For Use as a Study Aid for Math 476/567 Exam # 1, Fall 2015

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1 For Use as a Study Aid for Math 476/567 Exam # 1, Fall 2015 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Actuarial Science Program DEPARTMENT OF MATHEMATICS Math 476 / 567 Prof. Rick Gorvett Actuarial Risk Theory Fall, 2009 Exam # 1 (17 Problems Max possible points = 40) Thursday, September 24, 2009 You have 75 minutes (from 8:00 to 9:15 am) to complete this exam. The exam is closed-book, closed-note, except that you may refer to your two-sided 3x5-inch notecard. No clarification questions may be asked during the exam. Problems (1) through (14) are multiple choice, each worth two points. Circle the letter associated with the best answer to each question. Problems (15) through (17) are each worth four points. Please provide dollar answers to the nearest cent ($x,xxx.xx), and proportion and probability answers either as percentages to two decimal places (xx.xx%) or as numbers to four decimal places (0.xxxx). A standard normal distribution table is provided at the end of this exam. When using it, please use the value in the table closest to the one you need do not interpolate in the table. Please assume that holders of options always act rationally. Good luck! (1) You buy one 40-strike put option, you buy one 50-strike call option, and you sell two 60- strike call options. All these options are European, are on the same underlying asset, and have a common expiration date. Find the payoff from your position if the price of the underlying asset on the option expiration date is 68. (a) 0 (b) 1 (c) 2 (d) 3 None of (a) through (d) is correct. 1

2 (2) In which of the following option positions would you be exposed to a potentially unlimited loss? (a) (b) (c) (d) Long a put option Short a put option Long a call option Short a call option None of (a) through (d) involves a potentially unlimited loss (3) A three-month European put option with a strike price of 80 on ABC stock costs The price of ABC stock is currently 80, and ABC is a non-dividend-paying stock. The continuously-compounded risk-free rate of interest is 8%. Find the price, C, of a threemonth European call option with a strike price of 80 on ABC stock. (a) C 3.50 (b) 3.50 < C 3.75 (c) 3.75 < C 4.00 (d) 4.00 < C < C (4) XYZ stock is currently priced at 118, and pays a continuous annual dividend yield of 2.0%. A nine-month 120-strike European call option on XYZ stock costs the same amount as a nine-month 120-strike European put option on XYZ stock. Find the continuouslycompounded annual risk-free rate of interest, r. (a) r (b) < r (c) < r (d) < r < r 2

3 (5) You sell a one-year call option with a strike price of 50 for $8.75, and you buy a one-year put option with a strike price of 60 for $6.25. Both options are European, and are on the same underlying asset. On the option expiration date, the price of the underlying asset is 54. The continuously-compounded annual interest rate is 10%. Find X, your profit / (loss) on the option expiration date. (Include any investment income related to the option premiums.) (a) X (b) < X 0.00 (c) 0.00 < X 2.00 (d) 2.00 < X < X (6) Consider a one-year European 110-strike call option on 100 shares of ABC stock. You believe that, on the expiration date of the option one year from now, the stock price is equally likely to be anywhere between 80 and 120 per share. The continuously compounded annual risk-free rate of interest is 6%. Find the current price, C, of this call option. (a) C 117 (b) 117 < C 118 (c) 118 < C 119 (d) 119 < C < C (7) Consider a two-year, 90-strike American call on a non-dividend-paying stock. The current price of the stock is 100, and each year the price can increase or decrease by 20%. The continuously compounded risk-free interest rate is 8%. Find the current price, C, of the call. (a) C 26 (b) 26 < C 27 (c) 27 < C 28 (d) 28 < C < C 3

4 (8) Consider a one-year, 150-strike European call on a non-dividend-paying stock. The current price of the stock is 150, and the price one year from now will be either 130 or 175. The continuously compounded risk-free interest rate is 4%. The expected rate of return on the stock is 9%. Find the real-world probability of an increase in the price of the stock during a given year. (a) p 0.73 (b) 0.73 < p 0.75 (c) 0.75 < p 0.77 (d) 0.77 < p < p (9) A non-dividend-paying stock, currently priced at 90 per share, can either go up 20 or down 10 in any year. Consider a one-year American put option with an exercise price of 100. The continuously-compounded risk-free interest rate is 10%. Use a one-period binomial model approach to determine the current price of the put option. (a) 6.35 (b) 7.02 (c) 8.47 (d)

5 (10) Suppose a one-year 50-strike European call option on a share of stock has a premium of 6, and a one-year 60-strike European call option on the same share of stock has a price of 2. The current price of ABC stock is 54, and the continuously-compounded risk-free interest rate is 12.26%. An arbitrage opportunity is available with the following portfolio: short-sell one share of stock, buy one 50-strike call, buy one 60-strike call, and borrowing or lending at the risk-free rate. What is the minimum payoff, at the expiration of the call option, from this arbitrage portfolio? (a) 0 (b) 1 (c) 2 (d) 3 4 (11) You want to estimate a non-dividend-paying stock s annualized volatility,, using its prices over the last five months: Month Stock Price ($ / share) Calculate the annual historical volatility for this stock over this period. (a) 10% (b) 17% (c) 25% (d) 33% 41% 5

6 (12) Consider a non-dividend-paying stock with a three-month binomial period, a current price of S, and which can go up to 55, or down to 40, three months from now. Using the binomial model approach to constructing a binomial tree, find. (a) 0.33 (b) 0.33 < 0.35 (c) 0.35 < 0.37 (d) 0.37 < < (13) Use the Black-Scholes model to determine the price, C, of a one-year 40-strike European call option on a stock which pays a continuous annual dividend yield of 3%. The current price of the stock is 40, the continuously-compounded annual risk-free interest rate is 7%, and the annualized standard deviation of continuously-compounded stock returns is (a) C 3.00 (b) 3.00 < C 3.50 (c) 3.50 < C 4.00 (d) 4.00 < C < C (14) Suppose you purchase a one-year 100-strike European call, and a one-year 100-strike European put, both on a share of non-dividend-paying stock. The stock is currently priced at 95, and will be priced at either 120 or 75 one year from now. The continuously compounded annual rate of interest is 6%. Calculate the current combined cost of the call and put (C+P). (a) C+P 16 (b) 16 < C+P 18 (c) 18 < C+P 20 (d) 20 < C+P < C+P 6

7 (15) A non-dividend-paying stock currently has a price of 100. The continuously-compounded rate-free rate is 10%. Let u = 1.30 and d = Consider a one-year European call option with an exercise price of 105. Assuming a one-period binomial pricing model, find the values for and B, and find the premium for this call option. (16) Consider a one-year 80-strike European call option on a non-dividend-paying stock. The current stock price is 80, and the stock price one year from now will be either 60 or 100. The continuously compounded annual risk-free interest rate is 8%, and the continuously compounded annual rate of return on the stock is 14%. Find the expected (real-world) return on the option. (17) Find the current price of a one-year American put option on an underlying non-dividendpaying stock. Let S = 100, r = 0.06 annually, = 0.25 annually, K = 110, and h = 0.50 (the binomial interval is 6 months thus, you need a two-period, or two-step, binomial tree to price this option). Let u = , and d =

8 Standard Normal Distribution Values x