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business
options futures and other derivatives

Questions and Answers of Options Futures And Other Derivatives

Suppose that the daily volatility of the FTSE 100 stock index (measured in pounds sterling) is 1.3% and the daily volatility of the dollar sterling exchange cate is 0.9% Suppose further that the
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 dose to zero by
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
A financial institution owas a portfolio of options on the US dollar-sterling exchange rate. The delta of the portfolio is 56.0.The current exchange rate is 1.5000. Derive an approximate linear
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
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 test calculates a VaR estimate for the example in Table 185 assuming two factors. How does the estimate change if you assume (a) one factor and (b) three factors.
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 arro-coupon bond in the cash-low 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 bond.
Explain why the linese model can provide only approximate estimates of VaR for a portfolio containing options.
Explain how an interest rate swap is mapped into a portfolio of zero-coupon bands with standard maturities for the purposes of a Valk calculation.
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 emptions underlying (a) the linear
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
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 cate of interest is 12% per annum, and the volatility of the futures
Provide formulas that can be used for obtaining three random samples from standard normal distributions when the correlation between sample and sample is a
A company has issued a 3-year conversible 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 when the share price is greater
How would you use the antithetic variable method to improve the estimate of the European option in Business Saapshot 17.2 and Table 17.27
Use the binomial tree in Problem 17.19 to value a security that pays off' in 1 year where x is the price of copper.
As 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 anaum, 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.
A 2-month American put option on a stock indes has an exercise price of 480. The current level of the indes is 484, the risk-free interest rate is 10% per annum, the dividend yield on the index is 3%
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. Lise a
Use stratified sampling with 100 trials to improve the estimate of in Business Snap- shot 17.1 and Table 17.1
Calculate the price of a 9-month American call option on corn fatures when the current futures price is 198 cents, the strike price is 200 cents, the risk-free interest rate is 3% per annum, and the
The method for calculating Greek letter is similar to that used for trees. Delta, gamma, and thats can be calculated directly from the fr values on the grid. For veg is necessary to make a small
Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 9 bolds. 17.8 Use stratified sampling with 100 trials to improve the estimate of x
Calculate the price of a 9-month American call option on corn futures when the current futures price is 198 ceats, the strike price is 200 cents, the risk free interest rate is 8% per annum, and the
Which of the following can be estimated for an American option by constructing a single binomial tree delta, games, vega, theta, tho? 17.2 Calculate the price of a 3-month American put option on a
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 o, to a new
Data for a number of foreign currencies are provided on the author's website http://www.rotman.utoronto.ca/~bul Choose a currency and use the data to produce a table similar to Table 16.1.16.22. Data
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 $10. If the outcome is positive, the stock price is expected
"The Black Scholes model is used by traders as an interpolation tool." Discuss this view.16:18. A company's stock is selling for $4. The company has no outstanding debt. Analysts consider the
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
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,
What volatility smile is likely to be observed for 6-month options when the volatility is uncertain and positively combated to the stock prior?
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
What volatility smile is likely to be caused by jumps in the underlying asset price? Is the patters likely to be more pronounced for a 2-year option than for a 3-month option?
What volatility smile is likely to be observed whos (a) Boch tails of the stock price distribution are less heavy than those of the lognormal distribution? (b) The right tail is heavier, and the left
(In Table 15.2 the stock position is rounded to the nearest 100 shares.) Calculate the gamma and the of the position each week. Calculate the change in the value of the portfolio each week and check
Use the DeriveGem Application Builder functions to reproduce Table
(Note: DerivaGe produces a value of theta "per calendar day". The thela In equation (15.7) is "per year")
Use DerivaGem to check that equation (15.7) is satisfied for the option considered in Section
The delta, gamma, thela, and vega of a call futures option are the same as those for a call option on a stock paying dividends at rate q, with a replaced by rand & replaced by Fo Explain why the same
The formula for the price of a European call futures option in terms of the futures price F is given in Chapter 14 as where c=N(d) KN In Fa/K)+21/2 o and d-4-0 and K, r, 7, and respectively. (a)
Consider a 1-year Europeas 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
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 (dollar per euro) is 0.90,
Does a forward contract on a stock lodes have the same delta'as the corresponding futures contract? Explain your answer.
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
Repeat Problem 15.16 on the assumption that the portfolio has a beta of 1.5. Assume that the dividend yield on the portfolio is 4% per annum. 15.18 Show by substituting for the various terms in
A financial institution has just sold 1,000 7-month European call options on the Japanese yen. Suppose that the spot exchange rate is 0.30 cent per yes, the exercise price is 0.81 cent per yen, the
Repeat Problem 15 12 for a financial institution with a portfolio of short positions in pat and call options on a
A company uses delta hedging to hedge a portfolio of long positions in put and call options on a currency. Which of the following would give the most favorable result? (a) A virtually constant spot
In Problem 15,10, what initial position in 9-month silver futures is necessary for delta hodging? If silver itself in seed, what is the initial position? If 1-year silver futures are wed, what is the
What is the delta of a short position in 1,000 European call options on silver futures? The options mate in 3 months, and the futures contract underlying the option matures in 9 months. The current
The Black-Scholes price of an out-of-the-money call option with an exercise price of $40 is $4. A trader who has writtes the option plans to use a stop-loss strategy. The trader's plan is to buy at
Calculate the delta of an at-the-money 6-month European call option on a non- dividend paying stock when the risk-free interest rate is 10% per annum and the stock price volatility is 25% per annum.
Assuse the futures prices in Table 2.2 apply and that the risk-free rate is 1.1% per annum. Treat the options as American and use 100 time steps. The options mature on June 19, 2004 Can you draw any
Use the DerivaGem software to calculate implied volatilities for the July options on com futures in Table
Calculate the implied volatility of soybean futures prions from the following information concerning a European put on soybean future Current future price Exercise price Risk-free rate Time to
A futures price is currently 40. It is known that at the end of 3 months the price will be either 35 or 45. What is the value of a 3-month European call option on the futures with a strike price of
Suppose that the spot price of the Canadian dollar is US $0.75 and that the Canadian dollar US dollar exchange rate has a volatility of 4% per annum. The risk-free rates of interest in Canada and the
A stock Index currently stands at 300. It is expected to increase or decrease by 10% over each of the next two time periods of 3 months. The risk-free interest rate is 3% and the dividend yield on
The value of the DIX on February 4, 2004, was 104.7). Assume that the risk-free rate was 1.2% and that the dividend yield was 3.5%. The options expire on March 20, 2004. Are the quotes for the two
Use the DerivaGem software to calculate implied volatilities for the March 104 call and the March 104 put on the Dow Jones Industrial Average (DIX) in Table
If the price of currency A expressed in terms of the price of currency B follows the proce S=(-)+3dz where is the risk-free interest rate in currency A and is the risk-free interest rate in currency
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 7, and P is the price of an American put on the same futures contract with
Prove the results in equations (14.1), (14.2), and (14.3) using the following portfolios Parole A: one European call option plus an amount of cash equal to Ke Portfolio : shares, with dividends being
A corporation knows that in 3 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 kast 6.5%. What position in
Suppose that a futures price is currently 30. The risk-free interest rate in 5% per annum. A 3-month American call futures option with a strike price of 28 is worth 4. Calculate bounds for the price
Suppose that a futures price is currently 35. A European call option and a European option on the futures with a strike price of 34 are both priced at 2 in the market. The risk-free interest rate is
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 9-month European call on the futures a strike price of 26
In Problem 14.37, what is the value of a 6-month European put option on futures with a strike price of 607 If the pet were American, would it ever be worth exercising it early? Verify that the call
A futures price is currently 60. It is known that over each of the next two 3-mouth periods it will either rise by 10% or fall by 10%. The risk-free interest rate is 3% per annum. What is the value
Consider a 4-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 futures
Consider a 2-mouth 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 futures
Suppose you buy a put option contract on October gold futures with a strike price of $400 per ounce Each contract is for the delivery of 100 ounces What happens if you exercise when the October
Suppose that the portfolio has a beta of 2.0, that the risk-free interest rate is 5% per annum, and that the dividend yield on both the portfolio and the index is 3% per annum. What options should be
Consider a stock index currently standing at 250. The dividend yield on the index 4% per annum and the risk-free rate is 6% per annum. A 3-month European call option on the index with a strike price
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 6-month call option on the
The S&P 100 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 indes provides a dividend yield of 4% per annum Calculate the
A total return index tracks the retum, including dividends, on a certain portfolio. Explain how you would value (a) forward contracts and (b) European options on the index.
Calculate the value of a 5-month European pul 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
A futures price is currently 50. At the end of 6 months it will be either 56 or 46. The risk- free interest rate is 6% per annum. What is the value of a 6-month European call option with a strike
Calculate the value of an 8-month European put option on a currency with a strike price of 0.50. The current exchange rate is 0.52, the volatility of the exchange rate is 12%, the domestic risk-free
Calculate the value of a 3-month at-the-money European call option on a stock index when the index is at 250, the risk-free interest rate is 10% per annum, the volatility of the indes is 18% per
Over each of the next 2 months it is expected to increase or decrease in value by 2%. The domestic and foreign risk-free interest rates are 6% and 3%, respectively. What is the value of a 2-mouth
A currency is currently worth
A stock Index is currently 300, the dividend yield on the indes is 3% per annum, and the risk-free interest rate is 8% per annum. What is a lower bound for the price of a 6-month European call option
A portfolio is currently worth $10 million and has a beta of 10. The S&P 100 is currently standing at 500 Explain how a pet option on the S&P 100 with a strike of 480 can be used to provide portfolio
Consider an American call option when the stock price is S18, the exercise price is $20, the time to maturity is 6 months, the volatility is 30% per annum, and the risk-free interest rate is 10% per
Assume that the stock in Problem 13.29 is due to go ex-dividend in 1 months. The expected dividend is 50 cents. (a) What is the peice of the option if it is a European call? (b) What is the price of
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%, the volatility is 25% per annum. the time to maturity is 4
A financial institution plans to offer a security that pays off a dollar amount equal to at time T. (a) Use risk-neutral valuation to calculate the price of the security at time in terms of the stock
A stock price is currently $50. Assume that the expected retum from the stock is 18% and its volatility is 30%. What is the probability distribution for the stock price in 2 years? Calculate the mean
A company's stock price is $50 and 10 million shares are outstanding. The company's considering giving its employees 3 million at the money 5-year call options. Option exercises will be handled by

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