Stockholm School of Economics in Riga Financial Economics, Spring 2010 Jevgenijs Babaicevs

Problem Set IV: Valuation of Options - Basics Problem 1: Position Diagrams For each of the following cases below, draw a position diagram (i.e. show the gross value at the date of maturity) to illustrate how options can be used. Be very specific to state the individual contracts constituting the portfolio. a) You want to create an insurance against declining stock prices when you already own the stock. Can you achieve the same result with other financial instruments? b) You want to create a hedge against all changes of the stock price when you already own the stock. c) You want to gain from all changes in the stock price without owning the underlying stock. d) Suppose that short selling of the stock is not allowed. Describe how to construct a portfolio of bonds and options with the identically the same return profile as a short stock. Illustrate using a position diagram! e) You are discussing with your colleague whether to issue a loan to a small business with a high debt/equity ratio. To better understand all the risks involved, explain: (i) Why equity can be viewed as a call option on a firm? Express the position of an equityholder in terms of call options. (ii) How can debt be viewed as an option on a firm? Express the position of a debtholder in terms of put options. f) This summer you are offered an internship opportunity at the options-trading department at the investment bank CHASE. Prior to this, you have decided to train yourself up and draw position diagrams for each of the following strategies. Assume that all options are identical but the exercise prices. (i) Buy 1 call at E1 and 1 put at E2  E1  E2  . (ii) Long 1 call at E1 and short 1 call at E2  E1  E2  . (iii) Long 1 call at E1 and 1 call at E2 , short 1 call at E3 and 1 call at E4  E1  E3  E4  E2  .

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Problem 2: The Put-Call Parity Condition a) Assume that a call option is traded for $10 with an exercise price of $110. The price of the underlying security is $120. The expiration is one year and the continuously compounded interest rate is 10% per annum. (i) Do these figures represent equilibrium in the market? (ii) Using the figures above, demonstrate which transactions should be made in order to exploit the disequilibrium situation! b) A European put is sold for LVL 1. It has an expiration date in two month’s time and the risk free interest rate is 2% during this period. The underlying equity does not pay a dividend and its current price is LVL 160. The exercise price of the option is LVL 165. (i) Given these numbers, is it possible to exploit a risk-free arbitrage opportunity? Please notice that there are NO call options traded on this share! c) The common stock of Pirate Bank is selling for $90. A 26-week call option written on Pirate Bank’s stock is selling for $8. The call’s exercise price is $100. The continuously compounded risk-free interest rate is 10% per annum. (i) Suppose that puts on Pirate Bank’s stock are not traded but you are very eager to buy one. How can you achieve this? (ii) Suppose that puts are traded. What should a 26-week put with an exercise price of $100 sell for?

Problem 3: Understanding Options The corporate charter of Lidosta Derivatives contains a paragraph stating that the firm will be liquidated after exactly one year from today and the market value of total assets will be distributed to the firm’s debtholders and shareholders. Lidosta Derivatives is financed by: (i) A seasoned equity offering of 35 million LVL, and (ii) two zero-coupon loans each with a principal of 100 million LVL that will be repaid after exactly one year. The total market value today of equity and debt is 200 million LVL. Loan-AA is senior, i.e. it has priority at liquidation over Loan-BB that is subordinate. Loan-AA has a simple interest rate of 8.5% that is equal to the market interest rate. The simple risk free interest rate is 8%. a) Illustrate in a single position diagram the possible values of the three sources of finance after exactly one year. Be sure to state what you measure on the two axes, and indicate precisely the critical values on both axes. b) Loan-AA may be interpreted as a synthetic portfolio of one risk free asset and one put option. Indicate the exercise price of the put option, who is the buyer and the seller of the put option, and calculate the premium of the put option today. c) Loan-BB may be interpreted as a synthetic portfolio (positive vertical price spread) consisting of two call options with different exercise prices. Calculate the premiums today of the two call options.

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Problem 4: Introduction to the Binomial Option Pricing Model Assume that there are only two possible states of the world. In State 1, the stock price rises by 50% and in State 2, it drops by 25%. The current stock price is $100, the exercise price of the call option written on this stock is $105, and the simple risk-free rate for the period until expiration is 5%. a) Define and calculate the hedge ratio in this specific case? b) How much should you borrow to hedge the call? c) Use the one period binomial option pricing model to determine the value of the call.

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