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Exercise #1. A firm produces output using the technology y =11,000KL12 , where capital, K, is measured in machine-hours, labor, L, is measured in person-hours, and y denotes the yearly output. The hourly wage rate wl = 10, and the hourly rental rate of capital is wk = 20. (a) Show that this technology displays increasing returns to scale. (b) Compute the marginal products of labor and capital. (c) Suppose that at the end of 1999 the firm has signed a contract to rent K = 1,000 machine hours over the course of the year 2000. Derive the firm's short run cost function in the year 2000. (d) What is the firm's short run marginal cost function? What is the firm's short run average cost function? At which point do these two curves intersect? (e) On a diagram plot the firm's short run average and marginal cost curves. Exercise #2. The aggregate labor supply in the state of Bahnanas is Ls = 2,000w, where w is the hourly wage, measured in Bahnanas $ and Ls denotes the number of person- hours supplied by the workers of Bahananas in a year. The aggregate labor demand in the state of Bahananas is Ld = 12,000 2,000w, where Ld denotes the number of person-hours demanded by the firms of Bahnanas in a given year. (a) Compute the equilibrium hourly wage and the amount of hours worked in a given year. (b) The government of Bahnanas introduces a minimum wage law requiring firms to pay an hourly wage not lower than $4. Compute the equilibrium number of hours worked by the workers of Bahnanas. 1 (c) On a diagram that has the wage rate on the y-axis and the number of hours on the x-axis, plot the labor demand and supply functions and the equilibrium prices and quantities that you found in points (a) and (b). (d) Compute the deadweight loss induced by this policy, and show it on the diagram of point (c). Exercise #3. Short questions. (a) Compute the price and income elasticities of the following demand function xd (p,m) = 1 2 m p . (b) A firm produces output, denoted by y, using the following production function y = 10L, where L represents the labor input, measured in person-hours. The unit price of output is $10. Let w denote the hourly wage rate. Compute and plot on a diagram the inverse labor demand curve for the firm. (c) Explain how it can be possible that as tax rates decrease, government tax revenues increase (i.e. the Laffer curve).

11. You write a call option with strike $50 and buy a call with strike $60. The options are on the same stock and have the same maturity date. One of the calls sells for $3; the other sells for $9. (Assume zero interest rate.) (a) Draw the payo graph for this strategy at the option maturity date. (b) Draw the prot graph for this strategy. (c) What is the break-even point for this stratgy? Are you bullish or bearish on the stock? 12. Consider an increasingly popular deposit contract with payos linked to the performance on the S&P 500 Index on the U.S. stock market. For every dollar invested in the contract, the rate of return in one year is equal to 60% of the realized rate of return of the S&P 500 Index during this year if this rate of return is positive; otherwise, you get your money back. In essence, you are protected from the downside risk of the S&P 500 Index, while you are still able to participate in the upside potential of the stock market. The one year riskless interest rate is 10%. For simplicity, assume that the stocks in the index do not pay dividends. (a) Draw a graph for the payo one year from now for a one dollar investment in the contract with the horizontal axis being the re- 12 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS alized rate of return on the S&P 500 Index. Also write down the payo symbolically. (b) Show that the payo one year from now for a one dollar investment in this contract is the payo to a portfolio of a default-free bond and a European call option on the S&P 500 Index. (c) Suppose that the rates of return on the S&P 500 Index can take two possible values one year from now, 20% and -20% with prob- abilities 60% and 40%, respectively. Do you make money or lose money investing in this contract? If so, how much? 13. What is a lower bound for the price of 3-month call option on a non- dividend-paying stock when the stock price is $50, the strike price is $45, and the 3-month risk-free interest rate is 8%? Explain briey. 14. Draw position (payo) diagrams for each of the following trades. Each put or call option is written on 100 shares of the same stock and has the same 6-month maturity. The current stock price is $50 per share. (a) Buy 100 shares, buy a put with an exercise price of $40, sell a call with an exercise price of $60. (b) Same as (a), except that you borrow $4902. The semi-annual interest rate is 2%, so you will have to repay $49021:02 = $5000 after six months. (c) Buy a put and a call with exercise price of $50, sell a put with exercise price of $40, sell a call with an exercise price of $60. 15. Explain how you could generate the same payos as in part a of last question without purchasing any shares. 16. Ineable Corporations stock price is currently $100. At the end of 3 months it will be either $110 or $90.91. The risk-free interest rate is 2% per annum. What is the value of a 3-month European call option with a strike price of $100? Calculate your answer to this problem using (a) replication. (b) the risk-neutral method. 17. State whether the following statements are true or false. In each case, provide a brief explanation. (a) In a risk averse world, the binomial model states that, other things being equal, the greater the probability of an up movement in the stock price, the lower the value of a European put option. 13 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS (b) By observing the prices of call and put options on a stock, one can recover an estimate of the expected stock return. (c) An investor would like to purchase a European call option on an underlying stock index with a strike price of 210 and a time to ma- turity of 3 months, but this

Draw a diagram showing an investor's prot and loss with the terminal stock price for a portfolio consisting of: 18. (a) One share of stock and a short position in one call option (b) Two shares of stock and a short position in one call option (c) One share of stock and a short position in two call options (d) One share of stock and a short position in four call options You should take into account the cost from purchasing the stock and revenue from selling the calls. For simplicity ignore discounting when combining these costs and revenues with the terminal payo of the portfolio. For simplicity also assume that the current stock price is equal to the strike price, K, of the call. Denote the current call price by c, and the terminal stock price by ST . 19. Stock XYZ is worth S = $80 today. Every 6 months the stock price goes either up by u = 1:3 or down by d = 0:8. The riskless rate is 6% APR with semiannual compounding. The stock pays no dividends. (a) Compute the price of a European call with a maturity of 1 year and a strike price of X = $95. (b) Compute the price of an American call with a maturity of 1 year and a strike price of X = $95. (c) Compute the price of a European put with a maturity of 1 year and a strike price of X = $95. 20. In August 1998 the Bank of Thailand was reported as oering to foreign investors in troubled banks the opportunity to resell their shares back to the central bank within a period of ve years for the original purchase 14 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS price. \This is to guarantee that at least they will not lose any of the money they plan to invest," said the Deputy Governor. (The Wall Street Journal Europe, August 6, 1998, p.20.) Suppose that(a) the standard deviation of Thai bank shares was about 50 percent a year, (b) the interest rate on teh Thai baht was 15%, and (c) the banks were not expected to pay a dividend in this ve-year period. How much was this option worth? Assume an investment of 100 million baht. 21. Shares of ePet.com are traded at $60. In six months, share price could either be $66 or $54 with probability 0.6 and 0.4, respectively. The current 6-month risk-free rate is 6%. What is the price of a European put on 100 ePet shares with a strike price of $64 per share? Would your answer be dierent if the option is American? 22. Consider again ePet. You want to use ePet shares and the risk-free bond to replicate a payo in six months that equals the square of ePet's share price. That is, when ePet price goes up to $66, you have a payo of 662 = $4; 356 and when the price goes down to $64, you have a payo of 542 = $2; 916. Describe the strategy that gives these payos. What is the present value of these payos? 23. The price of the stock of NewWorld Chemicals Company is $80. The standard deviation of NewWorld's stock returns is 50%. The 1-year interest rate is 6%. (a) What should be the price of a call on one share of NewWorld with a maturity of 1 year and strike price of $85? Use the Black-Scholes formula. (b) What should be the price of a put on one share of NewWorld with the same maturity and strike price? 24. You are asked to price some options on ABC stock. ABC's stock price can go up by 15 percent every year, or down by 5 percent. Both out- comes are equally likely. The risk free interest rate is 5 percent per year for the next two years, and the current stock price of ABC is $100. (a) Find the risk neutral probabilities (b) What is the price of a European Call option

25. You are asked to price some options on KYC stock. KYC's stock price can go up by 15 percent every year, or down by 10 percent. Both out- comes are equally likely. The risk free rate is 5 percent, and the current stock price of KYC is 100. (a) Price a European Put option on KYC with maturity of 2 years and a strike price of 100. (b) Price an American Put option on KYC with the same character- istics. Is the price dierent? Why or why not? 26. IBM is currently trading at $90.29 per share. You believe that IBM will have an expected return of 7% with volatility of 26.1% per year, while annual interest rates are at 0.95%. What is the price of an European put on IBM with a strike price of $90 and maturity of 1 year? 27. Shares of Ontel will sell for either $150 or $80 three months later, with probabilities 0.60 and 0.40, respectively. A European call with an exercise price of $100 sells for $25 today, and an identical put sells for $8. Both options mature in three months. What is a price of a three-month zero-coupon bond with a face of $100? 28. 401.com's stock is trading at $100 per share. The stock price will either go up or go down by 25% in each of the next two years. The annual interest rate is 5%. (a) Determine the price of a two-year European call option with the strike price X = $110. (b) Determine the price of a two-year European put option with the strike price X = $110. (c) Verify that the put-call parity holds. (d) Determine the price of a two-year American put option with the strike price X = $110. (e) What is the replicating portfolio (at every node of the tree) for the American put option with the strike price X = $110? 29. For this problem assume that the risk-free rate of interest for one year loans is 5%. Google stock is selling today for $500 a share. Assume that in one year Google will either be worth $600 a share or $475 a 16 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS share and that Google will pay no dividends