Suppose that the observed spread between the yield on a 4-year zero-coupon riskless bond and the yield on a 4-year zero-coupon bond issued by a corporation is 1.35%. By how much (as a percentage) would the Black-Scholes-Merton model overstate the value of a 4-year European option sold by the corporation?

Suppose that the observed spread between the yield on a 4-year zero-coupon riskless bond and the yield on a 4-year zero-coupon bond issued by a corporation is 1.35%. By how much (as a percentage) would the Black-Scholes-Merton model overstate the value of a 4-year European option sold by the corporation?

EBK CONTEMPORARY FINANCIAL MANAGEMENT

14th Edition

ISBN:9781337514835

Author:MOYER

Publisher:MOYER

Chapter12: The Cost Of Capital

Section: Chapter Questions

Problem 10QTD

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The price S(t) of a share follows the geometric Brownian motion S(t) = Se µt+σW(t) . The price of the share at t = 0 is S = £25 and µ = 0.1. The continuously compounded interest rate is 8%. However, the volatility σ is not known. (a) Explain the definition of implied volatility. (b) A European call option on the above share with the strike price K = £23 and expiration time of 6 months has the market price £3.1. Does the equation defining the implied volatility have a solution? [6] Hint. You are not supposed to solve the equation defining the volatility to answer this question.
Suppose that the observed spread between the yield on a 2-year zero-coupon riskless bond and the yield on a 2-year zero-coupon bond issued by a corporation is 1.85%. By how much (as a percentage) would the Black- Scholes-Merton model overstate the value of a 2-year European option sold by the corporation?
A European call that will expire in one year is currently trading for $3. Assume the risk-free rate (based on continuous compounding) is 5%, the underlying stock price is $60 and the strike price is $55. a. Is there an arbitrage opportunity? b. Describe exactly what a trader should do to take advantage of the arbitrage opportunity assuming it exists. c. Determine the present value of the profit that the trader can earn assuming you identify an arbitrage opportunity. Use at least four decimal places for those questions that require a numerical answer.
(a) You hold a consol that pays a coupon C in perpetuity. The current interest rate is i, and the average expectation in the market is that this will remain unchanged. What will be the price of the consol today? (b) In the next period however, the interest rate changes unexpectedly to i 0 . What is the new price of the bond? If the bond is sold at the beginning of that next period, what is the yield from the consol? Does the yield increase or decrease if i 0 > i? (c) Suppose alternatively that the market expects that the interest rate will change to i 0 after the initial period. What is the initial value of the consol, and what is the yield from selling it after one period?
Consider a European call option and a European put option that have the same underlying stock, the same strike price K = 40, and the same expiration date 6 months from now. The current stock price is $45. a) Suppose the annualized risk-free rate r = 2%, what is the difference between the call premium and the put premium implied by no-arbitrage? b) Suppose the annualized risk-free borrowing rate = 4%, and the annualized risk-free lending rate = 2%. Find the maximum and minimum difference between the call premium and the put premium, i.e., C − P such that there is no arbitrage opportunities.
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You wish to create a synthetic forward rate agreement in which you would lock in a return between 150 and 310 days. The price of a 150-day zero coupon bond is 0.9823 and the price of 310-day zero coupon bond is 0.9634. Which one of the following method is the correct method to create the synthetic FRA? Explain. [Professor’s note: This is a challenging questions]. A) Borrow one 150-day bond and invest in 1.02 of the 310-day bondsB) Borrow two 150-day bonds and invest in 0.98 of the 310-day bonds C) Lend one of the 150-day bonds and borrow 1.02 of the 310-day bonds D) Lend two of the 150-day bonds and borrow 0.98 of the 310-day bonds
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Without using the Black‐Scholes model, compute the price of a European put option on a non‐dividend‐paying stock with the strike price is $70 when the stock price is $73, the risk‐free interest rate is 10% pa, the volatility is 40% pa, and the time to maturity is 6 months?
In this problem, we derive the put-call parity relationship for European options on stocks that pay dividends before option expiration. For simplicity, assume that the stock makes one dividend payment of $D per share at the expiration date of the option.a. What is the value of a stock-plus-put position on the expiration date of the option?b. Now consider a portfolio comprising a call option and a zero-coupon bond with the same maturity date as the option and with face value (X + D). What is the value of this portfolio on the option expiration date? You should find that its value equals that of the stock-plus-put portfolio regardless of the stock price.c. What is the cost of establishing the two portfolios in parts (a) and (b)? Equate the costs of these portfolios, and you will derive the put-call parity relationship.
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Mf2. Assume a one-period binomial model in which the initial stock price is S = 60 and in each period the stock price can go either up by a factor of u = 7 3 or down by a factor of d = 2 3 . Assume that the simple interest rate over one time period is r = 1 3 . (a) Determine the fair price of the European put option with strike K = 60. (b) Assume that instead of the price determined in part (a), the European call option is trading at 11. Prove that there is an arbitrage and explain how the arbitrage can be achieved