# 13. What is a lower bound for the price of 3-month call option on a non- dividend-paying...

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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 option is not actively traded. However, two otherwise identical call options are traded with strike prices of 200 and 220 respectively, hence the investor can replicate a call with a strike price of 210 by holding a static position in the two traded calls. (d) In a binomial world, if a stock is more likely to go up in price than to go down, an increase in volatility would increase the price of a call option and reduce the price of a put option. Note that a static position is a position that is chosen initially and not rebalanced through time. 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 on ABC, with strike 100 and maturity 2 years? (c) Describe the strategy to replicate the payo of the call using the stock and the risk-free bond. 15 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS (d) What is the price of an American option with the same charac- teristics? 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 for at least two years. A call option with an exercise price of $550 and one year to go until expiration is available for Google stock. What is the value of this call option? 30. A particular stock follows the price movement below. $25 $29 $23 $21 $26 $24 $31 today 1-month 2-months Figure 2: Stock Price Movement (a) For this part, suppose the interest rate is xed at 1% per month. What is the price of a put option with maturity two months, and strike of $26 ? (b) Again, suppose the interest rate is xed at 1% per month. What is the price of an exotic derivative that in 2-months has a pay o that is a function of the maximum price of the stock during the two month period given by max( ^ S $25; 0); where ^ S = max 0t2 St: and t is measured in months. 31. Intel stock is trading at $120 per share, and the company will not pay any dividends over the next year. Consider an Intel European call option and a European put option, both having an exercise price of 17 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS $124 and both maturing in exactly one year. The simple (annualized) interest rate for borrowing and lending between now and one year from now is 3% for each 6 month period (6.09% per year). Assume that there are no arbitrage opportunities. Is there enough information to determine which option has the higher market value? If so, which option, the call or the put, has higher market value? 32. Calculate the price of a three-month European put option a non-dividend paying stock with a strike price of $50 when the current stock price is $50, the risk free rate is 10% per annum, and the volatility is 30% per annum. What dierence does it make to your calculations if a dividend of $1.50 is expected in two months? Assume that the assumptions made to derive the Black-Scholes formula are valid. 33. It is possible to buy three-month call options and three-month put options on stock X. Both have an exercise price of $60 and both are worth $10. Is a six-month call with an exercise price of $60 more or less valuable than a similar six-month put? 18 c 2008, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS 1.6 Risk & Portfolio Choice 1. True or false or "it depends"? (a) Briey explain or qualify your answer: diversication can reduce risk only when asset returns are negatively correlated. (b) If the returns on all risky assets in the world were uncorrelated with each other, the expected return of each risky asset should be the same. 2. True or false or "it depends"? Optimal portfolios should exclude indi- vidual assets whose expected return and risk (measured by its standard deviation) are dominated by other available assets. 3. Is the following statement true or false? Explain. As more securities are added to a portfolio, total risk would typically be expected to fall at a decreasing rate. 4. You need to invest $10M in two assets: a risk-free asset with an ex- pected return of 5% and a risky asset with an expected return of 12% and a standard deviation of 40%. You face a cap of 30% on the port- folio's standard deviation (the isk budget"). What is the maximum expected return you can achieve on your portfolio? 5. Are the following statements true, false or uncertain? Briey explain your answer. (a) Diversication over a large number of assets completely eliminates risk. (b) Diversication works only when asset returns are uncorrelated. (c) A stock with high standard deviation may contribute less to port- folio risk than a stock with lower standard deviation. (d) Diversication reduces the expected return on the portfolio as its risk decreases. 6. Are the following statements true or false? Give brief but precise ex- planations for your answers. (a) Stock A has expected return 10% and standard deviation 15%, and stock B has expected return 12% and standard deviation 13%. Then, no investor will buy stock A. (b) Diversication means that the equally weighted portfolio is opti- mal. 7. Which statement about portfolio diversication is correct? 19 c 2008, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS (a) Proper diversication can reduce or eliminate systematic risk. (b) Diversication reduces the portfolio's expected return because it reduces the portfolio's total risk. (c) As more securities are added to a portfolio, total risk would typi- cally be expected to fall at a decreasing

11. Stock A and B have the following characteristics: E(r) A 8% 20% B 8% 40% Their correlation is 0. The risk-free interest rate is 2%. (a) Consider a portfolio, P, with 90% in stock A and 10% in the risk- free asset. What is the mean and standard deviation of portfolio P's return? (b) Consider another portfolio, Q, which consists of 80% of stock A and 20% of stock B. What is the mean and standard deviation of portfolio Q's return? (c) You need to choose a portfolio to invest all your wealth in. Be- tween portfolio P and Q, which one is better? Explain why. (d) Given that stock A dominates stock B (A has the same mean but lower risk), explain why you ever include stock B in your portfolio. 12. You can form a portfolio of two assets, A and B, whose returns have the following characteristics: Stock E[R] Standard Deviation Correlation A 0.10 0.20 0.5 B 0.15 0.40 If you demand an expected return of 12%, what are the portfolio weights? What is the portfolio's standard deviation? 13. Your have decided to invest all your wealth in two mutual funds: A and B. Their returns are characterized as follows: the mean returns are rA = 20% and rB = 15% the covariance matrix is rA rB rA 0.3600 0.0840 rB 0.0840 0.1225 21 c 2008, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS You want your total portfolio to yield a return of 18%. What proportion of your wealth should you invest in fund A and B? What is the standard deviation of the return on your portfolio? 14. In addition to the fund A and B in the previous question, now you decide to include fund C to your portfolio. Its expected return is rC = 10%. The covariance matrix of the three funds is rA rB rC rA 0.3600 0.0840 0.1050 rB 0.0840 0.1225 0.0700 rC 0.1050 0.0700 0.0625 Your portfolio now consists of fund A, B and C. You would like to have an expected return of 16% on your portfolio and a minimum risk (measured by standard deviation of the return). What portfolio should you hold? What is the return standard deviation of your portfolio? (Hint: You would need to use Excel Solver or some other optimization software to solve the optimal portfolio.) 15. You can only invest in two securities: ABC and XYZ. The correlation between the returns of ABC and XYZ is 0.2. Expected returns and standard deviations are as follows: Security E[R] (R) ABC 20% 20% XYZ 15% 25% a) It seems that ABC dominates XYZ in that it has a higher expected return and lower standard deviation. Would anyone ever invest in XYZ? Why? b) What is the expected return and standard deviation of a portfo- lio that invests 60% in ABC and 40% in XYZ? c) Suppose instead that you want your portfolio to have an expected return of 19.5%. What portfolio weights do you select now? What is the standard deviation of this portfolio? 16. You have the same data as the previous question. In addition, you have a risk-free security with a guaranteed return of 5%. The tangency portfolio has an expected return of ?? and standard deviation of ??. (a) What weights are placed on ABC and XYZ in the tangency port- folio?

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