to receive a fixed rate of Rk
Calculate the price of a cap on the three-month LIBOR rate in nine months’ time for a principal amount of $1,000. Use Black’s model and the following information:Quoted nine-month Eurodollar
Suppose that the LIBOR yield curve is flat at 8% with annual compounding. A swaption gives the holder the right to receive 7.6% in a five-year swap starting in four years. Payments are made annually.
Use the DerivaGem software to value a five-year collar that guarantees that the maximum and minimum interest rates on a LIBOR-based loan (with quarterly resets) are 7% and 5% respectively. The LIBOR
Consider an eight-month European put option on a Treasury bond that currently has 14.25 years to maturity. The bond principal is $1,000. The current cash bond price is $910, the exercise price is
Suppose that zero rates are as in Problem 21.14. Use DerivaGem (with LIBOR discounting) to determine the value of an option to pay a fixed rate of 6% and receive LIBOR on a five-year swap starting in
Explain why there is an arbitrage opportunity if the implied Black (flat) volatility for a cap is different from that for a floor. Do the broker quotes in Table 21.1 present an arbitrage opportunity?
A bank uses Black’s model to price European bond options. Suppose that an implied price volatility for a 5-year option on a bond maturing in 10 years is used to price a 9-year option on the bond.
Consider a four-year European call option on a bond that will mature in five years. The five-year bond price is $105, the price of a four-year bond with the same coupon as the five-year bond is $102,
What is the relationship between a regular call option, a binary call option, and a gap call option?
Suppose that c1 and p1 are the prices of a European average price call and a European average price put with strike price K and maturity T, c2 and p2 are the prices of a European average strike call
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