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business
options futures and other derivatives

Questions and Answers of Options Futures And Other Derivatives

Explain why the credit exposure on a matched pair of forward contracts resembles a straddle.AppendixLO1
"A long forward contract subject to credit risk is a combination of a short position in a no-default put and a long position in a call subject to credit risk." Explain this statement.AppendixLO1
Suppose that in Problem 20.17, the 6-month forward rate is also 1.50 and the 6-month dollar risk-free interest rate is 5% per annum. Suppose further that the 6-month dollar rate of interest at which
A company enters into a 1-year forward contract to sell $100 for AUD150. The contract is initially at the money. In other words, the forward exchange rate is 1.50. The 1-year dollar risk-free rate of
Suppose that a financial institution has entered into a swap dependent on the sterling interest rate with counterparty X and an exactly offsetting swap with counterparty Y. Which of the following
A company has issued 3- and 5-year bonds with a coupon of 4% per annum payable annually. The yields on the bonds (expressed with continious compounding) are 4.5% and 4.75%, respectively. Risk-free
A 4-year corporate bond provides a coupon of 4% per year payable semiannually and has a yield of 5% expressed with continuous compounding. The risk-free yield curve is flat at 3% with continuous
Show that the value of a coupon-bearing corporate bond is the sum of the values of its constituent zero-coupon bonds when the amount claimed in the event of default is the no-default value of the
Suppose that the LIBOR/swap curve is flat at 6% with continuous compounding and a 5-year bond with a coupon of 5% (paid semiannually) sells for 90.00. How would an asset swap on the bond be
Suppose that the probability of company A defaulting during a 2-year period is 0.2 and the probability of company B defaulting during this period is 0.15. If the Gaussian copula measure of default
Explain the difference between the Gaussian copula model for the time to default and CreditMetrics as far as the following are concerned: (a) the definition of a credit loss and (b) the way in which
What is meant by a "haircut" in a collateralization agreement. A company offers to post its own equity as collateral. How would you respond?AppendixLO1
Suppose that the measure BAB(T) in equation (20.9) is the same in the real world and the risk-neutral world. Is the same true of the Gaussian copula measure, PAB?AppendixLO1
Describe how netting works. A bank already has one transaction with a counterparty on its books. Explain why a new transaction by a bank with a counterparty can have the effect of increasing or
Verify (a) that the numbers in the second column of Table 20.4 are consistent with the numbers in Table 20.1 and (b) that the numbers in the fourth column of Table 20.5 are consistent with the
Explain the difference between an unconditional default probability density and a default intensity.AppendixLO1
Suppose that in Problem 20.1 the spread between the yield on a 5-year bond issued by the same company and the yield on a similar risk-free bond is 60 basis points. Assume the same recovery rate of
The spread between the yield on a 3-year corporate bond and the yield on a similar risk- free bond is 50 basis points. The recovery rate is 30%. Estimate the average default intensity per year over
Suppose that the parameters in a GARCH (1,1) model are a = 0.03, p = 0.95, and @= 0.000002. (a) What is the long-run average volatility? (b) If the current volatility is 1.5% per day, what is your
An Excel spreadsheet containing over 900 days of daily data on a number of different exchange rates and stock indices can be downloaded from the author's website: http://www.rotman.utoronto.ca/~hull
Suppose that in Problem 19.15 the price of silver at the close of trading yesterday was $8, its volatility was estimated as 1.5% per day, and its correlation with gold was estimated as 0.8. The price
Suppose that the price of gold at close of trading yesterday was $300 and its volatility was estimated as 1.3% per day. The price at the close of trading today is $298. Update the volatility estimate
Suppose that in Problem 19.12 the correlation between the S&P 500 Index (measured in dollars) and the FTSE 100 Index (measured in sterling) is 0.7, the correlation between the S&P 500 Index (measured
Suppose that the current daily volatilities of asset X and asset Y are 1.0% and 1.2%, respectively. The prices of the assets at close of trading yesterday were $30 and $50 and the estimate of the
The parameters of a GARCH(1, 1) model are estimated as w= 0.000004, = 0.05, and B=0.92. What is the long-run average volatility and what is the equation describing the way that the variance rate
Suppose that the daily volatilities of asset A and asset B, calculated at the close of trading yesterday, are 1.6% and 2.5%, respectively. The prices of the assets at close of trading yesterday were
Assume that S&P 500 at close of trading yesterday was 1,040 and the daily volatility of the index was estimated as 1% per day at that time. The parameters in a GARCH(1, 1) model are w = 0.000002, a =
The most recent estimate of the daily volatility of the US dollar/sterling exchange rate is 0.6% and the exchange rate at 4 p.m. yesterday was 1.5000. The parameter in the EWMA model is 0.9. Suppose
A company uses the GARCH(1,1) model for updating volatility. The three parameters are w,a, and B. Describe the impact of making a small increase in each of the parameters while keeping the others
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.AppendixLO1
The most recent estimate of the daily volatility of an asset is 1.5% and the price of the asset at the close of trading yesterday was $30.00. The parameter in the EWMA model is 0.94. Suppose that the
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 by
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 has a long position in a 2-year bond and a 3-year bond, as well as a short position in a 5-year bond. Each bond has a principal of $100 and pays a 5% coupon annually. Calculate the
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%
Suppose that in Problem 18.14 the vega of the portfolio is -2 per 1% change in the annual volatility. Derive a model relating the change in the portfolio value in 1 day to delta, gamma, and vega.
A bank has a portfolio of options on an asset. The delta of the options is -30 and the gamma is -5. Explain how these numbers can be interpreted. The asset price is 20 and its volatility is 1% per
The text calculates a VaR estimate for the example in Table 18.5 assuming two factors. How does the estimate change if you assume (a) one factor and (b) three factors.AppendixLO1
Some time ago a company entered into a forward contract to buy 1 million for $1.5 million. The contract now has 6 months to maturity. The daily volatility of a 6-month zero-coupon sterling bond (when
Suppose that the 5-year rate is 6%, the 7-year rate is 7% (both expressed with annual compounding), the daily volatility of a 5-year zero-coupon bond is 0.5%, and the daily volatility of a 7-year
Verify that the 0.3-year zero-coupon bond in the cash-flow mapping example in the appendix to this chapter is mapped into a $37,397 position in a 3-month bond and a $11,793 position in a 6-month
Explain the difference between value at risk and conditional value at risk.AppendixLO1
Suppose that a company has a portfolio consisting of positions in stocks, bonds, foreign exchange, and commodities. Assume that there are no derivatives. Explain the assumptions underlying (a) the
Suppose that the daily change in the value of a portfolio is, to a good approximation, linearly dependent on two factors, calculated from a principal components analysis. The delta of a portfolio
Suppose you know that the gamma of the portfolio in the previous question is 16.2. How does this change your estimate of the relationship between the change in the portfolio value and the percentage
Describe three ways of handling instruments that are dependent on interest rates when the model-building approach is used to calculate VaR. How would you handle these instruments when historical
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 DerivaGem Application Buider 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.
Answer the following questions concerned with the alternative procedures for construct- ing trees in Section 17.4: (a) Show that the binomial model in Section 17.4 is exactly consistent with the mean
The current value of the British pound is $1.60 and the volatility of the pound/dollar exchange rate is 15% per annum. An American call option has an exercise price of $1.62 and a time to maturity of
A 6-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 1-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 futures
An American put option to sell a Swiss franc for dollars has a strike price of $0.80 and a time to maturity of 1 year. The volatility of the Swiss franc is 10%, the dollar interest rate is 6%, the
Provide formulas that can be used for obtaining three random samples from standard normal distributions when the correlation between sample i and sample j is pi.j.AppendixLO1
A company has issued a 3-year convertible bond that has a face value of $25 and can be exchanged for two of the company's shares at any time. The company can call the issue when the share price is
How would you use the antithetic variable method to improve the estimate of the European option in Business Snapshot 17.2 and Table 17.2?AppendixLO1
When do the boundary conditions for S = 0 and S affect the estimates of derivative prices in the explicit finite difference method?AppendixLO1
Use the binomial tree in Problem 17.19 to value a security that pays off x2 in 1 year where x is the price of copper.AppendixLO1
The spot price of copper is $0.60 per pound. Suppose that the futures prices (dollars per pound) are as follows: 3 months 0.59 6 months 0.57 9 months 0.54 12 months 0.50 The volatility of the price
An American put option on a non-dividend-paying stock has 4 months to maturity. The exercise price is $21, the stock price is $20, the risk-free rate of interest is 10% per annum, and the volatility
Explain how equations (17.27) to (17.30) change when the implicit finite difference method is being used to evaluate an American call option on a currency.AppendixLO1
Suppose that Monte Carlo simulation is being used to evaluate a European call option on a non-dividend-paying stock when the volatility is stochastic. How could the control variate and antithetic
How can the control variate approach improve the estimate of the delta of an American option when the tree approach is used?AppendixLO1
A 2-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 3%
A 1-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 stock
A 3-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 in
Use a three-time-step tree to value a 9-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 9-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
Explain why the Monte Carlo simulation approach cannot easily be used for American- style derivatives.AppendixLO1
Use stratified sampling with 100 trials to improve the estimate of in Business Snap- shot 17.1 and Table 17.1.AppendixLO1
Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 9 holds.AppendixLO1
"For a dividend-paying stock, the tree for the stock price does not recombine; but the tree for the stock price less the present value of future dividends does recombine." Explain this
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 using the binomial tree approach?
Calculate the price of a 9-month American call option on corn futures when the current futures price is 198 cents, the strike price is 200 cents, the risk-free interest rate is 8% per annum, and the
Explain how the control variate technique is implemented when a tree is used to value American options.AppendixLO1
Calculate the price of a 3-month American put option on a non-dividend-paying stock when the stock price is $60, the strike price is $60, the risk-free interest rate is 10% per annum, and the
Which of the following can be estimated for an American option by constructing a single binomial tree: delta, gamma, vega, theta, rho?AppendixLO1
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 website: http://www.rotman.utoronto.ca/~hull Choose 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 website: http://www.rotman.utoronto.ca/~hull Choose a currency and use the data to produce a table similar to Table
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 3 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 1 month. The stock price is currently $20. If the outcome is positive, the stock price is expected
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. What
"The Black-Scholes model is used by traders as an interpolation tool." Discuss this view.AppendixLO1
A stock price is $40. A 6-month European call option on the stock with a strike price of $30 has an implied volatility of 35%. A 6-month European call option on the stock with a strike price of $50
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
Suppose that the result of a major lawsuit affecting Microsoft is due to be announced tomorrow. Microsoft's stock price is currently $60. If the ruling is favorable to Microsoft, the stock price is
A European call option on a certain stock has a strike price of $30, a time to maturity of 1 year, and an implied volatility of 30%. A European put option on the same stock has a strike price of $30,
Option traders sometimes refer to deep-out-of-the-money options as being options on volatility. Why do you think they do this?AppendixLO1
What problems do you think would be encountered in testing a stock option pricing model empirically?AppendixLO1
What volatility smile is likely to be observed for 6-month options when the volatility is uncertain and positively correlated to the stock price?AppendixLO1
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 value
Explain what is meant by "crashophobia".AppendixLO1
The market price of a European call is $3.00 and its price given by Black-Scholes model with a volatility of 30% is $3.50. The price given by this Black-Scholes model for a European put option with
Explain carefully why a distribution with a heavier left tail and less heavy right tail than the lognormal distribution gives rise to a downward sloping volatility smile.AppendixLO1
A European call and put option have the same strike price and time to maturity. The call has an implied volatility of 30% and the put has an implied volatility of 25%. What trades would you
What volatility smile is likely to be caused by jumps in the underlying asset price? Is the pattern likely to be more pronounced for a 2-year option than for a 3-month option?AppendixLO1
What volatility smile is observed for equities?AppendixLO1
What volatility smile is likely to be observed when: (a) Both tails of the stock price distribution are less heavy than those of the lognormal distribution? (b) The right tail is heavier, and the
Prove equations (28.25), (28.26), and (28.27).AppendixLO1

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