MORNING SESSION. Date: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES

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1 SOCIETY OF ACTUARIES Quaniaive Finance and Invesmen Core Exam QFICORE MORNING SESSION Dae: Wednesday, April 26, 2017 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Insrucions 1. This examinaion has a oal of 100 poins. I consiss of a morning session (worh 60 poins) and an afernoon session (worh 40 poins). a) The morning session consiss of 10 quesions numbered 1 hrough 10. b) The afernoon session consiss of 7 quesions numbered 11 hrough 17. The poins for each quesion are indicaed a he beginning of he quesion. 2. Failure o sop wriing afer ime is called will resul in he disqualificaion of your answers or furher disciplinary acion. 3. While every aemp is made o avoid defecive quesions, someimes hey do occur. If you believe a quesion is defecive, he supervisor or procor canno give you any guidance beyond he insrucions on he exam bookle. Wrien-Answer Insrucions 1. Wrie your candidae number a he op of each shee. Your name mus no appear. 2. Wrie on only one side of a shee. Sar each quesion on a fresh shee. On each shee, wrie he number of he quesion ha you are answering. Do no answer more han one quesion on a single shee. 3. The answer should be confined o he quesion as se. 4. When you are asked o calculae, show all your work including any applicable formulas. When you are asked o recommend, provide proper jusificaion supporing your recommendaion. 5. When you finish, inser all your wrien-answer shees ino he Essay Answer Envelope. Be sure o hand in all your answer shees because hey canno be acceped laer. Seal he envelope and wrie your candidae number in he space provided on he ouside of he envelope. Check he appropriae box o indicae morning or afernoon session for Exam QFICORE. 6. Be sure your wrien-answer envelope is signed because if i is no, your examinaion will no be graded. Tournez le cahier d examen pour la version française by he Sociey of Acuaries Prined in he U.S.A. 475 N. Maringale Road Exam QFICORE-Fron Cover Schaumburg, IL

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3 **BEGINNING OF EXAMINATION** 1. (7 poins) A complee marke conains one risk-free asse B and wo correlaed risky asses S 1, and S 2, which have he following diffusion processes: B0 1 db rbd ds1, 1d 1dW1, S1, ds2, d dw S2, dw dw d 1, 2, 2 2 2, where r, 1 1, 1, 2, 1 0 and 2 0 are consans, and W 1, and W 2, are sandard Wiener processes under he measure. Le Z S1, S2,. You are pricing a derivaive on Z wih a payoff of ZT a ime T where is a consan power. dz (a) (2 poins) Express he diffusion process of Z as d dw where W is Z a sandard Wiener process under by deriving,, and W in erms of,,,,,, and W. 2, W1, A ime, consider a self-financing porfolio valued a unis of B such ha T Z T almos surely. holding unis of Z and (b) (2 poins) Show, using Girsanov s heorem, ha by changing he measure o an equivalen risk-neural measure, e r is a -maringale. (c) (1 poin) Find he diffusion process for Z under he measure. (d) (2 poins) Calculae he value of he derivaive a ime. Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

4 2. (6 poins) The one-period spo rae r follows he ineres rae ree given below. = 0.24 The following insrumens are raded: A defaul-free savings accoun A defaul-free bond mauring a ime 3 wih value 1 (price a ime 0 = 0.84) A defaul-free bond mauring a ime 2 wih value 1 (price a ime 0 = 0.89) (a) (2.5 poins) Calculae he remaining wo risk-neural probabiliiesq ud and Q du. (b) (2 poins) Deermine he ime-0 price of a caple on wih noional of 1000, cap K 6% and expiry a ime 3 using: (i) (ii) Risk-neural measure Forward measure Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

5 2. Coninued Now assume he forward rae follows he sochasic differenial equaion (SDE) below under he real-world measure: df F d F dw where is a sandard Wiener process. Le C be he price of a caple on LIBOR rae L wih enor 1, cap rae K, noional amoun N 1, and mauriy a ime T. Your colleague calculaed C using he following formula: C FN d e KN d rt 1 2 where is he risk-free rae, d 1 2 F ln r T K 2 T and d d T. 2 1 (c) (1.5 poins) Idenify errors of his approach for calculaing C. Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

6 3. (8 poins) Le, FP, process. Le be a probabiliy space andw : 0 be a sandard Wiener X : 0 and : 0 Y be he processes defined by: X Y 0 0 Wds s W ds 2 s (a) (1 poin) Prove ha E W (b) (7 poins) Derive an expression in erms of for each of he following: (i) Var X (ii) Var Y (iii) CovX, Y Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

7 4. (5 poins) Sock price S follows geomeric mean revering process wih a sandard Wiener process W : 0 ds log S S d S dw where,, and are posiive consans. Le Y log S. (a) (2 poins) Prove by using Io s lemma ha for T. T 2 T T Ts YT e Y 1 e e dws. 2 (b) (1 poin) Derive he mean and he variance of Y T a ime T. (c) (2 poins) Derive T E S a ime T. Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

8 5. (7 poins) You are given he following informaion of a European call opion on a sock as of Sepember 30, 2016: Marke price of he call opion = $6.92 Call opion srike price = $100 Call opion s remaining ime o expiraion = 1 year Sock price = $100 Sock dividend yield = 0% Coninuously compound risk-free ineres rae = 2% for all mauriies All assumpions underlying he Black-Scholes opion pricing model hold You firmly believe ha he sock s acual volailiy will be 20%. (a) (1 poin) Calculae he call opion price ha is consisen wih your volailiy view. You can rade he call opion on 9/30/2016 only, bu you can rade he sock on any day when he marke is open. All your rades are o be conduced a he marke price. (b) (c) (1 poin) Propose a rading sraegy on 9/30/2016 involving he call opion and/or is underlying sock ha guaranees a profi if your volailiy view proves correc. (2.5 poins) Provide a mahemaical derivaion o suppor your proposed sraegy in par (b). (d) (0.5 poins) Deermine he presen value of expeced profi of your sraegy on 9/30/2016. (e) (1 poin) Deermine he ne cash posiion of your sraegy on 9/30/2016. (f) (1 poin) Ouline pros and cons of your sraegy. Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

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10 6. (7 poins) Suppose ha he price of a sock a ime follows he following SDE ds rs d S dw where W is a sandard Wiener process under he risk-neural measure wih filraion F, r is he consan risk-free rae, and 0 is consan. (a) (1 poin) Show ha S S e 2 r W 2 0 A European chooser opion is an opion where, a a specified fuure ime u, he holder can choose wheher he opion is a call or a pu, boh of which maure a ime T u and have he same srike price K. The value of he chooser opion a ime u equals he larger of he value of he call and he value of he pu. Denoe by CS,, K, T he value a ime of he European call wih srike price K and mauriy T PS,, K, T he value a ime of he European pu wih srike price K and mauriy T HS,, K, u, T he value a ime of he European chooser opion wih srike price K, choice o be made a ime u, and mauriy T u (b) (c) (1 poin) Show ha he value of he European chooser opion,,,, max,,,,,,, H S ukut C S ukt C S ukt e K S rtu u u u u (1.5 poins) Show ha rt rt HS0,0, K, u, TCS0,0, K, Te Emax 0, K S0e e 2 u Wu 2 where E is he expecaion under he risk-neural measure. Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

11 6. Coninued (d) (2.5 poins) Show ha,0,,, * rt * H S K u T S N d N d Ke N d N d S0 ln r T K 2 where d1 T d d T 2 1 d * 1 2 S0 ln rt u K 2 u d d u * * 2 1 (e) (f) (0.5 poins) Derive he dela of he chooser opion. (0.5 poins) Calculae he limis of delas of European chooser opions: (i) When he underlying sock price S 0 approaches o zero. (ii) When he underlying sock price S 0 approaches o infiniy. Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

12 7. (4 poins) For an opion on a sock, Gamma and Speed are he second- and hirdorder sensiiviy of he opion value o he underlying sock price movemen, respecively. Assuming ha he opion value follows he Black-Scholes pricing formula, you are o derive Gamma and Speed for a European call opion on a non-dividend paying sock. Noaions: K = srike price of he opion S = sock price T = remaining mauriy of he opion r = risk-free ineres rae = volailiy of he sock price (a) (1.5 poins) (i) (ii) Derive Gamma in erms of he noaions given above. Derive Speed in erms of Gamma and he noaions given above. (b) (1.5 poins) Skech he following wo graphs ha illusrae how Gamma and Speed change wih he underlying sock price, respecively, based on your derivaions in pars (a)(i) and (a)(ii) above. Explain he shape of each of your graphs. Gamma Speed 0 K Sock Price 0 K Sock price An inern saes ha we should reduce he absolue level of Speed, which would resul in a more sable Gamma hedge. (c) (1 poin) Criique he inern s saemen. Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

13 8. (5 poins) The Heah-Jarrow Moron (HJM) model s saring poin is o define zerocoupon bond prices PT, mauring a ime T in he risk-neural world, which a ime mus yield r he risk-free shor rae. PT, follows he process: where dp T, r P T, d v T, P T, dw vt, is he ime-dependen volailiy of, The insananeous forward rae, where, PT and W is a sandard Brownian moion. F T for ime T observed a under HJM follows: df, T m, T d s, T dw mt is he ime-dependen drif and st, is he volailiy. (a) (0.5 poins) Describe he pros and cons of using he HJM model. (b) (1.5 poins) Show ha,, T, (Hin: Express F, Tin erms of PT,.) Suppose ha st, mt st sudu a T e for some posiive consans and. (c) (1.5 poins) Show ha in his siuaion he diffusion coefficien of he bond price SDE is in he same form as he diffusion coefficien of he bond price SDE under he Hull-Whie model. An invesmen analys would like o model a number of swapions. He is considering he following one-facor models: One-facor HJM One-facor BGM (d) (e) (0.5 poins) Ouline he main feaures of he wo models under he risk-neural measure. (1 poin) Describe he advanages and disadvanages of using hese models o price and hedge exoic ineres rae derivaives. Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

14 9. (5 poins) Le Ω, F, Ρ, be a probabiliy space. (a) (1 poin) Sae he hree condiions for a sochasic process o be a maringale. Le B be a sandard Brownian moion in Ω, F, Ρ. (b) (2 poins) Demonsrae ha he process X B 2 is a maringale by showing ha each of he hree condiions in par (a) holds. (c) (1.5 poins) Derive he sochasic differenial equaion for he process Y B 6B c where is a given consan by using Io s lemma. (d) (0.5 poins) Deermine he value(s) of he consan c, if any, for he process Y in par (c) o be a maringale. Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

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16 10. (6 poins) You have been assigned o a eam in an ALM group o provide suppor on porfolio immunizaion. During he kick-off meeing, your colleague made he following saemens: I. The immunizaion arge rae of reurn is less han he yield o mauriy. II. Immunizaion is a se-and-forge sraegy. III. The liquidiy of securiies used o consruc an immunized porfolio is relevan. IV. If a porfolio is immunized agains a change in he marke yield a a given horizon by maching porfolio duraion o he horizon, he porfolio faces no risk excep for defaul risk. V. In order o have a clear picure of he economic surplus of he porfolio, one merely needs o focus on he duraion of he company s asses. VI. To assure muliple liabiliy immunizaion agains parallel rae shifs, he disribuion of duraions of he liabiliies mus have a wider range han he disribuion of duraions of individual porfolio asses. (a) (2 poins) Criique each of your colleague s saemens. The company has decided o consruc a porfolio consising of hree bonds in equal par amouns of $1 million each. The iniial values and duraions are: Marke Value (including accrued ineres) Duraion Bond 1 1,010, Bond 2 998, Bond 3 1,097, Afer one year, he porfolio values including a shif in he yield curve are shown in he able below. Marke Value (including accrued ineres) Duraion Bond 1 1,008, Bond 2 1,000, Bond 3 1,088, (b) (1.5 poins) Calculae he amoun of cash required o rebalance he porfolio in order o mainain he dollar duraion a he iniial level, assuming he proporion by par value is unchanged. Exam QFICORE Spring GO ON TO NEXT PAGE Quaniaive Finance and Invesmen Core

17 10. Coninued Alernaively, assume ha you wan o mainain he dollar duraion a he iniial level and also keep he oal porfolio par value a $3 million. You will achieve hese objecives by reducing he par value of one bond and simulaneously increasing he par values of he oher wo bonds by equal amouns. (c) (2.5 poins) Calculae he new weighs by par value for all hree bonds. **END OF EXAMINATION** Exam QFICORE Spring STOP Quaniaive Finance and Invesmen Core

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