I need help answering question 13 and 17 in this homework set. Thank you that is all I need currently.

HOMEWORK SET 4 Derivatives 1. What is a lower bound for the price of a 4 months call option on a non-dividend paying stock when the stock price is $28, the exercise price is $25 and the risk free interest rate is 8% per annum. 2. What is a lower bound for the price of a 1-month European put option on a non-dividend paying stock when the stock price is $12, the exercise price is $15 and the risk free interest rate is 6% per annum. 3. A European call and put option on a stock both have a strike price of $20 and an expiration date in 3 months. Both sell for $3. The risk free interest rate is 10% per annum, the current stock price is $19, and a $1.00 dividend is expected in 1 month. Identify the arbitrage opportunity open to the trader. 4. The price of a European call which expires in 6 months and has a strike price of $30 is $2.00. The underlying stock's price is $29 and a dividend of $.50 is expected in 2 months and in 5 months. The term structure is flat with all risk free interest rates being 10%. What is the price of a European put option that expires in 6 months and has a strike price of $30? 5. Three put options on a stock have the same expiration date and an exercise price of $55, $60, and $65. The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread could be created. Construct a table showing the profit from the strategy. For what range of stock prices would the butterfly spread lead to a loss? 6. Suppose calls were traded with the indicated exercise prices where E1 E 2 E 3 E 4 E 5 E 6 and E i 1 E i E j 1 E j for all i and j (equally spaced). Draw the payoff diagram of the following portfolio: {write 1 call, buy 1 call, buy 1 call, write 1 call, write 1 call, buy 1 call} E=E1, E=E2 E=E3 E=E4 E=E5 E=E6 7. The current price of the common stock of Internet Enterprises is $100. Over the course of a year, the stock's price will either increase by 100% or decrease by 50%. The stock pays no dividends. The current prices of one-period and two period zero coupon risk free bonds are $909.09 and $826.45 respectively ($1000 face value and the period here is 6 months). 1 A special European option has recently been created on the common stock of Internet with the following terms: On the expiration date the holder of the option has the right to sell the underlying asset for the highest stock price that occurred during the life of the option inclusive of the date on which it was issued. What is the current value of this newly issued option on Internet Enterprises.? Consider a two-period binomial tree. 8. A stock is presently selling for $100. over each of the next two months, the stock will either increase of decrease in value by 9%, and will not pay any dividends. The riskfree rate is 2% per month. Consider a call option on the stock with an exercise price of $90 and a maturity date two months hence: u 1.09; d 0.91; r f 0.02; E $90 (a) What is the value of the option if it is an American option? What is the value if it is a European option? (b) What position in stocks and bonds at time zero will have the same value as the call at t = 1. (c) If the expected return on the stock is 4% per month, what must the expected return on the call be over the first month? What will the expected return on the call be in the second month given that the stock has fallen to $91? 9. Consider a stock that provides an expected return of 15% per annum and has a volatility of 40% per annum. Suppose that a time interval of .01 year is used for the binomial model, Calculate u, d, and p. Show that they give correct values for the expected return and the variance of return during the time interval. Suppose the stock price starts at $50. What are the possible stock prices at the end of .03 year? What is the probability of each one occurring? 10. Assume a stock pays no dividends and is presently selling for $80. The continuously compounded return on the stock ln(St/So), is distributed , with = 25% per annum and 2 = 16% per annum. The risk free rate is 14% N t , t per annum. (a) Use the Black Scholes model to value a call option on the stock with an exercise price of $75 and a maturity date 3 months hence. S = $80, E = $75, T =1/4 yr, 2 = .16 (b) If you wished to replicate the payoff from the call by continually adjusting a position in the stock and a position in bonds, what position should you take today? (c) If the stock were to decrease in value overnight by $1.00, by how much would the call change in value? 2 11. The riskless rate of interest is five percent. What is the maximum price you would pay for a bond with one year to maturity, a maturity (par) value of $1,000, convertible into 50 shares of Common stock? The stock has a current value of $20 per share and the current price of a one-year call option on 100 shares of the stock is $200. Assume that the exercise price of the option is equal to the current stock price. 12. Your client, the Notsoblue Chip Corp., wants to issue $50 million of 10 year debt at par with an 8% annual coupon (50,000 bonds of $1,000 par value each, with principal to be repaid in year 10). Your client is willing to attach warrants to the bonds in order to raise the effective yield to 14%, the yield required by the market for this type of issue. Your problem is designing the bond-warrant package that will permit your client to implement its financing plan. For this calculation ignore underwriting fees and other expenses. So far you have determined that: --The warrants will expire in 7 years. --Yield on a seven year government note is 9.0% per annum. --The current stock price of Notsoblue Chip Corp. is $14 a share. --The volatility (standard deviation) of the rate of return of Notsoblue is 65% per annum. --Common shares outstanding: 5 million. --Each warrant will entitle the holder to buy one share of Notsoblue stock for an exercise price of $20. --The Company has no other warrants or convertibles outstanding. How many warrants do you need to attach to each bond? What is the value of each warrant? 13. You are the senior assistant to the Chairman of the Board of Hi-Teck Inc. In order to attract the chief operating officer of his choice from another firm, the Chairman offers the following incentive supplement to her base salary package: (i) 2% of any price appreciation in the price of the stock up to $60 per share; (ii) 3% of any price appreciation above $60 but less than $75 per share; (iii) 4% of any price appreciation above $75 per share; The price appreciation is to be computed with respect to the share price exactly 3 years from now, and there is no early payment of the supplement. The aggregate value of the supplement is based on the price appreciation of 10 million shares. The annual (not 3 continuously compounded) risk free rate is 10% and the stock volatility is 50% annualized. The current price of the stock is $55. Use the Black Scholes formula to value the supplement. Should you be concerned in your calculations that the new chief operating officer might increase the expected rate of price appreciation by wise actions? 14. Consider the following type of equity related contract. In exactly one year's time if the price of BBN Inc. Stock is between $30 and $60, you must pay the then going spot price to buy the stock. If the stock's price is above $60, you must pay an amount given by the formula: 60+.1(S-60) where S is the stock's price (S $60). If the stock is below $30, then you must pay $30. a. b. c. Draw a diagram showing the amount you must pay for the stock when the contract matures. Ignore the initial cost of the contract. You can construct this payoff by buying stock plus different options. Identify the options. Divide the one year time to expiration into two sub-periods of 6 months each. Assume a binomial process for the underlying stock. You have the following information: (i) (ii) (iii) d. 15. the annualized (not continuously compounded)risk free rate of interest is 5.85%. the volatility of the underlying asset (S.D. of its continuously compounded ROR) is 35% per year. The current price of the stock is $45. What is the value of a contract? How would you redesign this contract so that the net value of the options is zero? (Give a qualitative answer). In October 1990 Lac Minerals, a gold mining company, attached gold purchase warrants to a 20 year debt issue. Investment dealers have estimated that a 20 year straight debt issue would require a 12% annual coupon. The warrant provides the right to the holder to buy 0.5 ounces of gold at an exercise price of $230 at any time up to and include October 1995. Assume that at that time the riskless rate of interest was 10%, the standard deviation of the rate of return on gold was 30%, and the market price of gold was $300 per ounce (that is, $150 per 0.5 ounces). One package consists of 1 bond and 2 warrants. a. What was the Black-Scholes value of the gold purchase warrant? What coupon rate should be set on the bond in order for the total package to sell for $1,000? 4 16. (i) In class, we derived a relationship between European option prices, PUT-CALL PARITY, which states C=P+S-PV(E). Consider a security that pays off in terms of deviations from put-call parity. That is, the holder gets a cash payment equivalent to I C-P- S +PV(E)| (where || represents the absolute value). Assuming our no arbitrage and risk neutral pricing arguments hold true, what is the value of this security? (i) What is the value of an option written on a stock with infinite volatility? Hint: consider the Black-Scholes Formula. (ii) Suppose there exist two stocks with the same price and volatility. The Black-Scholes option pricing model will give options with the same strike price written on these two stocks the same price. Can the expected returns of the two stocks, as given by the CAPM, be different? 17. Consider a stock, XYZ, whose current price is $100 with volatility equal to 60% per annum. XYZ does not pay dividends. The annual (not continuously compounded) risk free interest rate is 12%. (i) (ii) (iii) Consider a standard European option on XYZ with 3 months to maturity and exercise price E=100. What is the Black-Scholes value of the call option? What is the value of a put on XYZ with the same E and same time to maturity as the call? The CBOE has recently approved the creation of a special European option on XYZ, which they are calling a "High-Five". This high-five, on expiration, gives the holder the right to sell the underlying stock (XYZ) for the highest stock price that occurred during the life of the option (inclusive of the date on which it was issued) minus $5.00. Assuming this option has a maturity of 3 months and there are four `trading periods' in this time period. What is the price of this High-Five? (Hint use the Binomial model and remember that the Binomial model uses the per period interest rate calculated on a simple compounding basis). (iv) If you were to attempt to replicate this option with a position in the XYZ stock and risk free debt. what position would you initially construct? How would it change if the up state were first observed (context of this question is still the four period binomial model. 5 18. The following are quotes from proxy statements regarding options included in managerial compensation packages (courtesy of David Yermack, NYU). These options are "issued" on a grant date and can be exercised early. Their grant-date (time-zero) values are routinely evaluated as Black-Scholes calls of assumed maturity. "The values shown are based on the Black-Scholes option pricing model. This is a complex mathematical formula used to value publicly traded options... . It assumes that options may be exercised immediately. REYNOLDS METALS CO. (1993) "Depending upon fluctuations in the market price of the common stock, optionees may decide to exercise earlier or later than these assumed periods resulting in BlackScholes values which would be lower or higher than those shown." KELLOGG CO. (1994) "Option grant date values were developed by using the Black-Scholes option pricing model and assuming that exercisability of the options (occurs) on the fourth anniversary of the grant date as a result of meeting performance conditions... . In the event that such performance conditions are met (and the options become exercisable) at a later date, the grant date value would be lower." AVERY DENNISON CORP. (1993) 1. 2. 3. Give a one-sentence comment on the validity of the statement in the first quote. comment briefly on the second and third quotes. For full credit, you should provide a formal (i.e., algebraic) statement of violation of arbitrage bounds, if any. Assume that the conditions for the Black-Scholes formula are valid. Are the options granted to managers at Reynolds Metals Co., and Avery Dennison Corporation worth more or less that the shareholders are told? 19 Consider the binomial approximation to a geometric Brownian motion process on a stock's return. The stock has an annualized expected rate of return of 12%, and an annualized volatility of 30%. Suppose our approximation assumes a period of t=.04 year (=2 weeks); what is the distribution of prices in one period under the binomial model? After 2 periods; after 3 periods? The current price of the stock is $50. 20 Consider the binomial approximation to a geometric Brownian motion process on a stock's rate of return. You have estimated g = 15%, and a = 25% (both annualized).Compare the distribution of prices in one year where At =1 year, A=6 months, and At=3 months. The current price of the stock is $100. 6