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business
options futures and other derivatives

Questions and Answers of Options Futures And Other Derivatives

With the notation used in this chaptera. What is N\x)1b. Show that SN'(di) = Ke~r{T~t}N'(d2), where S is the stock price at time t and(S/K) + (r - 02/2)(T - t)c. Calculate ddt/dS and dd2/dS.d. Show
A call option on a non-dividend-paying stock has a market price of $2j. The stock price is $15, the exercise price is $13, the time to maturity is three months, and the risk-free interest rate is
Consider an American call option on a stock. The stock price is $70, the time to maturity is eight months, the risk-free rate of interest is 10% per annum, the exercise price is $65, and the
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 Sj at time T, where ST 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(t < T) has
Assume that a non-dividend-paying stock has an expected return of /x and a volatility of a.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?
Prove that, with the notation in the chapter, a 95% confidence interval for ST is between
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 12.4 if a dividend of $1.50 is expected in two months?
Calculate the price of a three-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,
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 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 return
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 two years? (Hint: ST > 80 when
If S follows the geometric Brownian motion process in equation (11.12), what is the process followed by:a. y = 2S?b. y = S2 ?c. y = esld. y = er(T-t)/Sl In each case express the coefficients of dt
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 (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 2.0.a.
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 expected
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(x0 — x)dt + sx dz wherea, XQ, and s
Suppose that a stock price, S, follows geometric Brownian motion with expected return /J, and volatility a:dS = [MSdt + aSdz What is the process followed by the variable 5"? Show that 5" also follows
It has been suggested that the short-term interest rate, r, follows the stochastic process dr = a(b - r) dt + re dz wherea, b, and c are positive constants and dz is a Wiener process. Describe the
The process for the stock price in equation (11.8) is SS = fiS St where /x and a are constant. Explain carefully the difference between this model and each of the following:SS = fi St + aeVsi SS =
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 as and aG are the volatilities of S and G. Show that when the expected return of S increases by kas, the growth rate of G
Consider a variable S that follows the process dS =For the first three years, /x = 2 and a = 3; for the next three years, jix = 3 and a = 4. If the initial value of the variable is 5, what is the
Variables X, and X2 follow generalized Wiener processes with drift rates fix and ix2 and variances a\ and of. What process does Xx + X2 follow if:a. The changes in X{ and X2 in any short interval of
A company's cash position (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
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 follows a Markov process?Do you think that temperatures do, in fact, follow a Markov process?
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. Each month for the next two months it is expected to increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of
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 Section 10.2 is 42.6%. Show that if the option is a put rather
Using a "trial-and-error" approach, estimate how high the strike price has to be in Problem 10.15 for it to be optimal to exercise the option immediately.
A stock price is currently $40. Over each of the next two three-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
A stock price is currently $50. It is known that at the end of six months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculate the
A stock price is currently $25. It is known that at the end of two months it will be either $23 or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose ST is the
For the situation considered in Problem 10.11, what is the value of a six-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 three-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.
A stock price is currently $40. It is known that at the end of three months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the
A stock price is currently $80. It is known that at the end of four 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 stock price is currently $50. It is known that at the end of two months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of
Consider the situation in which stock price movements during the life of a European option are governed by a two-step binomial tree. Explain why it is not possible to set up a position in the stock
For the situation considered in Problem 10.5, what is the value of a one-year European put option with a strike price of $100? Verify that the European call and European put prices satisfy put-call
A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuous compounding.
A stock price is currently $50. It is known that at the end of six months it will be either $45 or $55. The risk-free interest rate is 10% per annum with continuous compounding. What is the the value
What is meant by the delta of a stock option?
Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.
A stock price is currently $40. It is known that at the end of one month it will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a
Suppose that the price of a non-dividend-paying stock is $32, its volatility is 30%, and the riskfree rate for all maturities is 5% per annum. Use DerivaGem to calculate the cost of setting up the
Draw a diagram showing the variation of an investor's profit and loss with the terminal stock price for a portfolio consisting of:a. One share and a short position in one call optionb. Two shares and
A diagonal spread is created by buying a call with strike price K2 and exercise date T2 and selling a call with strike price K{ and exercise date Tx (T2 > Ti). Draw a diagram showing the profit
Three put options on a stock have the same expiration date and strike prices of $55, $60, and $65.The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can be
One Australian dollar is currently worth $0.64. A one-year butterfly spread is set up using European call options with strike prices of $0.60, $0.65, and $0.70. The risk-free interest rates in the
What is the result if the strike price of the put is higher than the strike price of the call in a strangle?
A box spread is a combination of a bull call spread with strike prices K\ and K2 and a bear put spread with the same strike prices. The expiration dates of all options are the same. What are the
How can a forward contract on a stock with a particular delivery price and delivery date be created from options?
An investor believes that there will be a big jump in a stock price, but is uncertain as to the direction.Identify six different strategies the investor can follow and explain the differences among
Construct a table showing the payoff from a bull spread when puts with strike prices K\ and K2(K2 > Ki) are used.
A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs$4. Construct a table that shows the profit from a straddle. For what range of stock prices would
Use put-call parity to show that the cost of a butterfly spread created from European puts is identical to the cost of a butterfly spread created from European calls.
Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively.How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that
Explain how an aggressive bear spread can be created using put options.
Use put-call parity to relate the initial investment for a bull spread created using calls to the initial investment for a bull spread created using puts.
A call option with a strike price of $50 costs $2. A put option with a strike price of $45 costs $3.Explain how a strangle can be created from these two options. What is the pattern of profits from
What is the difference between a strangle and a straddle?
What trading strategy creates a reverse calendar spread?
Call options on a stock are available with strike prices of $15, $17^, and $20 and expiration dates in three months. Their prices are $4, $2, and $ j , respectively. Explain how the options can be
When is it appropriate for an investor to purchase a butterfly spread?
Explain two ways in which a bear spread can be created.
What is meant by a protective put? What position in call options is equivalent to a protective put?
Consider an option on a stock when the stock price is $41, the strike price is $40, the risk-free rate is 6%, the volatility is 35%, and the time to maturity is one year. Assume that a dividend of
Suppose that you are the manager and sole owner of a highly leveraged company. All the debt will mature in one year. If at that time the value of the company is greater than the face value of the
What is the result corresponding to that in Problem 8.22 for European put options?
Suppose that c,, c2, and c3 are the prices of European call options with strike prices Kx, K2, and K3, respectively, where A"3 > K2 > Kx and A"3 — K2 = K2 — Kx. All options have the same
A European call option and put option on a stock both have a strike price of $20 and an expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per annum, the current
Use the software DerivaGem to verify that Figures 8.1 and 8.2 are correct.
Even when the company pays no dividends, there is a tendency for executive stock options to be exercised early (see Section 7.12 for a discussion of executive stock options). Give a possible reason
Prove the result in equation (8.8). (Hint: For the first part of the relationship consider (a) a portfolio consisting of a European call plus an amount of cash equal to D + K and (b) a portfolio
Prove the result in equation (8.4). (Hint: For the first part of the relationship consider (a) a portfolio consisting of a European call plus an amount of cash equal to K and (b) a portfolio
Explain carefully the arbitrage opportunities in Problem 8.15 if the American put price is greater than the calculated upper bound.
The price of an American call on a non-dividend-paying stock is $4. The stock price is S31, the strike price is $30, and the expiration date is in three months. The risk-free interest rate is
Explain carefully the arbitrage opportunities in Problem 8.13 if the European put price is $3.
The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of SO.50 is expected in two months and again in five
Give an intuitive explanation of why the early exercise of an American put becomes more attractive as the risk-free rate increases and volatility decreases.
A one-month European put option on a non-dividend-paying stock is currently selling for $2.50.The stock price is $47, the strike price is $50, and the risk-free interest rate is 6% per annum.What
A four-month European call option on a dividend-paying stock is currently selling for $5. The stock price is $64, the strike price is $60, and a dividend of $0.80 is expected in one month. The
What is a lower bound for the price of a two-month European put option on a non-dividendpaying stock when the stock price is $58, the strike price is $65, and the risk-free interest rate is 5% per
What is a lower bound for the price of a six-month call option on a non-dividend-paying stock when the stock price is $80, the strike price is $75, and the risk-free interest rate is 10% per annum?
Explain why the arguments leading to put-call parity for European options cannot be used to give a similar result for American options.
Explain why an American call option is always worth at least as much as its intrinsic value. Is the same true of a European call option? Explain your answer.
"The early exercise of an American put is a tradeoff between the time value of money and the insurance value of a put." Explain this statement.
Give two reasons why the early exercise of an American call option on a non-dividend-paying stock is not optimal. The first reason should involve the time value of money. The second reason should
What is a lower bound for the price of a one-month European put option on a non-dividendpaying stock when the stock price is $12, the strike price is $15, and the risk-free interest rate is 6% per
What is a lower bound for the price of a four-month call option on a non-dividend-paying stock when the stock price is $28, the strike price is $25, and the risk-free interest rate is 8% per annum?
List the six factors affecting stock option prices.
Use DerivaGem to calculate the value of an American put option on a nondividend paying stock when the stock price is $30, the strike price is $32, the risk-free rate is 5%, the volatility is 30%, and

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