UvA-DARE (Digital Academic Repository) Essays on valuation and risk management for insurers Plat, H.J. Link to publication

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1 UvA-DARE Dgal Academc Reposory Essays o valuao ad rsk maageme for surers Pla, H.J. Lk o publcao Cao for publshed verso APA: Pla, H. J Essays o valuao ad rsk maageme for surers Geeral rghs I s o permed o dowload or o forward/dsrbue he ex or par of whou he cose of he auhors ad/or copyrgh holders, oher ha for srcly persoal, dvdual use, uless he work s uder a ope coe lcese lke Creave Commos. Dsclamer/Complas regulaos If you beleve ha dgal publcao of cera maeral frges ay of your rghs or prvacy eress, please le he Lbrary kow, sag your reasos. I case of a legmae compla, he Lbrary wll make he maeral accessble ad/or remove from he webse. Please Ask he Lbrary: hp://uba.uva.l/e/coac, or a leer o: Lbrary of he Uversy of Amserdam, Secreara, Sgel 425, 1012 WP Amserdam, he Neherlads. You wll be coaced as soo as possble. UvA-DARE s a servce provded by he lbrary of he Uversy of Amserdam hp://dare.uva.l Dowload dae: 24 Dec 2018

2 I rece years here has bee creasg aeo of he surace dusry for marke cosse valuao of surace lables ad he quafcao of surace rsks. Impora drvers of hs developme are he ew regulaory requremes resulg from he roduco of IFRS 4 Phase 2 ad Solvecy 2. Furhermore, valuao of surace lables ad measurg ad maagg he rsks are he corersoes of rug a surace compay successfully. Cosequely, he measureme of fuure cash flows ad s uceray becomes more ad more mpora. hs hess s a combao of papers o several ssues relaed o valuao ad rsk maageme for surers. Valuao of several embedded opos surace producs wll be deal wh. Furhermore, sochasc models for logevy, moraly ad geeral surace rsks are developed. All models ad coceps are drecly applcable he day-o-day busess of surace compaes. Essays o Valuao ad Rsk Maageme for Isurers Essays o Valuao ad Rsk Maageme for Isurers m x 3,5% 3,0% 2,5% 2,0% 1,5% 1,0% l m a x x 1 2 x, x x Rchard Pla 1976 holds a Maser s degree Acuaral Scece a he Uversy of Amserdam. He preseed hs research a varous eraoal cofereces ad publshed several arcles he joural Isurace: Mahemacs ad Ecoomcs. Rchard currely holds a poso of Seor Rsk Maager a Eureko / Achmea Holdg. He specalzes all aspecs of valuao ad rsk maageme. ISBN Rchard Pla 0,5% 0,0% year Rchard Pla

3 Essays o Valuao ad Rsk Maageme for Isurers

4 ISBN Rchard Pla, 2010 Publshed by Wöhrma Pr Servce, Zuphe, he Neherlads All rghs reserved. No par of hs book may be reproduced ay form or by ay meas, whou permsso wrg from he auhor.

5 Essays o Valuao ad Rsk Maageme for Isurers ACADEMISCH PROEFSCHRIF er verkrjgg va de graad va docor aa de Uverse va Amserdam op gezag va de Recor Magfcus Prof. Dr. D.C. va de Boom e oversaa va ee door he college voor promoes geselde commsse, he opebaar e verdedge de Ageekapel op dsdag 1 februar 2011, e uur door Hedrkus Jozef Pla gebore e Edam-Voledam

6 Promoecommsse Promoor: Prof. dr. A.A.J. Pelsser Overge lede: Prof. dr. r. M.H. Vellekoop Prof. dr. R. Kaas Prof. dr. A.J.G. Cars Prof. dr. A.M.B. De Wageaere Dr. K. Aoo Facule der Ecoome e Bedrjfskude

7 Preface hs hess s he resul of hree years of par-me research a he Quaave Ecoomcs deparme of he Uversy of Amserdam. he combao of dog research a he uversy ad my job a surer Eureko has bee ejoyable, valuable ad fruful. Oe of he reasos for hs s ha he lk bewee academa ad he surace dusry has become sroger he las few years, whch gves he opporuy o perform research ha s drecly applcable he dayo-day busess of surace compaes. Durg hese years of research I have receved suppor, oe form or he oher, from a umber of people. Frs of all I would lke o hak my supervsor Aoo Pelsser for hs gudace, ehusasm ad deas. Also, hs early work o surace coracs was a sprao for me o sar a PhD. Nex, I would lke o hak my co-auhors Alexader va Haasrech ad Kare Aoo. Alexader has always bee very ope ad helpful, whch provded a bass for havg may eresg dscussos abou valuao ad rsk maageme, as well as dscussos abou opcs ha were o relaed o work a all. Kare has a modes persoaly, bu hs cao hde he fac ha she s very good her work. Nex o hs, was very pleasa o work ogeher o a paper. I would also lke o hak he people from he acuaral deparme Rob Kaas, Agela va Heerwaarde, Mchel Jasse, Ja Kué, Wllem-Ja Wllemse, Mchel Vellekoop, Marc Goovears, Ages Joseph ad Jule omas for provdg a pleasa ad sprg amosphere a he uversy. Furhermore, I am graeful o Eureko ad Nespar for her facal suppor. A Eureko, I would lke o hak my maager Mar Sadford for gvg me he opporuy o perform a PhD ad my colleagues of he Group Rsk Maageme deparme for he excelle amosphere ad for akg o accou he fac ha I was oly par-me avalable for Eureko. Of course I would lke o hak my freds, famly ad famly--law for her eres ad for provdg he ecessary dsracos. Above all, I would lke o hak Ae-Mare ad my daughers Noa ad Mla for beg such a good reaso o go home o me ad o o hk abou valuao ad rsk maageme a all whle beg here.

8 Coes 1. INRODUCION AND OULINE VALUAION AND RISK MANAGEMEN FOR INSURERS OULINE Chaper 3: Valuao of Swap Rae Depede Embedded Opos Chaper 4: Valuao of Guaraeed Auy Opos usg a Sochasc Volaly Model for Equy Prces Chaper 5: O sochasc moraly modelg Chaper 6: Sochasc porfolo specfc moraly ad he quafcao of moraly bass rsk Chaper 7: Mcro-level sochasc loss reservg SOCHASIC PROCESSES RISK NEURAL SOCHASIC PROCESSES FOR VALUAION Margales ad Measures Affe Jump-Dffusos Gaussa eres rae models Sochasc volaly model for equy prces Sochasc processes for valuao of uhedgeable surace rsks REAL WORLD SOCHASIC PROCESSES FOR RISK MANAGEMEN ARIMA me Seres Models Posso processes ad reewal processes VALUAION OF SWAP RAE DEPENDEN EMBEDDED OPIONS INRODUCION SWAP RAE DEPENDEN EMBEDDED OPIONS HE UNDERLYING INERES RAE MODEL Mul-facor Gaussa models Valuao for oher eres rae models HE SCHRAGER-PELSSER RESUL FOR SWAPIONS ANALYICAL APPROXIMAION DIREC PAYMEN Deermg he expecao of R Deermg he varace of R Prcg formulas VALUAION FOR MORE COMPLEX PROFI SHARING RULES Compoudg prof sharg Prof sharg cludg he reur o a addoal asse Addoal maageme acos or oher complex feaures NUMERICAL EXAMPLES Example 1: 10-year average of 7-year swap rae, drec payme Example 2: 10-year average of 7-year swap rae, compoudg opo CONCLUSIONS...31 APPENDIX 3A: PROOF OF APPENDIX 3B: PROOFS OF 3.11 AND APPENDIX 3C: INPU EXAMPLE

9 4. VALUAION OF GUARANEED ANNUIY OPIONS USING A SOCHASIC VOLAILIY MODEL FOR EQUIY PRICES INRODUCION GUARANEED ANNUIY CONRAC HE SCHÖBEL-ZHU-HULL-WHIE MODEL CALIBRAION OF HE SZHW AND BSHW MODEL PRICING HE GUARANEED ANNUIY OPION UNDER SOCHASIC VOLAILIY AND SOCHASIC INERES RAES akg he equy prce as umérare Explc formula for he GAO prce EXENSION O WO-FACOR INERES RAE MODEL NUMERICAL EXAMPLES Comparso resuls SZHW model ad Black-Scholes Hull-Whe model Impac of dffere rsk drvers Comparso resuls of he wo-facor model wh Chu ad Kwok CONCLUSIONS...54 APPENDIX 4A: PRICING OF A COUPON BEARING OPION UNDER A WO-FACOR INERES RAE MODEL...55 APPENDIX 4B: MOMENS AND ERMINAL CORRELAION OF HE WO-FACOR GAUSSIAN INERES RAE MODEL 56 APPENDIX 4C: SPECIAL CASE: INDEPENDEN EQUIY PRICE PROCESS OR PURE INERES RAE GUARANEED ANNUIIES...58 C.1 Hull-Whe model...58 C.2 Gaussa wo-facor model...60 APPENDIX 4D: YIELD CURVE SHOCKS...60 APPENDIX 4E: MODEL SEUP OF HE CHU AND KWOK 2007 CASE ON SOCHASIC MORALIY MODELING INRODUCION LIERAURE REVIEW: CRIERIA AND MODELS Crera for sochasc moraly models Sochasc moraly models Problems wh modelg cohor effec A NEW SOCHASIC MORALIY MODEL he proposed model Idefably cosras FIING HE MODEL Fg mehodology Comparso of f qualy wh exsg models Fg he ARIMA processes U.S. Males MORALIY PROJECIONS U.S. MALES Smulao resuls U.S. Males Robusess of smulao resuls Comparso wh oher models RISK NEURAL SPECIFICAION OF HE MODEL Rsk eural dyamcs Calbrao of he marke prce of rsk PARAMEER UNCERAINY CONCLUSIONS...81 APPENDIX 5A: U.S. MALE - ESIMAES FOR A X AND -X...83 APPENDIX 5B: SIMULAION RESULS ENGLAND & WALES AND HE NEHERLANDS...84 APPENDIX 5C: SIMULAION RESULS ROBUSNESS ESS SOCHASIC PORFOLIO SPECIFIC MORALIY AND HE QUANIFICAION OF MORALIY BASIS RISK INRODUCION GENERAL MODEL FOR SOCHASIC PORFOLIO SPECIFIC MORALIY EXPERIENCE...89

10 6.2.1 he basc model Fg he basc model Addg sochasc behavor Combe he process wh he sochasc coury populao model APPLICAION O EXAMPLE INSURANCE PORFOLIOS NUMERICAL EXAMPLE 1: VALUE A RISK Sochasc coury populao moraly model Impac o Value a Rsk NUMERICAL EXAMPLE 2: HEDGE EFFECIVENESS / BASIS RISK CONCLUSIONS APPENDIX 6A: EXAMPLE 2-FACOR MODEL BASED ON NELSON & SIEGEL APPENDIX 6B: FURHER RESULS APPENDIX 6C: HEDGE PORFOLIOS MICRO-LEVEL SOCHASIC LOSS RESERVING INRODUCION DAA HE SAISICAL MODEL Poso Depede Marked Posso Process he Lkelhood Dsrbuoal assumpos ESIMAION RESULS PREDICING FUURE CASH FLOWS Predcg IBNR clams Predcg RBNS clams NUMERICAL RESULS CONCLUSIONS REFERENCES SAMENVAING SUMMARY IN DUCH...144

11 Chaper 1 Iroduco ad Oule Idvdual persos, compaes ad oher ees are exposed o several rsks ha poeally ca lead o udesrable facal cosequeces. For example, for a dvdual perso could be damage o a car, propery damage, lvg loger or shorer ha expeced, expeses relaed o healh ad several oher rsks. Compaes could be exposed o, amogs ohers, a lably clam, a compay buldg o fre, damage o he producs ad dsabled employees. hese rsks ca be rasferred by buyg a surace polcy a a surace compay. I exchage for hs he surace compay receves a premum from he polcyholder. he surace compay pools he rsks so ha he resuls o he dvdual polces compesae each oher. As a resul of wrg surace busess for decea, mos surers have o pay cosderable amous he fuure o her polcyholders. he compay holds a reserve o cover for hs, whch s based o a valuao of hese fuure surace lables. Besdes hs, he surace compay s exposed o several rsks, for whch holds addoal capal. As such, valuao of surace lables ad measurg ad maagg he rsks are wo major buldg blocks for rug a surace compay successfully. hs hess s a combao of papers o several ssues relaed o valuao ad rsk maageme for surers. I he remader of hs chaper some more backgroud s gve o valuao ad rsk maageme for surers, followed by a oule ad dscusso of he research preseed hs hess. 1.1 Valuao ad Rsk Maageme for Isurers A hs mome, mos surers are reporg her lables o a book value bass, where he ecoomc assumpos are ofe o drecly lked o he facal marke. Furhermore, regulaors requre addoal solvecy capal o be held by surers whch s a fxed perceage of he reserve, premums or clams ad hus o based o he acual rsks of he surer. However, rece years here has bee a creasg amou of aeo of he surace dusry for marke valuao of surace lables ad he quafcao of surace rsks. Impora drvers of hs developme are he roduco of IFRS 4 Phase 2 ad Solvecy 2. Wh he roduco of Solvecy 2 ad IFRS 4 Phase 2 boh expeced 2013 surers face major challeges. IFRS 4 Phase 2 wll defe a ew accoug model for surace coracs, based o marke values of lables. I he docume Prelmary Vews o Isurace 1

12 Coracs May 2007, dscusso paper he Ieraoal Accoug Sadards Board IASB saes ha a surer should base he measureme of all s surace lables for reservg o bes esmaes of he coracual cash flows, dscoued wh curre marke dscou raes. O op of hs, margs ha marke parcpas are expeced o requre for bearg rsk should be added o hs. he IASB s currely furher developg he sadards, of whch a cosulao paper wll appear Solvecy 2 wll lead o a chage he regulaory requred solvecy capal for surers. Uder Solvecy 2 he so-called Solvecy Capal Requreme SCR wll be rsk-based, ad marke values of asses ad lables wll be he bass for hese calculaos. he drecve 1 of Solvecy 2 prescrbes ha he reserve... shall be equal o he sum of he bes esmae ad a rsk marg ad ha he bes esmae wll correspod o he probably-weghed average of fuure cash-flows, akg accou of he me value of moey, usg he releva rsk-free eres rae erm srucure. Furhermore, saes ha he calculao of he bes esmae shall be based upo up-o-dae ad credble formao ad realsc assumpos, ad be performed usg adequae, applcable ad releva acuaral ad sascal mehods. he SCR ams o reflec all of he rsks a surace compay s exposed o: marke rsk, operaoal rsk, lfe uderwrg rsk, healh uderwrg rsk, o-lfe uderwrg rsk, couerpary defaul rsk ad agble asse rsk. CEIOPS 2, he advsg commee of he Europea Commsso o Solvecy 2, has developed a sadard formula ha leads o a requred solvecy marg ha s amed a coverg all rsks over a oe-year horzo wh a probably of 99,5%. However, surace compaes are ecouraged o develop her ow eral models o reflec he specfc rsks of he compay more accuraely. Gve he above, s clear ha he measureme of fuure cash flows ad s uceray hus becomes more ad more mpora. 1.2 Oule hs hess cosss of a colleco of papers ha each ackle a specfc ssue valuao or rsk maageme for surers. Frs chaper 2 wll cover some geeral coceps ha are used hroughou he hess, maly relag o sochasc processes of some kd. Lfe surace producs ofe have prof sharg feaures combao wh guaraees. Valuao of hese so-called embedded opos s oe of he key challeges marke valuao of he surace lables. Chaper 3 ad 4 are boh coverg he valuao of specfc embedded opos. I chaper 3 aalycal approxmaos for prces of swap rae depede embedded opos are developed. hese opos are very commo producs of Europea surers. Chaper 4 covers he valuao of Guaraeed Auy Opos, whch have bee wre by U.K. surace compaes for may years. he valuao of embedded opos s o oly a valuao ssue, s also a mpora aspec rsk maageme. Afer all, he rsk of varaos he 1 See Drecve of he Europea parlame ad of he coucl o he akg-up ad pursu of he busess of surace ad re-surace Solvecy 2 of he Europea parlame. 2 Commee of Europea Isurace ad Occupaoal Pesos Supervsors 2

13 prces of embedded opos s a rsk eleme ha has o be maaged by he surace compay, for example by hedgg hs rsk exposure. Impora rsks o be quafed for Lfe surers ad peso fuds are moraly ad logevy rsk. Chaper 5 ad 6 wll boh cover dffere aspec quafyg hese rsks. Chaper 5 wll roduce a ew sochasc moraly model for he populao of a coury. Chaper 6 wll focus o aoher sochasc model ha s he mssg lk o come o a full sochasc moraly model for specfc surace porfolos. he laer also gves he opporuy o quafy he bass rsk ha s volved whe surace porfolos are hedged wh srumes of whch he payoff depeds o coury populao moraly raes. he oher uderwrg rsks, relaed o he healh ad o-lfe busess, are reaed chaper 7. Usually, reservg ad rsk maageme for hs busess s based o acuaral echques ha are appled o aggregaed daa. hs chaper descrbes a ew sochasc reservg echque o he level of dvdual clams mcro-level. he remader of hs chaper coas a shor roduco o he subjecs covered he dffere chapers Chaper 3: Valuao of Swap Rae Depede Embedded Opos May lfe surace producs have prof sharg feaures combao wh guaraees. hese so-called embedded opos are ofe depede o or approxmaed by forward swap raes. I pracce, hese kds of opos are mosly valued by Moe Carlo smulao, a compuer esve calculao echque. However, for rsk maageme calculaos ad reporg processes, los of valuaos are eeded. herefore a more effce mehod o value hese opos would be helpful. I hs chaper aalycal approxmaos are derved for hese kds of opos. he aalycal approxmao for opos where prof sharg s pad drecly s almos exac whle he approxmao for compoudg prof sharg opos s also sasfacory. I addo, he proposed aalycal approxmao ca be used as a corol varae Moe Carlo valuao of opos for whch o aalycal approxmao s avalable, such as smlar opos wh maageme acos. hs cosderaely speeds up he calculao process for hese opos. Furhermore, s also possble o cosruc aalycal approxmaos whe reurs o addoal asses such as eques are par of he prof sharg rae Chaper 4: Valuao of Guaraeed Auy Opos usg a Sochasc Volaly Model for Equy Prces Guaraeed Auy Opos are opos provdg he rgh o cover a polcyholder s accumulaed fuds o a lfe auy a a fxed rae whe he polcy maures. hese opos were a commo feaure UK rereme savgs coracs ssued he 1970 s ad 1980 s whe eres raes were hgh, bu caused problems for surers as he eres raes bega o fall he 1990 s. Currely, hese opos are frequely sold he U.S. ad Japa as par of varable auy producs. 3

14 he las decade he leraure o prcg ad rsk maageme of hese opos evolved. Ul ow, for prcg hese opos geerally a process for equy prces s assumed where volaly s cosa. However, gve he log maures of he surace coracs a sochasc volaly model for equy prces would be more suable. I hs chaper explc expressos are derved for prces of guaraeed auy opos assumg sochasc volaly for equy prces ad eher a 1-facor or 2-facor Gaussa eres rae model. he resuls dcae ha he mpac of gorg sochasc volaly ca be sgfca Chaper 5: O sochasc moraly modelg he las deceum a vas leraure o sochasc moraly models has bee developed, maly for use rsk maageme. All well kow models have ce feaures bu also dsadvaages. I hs chaper a sochasc moraly model s proposed ha ams a combg he ce feaures from exsg models, whle elmag he dsadvaages. More specfcally, he model fs hsorcal daa very well, s applcable o a full age rage, capures he cohor effec, has a orval bu o oo complex correlao srucure ad has o robusess problems, whle he srucure of he model remas relavely smple. Also, he chaper descrbes how o corporae parameer uceray he model. Furhermore, a verso of he model s gve ha ca be used for prcg Chaper 6: Sochasc porfolo specfc moraly ad he quafcao of moraly bass rsk Chaper 5 wll descrbe several sochasc moraly models ha have bee developed over me, usually appled o moraly raes of a coury populao. However, hese models are ofe o drecly applcable o surace porfolos because: a For surers ad peso fuds s more releva o model moraly raes measured sured amous sead of measured umber of polces. b Ofe here s o eough surace porfolo specfc moraly daa avalable o f such sochasc moraly models relably. herefore, hs chaper a sochasc model s proposed for porfolo specfc moraly experece. Combg hs sochasc process wh a sochasc coury populao moraly process leads o sochasc porfolo specfc moraly raes, measured sured amous. he proposed sochasc process s appled o wo surace porfolos, ad he mpac o he hegh of he logevy rsk s quafed. Furhermore, he model ca be used o quafy he bass rsk ha remas whe hedgg porfolo specfc moraly rsk wh srumes of whch he payoff depeds o populao moraly raes Chaper 7: Mcro-level sochasc loss reservg he las deceum also a subsaal leraure abou sochasc loss reservg for he o-lfe surace busess has bee developed. Apar from few excepos, all of hese papers are based o daa aggregaed ru-off ragles. However, such a aggregae daa se s a summary of a uderlyg, much more dealed daa based ha s avalable o he surace compay. hs daa se a dvdual clam level as wll be referred o as mcro-level daa. I hs chaper s vesgaed wheher he use of such mcro-level clam daa ca mprove he reservg process. A realsc mcro-level daa se o geeral lably clams maeral ad jury from a Europea surace compay s modeled. Sochasc processes are specfed for he varous aspecs volved he developme of a clam: he me of occurrece, he delay bewee occurrece 4

15 ad he me of reporg o he compay, he occurrece of paymes ad her sze ad he fal seleme of he clam. hese processes are calbraed o he hsorcal dvdual daa of he porfolo ad used for he projeco of fuure clams. hrough a ou-of-sample predco exercse s show ha he mcro-level approach provdes he acuary wh dealed ad valuable reserve calculaos. A comparso wh resuls from radoal acuaral reservg echques s cluded. For our case-sudy reserve calculaos based o he mcro-level model are preferable: compared o radoal mehods, hey reflec real oucomes a more realsc way. 5

16 Chaper 2 Sochasc processes A he hear of mos valuao ad all rsk maageme calculaos are assumpos abou he sochasc processes of he releva varables. Sochasc processes requred for valuao are ofe of a dffere aure ha he sochasc processes requred for rsk maageme. For he valuao of embedded opos s mpora ha he uderlyg sochasc model s arbrage free. Arbrage free meas ha s o possble o geerae a o-zero payoff whou ay al vesme. A covee way o accomplsh hs s he use of a so-called rskeural model. he rsk-eural sochasc processes used hs hess are descrbed seco 2.1. For rsk maageme s more mpora ha he sochasc processes are as realsc as possble reflecg he dyamcs of he uderlyg sochasc varable. hs meas ha a real-world model s requred. he real-world sochasc processes used hs hess are descrbed seco Rsk Neural Sochasc Processes for Valuao I hs hess he opcs regardg valuao of embedded opos requre arbrage free sochasc processes for eres raes ad equy prces. he sochasc processes used are members of a more geeral class of models, he affe jump-dffusos. hs seco descrbes hs geeral class of models ad he specfc eres rae ad equy model used hs hess. hs wll be preceded by a shor roduco he oo of margales ad measures. he seco eds wh a shor dscusso abou sochasc processes for valuao of uhedgeable surace rsks Margales ad Measures he foudao of opo prcg heory s he assumpo ha arbrage opporues do o exs. Aoher mpora uderlyg cocep s compleeess of he ecoomy. If a ecoomy he payoffs of all dervave secures ca be replcaed by a self-facg radg sraegy, he ecoomy s called complee. If o arbrage opporues ad o rasaco coss exs a ecoomy, he value of a self-facg radg sraegy should be equal o he value of he correspodg dervave. If hs would o be he case, arbrage opporues exs. Harrso ad Kreps 1979 ad Harrso ad Plska 1981 brough he coceps of arbrage free ad compleeess ogeher wha s called he Fudameal heorem of Asse Prcg. Ay 6

17 asse whch has srcly posve prces for all fuure mes s called a umérare. Numérares ca be used o deomae all prces a ecoomy sead of Euro s or Dollars. A margale s a sochasc process wh a zero drf. Harrso ad Kreps 1979 ad Harrso ad Plska 1981 proved ha a couous ecoomy s complee ad arbrage free f for every choce of umérare here exss a uque equvale margale measure. I oher words, gve a choce of umérare, here s a uque probably measure such ha he relave prce processes are margales. hs mpora resul s very useful for opo valuao. For example, say ha prce a me of a opo H maurg a me relave o he prce of secury M s defed as V. he uder he releva measure Q M he process V s a margale. hs meas ha: M M H M H M H 2.1 V E V E H M E M M where E M [] s he expecao uder he releva measure. By choosg a covee umérare he opo prce calculao ca be smplfed cosderably some cases. Usually as a sarg po he rskless moey-marke accou s used as he umérare. Uder he uque probably measure correspodg o hs umérare he expeced reur o all asses s equal o he rsk-free rae. herefore, hs measure s called he rsk-eural measure, usually deoed as Q. Ofe sochasc processes eded o be used for valuao are defed he rskeural measure. However, somemes s more covee o chage o aoher measure. Cosder wo umérares N ad M wh he margale measures Q N ad Q M. Gema e al 1995 proved ha he Rado-Nkodym dervave ha chages he equvale margale measure Q M o Q N s gve by: N dq N / N 2.2 M dq M / M Grsaov s heorem saes ha f hs Rado-Nkodym dervave ca be wre as: 2.3 M exp s dw s 0.5 s ds where W M s a Browa moo uder he measure Q M. hs leads o: N M M N 2.4 W W s ds or dw dw d 0 So order o use Grsaov s heorem he process has o be foud ha yelds 2.3. A applcao of Io s Lemma shows ha d = dw M, showg ha s a margale 7

18 uder he measure Q M uder he codo Lemma o he rao 2.2 wll gve. 1 exp s ds. Now applyg Io s Affe Jump-Dffusos he sochasc processes used hs hess for eres raes ad equy prces are par of a broader class of models, called he affe jump-dffusos. A class of affe models was roduced frs he coex of eres raes by Duffe ad Ka Laer hs s geeralzed by Duffe e al 2000 ad Duffe e al he class of affe jump-dffusos provdes a flexble ad geeral model srucure combed wh aalycal racably. he laer feaure faclaes he calbrao ad smulao of such models. Well kow erm srucure models ha are members of hs class are, amogs ohers, he models of Hull ad Whe 1993, Cox e al 1985 ad Logsaff ad Schwarz Nex o he equy prce model of Black ad Scholes 1973 also he sochasc volaly models of Heso 1993, Schöbel ad Zhu 1999 ad he sochasc volaly wh jumps model of Baes 1996 are members of hs class. he class of affe jump-dffusos ca be defed as follows. Le X be a real-valued - dmesoal Markov process sasfyg: 2.5 dx X d X dw dz Where W s a sadard Browa moo,, x, ad Z s a pure jump process whose jumps have a fxed probably dsrbuo v ad arrve wh esy X. he jump mes of Z are he jump mes of a Posso process wh me-homogeeous esy. Posso processes are furher hghlghed seco 2.2. he process X s affe f ad oly f he dffuso coeffces are of he followg form: 2.6 x K0 K1x for K=K 0,K 1 x 2.7 x x H H x for H=H 0,H 1 x x x j 0 j 1 j 2.8 x l0 l1x for l=l 0,l rx 0 1x for = 0, 1 x where rx s he shor erm eres rae. Now ca be proved ha he characersc fuco of X, cludg he effecs of ay dscoug, s kow closed form up o he soluo of a sysem of Ordary Dffereal Equaos. Duffe e al 2000 show ha for u C he Fourer rasform u,x,, of X, codoal o flrao F, s gve by: 8

19 rx s ds ux,,, e e F e 2.10 ux A B X where A ad B sasfy he followg sysem of Rca equaos: da d KB B HB lb db d K B B H B l B wh boudary codos A = 0 ad B = u. he jump rasform s gve by: cz 2.13 c e dv z I geeral he soluos of Aad B have o be compued umercally, alhough he well kow models meoed above resul explc expressos for A ad B Gaussa eres rae models I hs hess he uderlyg eres rae model for he valuao s he class of mul-facor Gaussa models. Specal cases of hs class of models are he 1-facor ad 2-facor Hull-Whe model, whch are ofe used pracce. hese models are appealg because of her aalycal racably. he Gaussa eres rae models are also a specal case of he affe erm srucure models roduced by Duffe ad Ka he m-facor Gaussa model descrbes he sochasc process for he saaeous shor rae as follows 3 : 2.14 r 1 Y Q 2.15 dy CY d dw where W Q s a m-dmesoal Browa moo uder he rsk-eural measure ad C ad are m x m marces. C s a dagoal marx. he fuco s chose such a way ha he f of he model o he al erm srucure s perfec. he covarace marx of he Y-varables s equal o. he aalycal racably of hs model makes possble o oba bod prces aalycally, from whch swap ad zero raes ca be derved. he prce a me of a zero bod maurg a me s gve by: 3 See Brgo & Mercuro 2006 for a exesve explaao of ad prcg formulas for he 2-facor Gaussa model. 9

20 2.16 m D, A, exp B Y, 1 where B, 1 / A 1 exp A he expresso for A, s furher specfed for he 1-facor ad 2-facor case chaper Sochasc volaly model for equy prces I a semal paper Black & Scholes 1973 made a major breakhrough he prcg of equy opos. he uderlyg sochasc model for equy prces has become kow as he Black- Scholes model. he Black-Scholes model assumes he volaly o be cosa. However, pracce he volaly vares hrough me. For hs reaso a sgfca leraure has evolved o alerave models ha corporae sochasc volaly. Nex o leadg o more realsc dyamcs of he sochasc process for equy prces, hese models have he advaage ha hey provde a beer f of he model o acual marke opo daa. hs s a mpora feaure for beg able o adequaely prce more exoc opos such as embedded opos surace producs. Well kow sochasc volaly models are he models of Hull ad Whe 1987, Se ad Se 1991, Heso 1993 ad Schöbel ad Zhu he am chaper 4 s o combe a sochasc volaly model for equy prces wh a sochasc eres rae model. Va Haasrech e al 2009 show ha s possble o oba a explc expresso for he prce of Europea equy opos whe he Schöbel ad Zhu 1999 model s combed wh a sochasc Gaussa model for eres raes, explcly akg o accou he correlao bewee hose processes. ha makes hs combed model suable for valuao of he Guaraeed Auy Opos chaper 4. I he Schöbel ad Zhu 1999 model, he process for equy prce S uder he rsk-eural measure Q s: ds Q 2.17 r d v dws S0 S0 S dv v d dw Q v0 v v Here v, whch follows a Orse-Uhlebeck process, s he saaeous sochasc volaly of he equy S. he parameers of he volaly process are he posve cosas κ mea reverso, v 0 shor-erm mea, ψ log-erm mea ad τ volaly of he volaly Sochasc processes for valuao of uhedgeable surace rsks he valuao of surace lables also requres he valuao of uhedgeable surace rsks. For example, moraly models for he valuao of moraly or logevy lables or dervaves are gve by Dahl 2004, Schrager 2006, Cars e al 2006b ad Bauer e al he models of Dahl 2004 ad Schrager 2006 belog o he geeral class of affe 10

21 jump-dffusos defed paragraph ad as a resul allow for closed form expressos of he survval rae. Usually surace rsk models are calbraed o hsorcal daa ad are herefore defed he real world measure, deoed by P. Gve he echques meoed paragraph 2.1.1, oe could apply a chage of measure o rsk eural measure Q, uder whch he surace lably ca be valued. However, hs case oe crucal codo s o sasfed, beg he compleeess of he ecoomy. As explaed paragraph 2.1.1, he compleeess of he ecoomy forces he rsk eural measure Q o be uque. he marke for surace rsks s far from complee, meag ha he surace rsks are uhedgeable ad herefore a rage of possbles for Q exs. As meoed by Cars e al 2006a he choce of Q eeds o be cosse wh he lmed marke formao, bu beyod hs resrco he choce of Q becomes a modelg assumpo. A alerave mehod for valuao complee markes s he use of uly fucos ad he prcple of equvale uly, see Youg ad Zarphopoulou 2002, Youg ad Moore 2003 ad Youg hs prcple mples ha he maxmal expeced uly wh ad whou he specfc surace rsk are examed. he compesao a whch he surer s dffere bewee he wo alerave aleraves yelds he value of he uhedgeable surace rsk. However, hs approach s currely oly feasble for relavely smple producs. 2.2 Real World Sochasc Processes for Rsk Maageme As meo above, for rsk maageme s parcularly mpora ha he sochasc processes used realscally reflec he observed characerscs of he uderlyg sochasc varable. I chaper 5 ad 6 paramerc models are f o yearly observaos, leadg o me seres of fed varables. Sochasc processes have o be f o hese me seres, for whch he Auoregressve Iegraed Movg Average ARIMA models ca be used. hese are descrbed paragraph he sochasc processes eeded chaper 7 are of a dffere aure ad are descrbed paragraph ARIMA me Seres Models A semal work o he esmao ad defcao of ARIMA models s he moograph by Box ad Jeks Addoal deals ad dscusso of more rece opcs ca be foud for example Mlls 1990, Eders 2004 ad Hamlo A mpora ssue s wheher a me seres process s saoary, meag ha he dsrbuo of he varable of eres does o deped o me. If hs s o case, he frs sep would be o dfferece he me seres ul he dffereced me seres s saoary. Box ad Jeks foud ha usually oly oe or wo dfferecg operaos are requred. he geeral ARIMAp, d, q model for a me seres of a varable y ca be wre as: 2.19 y d y y * p q * * 0 y 1 j1 11

22 where he s ad s are he ukow parameers, he s are depede ad decally dsrbued ormal errors ad d represes he dfferecg, meag 0 y = y, 1 y = y y -1, 2 y = y y -1 - y -1 y -2, ec. he parameer p s he umber of lagged values of y, represeg he order of he auoregressve AR dmeso of he model, ad q s he umber of lagged values of he error erm, represeg he order of he movg average MA dmeso of he model. Box ad Jeks defe hree seps for he developme of a ARIMA model: 1 Model defcao ad model seleco: deermg he values for p, d, q. 2 Parameer esmao: eher by usg Maxmum Lkelhood or o-lear Leas Squares esmao. 3 Dagosc checkg: esg wheher he esmaed model mees he specfcaos of a saoary uvarae process. Ofe a exeso s eeded o allow he modelg of mulvarae me seres. hs requres a mulvarae geeralzao of he ARIMA process, see for example Verbeek Posso processes ad reewal processes he requred sochasc processes chaper 7 are of a dffere aure ha hose descrbed above. Posso processes ad he relaed reewal processes are covee coceps for modelg he developme process of dvdual clams. For a exesve overvew of hese echques, see Cook ad Lawless Posso Processes A Posso process descrbes suaos where eves occur radomly such a way ha he umbers of eves o-overlappg me ervals are depede. Posso processes are herefore Markov, wh a esy fuco: 2.20 H Pr N N 1 lm 0 Where N s he cumulave umber of eves occurrg over he me erval [0,] ad H s he process hsory. I he case where s cosa, =, he process s called homogeeous. Oherwse, s homogeeous. he above specfcao mples: 2.21 N N s ~ Posso u du s Poso Depede Marked Posso Process PDMPP I chaper 7 he dvdual clams process s modeled as a PDMPP. A marked Posso process wh esy ad posodepede marks s a process, Z ,..., N 12

23 where he clams coug process N s a homogeeous Posso po process wh esy, pos ad marks Z. he Z >0 are muually depede, are depede of he Posso po process N ad have me-depede probably assumpos. Reewal processes Relaed o he Posso process s he reewal process, whch he wag gap mes bewee successve eves are sascally depede: ha s, a dvdual s reewed afer each eve occurrece. Reewal models for wag mes are defed as processes for whch 2.23 H h N where h s he hazard rae ad N s he me sce he mos rece eve before. Ofe used models for he me o a eve, say, are he Expoeal, Webull ad he Gomperz dsrbuo. hese dsrbuos have he covee propery ha he hazard fuco has a smple form. he followg hazard fucos gu are mpled by hese dsrbuos: - ~ Expoeal hu = cosa hazard - ~ Webull, hu = u -1 - ~ Gomperz, hu = e u Oher possbles are a pecewse cosa specfcao for he hazard rae or he Cox proporoal hazard model see Cox

24 Chaper 3 Valuao of swap rae depede embedded opos* * hs chaper has appeared as: PLA, R. AND A.A.J. PELSSER 2008: Aalycal approxmao for prces of swap rae depede embedded opos surace producs, Isurace: Mahemacs ad Ecoomcs 44, pp Iroduco A mpora par of he marke valuao of lables s he valuao of embedded opos. Embedded opos are opos ha have bee sold o he polcyholders ad are ofe he more complex feaures surace producs. A embedded opo ha s very commo surace producs Europe, s a prof sharg rule based o a movg average fxed come rae, combao wh a mmum guaraee. hs fxed come rae s eher from a exeral source or could be he book value reur o a fxed come porfolo. For example, he Neherlads he prof sharg s ofe based o he so-called u-yeld, whch s more or less a average reur of several reasury raes. I oher pars of Europe, he book value reur o he fxed come porfolo s ofe he bass for he prof sharg. I pracce he exac raes are dffcul o deerme ad o projec forward, ad mpled volales from he marke are o avalable. herefore, ofe he euro swap rae s used as a proxy. So wha remas s he valuao of a opo o a movg or weghed average of forward ad hsorc swap raes. Mos surers use Moe Carlo smulaos for he valuao of her embedded opos. he advaage of hs s ha may kds of opos ca be valued wh cludg he more complex oes ad ha gves oe uform smulao framework ha s applcable for varous embedded opos. However, a mpora dsadvaage s he compuaoal me requres. Embedded opo calculaos are requred for Far Value reporg, Marke Cosse Embedded Value, Asse Lably Maageme, produc developme ad prcg, Ecoomc Capal calculaos ad Mergers & Acqusos. For mos of hese purposes several calculaos are requred. For he calculao of Ecoomc Capal for example or more smulaos 14

25 are used ad each of hese scearo's he marke value of lables ad hus he value of embedded opos has o be calculaed. Also for oher purposes, ofe sesves ad aalyss of chages are ecessary. If a surer he also exss of several busess us or legal ees, he oal compuaoal me ca be sgfca. herefore, aalycal soluos for he valuao of embedded opos would be very helpful. I hs chaper aalycal approxmaos are derved for he above meoed swap rae depede embedded opos. he uderlyg eres rae model s a mul-facor Gaussa model. hs model s very appealg because of s aalycal racably. Also, he model mplcly accous for he volaly skew o some exe, wha s mpora for hese kd of opos because hose are mos cases o a-he-moey. Because of hs he model s ofe used pracce mos cases he 1-facor or 2-facor Hull-Whe vara. Aalycal approxmaos are derved for he case of drec payme of prof sharg, as well as for he case of compoudg prof sharg. I case of very complex opos wh maageme acos, he aalycal approxmao for he drec payme case ca be used as a corol varae combao wh Moe Carlo smulao, reducg he compuaoal me o a grea exe. I could well be ha a surace compay has oher kds of embedded opos for whch o aalycal approxmaos are avalable. hese embedded opos probably have o be valued usg Moe Carlo smulao. Sce he mul-facor Gaussa models are ofe used pracce, he aalycal approxmao for he swap rae depede opos ca ha case be used cojuco wh he smulao model ha may be requred for he valuao of oher embedded opos. hs resuls a cosse uderlyg eres rae model for he valuao of embedded opos, despe he fac ha perhaps some of he opos are valued wh Moe Carlo smulaos ad ohers wh aalycal formulas. he bass for he aalycal approxmao s he resul of Schrager ad Pelsser 2006, who have developed a approxmao for swapo prces for affe erm srucure models of whch he mul-facor Gaussa models are a subse. hey deerme he dyamcs of he swap rae uder he releva swap measure ad hese dyamcs are approxmaed by replacg some low-varace margales by her me zero values. hs echque s already used exesvely he coex of Lbor Marke Models ad gve he resuls of Schrager ad Pelsser, also proves o work well a affe seg. By use of he Chage of Numérare echques developed by Gema e al 1995, he resul of Schrager ad Pelsser ca be used o derve aalycal approxmaos for swap rae depede opos. Mos of he exsg leraure o valuao of embedded opos surace producs focuses o U Lked producs, wh-profs producs or Guaraeed Auy Opos. For example, Grose ad Jorgese 2000, Schrager ad Pelsser 2004 ad Casella e al 2007 developed aalycal approxmaos for U Lked ype producs wh guaraees. Wlke e al 2003 use umercal echques o value Guaraeed Auy Opos, whle Sheldo ad Smh 2004 developed aalycal formulas for hese producs. Nelse ad Sadma 2002 ad Preul e al 2001 use umercal echques for valuao of Wh-Profs coracs. However, o our kowledge here has bee lle focus o prof sharg based o movg average fxed come raes, despe hs beg oe of he mos commo ypes of prof sharg 15

26 Europe. Our corbuo o he exsg leraure s ha we provde aalycal approxmaos for hese kds of prof sharg. Aalycal approxmaos for drec payme of prof sharg ad for compoudg prof sharg are gve, whle a combao wh reurs o oher asses such as eques s also possble. I addo, he proposed aalycal approxmao ca be used as a corol varae Moe Carlo valuao of opos for whch o aalycal approxmao s avalable, such as smlar opos wh maageme acos. hs poeally reduces he umber of smulaos requred o a grea exe. Some of he echques proposed hs chaper ca also be used for facal producs, such as opos o a average of Cosa Maury Swap CMS raes, callable CMS accrual swaps ad callable CMS rage oes. he remader of he chaper s orgazed as follows. Frs, seco 3.2 he characerscs of he swap rae depede embedded opos are descrbed. I seco 3.3 he uderlyg Gaussa eres model s gve. I seco 3.4 he Schrager-Pelsser resul for swapos s repeaed ad hs s appled o he drec payme case seco 3.5. I seco 3.6 possbles are gve for more complex embedded opos. he umercal examples are worked ou seco 3.7 ad coclusos are gve seco Swap rae depede embedded opos radoal o-lked lfe surace producs ofe guaraee a cera sured amou. Commo pracce was ad ofe sll s o calculae he prce of hs surace by dscoug he expeced cash flows wh a relavely low eres rae, called he echcal eres rae. Ofe hs s combed wh prof sharg, where some referece reur s pad ou o he polcyholder f hs exceeds he echcal eres rae, possbly uder subraco of a marg. here exs varous ypes of prof sharg, such as: - Prof sharg based o a exeral referece dex - Prof sharg based o he book or marke value reur o he uderlyg vesme porfolo - Prof sharg based o he performace ad profs of he surace compay - Prof sharg of he so-called wh-profs producs, where regular ad ermal bouses are gve hough he lfe of he produc, based o he reur of he uderlyg vesme porfolos. he erms of hese polces ofe coa maageme acos ha allow he surace compaes o reduce he rsks of hese producs. I mos cases where he prof sharg rae depeds o a cera fxed come rae, he exac prof sharg rae s eher very complex or o fully kow, or mpled volales from he marke are o avalable. I pracce, hese kds of opos are ofe valued usg a average forward swap rae as a approxmao for he prof sharg rae. he prof sharg payoff PS year s ha case: 3.1 PS L Max{ c R K,0} 16

27 where L s he prof sharg bass, c s he perceage ha s dsrbued o he polcyholder ad K s he srke of he opo. he srke equals he sum of he echcal eres rae R ad a marg. I mos cases, eher he marg or he c s used for he beefs of he surer. R s he prof sharg rae ad s a weghed average of hsorc ad forward swap raes. he prof sharg as descrbed 3.1 s a call opo o a rae R ad has o be valued usg opo valuao echques. he prof sharg s eher pad drecly or s beg compouded ad pad a he ed of he corac. Noe ha depeds o he specfc prof sharg rules wheher he swap rae s a good approxmao for he prof sharg rae. hs has o be verfed for each specfc prof sharg arrageme. Below wo examples are gve of prof sharg arragemes where he swap rae s ofe used as approxmao pracce. Example 1 book value reur o uderlyg porfolo Oe of he mos commo forms of prof sharg across he Europea lfe surace busess s he oe where he prof sharg rae s based o he book value reur of he uderlyg fxed come porfolo 4. o be able o value hs opo, assumpos have o be made abou he revesme sraegy. A example of how hs problem s ofe ackled pracce s o assume: - a cera average urover rae - a revesme sraegy favorg m-year maury asses. - he m-year swap rae beg a approxmao for he yeld o he m-year maury asses Gve hese assumpos he book value reur of he porfolo ca be modeled as follows: 3.2 R 1 R 1 y, m where y,+m s he m-year swap rae a me. he book value reur o me ca also be expressed erms of he curre book value reur R0, leadg o a expoeally weghed movg average: 3.3 R 1 R0 y, m 1 0 beg a weghed combao of forward swap raes ad he curre book value reur. Aoher approach ha s ofe used s approxmag he book value reur by a movg average of swap raes: R y m, 1 where = 1/ s he umber of fxgs of he movg average. 4 hs s commo pracce for example Frace, Germay, Ialy, Czech Republc, Swzerlad ad Norway. 17

28 Example 2 u-rae prof sharg he Neherlads I he Neherlads he mos commo form of prof sharg s based o a movg average of he so-called u-rae. he u-rae s he 3-mohs average of u-rae-pars, where he subseque u-raepars are weghed averages of a effecve reur o a baske of goverme bods. hs leads o a complcaed expresso, ad o mpled volales are avalable for goverme bods. herefore, s commo pracce he Neherlads o approxmae he u-rae or he u-yeld pars by a swap-rae 5. ha meas ha he prof sharg rae s approxmaed by a movg average of swap raes, as 3.4. Besdes he drec payme ad compoudg versos of 3.1, oher varas of hs prof sharg exs, such as: 1 Prof sharg cludg he reur o a addoal asse 2 Compoudg prof sharg wh addoal maageme acos or oher complex feaures. I case of 1, he uderlyg vesme porfolo also coas addoal o-fxed come asses. hs meas ha he prof sharg rae s a combao of a weghed movg average of swap raes ad he reur o addoal asses. he prof sharg rae could he be expressed as: FI S 3.5 R w y, k w r k k km j Sj k s j where w s he wegh addoal asse S j, S j FI S w wl k 1. r S s he reur o ha asse ad j I case of 2, he surer has added maageme acos or oher complexes o he prof sharg rules, maly o lower he rsk exposure for he surer. I he followg secos aalycal approxmaos are developed for prces of embedded opos where he prof sharg rae depeds o or s approxmaed by forward swap raes. Noe ha he developed formulas are approxmag swap rae depede embedded opos. Whe cosderg he resuls or usg he formulas oe always has o be aware of he fac ha he frs error s roduced whe he swap rae s beg used as a proxy for he prof sharg rae. 3.3 he uderlyg eres rae model he aalycal approxmaos hs chaper are based o a uderlyg mul-facor Gaussa eres rae model. hs model s descrbed paragraph Paragraph gves a 5 Hsorcal daa ha show ha u-rae pars have behaved smlarly as swap raes he pas, s avalable upo reques. 18

29 dscusso wheher smlar echques as developed hs chaper ca be used for aalycal valuao of he opos descrbed seco 3.2 gve oher uderlyg eres rae models Mul-facor Gaussa models As meoed paragraph 2.1.3, he uderlyg eres rae model for he valuao s he class of mul-facor Gaussa models. hese models are very appealg because of her aalycal racably. hs makes he model easy o mpleme, whle here are also more possbles for aalycal approxmaos or soluos for embedded opos I he swapo marke, he observed mpled Black volaly s varyg for dffere srke levels, leadg o he so-called volaly skew. hs volaly skew exss because he marke apparely does o beleve logormally dsrbued swap raes. Isead, he volaly skew seems o dcae a dsrbuo ha s closer o he ormal dsrbuo 6. herefore, he Gaussa models mplcly accou for he volaly skew o a cera exe. hs s also a appealg propery of hese models he coex of embedded opos surace producs, sce hese opos are mos cases o a-he-moey Valuao for oher eres rae models hs paragraph gves a dscusso wheher smlar echques as developed hs chaper ca be used for aalycal valuao of he opos descrbed seco 3.2 gve oher uderlyg eres rae models. Geeral affe models Schrager ad Pelsser 2006 developed approxmaos for swapo prces for geeral affe eres rae models. For o-gaussa affe models hey come o a approxmae soluo for swapo prces for whch oly a umercal egrao s ecessary. A approxmao for he characersc fuco of he swap rae uder he swap measure ad he mehod of Carr ad Mada 1999 s used for hs. As a frs sep hs process hey derve approxmae dyamcs for he swap rae smlar fasho as descrbed seco 3.4. Wh a addoal approxmao a square-roo process for he swap rae resuls. Dassos ad Nagaradjasarma 2006 develop explc prces for Asa opos, gve a uderlyg square roo process. hey also oba dsrbuoal resuls cocerg he square-roo process ad s average over me, cludg aalyc formulae for her jo desy ad momes. For he embedded opos dscussed hs chaper a suggesed approach would be o use he approxmae dyamcs for he swap rae from Schrager ad Pelsser 2006 ad combe hs wh he echques Dassos ad Nagaradjasarma Lbor Marke Model LMM As meoed seco 3.4, he approxmao echque used hs chaper s already used exesvely he coex of Lbor Marke Models. For example, Brgo ad Mercuro 2006 use 6 See Lev 2004 for a dscusso o hs ssue. 19

30 he echque for approxmao of swapo prces he LMM model. Gaarek 2003 uses o approxmae prces of Cosa Maury Swaps. Now whe usg hs echque, he resulg dsrbuo of he approxmae swap rae he LMM model s logormal. However, for he valuao of he embedded opos hs chaper he dsrbuo of he average swap rae s eeded. I case he swap rae s logormally dsrbued, he dsrbuo of he average swap rae s ukow. hs s a well kow problem he coex of valuao of Asa opos. Mehods for approxmae aalycal valuao of opos o he average of logormally dsrbued varables are proposed, amogs ohers, Levy 1992, Curra 1994 ad Rogers ad Sh Lord 2006 gves a overvew of exsg mehods, compares he qualy of hose umercally ad develops approxmaos ha ouperform he oher mehods. Swap Marke Model SMM I a sadard SMM as proposed by Jamshda 1997 each swap rae s modeled s ow swap measure, makg hard o apply for prcg of mos exoc eres rae producs. hs could be oe of he reasos ha he SMM has o bee dscussed exesvely facal leraure. he co-sldg SMM proposed by, amogs ohers, Peersz ad Va Regemorel 2006 seems promsg hough ad s applcable especally for Cosa Maury Swap CMS ad swap rae producs. I he SMM he swap rae s modeled drecly a logormal seg, so o approxmao of he dsrbuo of he swap rae he swap measure s ecessary. A prce for he prof sharg opos dscussed hs chaper ca be obaed by applyg he releva covexy ad mg adjusmes ad usg oe of he above meoed echques for approxmae aalycal valuao of Asa opos. 3.4 he Schrager-Pelsser resul for swapos Schrager ad Pelsser 2006 developed a approxmao for swapo prces for affe eres rae models. I hs seco her ma resul for he Gaussa models s repeaed. he swap rae y,n s he par swap rae a whch a perso would lke o eer o a swap corac wh a value of 0, sarg a me frs payou a me +1 ad lasg ul N. he swap rae a me s gve by: 3.6 y, N D, N k 1 D, Y k 1 D, k N D, P D, 1, N N Y where k 1 s he marke coveo for he calculao of he daycou fraco for he swap payme a k. Whe usg P +1,N as a umérare, all P +1,N rebased values mus be margales uder he measure Q +1,N, assocaed wh hs umérare. ha meas ha y,n s a margale uder hs so-called swap measure, whch s roduced by Jamshda Whe 20

31 applyg Io s Lemma o he model defed 2.14 ad 2.15 he followg dyamcs for he swap rae y,n uder he swap measure resul: y, N 1, N 3.7 dy, N dw Y Where dw +1,N s a m-dmesoal Browa moo uder he swap measure Q +1,N correspodg o he umérare P +1,N. Schrager ad Pelsser 2006 deerme he paral dervaves 3.7, whch are sochasc, ad approxmae hese by replacg low-varace margales by her me zero values. hs echque s already used exesvely he coex of Lbor Marke Models 7 ad gve he resuls of Schrager ad Pelsser, also proves o work well a affe seg. hs approxmao makes he swap rae volaly deermsc ad hus leads o a ormally dsrbued forward swap rae. he approach descrbed leads o a aalycal approxmao for he egraed varace of y,n assocaed wh a x N swapo over he erval [0, ] for he proof, see appedx 3a: m m A A jj 2 ˆ j e N, j C N, C N, 1 j1 A A jj ˆ j where s he eleme,j of ad N ~ 1 A P A N P Y A k P 3.9 C, N e D 0, e D 0, N y, N 0 k 1 e D 0, k A k 1 where D P, = D, / P +1,N, he bod prce ormalzed by he umérare. he resul s a easy o mpleme aalycal approach o calbrae Gaussa models o he swapo marke. A ce by-produc of he approach as opposed o oher approaches for approxmag swapo prces s ha he dyamcs of he swap raes are approxmaed. hese approxmae dyamcs ca be used for approxmag prces of oher swap-rae depede opos. 3.5 Aalycal approxmao drec payme Assume ha he prof sharg rae a me s a weghed average of -year maury swap raes wh weghs w k ad he averagg perod s from me s o me : 3.10 R wk yk, k k where w k =1. k s 7 See Aderse ad Adrease 1998, Gaarek 2003 ad Brgo ad Mercuro