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business
options futures and other derivatives

Questions and Answers of Options Futures And Other Derivatives

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
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? Lop58
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 23.4 are consistent with the numbers in Table 23.1 and (b) that the numbers in the fourth column of Table 23.5 are consistent with the
Explain the difference between an unconditional default probability density and a hazard rate. Lop58
How are recovery rates usually defined? Lop58
Should researchers use real-world or risk-neutral default probabilities for (a) calculating credit value at risk and (b) adjusting the price of a derivative for defaults? Lop58
Suppose that in Problem 23.1 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 hazard rate per year over the
• Apply EWMA and GARCH(1, 1) to data on the euro-USD exchange rate between July 27, 2005, and July 27, 2010. This data can be found on the author's website: www.rotman.utoronto.ca/~hull/data.
• The calculations for the four-index example at the end of Section 22.8 assume that the investments in the DJIA, FTSE 100, CAC 40, and Nikkei 225 are $4 million, $3 million, $1 million, and $2
• Suppose that the parameters in a GARCH (1,1) model are a = 0.03, = 0.95, and w= 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:
• Suppose that in Problem 22.17 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
• Use the spreadsheets on the author’s website.
• What is the effect of changing A from 0.94 to 0.97 in the EWMA calculations in the fourindex example at the end of Section
• At the end of Section 22.8, 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?
• Show that the GARCH (1,1) model o;3=co+au,i,_1+,8o,‘3_1 in equation (22.9) is equivalent to the stochastic volatility model dV = a(VL - V) dt + §V dz, where time is measured in days, V is the
• Suppose that in Problem 22.12 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
• Suppose that the daily volatility of the FTSE 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 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 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
• 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
• A company uses the GARCH(1,1) model for updating volatility. The three parameters are w, o, and B. Describe the impact of making a small increase in each of the parameters while keeping the
Suppose that the portfolio considered in Section 20.2 has (in $000s) 3,000 in DJIA, 3,000 in FTSE, 1,000 in CAC 40 and 3,000 in Nikkei 225. Use the spreadsheet on the author’s website to calculate
How much is gained from exercising early at the lowest node at the 9-month point in Example 20.4?
Explain how each can be interpreted.
Estimate delta, gamma, and theta from the tree in Example
Answer the following questions concerned with the alternative procedures for construct- ing trees in Section 20.4: (a) Show that the binomial model in Section 20.4 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
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 Swiss franc's volatility is 10%, the dollar interest rate is 6%, the Swiss
How would you use the antithetic variable method to improve the estimate of the European option in Business Snapshot 20.2 and Table 20.2?
Use the binomial tree in Problem 20.19 to value a security that pays off x in 1 year where x is the price of copper.
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 (20.27) to (20.30) change when the implicit finite difference method is being used to evaluate an American call option on a currency.
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 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
Use stratified sampling with 100 trials to improve the estimate of x in Business Snap- shot 20.1 and Table 20.1,
Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 9 holds.
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
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
Using Table 19.2, calculate the implied volatility a trader would use for an 11-month option with K/S0 : 0.98
An exchange rate is currently 1.0 and the implied volatilities of 6-month European options with strike prices 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3 are 13%, 12%, 11%, 10%, 11%, 12%, 13%. The domestic and
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 o1 to a new
Data for a number of stock indices are provided on the author’s website:http:,//www.rotman.utoronto.ca/~hu11/data Choose an index and test whether a three-standard-deviation down movement happens
Assume that the expected return from all traded assets is the risk-free interest rate.
Data for a number of foreign currencies are provided on the author’s website:http://wWw.rotn1an.utoronto.ca/~hu11/data 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% chance
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
Using Table 19.2, calculate the implied volatility a trader would use for an 8-month option with K/S0 : 1.04.
What volatility smile is likely to be observed for 6-month options when the volatility is uncertain and positively correlated to the stock price?
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-Merton
The market price of a European call is $3.00 and its price given by Black-Scholes- Merton model with a volatility of 30% is $3.50. The price given by this Black-Scholes- Merton model for a European
(In Table 18.2 the stock position is rounded to the nearest 100 shares.) Calculate the gamma and theta of the position each week. Calculate the change in the value of the portfolio each week and
(Note: DerivaGem produces a value of theta "per calendar day." The theta in equation (18.4) is "per year.")
Use DerivaGem to check that equation (18.4) is satisfied for the option considered in Section
The formula for the price c of a European call futures option in terms of the futures price Fo is given in Chapter 17 as where d c=e[FoN(d)-KN(d)] In(Fo/K)+T/2 T and d = d-o and K, r, T, and are the
A financial institution has the following portfolio of over-the-counter options on sterling:Type Position Delta Gamma Vega of option of option of option Call -1,000 0.50 2.2 1.8 Call -500 0.80 0.6
What is the equation corresponding to equation (18.4) for (a) a portfolio of derivatives on a currency and (b) a portfolio of derivatives on a futures price?
Repeat Problem 18.16 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 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 to buy
In Problem 18.10, what initial position in 9-month silver futures is necessary for delta hedging? If silver itself is used, what is the initial position? If 1-year silver futures are used, what is
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 meant by the gamma of an option position? What are the risks in the situation where the gamma of a position is highly negative and the delta is zero?
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
Calculate the price of a three-month European call option on the spot value of silver. The three-month futures price is $12, the strike price is $13, the risk-free rate is 4% and the volatility of
)
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
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
"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 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.
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 of
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
In Problem 17.12, what does the binomial tree give for the value of 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
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
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
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
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
Suppose you buy a put option contract on October gold futures with a strike price of $1,200 per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the October
Calculate the value of a five-month European put futures option when the futures price is $19, the strike price is $20, the risk-free interest rate is 12% per annum, and the volatility of the futures
Consider an American futures call option where the futures contract and the option contract expire at the same time. Under what circumstances is the futures option worth more than the corresponding
How does the put-call parity formula for a futures option differ from put-call parity for an option on a non-dividend-paying stock?
A futures price is currently 50. At the end of six months it will be either 56 or 46. The risk-free interest rate is 6% per annum. What is the value of a six-month European call option on the futures
"A futures price is like a stock paying a dividend yield." What is the dividend yield?
Why are options on bond futures more actively traded than options on bonds?
Explain the difference between a call option on yen and a call option on yen futures.
The USD/euro exchange rate is 1.3000. The exchange rate volatility is 15%. A US company will receive 1 million euros in three months. The euro and USD risk-free rates are 5% and 4%, respectively. The
Assume that the price of currency A expressed in terms of the price of currency B follows the process dS=(rB-FA)Sdt+oS dz, where r is the risk-free interest rate in currency A and B is the risk-free
Hedge funds earn a fixed fee plus a percentage of the profits, if any, that they generate (see Business Snapshot 1.2). How is a fund manager motivated to behave with this type of arrangement?
Suppose that the spot price of the Canadian dollar is US $0.95 and that the Canadian dollar/US dollar exchange rate has a volatility of 8% per annum. The risk-free rates of interest in Canada and the
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
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
Prove the results in equations (16.1), (16.2), and (16.3) using the portfolios indicated.
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.

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