ACC 372 Practice Problem Set #4 Winter 2000

1.
Text problems:
Chapter 21 9, 10, 15, 17, 20, 21, 22

2.
Assess whether each of the following statements is true, false, or uncertain. Justify your answer.
(a)
Options with higher exercise prices are worth more than options with lower exercise prices.
(b)
A portfolio consisting of a stock and a risk free bond will be less risky than investing entirely in the stock. Since a call option can be duplicated by a portfolio combining the underlying stock and a risk free bond, it follows that a call option is less risky than the underlying stock.

3.
A bearish put spread is an investment strategy involving purchasing a put option with a high strike price and selling a put option with a low strike price (and the same maturity). Suppose that a put option with a strike price of $50 sells for $2.52 and a put option with a strike price of $60 sells for $7.95. Construct a table showing the profits from buying a bearish put spread using these options for stock prices at maturity of $45, $50, $55, $60, and $65.

4.
An ``at-the-money'' option is an option which has a strike price equal to the current stock price. Assume that the underlying stock does not pay any dividends. Show that an at-the-money European call option is worth more than an at-the-money European put option with the same expiry date. What is the intuition for this result?

5.
Consider a European put option on a stock which will not pay any dividends before the option matures. Let the current stock price be St, the strike price of the option be X, and the continuously compounded effective annual risk free interest rate be r. Prove that the option price Pt must satisfy:

\begin{displaymath}X\mbox{e}^{-r(T-t)} \geq P_t \geq \max \left[X\mbox{e}^{-r(T-t)} - S_t, 0\right] \end{displaymath}

6.
Consider three put options with exercise prices X1, X2, and X3, where X1 < X2 < X3. Prove, assuming X2 - X1 = X3 - X2, that $P(X_2) \leq (P(X_1) + P(X_3))/2$.

7.
XYZ Company pays no dividends and its stock currently sells for $50 per share. The continuously compounded risk free annual interest rate is 10%. What is the lower bound for the price of a one year call option with an exercise price of $50? If the call option actually sells for $4, show how you can make arbitrage profits.

8.
Consider the following set of six month European call option values on a stock currently selling for $100 per share where the continuously compounded effective annual risk free interest rate is 10%:
Exercise Price Call Price
95 9.75
100 5.10
105 0.25
Are there any arbitrage opportunities available? If so, describe how to exploit them.

9.
A stock is currently selling for $90. The continuously compounded effective annual risk free interest rate is 6%. The price of a European call option on this stock which has an exercise price of $90 and expires in four months is $4.00. The price of a European put option with the same exercise price and maturity date is $2.10. Assume that the stock will not pay a dividend over the next four months. Show that there is an arbitrage opportunity and demonstrate how to exploit it.

10.
The June 1986 issue of oil-indexed notes by Standard Oil included an option. Part of the issue consisted of pure discount bonds which had an attached option feature. In addition to the $1,000 par value of the note, the company promised to pay an additional amount based on the price of oil at maturity. In the case of the note maturing in 1990, this additional amount was equal to the product of 170 and the excess (if any) of the price of a barrel of oil at maturity over $25. The contract also included the stipulation that if the price of oil was above $40, a price of $40 would be used in computing the payoff to the investor. Thus, for example, if the price of oil is $20 at maturity, the value of the option then is zero. If the price of oil is between $25 and $40, say $35, the value of the option is $(35-25) \times 170$. If the price of oil is above $40, say $50, the option is worth $(40-25) \times 170$. Identify the attached option feature in terms of simple call and/or put options.

11.
Consider a two-period binomial model. The stock price today is $25. After 6 months, it will either increase to $27.39 or decrease to $22.82. At that point the stock will pay a dividend of $2 per share, so that the possible ex-dividend prices are $25.39 and $20.82. If the stock rises, then over the next 6 months, it can either continue increasing from $25.39 to $27.81 or it can decline to $23.17. If the stock falls, then over the next 6 months, it can either rise from $20.82 to $22.81 or it can continue decreasing to $19.01. Compute the value today of an American call option on this stock with a strike price of $23 and a maturity date of 1 year. Assume that the effective annual risk free interest rate is 8%.

12.
Consider a binomial model where a year is divided into two six month periods. The stock price today is $50. After six months it can either increase to $54.77 or decrease to $45.64. If it rises to $54.77, then over the next six months it can either keep rising to $60 or fall back to $50. If it falls over the first six months to $45.64, then it can either rise back to $50 or keep falling to $41.67 after a year. The effective annual risk free interest rate is 10% per year.
(a)
Compute the prices today of the following three securities:
i.
A security which pays $1 after one year if the stock price is $41.67.
ii.
A security which pays $1 after one year if the stock price is $50.
iii.
A security which pays $1 after one year if the stock price is $60.
(b)
Based on your answer to part (a), compute the prices today of the following securities:
i.
European call options with strike prices of $48 and $52 which mature in one year.
ii.
European put options with strike prices of $48 and $52 which mature in one year.
iii.
A risk free bond which pays $1 in one year.
iv.
The underlying stock.

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