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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 riskfree rate of interest is 5% and the price of a oneyear
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 nondividend
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 3period 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 3period 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 3period 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 nondividend 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 BlackScholes 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 BlackScholes 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 nondividend paying stock has a current price of 60 and expires in one
year. You are given the following, based on BlackScholes 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 BlackScholes 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 nondividend
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 DeltaGamma 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‘. .. v7 . . ‘ 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» 09515w4’19526 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 woomm ' 0.975 0.990m
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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