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options futures and other derivatives

Questions and Answers of Options Futures And Other Derivatives

What is the put-call parity relationship for European currency options?
A total return index tracks the return, including dividends, on a certain portfolio. Explain how you would value (a) forward contracts and (b) European options on the index.
An index currently stands at 1,500. European call and put options with a strike price of 1,400 and time to maturity of six months have market prices of 154.00 and 34.25, respectively. The six-month
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
Consider again the situation in Problem
Suppose that a portfolio is worth $60 million and the S&P 500 is at 1,200. If the value of the portfolio mirrors the value of the index, what options should be purchased to provide protection against
Does the cost of portfolio insurance increase or decrease as the beta of a portfolio increases? Explain your answer.
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 7 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
An 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 index provides a dividend yield of 4% per annum. Calculate the value of
Consider a stock index currently standing at 250. The dividend yield on the index is 4% per annum, and the risk-free rate is 6% per annum. A three-month European call option on the index with a
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 six-month call option on the
Show that the formula in equation (16.12) for a put option to sell one unit of currency A for currency B at strike price K gives the same value as equation (16.11) for a call option to buy K units of
Calculate the value of an eight-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
Calculate the value of a three-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 index is 18% per
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.
A currency is currently worth $0.80 and has a volatility of 12%. The domestic and foreign risk-free interest rates are 6% and 8%, respectively. Use a two-step binomial tree to value (a) a European
A stock index is currently 300, the dividend yield on the index is 3% per annum, and the risk-free interest rate is 8% per annum. What is a lower bound for the price of a six- month European call
"Once we know how to value options on a stock paying a dividend yield, we know how to value options on stock indices and currencies." Explain this statement.
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
A company has granted 1,000,000 options to its employees. The stock price and strike price are both $20. The options last 10 years and vest after 3 years. The stock price volatilityis 30%, the
A company has granted 2,000,000 options to its employees. The stock price and strike price are both $60. The options last for 8 years and vest after 2 years. The company decides to value the options
What is the (risk-neutral) expected life for the employee stock option in Example 15.2?What is the value of the option obtained by using this expected life in Black-ScholesMerton?
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
A company has granted 500,000 options to its executives. The stock price and strike price are both $40. The options last for 12 years and vest after 4 years. The company decides to value the options
In a Dutch auction of 10,000 options, bids are as follows:A bids $30 for 3,000 B bids $33 for 2,500 C bids $29 for 5,000 D bids $40 for 1,000 E bids $22 for 8,000 F bids $35 for 6,000.What is the
The notes accompanying a company's financial statements say: "Our executive stock options last 10 years and vest after 4 years. We valued the options granted this year using the Black-Scholes Merton
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?
What is the price of the option if it is a European call?
Explain how you would do the analysis to produce a chart such as the one in Figure
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 US 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.
Explain why employee stock options on a non-dividend-paying stock are frequently exercised before the end of their lives, whereas an exchange-traded call option on such a stock is never exercised
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?
Why was it attractive for companies to grant at-the-money stock options prior to 2005? What changed in 2005?
Consider an American call option when the stock price is $18, 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 14.29 is due to go ex-dividend in 1 months. The expected dividend is 50 cents. (a) What is the price 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. and the time to maturity
A financial institution plans to offer a security that pays off a dollar amount equal to S at time T, where Sy is the price at time 7 of a stock that pays no dividends. (a) Use risk-neutral valuation
30.3, 30.6, 33.0, 32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2 Estimate the stock price volatility. What is the standard error of your estimate?
Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows: 30.2, 32.0, 31.1, 30.1,
A stock price is currently $50. Assume that the expected return from the stock is 18% and its volatility is 30%. What is the probability distribution for the stock price in 2 years? Calculate the
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
A company has an issue of executive stock options outstanding. Should dilution be taken into account when the options are valued? Explain your answer.
Show that S-2/02 could be the price of a traded derivative security.
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
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
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
Show that the Black-Scholes-Merton formulas for call and put options satisfy put-call parity.
With the notation used in this chapter: (a) What is N'(x)? (b) Show that SN'(d) = Ke-N' (d2), where S is the stock price at time I and ?
A call option on a non-dividend-paying stock has a market price of $24. 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
Consider an American call option on a stock. The stock price is $70, the time to maturity is 8 months, the risk-free rate of interest is 10% per annum, the exercise price is $65, and the volatility
What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per
What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30%
Consider a derivative that pays off S at time T, where Sr is the stock price at that time. When the stock price follows geometric Brownian motion, it can be shown that its price at time t?
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
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?
Using the notation in this chapter, prove that a 95% confidence interval for ST is between and Se-2/2)7+1.960T Spe-/2)7-1.960-7
A stock price follows geometric Brownian motion with an expected return of 16% and a volatility of 35%. The current price is $38. (a) What is the probability that a European call option on the stock
A stock price is currently $40. Assume that the expected return from the stock is 15% and that its volatility is 25%. What is the probability distribution for the rate of return (with continuous
What is implied volatility? How can it be calculated?
What difference does it make to your calculations in Problem 14.4 if a dividend of $1.50 is expected in 2 months?
Calculate the price of a 3-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and
Explain the principle of risk-neutral valuation.
The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price change in one trading day?
What does the Black-Scholes-Merton stock option pricing model assume about the probability distribution of the stock price in one year? What does it assume about the continuously compounded rate of
Stock A, whose price is $30, has an expected return of 11% and a volatility of 25%. Stock B, whose price is $40, has an expected return of 15% and a volatility of 30%. The processes driving the
A stock price is currently 50. Its expected return and volatility are 12% and 30%, respectively. What is the probability that the stock price will be greater than 80 in 2 years? (Hint: ST > 80 when
If S follows the geometric Brownian motion process in equation (13.6), what is the process followed by (a) y = 2S (b) y = $ (c) y=es (d) y=e(-1)/S. In each case express the coefficients of dt and dz
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
A company's cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 0.1 per month and a variance rate of 0.16 per month. The initial cash position is
Suppose that a stock price has an expected return of 16% per annum and a volatility of 30% per annum. When the stock price at the end of a certain day is $50, calculate the following: -(a) The
A stock whose price is $30 has an expected return of 9% and a volatility of 20%. In Excel, simulate the stock price path over 5 years using monthly time steps and random samples from a normal
Suppose that x is the yield to maturity with continuous compounding on a zero-coupon bond that pays off $1 at time T. Assume that x follows the process dx = a(x-x) dt + sx dz wherea, xo, and s are
Suppose that a stock price S follows geometric Brownian motion with expected return j and volatility : dS = S dt + os dz What is the process followed by the variable S"? Show that S" also follows
It has been suggested that the short-term interest rate r follows the stochastic process dr = a(br) dt + redz wherea, b, c are positive constants and dz is a Wiener process. Describe the nature of
The process for the stock price in equation (13.8) is AS US At+oSeAT where u and are constant. Explain carefully the difference between this model and each of the following: $ = + AS =S At+AI
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
Suppose that G is a function of a stock price S and time. Suppose that s and are the volatilities of S and G. Show that, when the expected return of S increases by os, the growth rate of G increases
Consider a variable S that follows the process dS = u dt + odz For the first three years, ,ii = 2 and 0 : 3; for the next three years, ,ii = 3 and 0 = 4. If the initial value of the variable is 5,
Variables X1 and X2 follow generalized Wiener processes, with drift rates /L1 and /22 and variances of and 02. What process does X1 + X2 follow if:(a) The changes in X1 and X2 in any short interval
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
Can a trading rule based on the past history of a stock’s price ever produce returns that are consistently above average? Discuss.
What would it mean to assert that the temperature at a certain place followsa, Markov process? Do you think that temperatures do, in fact, follow a Markov process?
Footnote 1 shows that the correct discount rate to use for the real-world expected payoff in the case of the call option considered in Figure 12.1 is 42.6%. Show that ifthe option is a put rather
Repeat Problem 12.20 for an American put option on a futures contract. The strike price and the futures price are $50, the risk-free rate is 10%, the time to maturity is 6 months, and the volatility
Consider a European call option on a non-dividend-paying stock where the stock price is$40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time
A stock price is currently $30. During each 2-month period for the next 4 months it will increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value
Using a “trial-and-error” approach, estimate how high the strike price has to be in Problem 11.17 for it to be optimal to exercise the option immediately.
A stock price is currently $40. Over each of the next two 3-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding. A(a)
A stock price is currently $50. It is known that at the end of 6 months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum.Calculate the value
Calculate u,d, and p when a binomial tree is constructed to value an option on a foreign currency. The tree step size is 1 month, the domestic interest rate is 5% per annum, the foreign interest rate
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
For the situation considered in Problem 12.12, what is the value of a 6-month European put option with a strike price of $51? Verify that the European call and European put prices satisfy put-call
A stock price is currently $50. Over each of the next two 3-month periods it is expected to go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with continuous compounding. What is
A stock price is currently $40. It is known that at the end of 3 months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of
A stock price is currently $80. It is known that at the end of 4 months it will be either $75 or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a

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