Chapter 5: Q27I (page 555)
According to the Black-Scholes formula, what will be the value of the hedge ratio of a call option as the stock price becomes infinitely large? Explain briefly.
The probability of exercise N(d1) approaches 1.0
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A bearish spread is the purchase of a call with exercise price X 2 and the sale of a call with exercise price X 1, with X 2 greater than X 1. Graph the payoff to this strategy and compare it to Figure 15.10 .
Use the following case in answering Problems 10 – 15 : Mark Washington, CFA, is an analyst with BIC. One year ago, BIC analysts predicted that the U.S. equity market would most likely experience a slight downturn and suggested delta-hedging the BIC portfolio.
As predicted, the U.S. equity markets did indeed experience a downturn of approximately 4% over a 12-month period. However, portfolio performance for BIC was disappointing, lagging its peer group by nearly 10%. Washington has been told to review the options strategy to determine why the hedged portfolio did not perform as expected.
Which of the following best explains a delta-neutral portfolio? A delta-neutral portfolio is perfectly hedged against:
a. Small price changes in the underlying asset.
b. Small price decreases in the underlying asset.
c. All price changes in the underlying asset.
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 the stock-plus-put position on the expiration date of the option?
b. Now consider a portfolio consisting of a call option and a zero-coupon bond with the same expiration 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 cost of these portfolios, and you will derive the put-call parity relationship, Equation 16.3.
Consider the following options portfolio: You write a January 2012 expiration calloption on IBM with exercise price \(170. You also write a January expiration IBM putoption with exercise price \)165.
a. Graph the payoff of this portfolio at option expiration as a function of IBM’s stockprice at that time.
b. What will be the profit/loss on this position if IBM is selling at \(167 on the optionexpiration date? What if IBM is selling at \)175? Use The Wall Street Journal listingfrom Figure 15.1 to answer this question.
c. At what two stock prices will you just break even on your investment?
d. What kind of “bet” is this investor making; that is, what must this investor believeabout IBM’s stock price in order to justify this position?
A stock index is currently trading at 50. Paul Tripp, CFA, wants to value two-year indexoptions using the binomial model. In any year, the stock will either increase in value by20% or fall in value by 20%. The annual risk-free interest rate is 6%. No dividends arepaid on any of the underlying securities in the index.
a. Construct a two-period binomial tree for the value of the stock index.
b. Calculate the value of a European call option on the index with an exercise price of 60.
c. Calculate the value of a European put option on the index with an exercise price of 60.
d. Confirm that your solutions for the values of the call and the put satisfy put-call parity
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