Chapter 5: Q2B (page 588)
The current level of the S&P 500 is 1,200. The dividend yield on the S&P 500 is 2%. The risk-free interest rate is 1%. What should a futures contract with a one-year maturity be selling for?
Answer
$1,188
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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.
Suppose you are attempting to value a one-year maturity option on a stock with volatility (i.e., annualized standard deviation) ofσ= .40. What would be the appropriate values for u and d if your binomial model is set up using the following?
a. 1 period of one year
b. 4 sub-periods, each 3 months
c. 12 sub-periods, each 1 month
Joan Tam, CFA, believes she has identified an arbitrage opportunity for a commodity as indicated by the information given in the following exhibit:
a. Describe the transactions necessary to take advantage of this specific arbitrage opportunity.
b. Calculate the arbitrage profit.
Return to Problem 35. Value the call option using the risk-neutral shortcut described in the box on page 533. Confirm that your answer matches the value you get using the two-state approach.
Question: You are attempting to value a call option with an exercise price of \(100 and one year to expiration. The underlying stock pays no dividends, its current price is \)100, and you believe it has a 50% chance of increasing to \(120 and a 50% chance of decreasing to \)80.
The risk-free rate of interest is 10%. Calculate the call option’s value using the two-state stock price model.
Reconsider the determination of the hedge ratio in the two-state model (Section 16.2), where we showed that one-third share of stock would hedge one option. What would be the hedge ratio for each of the following exercise prices: \(120, \)110, \(100, \)90? What do you conclude about the hedge ratio as the option becomes progressively more in the money?
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