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20. In August 1993 the Bank of Thailand was reported as offering to foreign investors in troubled banks the opportunity to resell their share; back to the central bank within a period of ve years for the original purchase 14 @ m.hdmw.lalndJi-n.\"'hq 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 We\" Street Journal Europe. August IS, 1.993. p.20.) Suppose thatia} the stande 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 ofePetcom are traded at $60. In sir: months, share price could either be $615 or $54 with probability 0.5 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 $34 per share? Would your answer be different if the option is American? 22. Consider again ePet. You want to use ePet shares and the risk-free bond to replicate a payoff in six months that equals the square ofePet's share price. That is, when ePet price goes up to $66. you have a payoff of 66* = $4,356 and when the price goes down to $64, you have a payoff of 543 = $2. 916. Describe the strategy that gives these payoffs. What is the present value of these payoffs? 23. The price of the stock out New-World Chemicals Company is $30. The stande deviation of NewWorld's stock returns is 50%. The 1-year interest rate is 6%. (a) What should he the price of a call on one share of New-World with a maturity of 1 year and strike price of $85? Use the Black-Scholes formula. {b} What should he the price of a put on one share of New-World with the same maturity and strike price? 24. You are asked to price some options on ABC stock. ABC's stock price (\"311 go up by 15 peroent every year, or down by 5 peroerrt. 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 1m and maturity 2 years? {c} Describe the strategy to replicate the payoff of the call using the stock and the risk-free bond. 15 5| m.a.smw.r.uaii.w.q 1.5 Options 1 QUESTIONS {d} What is the price of an American option with the same charac- teristits'? 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 percent1 and the current stock price of KYO is 100. a use a LII an t 10]] on w1t. matunl: ears ll: ] P ' E ope Pu opt KYC ' l1 ' y of 2 y and a strike price of 100. [h] Price an American Put option on KYC with the same character- istics. Is the price different? Why or why not? 25. IBM is currently trading at $90.29 per share. 1r'ilirru believe that IBM 1will have an expected return of 7% with volatility of 20.1% per year1 while annual interest rates are at 0.95%. What is the price of an European put. an IBM with a strike price of $90 and maturity of 1 year? 2?. Shares of Date] will sell for either $150 DI $30 three months latter1 with probabilities can and use respectively. a EuIp-ean call with an Exercise price of $100 sells for $25 today1 and an identical put sells for $3. Both options mature in three months. What is a price of a three-month zero-coupon bond with a face of $100? 23. 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. [h] 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 Amecan 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 $000 a share or $4T5 a 16 a m.a=amw.r.mcnn.sisq 1.5 Options 1 QUESTIONS share and that Google will pay no dividends for at least tvm 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. $31 $29 $24 $25 $26 $23 $21 today 1-month 2-months Figure 2: Stock Price Movement (a) For this part, suppose the interest rate is fixed 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 fixed at 1% per month. What is the price of an exotic derivative that in 2-months has a pay off 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 St. Ost52 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 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 difference 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?1. True or false or " it depends"? (a) Briefly explain or qualify your answer: diversification 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 "risk budget"). What is the maximum expected return you can achieve on your portfolio? 5. Are the following statements true, false or uncertain? Briefly explain your answer. (a) Diversification over a large number of assets completely eliminates risk. (b) Diversification 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) Diversification 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) Diversification means that the equally weighted portfolio is opti- mal. 7. Which statement about portfolio diversification is correct? 19 2005, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS (a) Proper diversification can reduce or eliminate systematic risk. (b) Diversification 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 rate. (d) The risk-reducing benefits of diversification do not occur mean- ingfully until at least 30 individual securities are included in the portfolio. 8. Which of the following portfolios can not be on the Markowitz efficient frontier? Explain briefly. Portfolio|Expected Return|Standard Deviation 10% 15% 10.5% 16.5% 11.5% 18.5% 12.5% 20%9. You have decided to invest all your wealth in two mutual funds: A and 13. Their returns and risks are as follows: a the mean returns are 1'34 = 15% and F3 = 11% o the covariance matrix is You want your total portfolio to yield a return of 12%. What pro- portions of your wealth should you invest in A and B? What is the stande deviation of the return on your portfolio? 10. There are only two securities (A and B, no risk free asset] in the market. Expected returns and standard deviations are as follows: Security Expected return standard Deviation StockA 25% 20% (a) The correlation between stoclrs A and B is 0.3. Compute the expected return and standard deviation of a portfolio that has 0% of A, 10% of A, 20% of A, etc, until 100% of A. Plot the portfolio frontier formed by these portfolios. 20 a \".mmw.r.mdrin.m 1.6 Risk dc: Portfolio Choice i QUESTIONS [b] Repeat the previous question, assuming that the correlation is 0.8. (cl Explain intuitively why the portfolio frontier is different in the two cases. 11. Stock A and B have the following characteristics: II." A 3% 20% B 3% 40% free Their correlation is 0. The risk- interest rate is . (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 30% of stoclr 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 stool: 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 131 whose returns have the following characteristics: 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 FA = 20% and Fs = 15% . the covariance matrix is TA TA 0.3600 0.0840 0.0840 0.1225 21 9 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 fo = 10%. The covariance matrix of the three funds is TA TB rc TA 0.3600 0.0840 0.1050 TB 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