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Unformatted text preview: Chapter 11 — Questions Question 1 A perpetual American call option has a strilte price of $33. The underlying stoclt pays dividends at a continuously compounded rate of 11%. The volatility of the underlying
stoclt is zero. The continuously compounded l'iﬂkvfI'EE interest rate is 14%. Calculate the lowest stock price for which early exercise of the American call option is
optimal. A. 33.31 E 33.43 C 34.51 D 33.13 E 33.52 Question 2 The current exchange rate is ¥13ID per dollar. The continuously compounded dollar
denominated interest rate is set. The continuously compcanded yen—denominated
interest rate is 3%. An at—themoney American call option is yendenominated and allows its owner to buy $1.
The American call option is a perpetual option. The volatility is zero. Calculate the lowest exchange rate for which early exercise of the American call option is
optimal. A 115133.35 B ¥135.3{} C ¥133.ﬂﬁ D 35111.53 E ¥12£LUD lQuestion 3 A perpetual American call option has a strike price of $55. The underlying stoclt pays
dividends at a continuously compounded rate of 5%. The current price of the underlying
stock in SW]. The volatility of the underlying stock is zero. The continuously compounded risk—free interest rate is Tit. Calculate the amount of time {in years) until the American call option is exercised.
A. 3.33 B 1.35 C 15.3?r D 134.31 E Never lQuestion 4 A perpetual American call option has a strike price of $43. The underlying stock pays
dividends at a continuously compounded rate of 3%. The current price of the underlying
stock is $53. The volatility of the underlying stock is zero. The continuously compounded risk—free interest rate is 3%. Calculate the value of the American call option.
A. 2.33 B 3.04 C 11.5'iIr D 12.15 E 15.53 [ED ActuarislBrcweom 2010  Page Ell1 Question 5
Determine which of the following statements is TRUE. A An increase in the volatility of a stock maltes it more liltely that an intheInoney
American call option on the stock will be exercised early. E An increase in the volatility of a stock makes it more likely that an inthsmoney
American put option on the stock will be exercised early. 0 An increase in the volatility of a stock increases the exercise boundary for an
American call option. D As the time until maturity for an American call option decreases, its exercise
hnundary increases. E As the time nntil maturity for an American pnt option decreases. its exercise
boundary decreases. ' Question 6
An investor has owned the following 5 options for over a year: American call option on Stock )1
American call option on Stock Y
American call option on Stock 55
American put option on Stool: K
American put option on Stock Y Stocks X, Y, and Z pay con tinnous dividends. One year ago, it was optimal to exercise all 5 of the options. Unfortunately for the
investor. she was hnsy studying for an exam, and she failed to notice that it was optimal
to exercise the options. She still owns them today. (liver the course of the last year, the
following changes have occurred: The volatility for Stock X has increased.
The volatility for Stool: Y has decreased.
The volatility for Stool: E has increased,
The price of Stock X is the same. The price of Stock Y has decreased. The price of Stool: E has decreased. The investor does not expect the volatility to change in the future. It is now optimal to exercise only one of the options. Determine which option should be
exercised. A American call option on Stock X . E American call option on Stock Y
(3 American call option on Stock D American put option on Stock X
E American put option on Stock Y
@Actuarilerswnom EDl'll _ FREE 51112 Exam MFEEF Questions . ' I Chapter 1 l — Binomial Dption Pricing: 11 Question '1' The current price of a nondividendpaying stock is 000. The stock’s volatility is 00%. The
stock’s price follows a lperiod standard binomial model, and each period is one year. The continuously compounded risk—free rate of return is 0%. A European call option on the stoolE expires in 1 year and has a strike price of $40.
Ualculate the value of the call option. _ A 4.10 B 4.52. C 0.11 D 9.07 E 10.00 lQuestion 0 The current price of a nonrlitlidendpayring,r stock is 000. The stoclt’s volatility is 00%. The
stock’s price follows a 1period standard binomial model, and each period is one year. The continuously compounded riskfree rate of return is 0%. The real probability that the
stock price increases is 40%. A European call option on the stoclt expires in 1 year and has a strike price of $40. The expected return on the call option, expressed as a continuously compounded rate of
return, is y. Calculate y. A 0.0% ‘ B 10.1% C 14.2% D 10.0% E 01.0% Question 0 The current price of a nondividendpayinp,r stock is $00. The stock’s volatility is 00%. The
stock‘s price follows a laperiod standard binomial model, and each period is one year. The continuously compounded riskfree rate of return is 0%. The real probability that the
stock price increases ia 40%. A European put option on the stock expires in 1 year and has a strike price of $40. The expected return on the put option, expressed as a continnously componnded rate of
return, is y. Calculate y. A "0.2% B —l.4% C 1.0% D 0.0% E 15.0% Question 10 The current price of a stock is $02. The stock's conth'iuously compounded dividend yield is
4%. The stock’s volatility is 24%. The stock's price follows a 1—period standard binomial
model, and each period is one year. The continuously compounded riskfree rate of return is 0%. The continuously
compounded expected return on the stool: is 12%. a European pnt option on the stock expires in 1 year and has a strike price of $04. The expected return on the put option, expressed as a continuoualy compounded rate of
return, is y . Calculate y. a are a 4.3a. c 4.3% ' n 2.3% a 4.3% ﬁﬁctuarialﬂrewnom 0010 Page Q1143 Question 1 1 The current price of a stock is 333, its continuously compounded dividend yield is 2%, and
its volatility is '27%._ The stoclt’s price follows a 1period standard binomial model, and
each period is one year. The continuously compounded riskfree rate of return is 2%. A European call option on the stock expires in 1 year and has a strike price of $43. The
continuously compounded expected return on the call option is 34.333%. Calculate the probability that the stock price decreases over the next period.
A 23.5% E 42.3% C 43.3% D 33.T% E 32.2% Question 12 The current price of a stock is $33, its continuously compounded dividend yield is 2%, and
its volatility is 2T%._ The stock’s price follows a 1—period standard binomial model, and
each period is one year. The continuously componnded riskfree rate of return is T%. A European call option on the stock expires in 1 year and has a strike price of $43. The
continuously compounded expected return on the call option is 34.333%. The continuously compounded expected return on the stock is o . Calculate o .
A 7.3% B 12.3% C 14.3% D 23.3% E 23.3% Question 13 The current price of a stock is $53. The stock’s continuously compounded dividend yield is
5%. The stock’s volatility is 33%. The stock‘s price follows a. 2period standard binomial model, and there are 2 periods per year. The continuously compounded riskfree rate of
return is 3%. Au European call option on the stock expires in 1 year and has a strike price of $47. The
continuously compounded expected return on the stock is 13%. If the stock price increases during the fast period, then the continuously componnded
expected return on the call option during the second period is y . Calculate y . A 13.3% E 21.2% D 22.2% D 23.2% E 23.1% Qu action 14 The current price of a stock is $33. The stock’s continuously compounded dividend yield is
3%. The stock’s volatility is 33%. The stock’s price follows a 2pcriod standard hinomial model, ond there are 2 periods per year. The continuously compounded riskfree rate of
return is 3%. An American put option on the stock expires in 1 year and has a strike price of 34?. The
coutinuonsly compounded expected return on the stock is 13%. — if the stock price decreases dnring the ﬁrst period, then the continuously compounded
expected return on the put option during the second period is y . Calculate y . A —23.'l'% E —lﬁ.3% C —11.3% D —13.3% E 43.4% ﬂActuaiialﬂrewrom 2313 Page 311—4 Exam MFEJBF Questions Chapter 11 — Binomial ﬂption Pricing: IE Question 15 The current price of a stock is $60. The stock’s continuously compounded dividend yield is
10%. The stock’s volatility is 412%. The stock’s price follows a 2period binomial model,
and each period is one year in length. The price evolution of the stcclt follows the
textbook’s standard binomial pricing model with each period being,Ir 1 year in length. The continuously compounded riskfree rate of return is 11%.
an American call option on the stock expires 2 years and has a strike price of $40.
The continuously compounded expected return on the stock is 24%. Calculate the continuously compounded expected return on the call option during the first
period. A 16.2% B 218% U 318% D 38.2% E 38.8% Qn cation 15 The current price of a stock is 3E3. The stock does not pay dividends, and its volatility is
30%. The continuously compounded expected return on the stock is 15%. The stock’s
price follows a 3period binomial model as shown below, and each period is US year in
length. Stock Prices 1 1 4."?493
93.53139
T153335 31.1533
ESﬂUU‘U 55.45 12
54.4125 57.3334
415.9953
40. 5399 The continuouslyr compounded risk—free rate of return is 3%. an American call option on the stock expires in 1 year and has a strike price of $59.
Determine the node at which the expected return on the American call option is loWest.
A The node at which the stock price is $63.0Dﬂﬂ. B The node at which the stock price is SETH.9385.
G The node at which the stock price is $544126.
D The node at which the stock price is $93.93D9.
E The node at which the stock price is $63.4512. Q Actuariclﬂrewnom min I Page ﬁll5 Exam l'rIFEI'EF Questions Chapter 11 — Binomial Option Pricing: TI. Question 20 IThe current price of a steel: is $33. The stock dDEs not pay dividends, and its volatility is
30%. The stock’s price follows a 1period Con~RossR.uhinstehi binomial model, and each
period is one year. The continuously compounded riskfree rate of return is 3%. A European put option on the stock expires in 1 year and has a strike price of 343. Calculate the value of the put option.
ft 4.13 B 5.03 C 5.40 D 3.74 I E 3.02 Question 21 The current price of a stock is $100. The stock‘s continuously compounded dividend yield
is 5%. The stock’s yolsﬁlity is 30%. The stock’s price follows a 1period CoxRoss
Ruhinstein hinomisd model, and each period is 3 months long. The connnuously compounded riskfree rate of return is 3%. A European call option on the stock expires in 3 months and has a strilte price of $35.
Calculate the veins of the call option. A 10.12 E 10.34 C 10.?5 D 10.3? E 11.33 Question 2.2. The current price of a stock is $100. The stock's couthiuously compounded dividend yield
is 5%. The stock‘s volatility is 30‘} . The stock’s price follows a 1period CoxRoss
Ruhiustein binomial model, and each period is 3 months long. The continuously compounded riskFree rate of return is 3%. A Enropean call option on the stool: expires in 3 months and has a strike price of $35.
The continuously compounded expected return on the call option is 31.53%. Calculate the continuously compounded expected return on the stock.
A 5.0% E 10.0% C 15.0% D 20.0% E 25.0% Question 23 The current price of a stock is 333. The stool: does not pay dividends, and its volatility is
30%. The stock’s price follows a 1period Jerrow and Rudd hinomiol model, and each
period is one year. The continuously compounded riskfree rate oi'return 1's 7%.
ii European call option on the stool: expires in 1 year end has a strike price of $40.
l[.lalcnlate the veins of the call option. ii 4.53 B 4.33 C 5.“?4 I] 5.33 E 3.15 @ActuerieJErewrom 2310 Page 0117 Exam MEEFBF Questions . Chapter 11 — Binomial l5'ption Pricing: 1T Question 24 The current price of a stock is $35. The stock does not pa].T dividends, and its volatility is
55%. rI'he stock’s price follows a 1period Jarrow and Rudd binomial model, and each
period is one year. The continuously compounded riskvfree rote of return is 'i'%. A European put option on the stock expires in '.'. year and has a strike price of $45.
Calculate the value of the put option. A 4.15 E 5.55 C 5.15 D 5.74 E 5.52 Question 25 The current price of a stock is $155. The stock’s continuously compoimdcd dividend yield
is 5%. The stock*s volatility is 35%. The stock’s price follows a lpericd J arrow and Rudd
binomial model, and each period is 5 months long. The continuously compounded riskfree rate of return is 5%. A European call option on the stock expires in 5 months and has a strike price of $95.
Calculate the value of the call option. A 5.55 E 5.75 C 15.17 D 15.55 E 11.55 Question 25 The current price of a stock is $155. The stock’s continuously compounded dividend yield
is 5%. The stock’s volatility is 55%. The stock’s price follows a Lpe‘l'lﬂd JBTIDW and Rudd
binomial model, and each period is 3 months long. The continuously compounded riskfree rate of return is 5%. A European call option on the stock expires in 3 months and has a strike price of $55.
The continuously compounded expected return on the call option is 52.51%. Calculate the continuously compounded expected return on the stock.
A 5.5% E 5.5% C 15.5% D 15.5% E 25.5% Question 2'? The current price of a stock is 553. The stock’s continuoust compounded dividend yield is
4.5%. The stock’s volatility is RTEt. The stock’s price follows a l—period Jenner and Rudd
binomial model, and each period is 5 months long. The continuously compounded riskfree rate of return is 12%. A European call option on the stock expires in 5 months and has a strike price of $55.
Calculate the option’s delta. I A 5.254 H 5.255 C 5.255 D 5.272 E 5.252 It: notoarialﬂrswnom 2515 Page 5113 Exam MEJEF Questions Chapter 1.1 — Binomial Dptien Pricing: II Qu esti on 25 The current price of a stock is $53. The stock's continuoule compounded dividend yield is
1%. The stock's volatility is 35%. The stock's price follows a 4period J arrow and Rudd
binomial model, and each period is 3 months long. The continuoule compounded risksfree rate of return is 5%. A European put option on the stock expires in 1 year and has a strike price of $34.
Calculate the price of the put option. A [LEE E [LSD C 0.62 D 3.54 E [1.56 Question 29 The table below lists 6 weeks of Wednesday closing prices for Stock X. No dividends were
paid during the months that the data was collected. llllblﬂﬂﬂﬁ 93 Use the stock plices to estimate the annual volatility of Stock III.
A. [121311 E DDTE C 13.4134 D H.435 E H.561 Question 30 The table below lists 7 months of closing prices for Stock Z. No dividends were paid
during the months that the data was collected. Month Use the stock prices to estimate the annual volatility of Stock Z.
J—‘L [1.911 E DJJET U CLUEti I} 0.193 E 0.402 (Cl Actuarilerewnmn 2011] Page ﬁll9 Exam MFEIBF‘ Questions Chapter 11 — Binomial lOption Pricing: II
Question 31 In 6 months, the economj,T will either he in a high state or a lov;r state. The table helov:r
contains assumptions regarding equity cash ﬂows, probabilities, and utility values in the
high and law states of the economy. Gash Elev;r to risk—Eree bond “ The stock does not pay dividends. A European call option on the stock expires in 5
months and has a strike price of $1312]. Calculate the price of the call option. ' A $32.45 B $33.21 C $34.55 D $36.9ﬂ E $40.21?! Question 32 In 6 months, the economy,T will either he in a high state or a low state. The table below
contains assumptions regarding equitj.T cash ﬂows, prohahilities, and utility values in the
high and low states of the economy. an ax
Cash ﬂow to stock 551] The stock does not pay dividends. A European call option on the stock expires in 6 months and has a strike price of $130.
The expected return of the call, expressed as an annual effective rate nf return, is reap. A European put option on the stock expires in E months and has a strike price of 51m.
The expected return of the put, expressed as an annual eEEeetive rate of return, is he“; . Find fess. — rpm.  A 14% E 18% 23% D Elie E 23% til ﬁclaiarialﬂrewmom 2D ltl P's go ﬁll1E! Exam l‘u‘l‘E‘EiElF Quiestiens {Jhapter 11 — Iiino miel ICiptien Pricing: II Question 3 3 In 1 year. the economy will either he in a high state or a low state. The tahle llﬂlﬂw
contains assump tions regarding equity cash flows, probabilities. and utilitjtr values in the high and lev;r states of the economy.
— Utilityr 1ii'alue of $1 [LET 0.95 The stock pays continuously,T compounded dividends at an annual rate of 6%. Calculate
the expected return on the stock. expressed as an effective annual rate. A 3.4% B 3.3% C 10.0% D 11.8% E 12.5% lQuestion 34 There are 2 scenarios fer the state of the economy in 1 year. The prehahility ef Scenario 1
occurring is 3ﬂ%. In Scenario 1. Stock X has a price of $50. In Scenario 2. Stock K has a
price of $100. The continuously compounded expected return fer Steel: K is 1%. Stock K
does net pay dividends. A riskfree asset that matures for $1 in 1 year has a current price of $0.92. Q2 is the
price of a security that pays $1 only if Scenario 2 occurs. Determine Q2. A 3.255 B 11303 C 43.612 D 11.565 E {1723 lQuestion 35 At time T. the economy 1will he in one of three states. The table below contains
assumptions regarding equity cash flows. probabilities. and utility values in the possible
states of the ecenemy. The utility values in the last line of the table below are expressed in terms of dollars today.
—  m Utility Value of 5.1 0.5554 Steelt A does not pay dividends. and its current price is $101]. A European call option on
Steel: B has a strike price of $66 and expires at time T. Calculate the value of the
Eurepean call option. A 5.14 E ELTQ C 6.43 D THE E 5.5“? $1
$4 £3 Aetuerielﬂrcwrem 2010 Page Qtl‘ll Qu eation 36 In 1 veer, the economy 1will either he in a high etate or a loin,T etate. The table below containa aaaumptiona regarding equity caah ﬂows, probabilities, and utility values in the
high and low atatea of the economy. High State in 1 Year Lari».r State in 1 Year
Cash ﬂow to rickfree bond $110 5110 Stock price  $7 a $125 Utility l.Talue of $1 {1.85 0.95 The stock pave continuoual}r compounded dividends at an annnal rate of 5%. Calculate
the amount h_v which the atock price exceeda the price of the rickfree bond. A —$3.35 B $13.31] C 50.13 D $1.61 E $1311 Question 3'? In 2 veara. the economv will either he in a high etate or a lnvv etate. The table helov‘.r containa aaeumptione regarding equityr caeh ﬂovva, prohahilitiee, and utility,r values in the
high and low etatea of the economy. ' _
m
m The etoclrt does not pay dividends, and its currcut value is $36.53. The atock price at time
1'. ia denoted hv 843.). In 2 veara, a derivative has a pavoff of l_o([S(2]]Bl . Calculate the current price of the derivative. A $13.13 B $113.23 C $111.41} D $113.51] E $111.81} ﬂ ActuarialErew .com 201i] Page ﬁll12 Question 33 In ‘2 years, the economy will either he in a high state or a lot».T state. The table helow
contains assumptions regarding equity cash ﬂows, pi'ohahilities, and utility values in the
high and 10W states of the economy. ' Cash flow to riskfree bond $1 $1
Prohahility The stock does not pay dividends, and its cnrrent value is $36.53* The stock price at time
[8(2)]3] . Calculate the t is denoted by 51ft). In 3 years, a derivative has a payoﬁf of hi current price of the deriVatiye. The riskfree interest rate is constant across time.
A $9.40 B $9.51] C $9.9ﬂ D $113.00 E $10.10 Question 39 The current price of a stoolt is $40, and its contiiilitlnlisl}r compounded dividend yield is
11%. The continuously compounded rial:vﬂ'ee rate of return is 15%. A European put option on the stock expires in 3 months and has a strike price of $38. The
delta of the put option is —EI.262. The amount of cash that is lent in the replicating
portfolio is $1136. The option is priced using a oneperiod hinomial model. The binomial model is known to
he either the CovaossRuhinstein model or the Jerrow~Radd model Calculate the volatility, or.
A [LED E 1125 C [131] D [1.35 E [1.40 lQuestion 4] The risksfree rate, dividend rate+ volatility, time interval length, and type of model are
listed for five models helow. n t Three of Model
 .2 . . CoxRossRuhinstein
 Lh
:
III‘I
C:
o:
c: C: M G
Cit—ID
EPIC"N
":ICJUT Dd
:3
[Cl
_.  . . . Cox Ross Rubinstein
 JamsRudd
  0 Determine which of the models permits arhitrage.
A ModelA B Model B C Model C D ModelD E Model E 121
{I}
4:.
C: D C3
.vll—l is. I: :1 D C1 DJ I1 C) [—l D D tn
:2:
HD
mm
1:: Ed
D
J
c: @ActuaﬁalBrewroin 201D Page ﬁll13 Question 41 For a one~period binomial model for the price of a stock, you are given: (i) The period is one year. (ii) The steel: pays no dividends. (iii) a. = 1.4TT. where u. is one plus the rate of capital gain on the stool: per period if the
stool: price goes up. (iv) o! = 0.?63 . where d is one minus the rate of capital loss on the stool: per period if
the stool: price goes down. {v} The continuously1 compounded annual expected return on the steel: is 10%.
Calculate the true prohahility ot the stool: price going up.
A 0.42 E 0.48 C 0.50 D 0.52 E 0.53 Question 42 The current price of a stool: is $40. The steel: pays dividends at a continuoust
compounded rate of 4%. The volatility of the stool: is 32%. The continuously compounded riskfree interest rate is 11%. A 2—year American put option on the stoclt has a strike price of $42. The option is priced
using a 2period J arrow and Rudd hinomial model. Estimate the put option‘s gamma. A 0.0281 E 0.0344 G 0.0343 D 0.0350 E 0.03'1'3 Question 43 rI‘he tahle helour lists Tr months of ex—diiridend closing prices for Stool: Z. The only;
dividend paid during the T months is a $13 divideud paid at the end of month 5. 1
4
“ Use the steel: prices to estimate the annual volatility of Stool: Z.
A 0.069 E 0.130 C 0.193 D 0.224 E 0.240 «El AotuarialBreWan 2010 ' Page 1311141 Exam MFEIBI" Questions Chapter 11 — Binomial Dption Pricing: H Question 44 In 1 pear1 the eeonol'ns:r will either be in a high state or a low state. The table below
contains assumptions regarding equity cash .ﬂows, probabilities, and utilityr values in the high and low states of the economy. UtilityT Value of “351 0.92 1.05 The stool: does not pay dividends. A European call option on the stock expires in 1 year and has a. strilte price of Hg . The
expected return of the tell1 expressed as an annual effective rate ot'retum, is I’Caﬂ. A European put option on the stool: expires in 1 year and has a strike price of KP. The
expected return of the put, expressed as an annual eJ'l'eetive rate of return, is rpm . You are given that: 60 {Kg {loo
60 digs (1130 A 3.5% E 3.9% e set D 13.2% E 13.5% ﬁﬂotusrialﬂrewnom 2010' ‘ Page ﬁll1.5 Exam MFEFBF Questions Chapter 1.1 — Binomial lElption Pricing: H lQuestion 45 In 1 veer, the economy will either be in a high state or a low state. The table below
contains assumptions regarding equity cash ﬂows, probabilities, and utility values in the
high and low states of the economy. — The stock does not pay dividends. The expected returns of the Eve options below are calculated as annual effective rates of
return. Determine which option has the highest expected return. A Call option with a strilte price of 5540
E Dell option with a strike price of $65
D Dell option with a strilre piice of $110
D Put option with a strike price of $‘l’5
E Pnt optiou with a strike price of $120 Question 45 The following oneperiod binomial stock price model was used to calculate the price of a
one~_vear $10‘5L‘L‘llle put optiou on the stock; 3 .= lfi
So 2 10 < If
He = 7
You are given:
(i) The period is one veer.
(ii) The true probability of an np~move is Bill).
(iii) The stock pays no dividends. (iv) The price of the one~3rear put is $1.69. based on the stock prices shown shove. Upon review. you realize that there was an error in the model construction and He, the
veins of the stock on a doWnmcrve, should have been $5 rather than $T. The other
assumptions listed above were correct, and the assumption used for the rislofree rate of
return was correct as well. ' Recalculate the price of the put option.
A $1.35 B $1.70 C $2.433 D $2.2ﬁ E $2.53 '52:! Actuuislﬂrswrnm Eﬂlﬂ Page ﬁll113 ...
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This note was uploaded on 09/20/2011 for the course MATH Math 174 taught by Professor Kong during the Summer '10 term at UCLA.
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