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act370h-m08final exam

# act370h-m08final exam - UNIVERSITY OF TORONTO FACULTY OF...

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Unformatted text preview: UNIVERSITY OF TORONTO FACULTY OF ARTS & SCIENCE APRIL/MAY EXAMINATIONS, 2008 ACT370TH1\$ DURATION - 2 HOURS AID ALLOWED: Calculator. NAME STUDENT NUMBER This exam hale multiple choice questions. Multiple choice questions are 5 points each. Full marks will be given for a correct answer, and part marks may be given if the answer is incorrect but some of the written work is correct. Mark in the answers to the multiple choice questions in the appropriate spaces below. 1. 2. 3. ' 4. 5. 6. 7. 8. 9. 10. ll. 12. Total Marks: ACT370 - Final Exam, 2008 Questions 1 and 2 relate to the following information. An "exchange call option" gives the owner of the option the right to give up one share of Stock A in exchange for receiving one share of Stock B. Stock A currently has a price of \$56 and Stock B has a current price of \$52. The continuously compounded risk-free rate of interest is 5% and the price of a one-year European exchange call option is \$7. 1. Suppose that neither Stock A nor Stock B pays any dividends. Find the price of a European exchange put option expiring in one year which gives the owner the right to give up one share of Stock B in exchange for receiving one share of Stock A. A)\$3 B)\$5 C)\$7 D)\$9 E)\$11 2. Suppose that Stock A pays continuous dividends at a rate of 3% and Stock B will pay a single dividend in 6 months of \$D. Find the value of D if the one year European exchange put option has a current price of \$10. A) Less than .5 B) At least .5 but less than .75 C) At least .7 5 but less than 1 D) At least 1 but less than 1.25 E) At least 1.25 CONT'D. 3. You are given call option prices of \$12 and \$8 for European options with strike prices of \$50 and \$60, respectively. The options expire at the same time and are on a non-dividend paying stock. According to convexity of option prices, how many of the following prices are feasible prices for a call option on the same stock and expiring at the same time with a strike price of \$53? Feasible means that there are no arbitrage opportunities. 1. \$10 11. \$10.50 111. \$11 IV. \$11.50 A) None B) One C) Two D) Three B) All mn‘m‘r'ﬁ Questions 4 and 5 relate to the following information. In a binomial branching model, the stock price at time 0 is 120. At time 1, the stock price will be either 144 or 100. The annual effective risk free rate of interest from time 0 to time 1 is z' 2 .10 . 4. A call option on the stock with exercise price 110 expires at time 1. Find the number of shares of stock in the replicating portfolio at time 0. A) Less than .2 B) At least .2 but less than .4 C) At least .4 but less than .6 D) At least .6 but less than.8 E) At least .8 5. Another stock is available that has a value of 100 at time 0. The stock price at time 1 is either B (when the ﬁrst stock is 100) or 1.23 (when the ﬁrst stock is 144). Find the value of B that results in no arbitrage opportunities. A) Less than 80 B) At least 80 but less than 85 C) At least 85 but less than 90 D) At least 90 but less than 95 E) At least 95 'CON'm‘ ' Questions 6 to 8 relate to the following information. A 3-period binomial tree is constructed. The continuously compounded risk—free rate of interest per period is r, and you are given that 8T 2 1.05. You are also given that u = 1.1 and d = .9 for each period. The current stock price is 100. / @ <: >@ ®\@/ <6) \‘0/ 6. A European derivative that expires at the end of 3 periods pays 1 if the stock is at either node 7 or node 8 and pays 0 otherwise. Find the price of the derivative at time 0. A) Less than .2 B) At least .2 but less than .4 C) At least .4 but less than .6 D) At least .6 but less than .8 B) At least .8 7. It is possible to replicate the payoff at time 3 in Question 6 using a combination (some short some long) of European call options expiring at time 3 with strike prices of 100 and 120. Find the number of units of the call with strike 120 needed in this combination (round to the nearest .1 in units). A) .2 units, short B) .2 units, long C) .4 units short D) .4 units, long E) .6 units short 8. We consider a 3-period American call option with a strike price of 110. Suppose that the stock pays a dividend of amount D at time 2 (nodes 4, 5 and 6), and if the option is exercised at time 2, the dividend is included with the stock. The 3-period binomial tree is constructed using u = 1.1 and d = .9 applied to the prepaid forward stock price starting at time 0. Find the minimum value of D so that it is optimal to exercise the option at node 4. It will be assumed that D < 110.25. A) Less than 5 B) At least 5 but less than 6 C) At least 6 but less than 7 D) At least 7 but less than 8 B) At least 8 but less than 9 CONT’D. CONT’Dw CONT’I') 9. A call option on a non-dividend paying stock has a strike price of 100 and expires in one year. You are given the following: — current stock price is 100 - current option price is 19.384 based on the Black-Scholes option pricing model - continuously compounded risk free interest rate is 8% — the call delta is .6554 For the same stock, ﬁnd the price of a similar call option with a strike price of 110 using the Black-Scholes option pricing model. A) Less than 12 B) At least 12, but less than 13 C) At least 13, but less than 14 D) At least 14, but less than 15 E) At least 15 CONT’D. 10. A call option on a non-dividend paying stock has a current price of 60 and expires in one year. You are given the following, based on Black-Scholes option pricing: - the option I‘ is .018817 — the volatility of the underlying stock is a = .45 - the option Vega for a .01 change in a is .182313 - d1 > 0 Find the value of the call option A using Black-Scholes option pricing. A) Less than .50 B) At least .50, but less than .55 C) At least .55, but less than .60 D) At least .60, but less than .65 E) At least .65 comp. 11. 8'1, the price of a stock at time 1, has a lognormal distribution under the risk neutral measure and under the physical measure. You are given the following: — under the physical measure E[ln 5’1] 2 4.396227 and E[S’1] = 85.9724 — under the risk neutral measure E [ln 5'1] = 4.369227. Find E [31] under the risk neutral measure. A) Less than 80 B) At least 80, but less than 81 C) At least 81, but less than 82 D) At least 82, but less than 83 B) At least 83 CUNT’B. 12. You are given the following information regarding European call options on a non-dividend paying stock with a current price of 60 and a volatility of a = .45 . The continuously compounded risk free rate is 6%. Call Option 1 Call Option 2 Strike Price 62 65 Time to Expiry (years) .25 .5 Option Price 4.9101 6.3348 Delta .5133 .5007 Gamma .0295 .0209 A market maker has a long position in one European put option with strike price 62 expiring in .25 years. The market maker wishes to construct a Delta-Gamma hedge using shares of stock and put options with strike price 65 expiring in .5 years. How many shares of stock are needed in the hedging portfolio? A) Short at least 1 share B) Short between 0 and l shares C) No shares needed D) Long between 0 and 1 shares E) Long at least one share nnMT’D. CONT’D. NORMAL DISTRIBUTION TABLE Entries represent the area under the standardized normal distribution from -oo to z, Pr(Z<z) The value of z to the ﬁrst decimal is given in the left column. The second decimal place is given in the top row. 0.01 0.02 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.6950 0.6985 0.7019 0.7054 0.7088 _ 0.7123 0.7157 0.7190 0.7224 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.761 1 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 .0 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 .1 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 .2 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 ' 0.9015 .3 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 .4 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1 .5 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 ’ 1 .6 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9649 0.9656 ‘ ‘ 0.9664 ‘1‘ 0.9671‘ ’ ' 0.9678 ‘ 0.9686 0.9693 0.9699 0.9706 1.9 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 0.9772 0.9778 0.9783 0.9788‘ 0.9793“1 0.9798 0.9803 ' 0.9808 0.9812 0.9817 2.0 2.1 0.9821 0.9826 0.9830 0.9834 0.98381 ' 0.9842 30.9848 «4 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969, 0.9970 0.9971. 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9932 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 0 ,4 07389 r742». 174‘. .. v-7 . . ‘ 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 .. 0.9989 0.9990 0.9990 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.4 0.9997 0.9997 0.9997 0.9997. 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.9 1.0000 1.0000 1.0000 1,0000) 1.0000 1,0000 1.0000 1,0000 1.0000 1.0000 09.2.” 09495 «439505» 09515-w4’19526 095.»: .793... n4 . ,. r .» Values of z for selected values of Pr Z<z _ 0.842 1.036 1.282 1.645 1.960 2.326 2.576 m- 0.800 0.850 woo-mm ' 0.975 0.990-m t fix-.77 I - ' 5‘34 ‘l v7.71» .14.) U E1.“ . 0 ,. 3 -5 .1 05.1.4 '13" O. . 5 J3 m“ Um I 0 2,—0.9.3’ﬁ4i 8341.1 v. L2. .. .3 93.95 ’ 3.9. 59. 0‘ \$1560 *5 0.5.5.191 O 9.62 .33 ﬂ 0’: ‘1’; .7. 3"." ‘5 on“? - r gr“:- " 57' .r ' 7:370 ' .70... u 3-. ...
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act370h-m08final exam - UNIVERSITY OF TORONTO FACULTY OF...

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