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Chapter 5: Q16-2CP (page 558)

Michael Weber, CFA, is analyzing several aspects of option valuation, including the determinants of the value of an option, the characteristics of various models used to value options, and the potential for divergence of calculated option values from observed market prices.

a. What is the expected effect on the value of a call option on common stock if (i) the volatility of the underlying stock price decreases; (ii) the time to expiration of the option increases.

b. Using the Black-Scholes option-pricing model, Weber calculates the price of a three-month call option and notices the option’s calculated value is different from its market price. With respect to Weber’s use of the Black-Scholes option-pricing model, (i) discuss why the calculated value of an out-of-the-money European option may differ from its market price; (ii) discuss why the calculated value of an American option may differ from its market price.

Short Answer

Expert verified

a. (i) Decline in option price (ii) the option price increases.

b. (i) Black Scholes model falsely assumes the volatility, risk free rate and dividend as constant (ii) Because Black Scholes model doesn’t help calculate the value of early exercise

Step by step solution

01

Calculation of expected effect on the value of a call option ‘a’

Decline in option price accompanies the decrease in stock price volatility.

With increase in time to expiration, the option price increases.

02

Explanation on Black Scholes option price model ‘b’

Besides different assumption on implied volatility, Black Scholes model falsely assumes the volatility, risk free rate and dividend as constant.

Since the Black Scholes model doesn’t help calculate the value of early exercise, an American option may be exercised early. This will make the market price higher than the price calculated by the Black Scholes’ model.

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Most popular questions from this chapter

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?

The one-year futures price on a particular stock-index portfolio is 1,218, the stock index currently is 1,200, the one-year risk-free interest rate is 3%, and the year-end dividend that will be paid on a \(1,200 investment in the index portfolio is \)15.

a. By how much is the contract mispriced?

b. Formulate a zero-net-investment arbitrage portfolio, and show that you can lock in riskless profits equal to the futures mispricing.

c. Now assume (as is true for small investors) that if you short-sell the stocks in the market index, the proceeds of the short sale are kept with the broker and you do not receive any interest income on the funds. Is there still an arbitrage opportunity (assuming you don’t already own the shares in the index)? Explain.

d. Given the short-sale rules, what is the no-arbitrage band for the stock-futures price relationship? That is, given a stock index of 1,200, how high and how low can the futures price be without giving rise to arbitrage opportunities?

Imagine that you are holding 5,000 shares of stock, currently selling at \(40 per share. You are ready to sell the shares but would prefer to put off the sale until next year due to tax reasons. If you continue to hold the shares until January, however, you face the risk that the stock will drop in value before year-end. You decide to use a collar to limit downside risk without laying out a good deal of additional funds. January call options with a strike price of \)45 are selling at \(2, and January puts with a strike price of \)35 are selling at \(3. What will be the value of your portfolio in January (net of the proceeds from the options) if the stock price ends up at (a) \)30? (b) \(40? (c) \)50?

Compare these proceeds to what you would realize if you simply continued to hold the shares.

You establish a straddle on Walmart using September call and put options with a strike price of \(50. The call premium is \)4.25 and the put premium is \(5.

a. What is the most you can lose on this position?

b. What will be your profit or loss if Walmart is selling for \)58 in September?

c. At what stock prices will you break even on the straddle?

estion: A member of an investment committee interested in learning more about fixed-income investment procedures recalls that a fixed-income manager recently stated that derivative instruments could be used to control portfolio duration, saying, “A futures like position can be created in a portfolio by using put and call options on Treasury bonds.”

a. Identify the options market exposure or exposures that create a “futures-like

position” similar to being long Treasury-bond futures. Explain why the position you created is similar to being long Treasury-bond futures.

b. Explain in which direction and why the exposure(s) you identified in part (a) would affect portfolio duration.

c. Assume that a pension plan’s investment policy requires the fixed-income manager to hold portfolio duration within a narrow range. Identify and briefly explain circumstances or transactions in which the use of Treasury-bond futures would be helpful in managing a fixed-income portfolio when duration is constrained.
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