1. Assume that a (European) call option exists on this stock having on exercise price of $155.

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1 MØA 155 PROBLEM SET: Binomial Option Pricing Exercise 1. Call option [4] A stock s current price is $16, and there are two possible prices that may occur next period: $15 or $175. The interest rate on risk-free investments is 6% per period. 1. Assume that a (European) call option exists on this stock having on exercise price of $155. (a) How could you form a portfolio based on the stock and the call so as to achieve a risk-free hedge? (b) Compute the price of the call. 2. Answer the above two questions if the exercise price was $18. Exercise 2. Calls, hedge [6] A stock s current price is $1. There are two possible prices at the end of the year: $15 or % 75. A call option to buy one share at $1 at the end of the year sells at $2. Suppose that you are told that a) writing 3 calls, b) buying 2 stocks and c) borrowing $14 today is a perfect hedge portfolio. 1. What is the risk free rate of interest? Exercise 3. Call (RWJ 21.16) [5] You bought a 1-share call contranct three weeks ago. The expirations date of the call is five weeks from today. On that date, the price of the underlying stock will be either $12 or $95. The two states are equally likely to occur. Currently, the stock sells for $96; its strike price is $112. You are able to purchase 32 shares of the stock. You are able to borrow money at 1% per annum, annually compounded. 1. What is the value of your call contract? Exercise 4. Stock A is expected next period to have either a price of 1 or 3. The current stock price for A is 2 and the per period risk free interest rate is 2%. What is the price of a one period call option on A with exercise price of 15? (a). (b) 7.65 (c) 8.72 (d) 15. (e) I choose not to answer. Exercise 5. A stock will next period either be priced at 12% or 8% of todays price. The stock is currently priced at 5. The risk free interest rate is 6%. Consider a call option written on this stock with exercise price equal to 6. What is todays price on this option? (a). (b) 2.3 1

2 (c) 5. (d) 1. (e) I choose not to answer. Exercise 6. HAL [6] You are interested in the computer company HAL computers. Its stock is currently priced at 9. The stock price is expected to either go up by 25% or down by 2% each six months. The annual risk free interest rate is 2%. Your broker now calls you with an interesting offer. You pay C now for the following opportunity: In month 6 you can choose whether or not to buy a call option on HAL computers with 6 months maturity (i.e. expiry is 12 months from now). This option has an exercise price of $9, and costs $1,5. (You have an option on an option.) 1. If C is the fair price for this compound option, find C. 2. If you do not have any choice after 6 months, you have to buy the option, what is then the value of the contract? 2

3 Empirical Solutions MØA 155 PROBLEM SET: Binomial Option Pricing Exercise 1. Call option [4] 1. We have the following payoff next period: S u = S d = 15 Call option with exercise price 155. C u = max(, ) = 2 C C d = max(, ) = 2. How to get a risk free hedge by buying m call options? Need payoff in each period to be equal S d + mc d = S u + mc u m = S d S u = = 1.25 C u C d 2 Need to sell 1.25 options to create a risk free hedge. 3. To price the option, we find the parameters of the binomial option pricing model. Start by finding u and d: S u = us = u16 = 175 Solve for u and d: Then find the risk-neutral probabilities: q = (1 + r f ) d u d S d = ds = d16 = 15 u = = 1.1 d = =.973 = = =.685 3

4 Finding terminal payoffs C u = max(s u K, ) = max( , ) = 2 Then find price C d = max(s d K, ) = max(15 155, ) = C = qc u + (1 q)c d = = r f With an exercise price of 18, the call will never be exercised. It is worthless. Exercise 2. Calls, hedge [6] 1. A perfect hedge implies that total payoffs are zero in each state. Hence, Hence, Exercise 3. Call (RWJ 21.16) [5] Payoffs t = t = 1 S = 75 S = 15 3C 6 15 Buy stock Borrow (1+r) 14(1+r) 14(1 + r) = 15( for it to be perfect hedge ) r = = 1 14 = 71 7 % 1. Ignoring all the extraneous information, let us find the price of one Call. S = 96 S u = 12 S d = 95 Find u and d: Value.5974 Exercise 4. The stock has price movements: u = = 1.25 d = = S u = 3 S = 2 S d = 1 4

5 The Call has payoffs: C =? C u = (3 15) = 15 C d = u = 3/2 = 1.5, d = 1/2 =.5, q = =.52, C = (.52 15) = (b) is correct Exercise 5. The stock price can never exceed the exercise price, current value equals zero. is correct. Exercise 6. HAL [6] Stock price movement, u = 1.25, d =.8. S uu = S = 9 S u = 1125 S d = 72 S ud = 9 S dd = 576 Start by calculating the value of an option at time 1, with exercise price 9. q = er t d u d = e = q =.322 C u = e r t (qc uu + (1 q)c ud ) = e.2.5 ( ) = 3, C d = You are now offered to pay C for the ability to choose to pay $1,5 in month 6 for this option. This can be analyzed on a tree. Clearly, you will not pay 1,5 in the down state, but will pay it in the up-state. 5

6 3, , 5 = C =? Calculate the value the usual way C = e r t (q (1 q) ) = e.2.5 ( ) = This is the current value of the option on an option. For the second part of the question, if you do not have any choice, this is the picture 3, , 5 = C =? 15 C = e r t (q (1 q) ( 15)) = e.2.5 ( ( 15)) =