MORNING SESSION. Date: Wednesday, October 31, 2018 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES

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1 Quaniaive Finance and Invemen Core Exam QFICORE MORNING SESSION Dae: Wedneday, Ocober 31, 18 Time: 8:3 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Inrucion 1. Thi examinaion ha a oal of 1 poin. I coni of a morning eion (worh 6 poin) and an afernoon eion (worh 4 poin). a) The morning eion coni of 9 queion numbered 1 hrough 9. b) The afernoon eion coni of 7 queion numbered 1 hrough 16. The poin for each queion are indicaed a he beginning of he queion.. Failure o op wriing afer ime i called will reul in he diqualificaion of your anwer or furher diciplinary acion. 3. While every aemp i made o avoid defecive queion, omeime hey do occur. If you believe a queion i defecive, he upervior or procor canno give you any guidance beyond he inrucion on he exam bookle. Wrien-Anwer Inrucion 1. Wrie your candidae number a he op of each hee. Your name mu no appear.. Wrie on only one ide of a hee. Sar each queion on a freh hee. On each hee, wrie he number of he queion ha you are anwering. Do no anwer more han one queion on a ingle hee. 3. The anwer hould be confined o he queion a e. 4. When you are aked o calculae, how all your work including any applicable formula. When you are aked o recommend, provide proper juificaion upporing your recommendaion. 5. When you finih, iner all your wrien-anwer hee ino he Eay Anwer Envelope. Be ure o hand in all your anwer hee becaue hey canno be acceped laer. Seal he envelope and wrie your candidae number in he pace provided on he ouide of he envelope. Check he appropriae box o indicae morning or afernoon eion for Exam QFICORE. 6. Be ure your wrien-anwer envelope i igned becaue if i i no, your examinaion will no be graded. Recognized by he Canadian Iniue of Acuarie. Tournez le cahier d examen pour la verion françaie. 18 by he Sociey of Acuarie Prined in he U.S.A. 475 N. Maringale Road Exam QFICORE-Fron Cover Schaumburg, IL

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4 **BEGINNING OF EXAMINATION** 1. (6 poin) Le W be a andard Brownian moion. EWWW for u. (a) (1.5 poin) Show ha u Le M vdw for. v (b) (1.5 poin) Show ha, M (c) ( poin) Show ha Ee 3 Cov M M for e for any given conan. Therefore 3 M N ~,. 3 M ha he ame momen generaing funcion a N, 3 3 and hu Define X if 3 M if. Then X ~ N,. (d) (1 poin) Show ha X i acually no a Brownian moion. Var X X uing he reul of par (b). Hin: Derive u Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

5 . (5 poin) Le W be a andard Wiener proce and denoe he ime. (a) (b) (.5 poin) Sae he condiion for a proce o be a maringale. (1 poin) Show ha W i no a maringale by idenifying a condiion in par (a) ha i no aified. (c) (1 poin) Deermine a deerminiic funcion f uch ha W f () i a maringale. Le X W uw du. u (d) (.5 poin) Show ha X i a maringale by verifying ha all he condiion in par (a) hold. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

6 3. (8 poin) Aume ha he ock price S follow he following ochaic differenial equaion: ds S d S dw where, are poiive conan, and W i a andard Wiener proce under a realworld meaure. Given, your colleague ae ha for ln S ln S r W W where i he conan rik-free inere rae. W i a andard Wiener proce under he rik-neural meaure equivalen o. (a) ( poin) Show ha your colleague aemen i correc. Hin: Expre W W X and deermine X. For, denoe by E ( W ) W d and Var ( W ) W d he expecaion and he variance of ( W W ) d a ime, repecively. (b) (.5 poin) Calculae E ( W ) W d. (c) (.5 poin) Show ha Var ( W ) W d 3 3 Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

7 3. Coninued Le 1 ln Su du G e for and G S (d) (.5 poin) Show ha for, 1 u ln G ln G ln S du (e) (1 poin) Show ha he diribuion lng for given informaion up hrough ime under he meaure i: lng N lng ln S, r 3 Now conider a forward conrac on payoff a mauriyt : 3 G ruck a ome level K. Thi conrac ha a F G K T T (f) (1.5 poin) Derive K uch ha he forward conrac ha zero value a ime. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

8 4. (9 poin) In he Ho-Lee model he rik neural proce for he hor rae r i a follow: dr d dw where i deerminiic and ime-dependen, i conan, and W i a andard Wiener proce under he rik-neural meaure. (a) (1.5 poin) Show ha for T T T T r d T r T d W W d Le E be he expecaion a ime under he meaure. Conider he rik-free zero-coupon bond mauring a ime T. Le, be i price a ime T. Z T E e T rd (b) (1 poin) Derive he following formula uing he reul of par (a): where Z T, rt AT, e 1 A T T d T 6 T, 3 T Hin: The variance of W W d a ime equal T. (c) (1.5 poin) Show ha E Z, T e T 1 r T Td d T T 6 Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

9 4. Coninued f TT be he forward rae from ime T 1 o T oberved a ime. Le,, 1 1 df, T1, T T1 T d dw. (d) ( poin) Show ha Le F f, T, T, and F E f T, T, T ime he expeced po rae from T 1 o T a of (e) (1.5 poin) Derive he convexiy adjumen F F uing he reul of par (d). Aume he hor rae now follow a model oher han Ho-Lee. (f) (1.5 poin) Deermine and juify wheher E ZT, T 1 han e F T T. 1 i larger or maller Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

10 5. (6 poin) You manage a porfolio of variable annuiie inveed in he Ruell Index. The variable annuiie have a minimum guaranee ha proec he iniial depoi amoun. The Chief Economi a your company expec upcoming urbulence in he marke and expec a drop of up o 3%. You are aked o hedge he minimum guaranee uing opion baed on he claical Black-Schole model. Aume he following: Rik-free inere rae 5% per annum There i no dividend yield on underlying porfolio The curren Ruell Index level i 1 Ruell Index implied volailiy i conan a 4% per annum The muliplier for an opion conrac on he Ruell Index i 1 The ime o mauriy of opion relaed o he Ruell Index i 1 year There are 5 day in 1 year. The dela of he variable annuiy porfolio i $.4 million. (a) (b) (c) (1 poin) Conruc a aic hedge for he porfolio uing a bear pu pread and juify your rike level. ( poin) Calculae he number of opion on he Ruell Index needed o implemen he bear pu pread aic hedge in par (a) ha would mach he dela of he liabiliy uing he Black-Schole model. (.5 poin) Criique he dela-neural approach ued in par (b) and ugge an alernaive. Due o he low deciion-making proce, one day ha elaped. You are given he following for he bear pu pread: Yeerday marke price wa $6.7. Yeerday implied volailiy (andard deviaion) wa 4% per annum. Yeerday Vega wa Today marke price i $6.93. (d) (1 poin) Eimae oday implied volailiy (andard deviaion) of he Ruell Index underlying he bear pu pread baed on he informaion given above. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

11 5. Coninued Baed on hiorical daa, he rader eimae he acual volailiy (andard deviaion) of he Ruell Index o be 8%. You are dela hedging he bear pu pread wih oday implied volailiy (andard deviaion) eimaed in par (d) above and aume he Gamma of he pread i.15. (e) (1.5 poin) Deermine he one-day mark-o-marke profi or lo on your delahedged porfolio of one conrac of bear pu pread baed on hiorical daa, Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

12 6. (6 poin) BNR ell a wo-year equiy-linked invemen guaranee produc. I guaranee can be conidered a a hor wo-year European pu opion on an underlying equiy wih he guaraneed level a he rike price K. The liabiliy may be evaluaed uing he Black-Schole opion pricing formula. BNR ue a replicaing porfolio o hedge he equiy rik of i liabiliy. The porfolio coni of he underlying equiy index and a rik-free ae. The replicaing porfolio i rebalanced every monh. The parameer for he Black-Schole opion value calculaion are given below: A ime : Rik-free rae r 6% (annual coninuou rae) Dividend yield % S = 1 K = 1 N d = N d =.6114 (a) (b) (1 poin) Calculae he implied volailiy for he pu opion. ( poin) Calculae he amoun inveed in he rik-free ae and he equiy index in he replicaing porfolio a ime. Afer one monh, he equiy index value i (c) (1 poin) Calculae he value of he hedging porfolio righ before rebalancing. The hedge error can be defined a he difference beween he value of liabiliy and he value of hedging porfolio before rebalancing. (d) (e) (1.5 poin) Calculae he hedge error auming ha volailiy of he ock and he rik-free inere rae have no changed. (.5 poin) Explain wo poible ource of he hedge error. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

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14 7. (8 poin) Aume ha he po rae r r (SDE): dr d dw, where r, and r, follow a ochaic differenial equaion are funcion of r and uch ha, and W i a andard Wiener proce under he meaure. Le V Vr,; T TT. be he price a ime of a zero-coupon bond of $1 mauring a ime Conider rt,, V 1 V rv r V r. (a) (.5 poin) Show ha rt,, i independen of he mauriy T by conrucing a elf-financing and inananeouly rik-free porfolio wih 1 uni of a bond mauring a ime T 1 and uni of a bond mauring a ime T. Thu we have where r, V 1 V V rv, r r i a funcion of r and defined by. Le Z XV where X exp( rd). V (b) (1.5 poin) Show by uing Io Lemma ha dz X d dw. r Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

15 7. Coninued Denoe by E he expecaion a ime under a meaure. (c) (1.5 poin) Show by uing Giranov heorem ha for an appropriae meaure equivalen o : T V r,; T [exp( r d)]. (d) (1 poin) Show ha dr d dw where W i a andard Wiener proce under he meaure. For a pecific choice of parameerizaion of r you are given where V r,; T exp A, T B, T * r, T AT, T 6 for ome conan and, and BT, T. 3 (e) (1.5 poin) Deermine he drif and he diffuion of he SDE for r under he meaure in erm of and. Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

16 8. (7 poin) (a) (1.5 poin) Decribe he properie of he following wo model for inere rae: (i) (ii) The one-facor Vaicek model The Cox-Ingeroll-Ro model The Vaicek erm rucure model i decribed by he following ochaic differenial equaion (SDE): dr r d dw where,, are conan and W i a Brownian moion. (b) (1 poin) Show ha he oluion o he SDE i r r e e e dw Nex conider he correponding Cox-Ingeroll-Ro model below:. dr r d r dw I can be hown a in par (b) ha he oluion o he SDE aifie he following: r r e e e rdw Now you are o compare he oluion r and r in erm of heir expecaion and variance. (c) (d) (e) (1 poin) Deermine he expecaion of r and r, repecively. (.5 poin) Deermine he variance of r and r, repecively. (1 poin) Show ha he expecaion and he variance in par (c) and (d) have he following limi a (i) lim E r lim E r Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

17 (ii) limvar r and limvar r Exam QFICORE Fall GO ON TO NEXT PAGE Quaniaive Finance and Invemen Core

18 9. (5 poin) The following how he ABC company daily ock price daa: Day ABC Sock Price P $1. $1.4 $9.8 $1.3 Uing he maximum likelihood mehod and ome manual ep in compiling he daa, your aian eimaed a GARCH(1,1) model for he volailiy of ABC ock price a below: u P 1,, 3, P 1 u, 3 w.5u.8, 4, 5, (a) (.5 poin) Idenify a poenial iue wih he above model. You confirmed ha he above GARCH(1,1) model wa eimaed correcly. (b) (c) (1 poin) Propoe a impler alernaive model ha addree he iue you idenified in par (a) and ouline he pro and con of your propoed model v. he original model. ( poin) Deermine he parameer of your propoed model in par (b) by uing he maximum likelihood mehod and he available ock price daa. Aume ha here are 5 rading day per year. (d) (.5 poin) Calculae he forecaed Day 8 annualized volailiy a of Day 4 uing he model you eablihed in par (c). (e) (1 poin) Compare and conra your model o IGARCH(1,1) model. **END OF EXAMINATION** Exam QFICORE Fall STOP Quaniaive Finance and Invemen Core

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