Solve the following attachment with calculations

50. Assume the Black-Scholes framework. You are given the following information for a stock that pays dividends continuously at a rate proportional to its price. (i) The current stock price is 0.25. (ii) The stock's volatility is 0.35. (iii) The continuously compounded expected rate of stock-price appreciation is 15%. Calculate the upper limit of the 90% lognormal confidence interval for the price of the stock in 6 months. (A) 0.393 (B) 0.425 (C) 0.451 (D) 0.486 (E) 0.529 51-53. DELETED 54. Assume the Black-Scholes framework. Consider two nondividend-paying stocks whose time-f prices are denoted by S,(t) and S2(t), respectively. You are given: (i) $1(0) = 10 and $2(0) = 20. (ii) Stock 1's volatility is 0.18. (iii) Stock 2's volatility is 0.25. (iv) The correlation between the continuously compounded returns of the two stocks is -0.40. (v) The continuously compounded risk-free interest rate is 5%. (vi) A one-year European option with payoff max { min[251(1), $2(1)] - 17, 0} has a current (time-0) price of 1.632. Consider a European option that gives its holder the right to sell either two shares of Stock 1 or one share of Stock 2 at a price of 17 one year from now. Calculate the current (time-0) price of this option. IFM-01-18 Page 58 of 105 (A) 0.67 (B) 1.12 (C) 1.49 (D) 5.18 (E) 7.86 55. Assume the Black-Scholes framework. Consider a 9-month at-the-money European put option on a futures contract. You are given: (i) The continuously compounded risk-free interest rate is 10%. (ii) The strike price of the option is 20. (iii) The price of the put option is 1.625. If three months later the futures price is 17.7, what is the price of the put option at that time?46. You are to price options on a futures contract. The movements of the futures price are modeled by a binomial tree. You are given: (i) Each period is 6 months. (ii) uld =4/3, where u is one plus the rate of gain on the futures price if it goes up, and d is one plus the rate of loss if it goes down. (iii) The risk-neutral probability of an up move is 1/3. (iv) The initial futures price is 80. (v) The continuously compounded risk-free interest rate is 5%. Let Cr be the price of a 1-year 85-strike European call option on the futures contract, and Cr be the price of an otherwise identical American call option. Determine Cu - C. (A) 0 (B) 0.022 (C) 0.044 (D) 0.066 (E) 0.088 47. Several months ago, an investor sold 100 units of a one-year European call option on a nondividend-paying stock. She immediately delta-hedged the commitment with shares of the stock, but has not ever re-balanced her portfolio. She now decides to close out all positions. You are given the following information: (i) The risk-free interest rate is constant. (ii) Several months ago Now Stock price $40.00 $50.00 Call option price $ 8.88 $14.42 Put option price $ 1.63 $ 0.26 Call option delta 0.794 The put option in the table above is a European option on the same stock and with the same strike price and expiration date as the call option. IFM-01-18 Page 56 of 105 Calculate her profit. (A) $11 (B) $24 (C) $126 (D) $217 (E) $240 48. DELETED 49. You use the usual method in Mcdonald and the following information to construct a one-period binomial tree for modeling the price movements of a nondividend- paying stock. (The tree is sometimes called a forward tree). (i) The period is 3 months. (ii) The initial stock price is $100. (iii) The stock's volatility is 30%.42. Prices for 6-month 60-strike European up-and-out call options on a stock S are available. Below is a table of option prices with respect to various H, the level of the barrier. Here, S(0) = 50. H Price of up-and-out call 60 0 70 0.1294 80 0.7583 90 1.6616 4.0861 Consider a special 6-month 60-strike European "knock-in, partial knock-out" call option that knocks in at H1 = 70, and "partially" knocks out at H2 = 80. The strike price of the option is 60. The following table summarizes the payoff at the exercise date: H1 Hit Hi Not Hit H2 Not Hit H2 Hit 0 2 x max [S(0.5) - 60, 0] max[S(0.5) - 60, 0] Calculate the price of the option. (A) 0.6289 (B) 1.3872 (C) 2.1455 (D) 4.5856 (E) It cannot be determined from the information given above. 43. DELETED IFM-01-18 Page 54 of 105 44. Consider the following three-period binomial tree model for a stock that pays dividends continuously at a rate proportional to its price. The length of each period is 1 year, the continuously compounded risk-free interest rate is 10%, and the continuous dividend yield on the stock is 6.5%. 585.9375 468.75 375 328.125 300 262.5 210 183.75 147 102.9 Calculate the price of a 3-year at-the-money American put option on the stock.40. The following four charts are profit diagrams for four option strategies: Bull Spread, Collar, Straddle, and Strangle. Each strategy is constructed with the purchase or sale of two l-year European options. Bauch Price Match the charts with the option Strategies. Bull Spread Straddle Strangle Collar (A) II III IV (B) IV IV I (D) IV III I (E) IV III I IFM-01-18 Page 52 of 105 41. Assume the Black-Scholes framework. Consider a I-year European contingent claim on a stock. You are given: (@) The time-0 stock price is 45. (i) The stock's volatility is 25. (ili) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 3%. (iv) The continuously compounded risk-free interest rate as 7%. (v) The time-1 payoff of the contingent claim is as follows:31. You compute the current delta for a 50-60 bull spread with the following information: (i) The continuously compounded risk-free rate is 5%. (ii) The underlying stock pays no dividends. (iii) The current stock price is $50 per share. (iv) The stock's volatility is 20%. (iv) The time to expiration is 3 months. How much does delta change after 1 month, if the stock price does not change? (A) increases by 0.04 (B) increases by 0.02 (C) does not change, within rounding to 0.01 (D) decreases by 0.02 (E) decreases by 0.04 32. DELETED IFM-01-18 Page 50 of 105 33. You own one share of a nondividend-paying stock. Because you worry that its price may drop over the next year, you decide to employ a rolling insurance strategy, which entails obtaining one 3-month European put option on the stock every three months, with the first one being bought immediately. You are given: (i) The continuously compounded risk-free interest rate is 8%. (ii) The stock's volatility is 30%. (iii) The current stock price is 45. (iv) The strike price for each option is 90% of the then-current stock price. Your broker will sell you the four options but will charge you for their total cost now. Under the Black-Scholes framework, how much do you now pay your broker? (A) 1.59 (B) 2.24 (C) 2.86 (D) .48 (E) 3.61