What is the put–call parity relationship for European currency options?
> How can the control variate approach improve the estimate of the delta of an American option when the tree approach is used?
> 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% per annum, and the volatility of the index is 25% per
> 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 price is 40% per annum. Use the DerivaGem software wit
> 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 1.5 months. Use a three-step binomial tree to calculate
> Use a three-time-step binomial 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 volatility is 35% per annum. Estimate the delta of 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 three-step binomial tree to calculate the option price.
> Which of the following can be estimated for an American option by constructing a single binomial tree: delta, gamma, vega, theta, rho?
> Explain what is meant by ‘‘crashophobia.’’
> 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 put option with the same strike price and time to maturit
> ‘‘Resecuritization was a badly flawed idea. AAA tranches created from the mezzanine tranches of ABSs are bound to have a higher probability of default than the AAA-rated tranches of ABSs.’’ Discuss this point of view.
> Using Table 20.2, calculate the implied volatility a trader would use for an 8-month option with K=S0 ¼ 1:04. Table 20.2 Volatility surface. K/So 0.90 0.95 1.00 1.05 1.10 1 month 14.2 13.0 12.0 13.1 14.5 3 month 14.0 13.0 12.0 13.1 14.2
> ‘‘The Black–Scholes–Merton model is used by traders as an interpolation tool.’’ Discuss this view.
> 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 price is expected to jump to $75. If it is unfavorable, the s
> 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, a time to maturity of 1 year, and an implied volatilit
> Option traders sometimes refer to deep-out-of-the-money options as being options on volatility. Why do you think they do this?
> Explain the problems in testing a stock option pricing model empirically.
> 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 two scenarios: (a) Stock price increases steadily from $
> 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 is to buy at $40.10 and to sell at $39.90. Estimate the expect
> Why did portfolio insurance not work well on October 19, 1987?
> ‘‘The procedure for creating an option position synthetically is the reverse of the procedure for hedging the option position.’’ Explain this statement.
> 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 has an implied volatility of 28%. The 6-month risk-free
> 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?
> 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 type of option position is appropriate?
> A bank’s position in options on the dollar/euro exchange rate has a delta of 30,000 and a gamma of . Explain how these numbers can be interpreted. The exchange rate (dollars per euro) is 0.90. What position would you take to make the position delta neut
> 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 than 5% within 1 year. Making whatever estimates you fin
> What does it mean to assert that the delta of a call option is 0.7? How can a short position in 1,000 options be made delta neutral when the delta of each option is 0.7?
> Repeat Problem 19.16 on the assumption that the portfolio has a beta of 1.5. Assume that the dividend yield on the portfolio is 4% per annum. Data from Problem 19.16: A fund manager has a well-diversified portfolio that mirrors the performance of the S&P
> 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 insurance against a reduction of more than 5% in the val
> 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 financial institution has just sold 1,000 7-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, the risk-free interest rate in the United States is 8% per a
> In Problem 19.10, 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 the initial position? Assume no storage costs for silver.
> The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, the strike price is $30, and the expiration date is in 3 months. The risk-free interest rate is 8%. Derive upper and lower bounds for the price of an American put
> What is the delta of a short position in 1,000 European call options on silver futures? The options mature in 8 months, and the futures contract underlying the option matures in 9 months. The current 9-month futures price is $8 per ounce, the exercise pr
> Explain how a stop-loss trading rule can be implemented for the writer of an out-of-themoney call option. Why does it provide a relatively poor hedge?
> Suppose you sell a call option contract on April live cattle futures with a strike price of 130 cents per pound. Each contract is for the delivery of 40,000 pounds. What happens if the contract is exercised when the futures price is 135 cents?
> Calculate the value of a five-month European futures put 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 price is 20% per annum.
> 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 American option on the underlying asset?
> 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 in exchange-traded options should it take to hedge?
> 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 the same strike price and exercise date, then where
> 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 risk-free interest rate is 10% per annum. Identify
> 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 65?
> 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 of 26?
> Suppose that the principal assigned to the senior, mezzanine, and equity tranches is 70%, 20%, and 10% for both the ABS and the ABS CDO in Figure 8.3. What difference does this make to Table 8.1? Table 8.1 Estimated losses to tranches of ABS CDO in F
> 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 futures with a strike price of 60. If the call were Ame
> Explain the difference between a call option on yen and a call option on yen futures.
> Show that the formula in equation (17.12) for a put option to sell one unit of currency A for currency B at strike price K gives the same value as equation (17.11) for a call option to buy K units of currency B for currency A at strike price 1=K.
> Explain how corporations can use range forward contracts to hedge their foreign exchange risk when they are due to receive a certain amount of a foreign currency in the future.
> 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 equations (17.1), (17.2), and (17.3) using the portfolios indicated.
> Consider again the situation in Problem 17.16. 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 3% per annum. What options should be purchased to prov
> Suppose that a portfolio is worth $60 million and a stock index stands at 1,200. If the value of the portfolio mirrors the value of the index, what options should be purchased to provide protection against the value of the portfolio falling below $54 mil
> Does the cost of portfolio insurance increase or decrease as the beta of a portfolio increases? Explain your answer.
> 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 probability that the exchange rate in 3 months will be (a) less
> Would you expect the volatility of a stock index to be greater or less than the volatility of a typical stock? Explain your answer.
> 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.
> 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 strike price and exercise date, then where S0 is the s
> A portfolio is currently worth $10 million and has a beta of 1.0. An index is currently standing at 800. Explain how a put option on the index with a strike price of 700 can be used to provide portfolio insurance.
> On May 31 a company’s stock price is $70. One million shares are outstanding. An executive exercises 100,000 stock options with a strike price of $50. What is the impact of this on the stock price?
> Explain how you would do an analysis similar to that of Yermack and Lie to determine whether the backdating of stock option grants was happening.
> In what way would the benefits of backdating be reduced if a stock option grant had to be revalued at the end of each quarter?
> Why did some companies backdate stock option grants in the United States prior to 2002? What changed in 2002?
> ‘‘Granting stock options to executives is like allowing a professional footballer to bet on the outcome of games.’’ Discuss this viewpoint.
> ‘‘Stock option grants are good because they motivate executives to act in the best interests of shareholders.’’ Discuss this viewpoint.
> Suppose that x is the yield on a perpetual government bond that pays interest at the rate of $1 per annum. Assume that x is expressed with continuous compounding, that interest is paid continuously on the bond, and that x follows the process where a,
> What are the main differences between a typical employee stock option and an American call option traded on an exchange or in the over-the-counter market?
> 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 the value of each option on the grant date to be $5. At our
> Why was it attractive for companies to grant at-the-money stock options prior to 2005? What changed in 2005?
> A stock price follows geometric Brownian motion with an expected return of 16% and avolatility 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 maturity date in 6
> What is implied volatility ? How can it be calculated?
> Explain the principle of risk-neutral valuation.
> A company’s stock price is $50 and 10 million shares are outstanding. The company is considering giving its employees 3 million at-the-money 5-year call options. Option exercises will be handled by issuing more shares. The stock price volatility is 25%,
> Use the result in equation (15.17) to determine the value of a perpetual American put option on a non-dividend-paying stock with strike price K if it is exercised when the stock price equals H where H < K. Assume that the current stock price S is greater
> 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 derivative that pays off $100 if the price of a stock at time
> Consider an American call option on a stock. The stock price is $50, the time to maturity is 15 months, the risk-free rate of interest is 8% per annum, the exercise price is $55, and the volatility is 25%. Dividends of $1.50 are expected in 4 months and
> A stock price is currently $25. It is known that at the end of 2 months it will be either $23 or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose ST is the stock price at the end of 2 months. What is the value of a d
> 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 given by Black’s approach understate or overstate the true
> 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 volatilities, assuming no dividends. Are the option prices con
> With the notation used in this chapter: (a) What is N (x)? (b) Show that , where S is the stock price at time t and (c) (d) where c is the price of a call option on a non-dividend-paying stock. (e) Show that (f ) Show that c satisfies the Black&aci
> A call option on a non-dividend-paying stock has a market price of $212. The stock price is $15, the exercise price is $13, the time to maturity is 3 months, and the risk-free interest rate is 5% per annum. What is the implied volatility?
> Assume that a non-dividend-paying stock has an expected return of and a volatility of . An innovative financial institution has just announced that it will trade a security that pays off a dollar amount equal to ln ST at time T, where ST denotes the va
> A portfolio manager announces that the average of the returns realized in each year of the last 10 years is 20% per annum. In what respect is this statement misleading?
> It has been suggested that the short-term interest rate r follows the stochastic process where a, b, c are positive constants and dz is a Wiener process. Describe the nature of this process.
> The process for the stock price in equation (14.8) is Where µ and σ are constant. Explain carefully the difference between this model and each of the following: Why is the model in equation (14.8) a more appropriate model of stoc
> Stock A and stock B both follow geometric Brownian motion. Changes in any short interval of time are uncorrelated with each other. Does the value of a portfolio consisting of one of stock A and one of stock B follow geometric Brownian motion? Explain you
> Consider a variable S that follows the process For the first three years,and ; for the next three years, and . If the initial value of the variable is 5, what is the probability distribution of the value of the variable at the end of year 6?
> The one-day 99% VaR is calculated for the four-index example in Section 22.2 as $253,385. Look at the underlying spreadsheets on the author’s website and calculate: (a) the one-day 95% VaR, (b) the one-day 95% ES, (c) the one-day 97% VaR, and (d) the one
> Variables X1 and X2 follow generalized Wiener processes, with drift rates and and variances and . What process does X1 +X2 follow if : (a) The changes in X1 and X2 in any short interval of time are uncorrelated? (b) There is a correlation between t
> A company’s cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 0.5 per quarter and a variance rate of 4.0 per quarter. How high does the company’s initial cash position have to be for the company to
> The following table gives data on monthly changes in the spot price and the futures price for a certain commodity. Use the data to calculate a minimum variance hedge ratio. (Do not make an adjustment for daily settlement.) Spot price change Futures p
> A portfolio manager has maintained an actively managed portfolio with a beta of 0.2. During the last year, the risk-free rate was 5% and equities performed very badly providing a return of 30%. The portfolio manager produced a return of 10% and claims th
> A company wishes to hedge its exposure to a new fuel whose price changes have a 0.6 correlation with gasoline futures price changes. The company will lose $1 million for each 1 cent increase in the price per gallon of the new fuel over the next three mon
> A company has derivatives transactions with Banks A, B, and C that are worth + $20 million, ─ $15 million, and ─ $25 million, respectively, to the company. How much margin or collateral does the company have to provide in each of the following two situ
> What position is equivalent to a long forward contract to buy an asset at K on a certain date and a put option to sell it for K on that date.
> A company enters into a short futures contract to sell 5,000 bushels of wheat for 750 cents per bushel. The initial margin is $3,000 and the maintenance margin is $2,000. What price change would lead to a margin call? Under what circumstances could $1,50
> One orange juice futures contract is on 15,000 pounds of frozen concentrate. Suppose that in September 2017 a company sells a March 2019 orange juice futures contract for 120 cents per pound. At the end of December 2017, the futures price is 140 cents; a
> A trader buys a European call option and sells a European put option. The options have the same underlying asset, strike price, and maturity. Describe the trader’s position. Under what circumstances does the price of the call equal the price of the put?
> Suppose that, for a particular three-year derivative entered into by a bank, two outcomes, A and B, are equally likely. Under outcome A, the values of the derivative at the midpoint of the first, second, and third years are 3, 5, and 7, respectively. Und
> Explain carefully the difference between hedging, speculation, and arbitrage.
> Describe how foreign currency options can be used for hedging in the situation considered in Section 1.7 so that (a) ImportCo is guaranteed that its exchange rate will be less than 1.4700, and (b) ExportCo is guaranteed that its exchange rate will be at
> Suppose that in the situation of Table 1.1 a corporate treasurer said: ‘‘I will have £1 million to sell in 6 months. If the exchange rate is less than 1.42, I want you to give me 1.42. If it is greater tha
