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# Ch%2011%20Q - Chapter 11 — Questions Question 1 A...

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Unformatted text preview: Chapter 11 — Questions Question 1 A perpetual American call option has a stril-te 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'iﬂkv-fI'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—the-money American call option is yen-denominated 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 ActuarislBrc-weom 2010 - Page Ell-1 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 in-the-Inoney 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 in-ths-money 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 @Actuariler-swnom EDl'll _ FREE 5111-2 Exam MFEEF Questions . ' I Chapter 1 l — Binomial Dption Pricing: 11 Question '1' The current price of a nondividend-paying stock is 000. The stock’s volatility is 00%. The stock’s price follows a l-period 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 stool-E 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 nonrlitl-idend-payring,r stock is 000. The stoclt’s volatility is 00%. The stock’s price follows a 1-period standard binomial model, and each period is one year. The continuously compounded risk-free 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 nondividend-payinp,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 risk-free 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 risk-free 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 1-period standard binomial model, and each period is one year. The continuously compounded risk-free 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 risk-free 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. 2-period standard binomial model, and there are 2 periods per year. The continuously compounded risk-free 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 2-pcriod standard hinomial model, ond there are 2 periods per year. The continuously compounded risk-free 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% ﬂActuai-ialﬂrewrom 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 2-period 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 risk-free 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 3-period 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 ﬁll-5 Exam l'rIFE-I'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 1-period Con~Ross-R.uhinstehi binomial model, and each period is one year. The continuously compounded risk-free 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 1-period Cox-Ross- Ruhinstein hinomisd model, and each period is 3 months long. The connnuously compounded risk-free 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 1-period Cox-Ross- Ruhiustein binomial model, and each period is 3 months long. The continuously compounded risk-Free 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 1-period Jerrow and Rudd hinomiol model, and each period is one year. The continuously compounded risk-free 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 011-7 Exam MEEFBF Questions . Chapter 11 — Binomial l5'ption Pricing: 1T Question 2-4 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 1-period Jar-row 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 2-5 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 l-pericd J arrow and Rudd binomial model, and each period is 5 months long. The continuously compounded risk-free 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 2-5 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 risk-free 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 RTE-t. The stock’s price follows a l—period Jenner and Rudd binomial model, and each period is 5 months long. The continuously compounded risk-free 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 511-3 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 4-period 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 CLUE-ti I} 0.193 E 0.402 (Cl Actuarilerewnmn 2011] Page ﬁll-9 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 ﬁll-1E! 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 risk-free 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 Steel-t 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 rick-free 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 rick-free 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- ActuarialE-rew .com 201i] Page ﬁll-12 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 risk-free 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 risk-free 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 stool-t 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 one-period hinomial model. The binomial model is known to he either the Covaoss-Ruhinstein 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 . . Cox-Ross-Ruhinstein - Lh- :- III‘I C: o: c: C: M G Cit—ID EPIC-"N ":ICJUT Dd :3 [Cl |_|. - . . . Cox Ross Rubinstein - Jams-Rudd - - 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 .v-ll—l is. I: :1 D C1 DJ I-1 C) [—l D D tn :2: HD mm 1:: Ed D --J c: @ActuaﬁalBrewroi-n 201D Page ﬁll-13 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 risk-free 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 2-period 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 1311-141 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 ﬁll-1.5 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 one-period 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 doWn-mcrve, 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:! 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Ch%2011%20Q - Chapter 11 — Questions Question 1 A...

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