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Corporate Finance
What trading position is created from a long strangle and a short straddle when both have the same time to maturity? Assume that the strike price in the straddle is halfway between the two strike
A bank decides to create a five-year principal-protected note on a non-dividend-paying stock by offering investors a zero-coupon bond plus a bull spread created from calls. The risk-free rate is 4%
When is it appropriate for an investor to purchase a butterfly spread?
What trading strategy creates a reverse calendar spread?
What is the difference between a strangle and a straddle?
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
Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.
What is meant by the delta of a stock option?
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.
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
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?
Suppose that a stock price, S, follows geometric Brownian motion with expected return μ and volatility σ:dS = μS dt + σS dzWhat is the process followed by the variable Sn? Show that Sn also
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 dzwhere a, x0,
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 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 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 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
If follows the geometric Brownian motion process in equation (14.6), what is the process followed by(a) y = 2S,(b) y = S2,(c) y = eS,(d) y = er(T-t)/S. In each case express 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 two years?
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
Can a trading rule based on the past history of a stock's price ever produce returns that are consistently above average? Discuss.
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
Variables X1 and X2 follow generalized Wiener processes with drift rates μ1 and μ2 and variances σ12 and σ12. What process does X1 + X2 follow if: (a) The changes
Consider a variable,, that follows the process dS = μ dt + σ dzFor the first three years, μ = 2 and σ = 3; for the next three years, μ = 3 and σ = 4. If the initial value of the variable
Suppose that G is a function of a stock price, S and time. Suppose that σS and σG are the volatilities of S and G. Show that when the expected return of S increases by λσS, the growth rate of G
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
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
It has been suggested that the short-term interest rate, r, follows the stochastic process where a, b, and c are positive constants and dz is a Wiener process. Describe the nature of this
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 probability distribution of
Consider a derivative that pays off SnT at time T where ST is the stock price at that time. When the stock pays no dividends and its price follows geometric Brownian motion, it can be shown that its
A call option on a non-dividend-paying stock has a market price of $2.50. The stock price is $15, the exercise price is $13, the time to maturity is three months, and the risk-free interest rate is
With the notation used in this chapter (a) What is N'(x)?(b) Show that SN'(d1) = Ke–r(T–t)N'(d2), where S is the stock price at time t (c) Calculate (d) Show that whenwhere c is
The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price change in one trading day?
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
A company has an issue of executive stock options outstanding. Should dilution be taken into account when the options are valued? Explain you answer.
A company's stock price is $50 and 10 million shares are outstanding. The company is considering giving its employees three million at-the-money five-year call options. Option exercises will be
If the volatility of a stock is 18% per annum, estimate the standard deviation of the percentage price change in (a) One day, (b) One week, (c) One month.
A financial institution plans to offer a security that pays off a dollar amount equal to S2T at time T.(a) Use risk-neutral valuation to calculate the price of the security at time in
Explain the principle of risk-neutral valuation.
The appendix derives the key result: E[max(V−K,0)]=E(V)N(d1) −KN(d2). Show thatE[max(K−V,0)]=KN(−d1) −E(V)N(−d2)and use this to derive the Black-Scholes-Merton formula for the price of
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,
What difference does it make to your calculations in Problem 15.4 if a dividend of $1.50 is expected in two months?
What is implied volatility? How can it be calculated?
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
Using the notation in the chapter, prove that a 95% confidence interval for ST is between
Why was it attractive for companies to grant at-the-money stock options prior to 2005? What changed in 2005?
The notes accompanying a company's financial statements say: "Our executive stock options last 10 years and vest after four years. We valued the options granted this year using the
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
What is the (risk-neutral) expected life for the employee stock option in Example 16.2? What is the value of the option obtained by using this expected life in Black-Scholes-Merton?
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?
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
"Stock option grants are good because they motivate executives to act in the best interests of shareholders." Discuss this viewpoint.
"Granting stock options to executives is like allowing a professional footballer to bet on the outcome of games." Discuss this viewpoint.
Why did some companies backdate stock option grants in the US prior to 2002? What changed in 2002?
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?
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
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?
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 of 700 can be used to provide portfolio
"Once we know how to value options on a stock paying a dividend yield, we know how to value options on stock indices, currencies, and futures." Explain this statement.
The Dow Jones Industrial Average on July 20, 2016 was 18,580 and the price of a September 185 (European) call option on the index was $3.35. Use the DerivaGem software to calculate the implied
Assume that the price of currency A expressed in terms of the price of currency B follows the processdS = (rB – rA)S dt + σS dzwhere rA is the risk-free interest rate in currency A and rB is the
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
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
Explain how corporations can use range forward contracts to hedge their foreign exchange risk when they are due to receive a certain amount of the foreign currency in the future.
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
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
Explain the difference between a call option on yen and a call option on yen futures.
Why are options on bond futures more actively traded than options on bonds?
"A futures price is like a stock paying a dividend yield." What is the dividend yield?
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 with a strike
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?
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
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
Suppose you buy a put option contract on October gold futures with a strike price of $1400 per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the October
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
Explain how a stop-loss trading rule can be implemented for the writer of an out-of-the-money call option. Why does it provide a relatively poor hedge?
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
Repeat Problem 19.12 for a financial institution with a portfolio of short positions in put and call options on a currency.
What is the equation corresponding to equation (19.4) for (a) a portfolio of derivatives on a currency and (b) a portfolio of derivatives on a futures contract?
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?
Use the put-call parity relationship to derive, for a non-dividend-paying stock, the relationship between:(a) The delta of a European call and the delta of a European put.(b) The gamma of a European
The formula for the price of a European call futures option in terms of the futures price, F0, is given in Chapter 18 as
Calculate the delta of an at-the-money six-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.
What is meant by the gamma of an option position? What are the risks in the situation where the gamma of a position is large and negative and the delta is zero?
"The procedure for creating an option position synthetically is the reverse of the procedure for hedging the option position." Explain this statement.
Why did portfolio insurance not work well on October 19, 1987?
What volatility smile is likely to be observed when (a) Both tails of the stock price distribution are less heavy than those of the lognormal distribution? (b) The right tail is heavier, and the
Explain the problems in testing a stock option pricing model empirically.
What volatility smile is observed for equities?
An exchange rate is currently 1.0 and the implied volatilities of six-month European options with strike prices 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, and 1.3 are 13%, 12%, 11%, 10%, 11%, 12%, and 13%. The
What volatility smile is likely to be caused by jumps in the underlying asset price? Is the pattern likely to be more pronounced for a two-year option than for a three-month option?
A European call and put option have the same strike price and time to maturity. The call has an implied volatility of 30% and the put has an implied volatility of 25%. What trades would you do?
Explain carefully why a distribution with a heavier left tail and less heavy right tail than the lognormal distribution gives rise to a downward sloping volatility smile.
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
Explain what is meant by crashophobia.
Which of the following can be estimated for an American option by constructing a single binomial tree: delta, gamma, vega, theta, rho?
A nine-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
Suppose that Monte Carlo simulation is being used to evaluate a European call option on a non-dividend-paying stock when the volatility is stochastic. How could the control variate and antithetic
Ho do equations (21.27) to (21.30) change when the implicit finite difference method is being used to evaluate an American call option on a currency?
An American put option on a non-dividend-paying stock has four 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
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