PRICING OF EXOTIC OPTIONS ON LIFE INSURANCE CONTRACTS Rami Yosef *

Transcription

1 PRICING OF EOIC OPIONS ON LIFE INSURANCE CONRACS Rami Yosf * ABSRAC W considr a Europan and Amrican pu opion dfind on pur ndowmn insuranc and risk insuranc conracs rspcivl. s xoic opions giv oldr of opion or bnficiar of said oldr opporuni o xrcis opions and arn diffrnc bwn fuur valu of insuranc bnfi discound b a fixd inrs ra- srik pric of opion wic dfind in opion conrac and fuur valu of insuranc bnfi discound b ral inrs ras wic opion wrir acivs on invsmns roug xrcis da. randomnss of inrs ra is modulad b wo socasic procsss: Ornsin-Ulnbck procss and Vasick procss. In ac cas considrd an xplici xprssion of valu of opion conrac is givn as ar numrical xampls. INRODUCION W propos an xoic opion dfind on lif insuranc conracs wrb insurd paris can bu Europan and Amrican pu opions on ir insuranc bnfi. kind of opions proposd r is a p of gambl bwn opion wrir and insurd paris wo purcas opions on inrs ra opion wrir will aciv b xrcis da of opions. W firs considr a Europan pu opion dfind on a pur ndowmn insuranc conrac. In a pur ndowmn insuranc conrac insurd will av insuranc bnfi onl b surviving roug mauri da of polic conrac. In a Europan opion conrac opion can b xrcisd onl on xrcis da of opion conrac. W suggs a combinaion of s wo ps of conracs o form an xoic-europan opion maning a opion oldrs can xrcis opion onl if surviv roug xrcis da. In is opion an invsor inrsd in buing is conrac as a subjciv viw on inrs ra a inks opion wrir will aciv on is invsmns and is willing o gambl on i. On or and opion wrir inks a could aciv a igr inrs ra an is wrin in opion conrac so slls a pu opion in wic as a commimn o pa opion oldr a xrcis da if opion oldr survivs unil is da an amoun of mon B wic is bnfi insuranc dfind in opion conrac discound b diffrnc bwn fixd inrs ras subjciv inrs ra of opion bur and ral inrs ras acivd on invsmn. If opion wrir acivs a igr inrs ra an on wrin in opion conrac valu of opion conrac is zro. us opion oldrs gain from olding is opion onl if wo condiions ar m: nd o surviv roug xrcis da plus opion wrir as o aciv a lowr inrs ra an on dfind in opion conrac. * Scool of Managmn Dparmn of Businss Adminisraion Bn-Gurion Univrsi of Ngv Br-Sva 845 Isral. Pon: mail: [email protected]

2 No a is p of opion is a pu opion bcaus wn xrcising opion conrac opion oldr can sll o opion wrir for a srik pric naml a fuur valu of dfind insuranc bnfi discound b a fixd subjciv inrs ra and gain diffrnc bwn srik pric of opion and fuur valu of insuranc bnfi discound b ral inrs ra acivd b opion wrir a xrcis da. Also no a i is no paricularl difficul o monior inrs ras opion wrir acivs b xrcis da of opion: in Isral. For xampl insuranc companis ar obligad o rpor ac mon o govrnmn insuranc suprvisor inrs ra acivd for ac p of lif insuranc conrac. rsul is full ransparnc of inrs ras acivd. Morovr no a fixd inrs ra dfind in opion conrac sould b igr an inrs ra of long-rm govrnmn or corpora bonds; orwis no on will purcas is p of opion conrac. Givn aformniond circumsancs opion wrirs nd o invs in sock mark or in or drivaivs in ordr o aciv a igr inrs ra an on dfind in opion conrac o nsur a i is no xrcisd. Nx w considr an Amrican pu opion dfind on a risk insuranc conrac. In a risk insuranc conrac if insurd dos no surviv roug mauri da of polic conrac bnficiar rcivs sum assurd from insuranc compan as dfind in insuranc conrac afr da occurrnc. In an Amrican pu opion opion can b xrcisd a an poin during lif of opion. W suggs a a combinaion of s wo conracs consius an xoic opion. is mans a if opion oldr dis prior o mauri da bnficiar could xrcis opion conrac and rciv diffrnc bwn commimn of opion wrir wic is fuur valu of bnfi insuranc discound b a fixd inrs ra- srik pric of opion conrac and fuur valu of insuranc bnfi discound b ral inrs ras wic opion wrir acivs on is invsmns. us bnficiar of opion oldr onl gains from xrcising is opion if opion wrir acivs a lowr inrs ra an on dfind in opion conrac. No a is p of Amrican pu opion is no a pical on in sns a ownr of opion conrac dos no coos wn o xrcis opion sinc xrcis da dpnds on da of opion oldr wic is supposd o b random unlss w allowd suicid - wic in our cas w do no. Bu suc an Amrican opion can b viwd in sns a xrcis da could b an da unil nd of rm of opion conrac For bo ps of opion conracs considrd r w us wo kinds of socasic procsss o modula randomnss of inrs ras: Ornsin-Ulnbck procss and Vasick procss. In ac on of s socasic procsss w valua prics of s xoic opions. Acuaris and financ rsarcrs av long bn awar of random naur of inrs ras paricularl wn daling wi long-rm conracs. Rcn sudis also ingra mamaics of financ as a par of mamaics of insuranc. Saring wi uni-linkd lif insuranc Brnnan and Scwarz 976 rcognizd opion srucur of a uni-linkd lif insuranc conrac wi a guaran. Bris and d Varnn 994 dal wi bonus opion of polic-oldr and bankrupc opion of ownrs of insuranc compan in rms of coningn claims

3 analsis. Or rcn sudis a dal wi bonus opion ar Milrsn and Prsson 998 or Grosn and Jørgnsn. Or conxs in wic wo or mor socasic procsss govrn lif of a pu opion a av bn sudid in liraur ar pricing of pu opions on dfaulabl bonds or swaps and pricing of Asian xcang ra opions undr socasic inrs ras. sud of opions in or conxs in wic wo or mor socasic procsss govrn lif of dfaulabl bonds or swaps as a long isor bu sminal papr in is fild is mos likl on wrin b Duff and Singlon 997. r risklss insananous inrs ra is adjusd b firm issuing bond or swap dfaul azard o ild a modl a formall rsmbls dfaul-fr cas and a can b rsolvd in a similar mannr. adjusmn owvr involvs sum of wo azard-lik rms a impl indpndnc dspi fac a som p of rlaionsip probabl xiss bwn dfaul azard and insananous inrs ra. Similarl Asian opions ar wrin on xcang ra in a wo-currnc conom. In valuing s opions bo socasic naur of forign and domsic zro-coupon bond prics and xcang ra procss ar modld. A rcn ramn of problm is givn b Nilsn and Sandmann in wic wo counris' zro-coupon bond pric procsss ar assumd o b indpndn gomric-brownian moions bu xcang ra procss is modld b a socasic diffrnial quaion a is a gomric Brownian moion basd on diffrnc of sor-rm inrs ra procsss in wo counris. Bo discr and coninuous-im socasic modls for inrs ra procsss av bn prsnd in acuarial liraur primaril Gaussian auorgrssiv procsss. Panjr and Bllous 98 provid a oroug rviw of auorgrssiv procsss of ordr AR and of ordr AR wi consan volaili varianc. sow ow forc of inrs ma b modld according o an AR or AR procss lading o formula for momns of cumulaiv forc of inrs and annui crain funcion wic is prsn valu of $ n ars nc. 994 and rfrncs rin discusss modling forc of inrs vrsus modling accumulad forc of inrs using a coninuous-im auorgrssiv procss of ordr : procss wi a suprimposd linar rnd and Winr procss wi linar rnd. Mor rcnl Milvsk and Promislow modld sor-ra procss islf as a Cox-Ingrsoll-Ross CIR procss. CIR procss is an AR procss in coninuous im wi random volaili a is proporional o squar roo of insananous inrs ra jus prior o im. Addiionall acuarial liraur as also considrd pu opions dfind on pnsion insuranc. Hisoricall sud of pu opions on pnsion plans could b rgardd as an xnsion of ''pnsion pu opion'' approac of Sarp 976 and Bickslr and Cn o a ''pnsion call'' modl a dscribs gnral pnomnon of unwillingnss of fund sponsors o rmina ovr-fundd plans. No a in our cas of Europan pu opion dfind on pur ndowmn insuranc insad of rciving sum assurd as a lump sum i can b rcivd as an annui. In is cas i can b considrd as a Europan pu opion dfind on a pnsion annui. A pnsion pu opion as dscribd b Sarp 976 is sponsor's rig o abandon an undrfundd pnsion plan. If sponsor xrciss pnsion pu opion i lavs rsponsibili of sorfall o ir bnficiaris or o PBGC Pnsion Bnfi Guaran Corporaion in sor rm. Sarp 976 argus a o prsrv valu of pnsion pu opion; sponsor would no b moivad o

4 rmina an undr-fundd plan. Consqunl valu of arl rminaion of a dfind bnfi plan o sponsor is similar o xrcis valu of a call opion on pnsion ass porfolio o insurd. xrcis pric of call opion is qual o vsd bnfi a im plan rminaion. Or rcn sudis combining call opions on pnsion annui insuranc plans wr conducd b Balloa and Habrman 3 and Yosf Bnzion and Gross 4. rmaindr of is papr is srucurd as follows: In Scion w prsn a Europan pu opion dfind on a pur ndowmn insuranc conrac and find an xplici xprssion for valu of is opion conrac in cas of and Vasick procsss wic modula randomnss of inrs ra procss. Furrmor som imporan faurs of s procsss ar providd. In Scion 3 w solv cas of an Amrican opion conrac dfind on risk insuranc conracs in wo cass of socasic procsss prsnd abov. Numrics and conclusions ar givn in Scion 4. No a w mak no amp o facor in xpnss profis and or adminisraiv cargs bu rar assum a vring is prsnd on a n basis. EOIC EUROPEAN PU OPION ON PURE ENDOWMEN INSURANCE main purpos of is scion is o valua xoic Europan pu opion dfind on pur ndowmn insuranc as prsnd abov undr socasic srucur of inrs ras. As aformniond in is opion an invsor inrsd in buing is conrac as a subjciv viw on inrs ra a inks opion wrir will aciv on is invsmns and is willing o gambl on i. On or and opion wrir inks a could aciv a igr inrs ra an is wrin in opion conrac so slls a pu opion in wic as a commimn o pa o opion oldr a xrcis da in cas a opion oldr survivs unil is da an amoun of mon B wic is bnfi insuranc dfind in opion conrac discound b diffrnc bwn fixd inrs ras and ral inrs ras acivd on invsmn. If opion wrir will aciv a igr inrs ra an on wrin in opion conrac valu of opion conrac is zro. W can wri is xoic Europan pu opion conrac wr morali and inrs ra ar socasic b: P E p Pr > B wr - im from o nd of polic conrac. - random variabl a dscribs oal lifim of an individual. - consan risk-fr inrs innsi. - fixd subjciv inrs ra dfind in opion.

5 - consan facor. - random inrs procss. B - bnfi insuranc dfind in opion conrac. P p - dnos prsn valu of is xoic pu opion dfind on pur ndowmn insuranc - a im. No a formula basd on assumpion a im-a-da random variabl is socasicall indpndn of mark ras som masur. Also no a srik pric of is opion conrac is fuur valu of bnfi insuranc discound b fixd inrs ra: B. In conras is ral inrs ra a opion wrir acivs on is mark invsmns roug xrcis da. is inrs ra cangs randoml according o socasic srucur of inrs ra procss. As aformniond w assum a socasic srucur of inrs ra follows wo ps of socasic procsss: procss and Vasick procss. s wo procsss av inrsing baviors. procss as an advanag in a is sampl funcions nd o rvr o iniial posiion a propr a sms appropria for man inrs ra scnarios. fini dimnsional disribuions ar normal and procss as a Markovian propr s Bkaman and Fulling Vasick modl as a ndnc o flucua around a fixd inrs ra > wi an vnuall sabilizing volaili. conncion bwn s wo procsss and mor abou and Vasick procsss will b dscribd in following subscions. HE ORNSEIN-UHLENBECK PROCESS L B b a sandard Brownian moion and l b uniqu soluion of socasic diffrnial quaion: d db x wr > >. is rmd Ornsin-Ulnbck procss. I is wll known a soluion of is a Markov procss wi coninuous sampl pas and Gaussian incrmns. B Karlin and alor 98 p. 33 N x wr / [ ]. Dno b E Laplac ransform L of ; i can n b wrin b x. 3 No a procss as following propris assuming is a random variabl:

6 . min s s Var s Cov Var Var x E E If iniial random variabl as a normal disribuion wi man zro and varianc n is a saionar zro-man Gaussian procss wi covarianc funcion s s ρ s Bkman and Fulling99. Now for valuaion of w firs prov following lmma: Lmma L follow procss as dscribd in n for > siz E can b wrin b: Φ Φ x x wr is cumulaiv disribuion funcion of normal disribuion is L of procss wic is givn in 3 a Φ and wr. / Proof. Dno b cumulaiv disribuion funcion of procss a df a poin n for >

7 df df df E < >. Now sinc w can solv s wo ingrals and g x N Φ Φ Φ x x df x wr. Dno b pric of is xoic Europan pu opion dfind on pur ndowmn insuranc undr procss. us w can rwri pric of xoic Europan pu opion prsnd in b: P p Pr B x x P p > Φ Φ 4

8 wr / is L of procss givn in 3 a poin > and wr. HE VASICEK MODEL W ar now inrsd in valuaing wr inrs ra is modulad b Vasick procss. Dno b Vasick procss a is dfind as a diffusion procss saisfing socasic diffrnial quaion: d γ d db 5 wr γ > and B is sandard Brownian moion wi drif and varianc pr uni im s Baxr and Rnni 996 p.53. In rms of a socasic ingral soluion of 5 is givn b s γ γ db s. 6 According o 6 as a drif owards γ of sa-dpndn siz γ wic is us proporional o disanc from γ. No a w can rprsn Vasick procss b γ 7 wr is procss givn in. us from 7 and 3 w can wri L of Vasick procss b: x γ E R. 8 Lmma L follow Vasick procss as dscribd in 5. W can n wri siz E b:

9 ] [ Φ Φ x x γ γ wr is cumulaiv disribuion funcion of normal disribuion is L of Vasick procss givn in 8 a Φ and wr. / Proof. Sinc and using 7 x N as a normal disribuion wi man and varianc x γ. So following proof of Lmma rsul is as follows. pric a im zro of s xoic Europan pu opions dfind on pur ndowmn insuranc undr Vasick procss can b wrin b Vasick P p B x x P Vasick p Pr ] [ > Φ Φ γ γ wr is L of Vasick procss givn in 8 a poin > and wr. /

10 EOIC AMERICAN PU OPION ON RISK INSURANCE is scion xamins pricing valu of a pu opion dfind on risk insuranc undr socasic srucur of inrs ras. As prviousl mniond is p of opion givs bnficiar of opion oldr opporuni o xrcis is opion for a srik pric dfind in opion conrac onl in cas of da of opion oldr prior o xrcis da of opion. Sould opion oldr surviv roug xrcis da wor of is opion is zro. No a da of opion oldr can occur a an im prior o xrcis da maning a opion could b xrcisd b bnficiaris of opion oldr an im prior o xrcis da. Now valu of is Amrican pu opion can b wrin b: P ri E B wr B as dfind in abov subscion and wr P ri dnos prsn valu of is pu opion a im. W now urn our anion o calculaing valu of opion conrac prsnd in undr procss dnod b P ri and undr Vasick procss dnod b Vasick P ri. W bgin wi Vasick procss. Lmma 3 L follow Vasick procss as dscribd in 5. n for > w can wri b: x Vasick γ P B d ri Φ F [ x γ ] B Φ df % wr df is cumulaiv disribuion funcion of Φ is cumulaiv disribuion funcion of normal disribuion and procss givn in 5. Proof. L τ min[ ]. n for > % is L of Vasick

11 Vasick τ τ τ P ri BE τ τ τ BEτ E τ { τ τ. τ > } Now siz of condiional xpcaion wi rspc o τ could b wrin b: E τ τ τ τ us following Lmma w g τ Φ τ Φ τ { } { } τ τ τ τ τ df τ > τ > τ τ x τ τ γ τ [ x τ τ γ τ τ τ ]. Now givn a τ for < and a valu of is opion is for > aking xpcaion wi rspc o τ w g rquird rsul of wic is providd in. W ar now inrsd in valuaing procss. P ri wic is valu of is opion undr Lmma 4 L follow procss as dscribd in. n for > w can wri b: x P B Φ df ri x [ ] B Φ df wr df is cumulaiv disribuion funcion of Φ is cumulaiv disribuion funcion of normal disribuion and procss givn in 3. is L of

12 Proof. Following proof of Lmma 3 and ling γ w obain rquird rsul. NUMERICS AND CONCLUSIONS W now considr wo cass of random variabl of oal lifim of an individual. firs cas is an xponnial lifim wr Pr τ * > posiiv consans τ. scond cas is Gomprz s low: μ wc for posiiv consans w and c. In is cas w can wri survival lifim as: Pr > w ln c c ag c ag ag for wr paramr ag rfrs o prsn ag of insurd. In ac cas w will find prics of pu opions Vasick p ri P and P i.. formulas 4 9 and formulas rspcivl undr svral assumpions of paramrs. W compar s rsuls o prics wr probabili of insurd o surviv roug xrcis da is i.. Pr >. Vasick P ri P p Now suppos consan paramrs of procsss ar:.5... γ.7 x.5 and a bnfi insuranc is B $. rsuls will b givn for a scnario of fixd inrs ra for som posiiv τ and for ag paramr in following abls is givn in ars. Also suppos a consan paramrs of Gomprz s low ar: w -4 c.. abl prsns rsuls for Europan pu opion dfind on pur ndowmn insuranc i.. formulas 4 9 and abl prsns rsuls for Amrican pu opion dfind on risk insuranc i.. formulas. abl oulins prics of Europan pu opion dfind on pur ndowmn insuranc and snsiivi o consan paramrs. As mniond in Scion srik pric for is opion on xrcis da is B maning a prsn valu of gain from xrcising is BE. W can look for xampl a firs cas wr consans ar.3 τ. 5. No a assuming. 3 is qual o assuming a fixd inrs ra of 3.455% pr ar. prics of is opion conrac in cas of procss wr survival lifim is Pr > 5 is $.83 and in cas of Vasick procss is $.88. pric of is opion in xponnial cas is $.78 in procss and $.787 in Vasick procss. is mans a dclin of abou 5% of pric of opion from crain lifim cas and a dclin of abou.% comparing Gomprz s cas wi insurd ag 3 and 3% o insurd ag 4. Furr no a opion oldr pas a im zro $.78 in procss and will rciv an amoun of mon wi a prsn valu of $.83. is mans a inrs ra on invsmns is 5.4% in cas of survival roug xrcis da.

13 ABLE Prics of Europan pu opions on pur ndowmn insuranc: Formulas 4 9. p Pr > τ * 4 ag.. ln. Vasick p P p Vasick p τ ag P P P P p Vasick P p abl indicas prics of Amrican pu opion dfind on risk insuranc. If w ak for xampl s consan paramrs.3 τ.5 3 w can s prics of is opion conrac in cas of procss of a crain lifim is $4.497 and in cas of Vasick procss is $4.76. pric of is opion in xponnial cas is $.356 in procss and $.358 in cas of Vasick procss consiuing a rmndous diffrnc bwn prics. is diffrnc drivs from lifim probabili of ovr 3 ars. Addiionall diffrncs xis bwn opion prics undr Gomprz low of morali and xponnial cas prdominanl du o lack of mmor wi rgard o ag of insurd paris aribud o xponnial lifim.

14 ABLE Prics of Amrican pu opions on risk insuranc: Formulas. Pr > τ * 4 ag.. ln. τ ag P p Vasick Vasick Vasick Pp Pp Pp p P p P o conclud w no a suggsion of s xoic opions could lad insuranc companis if w ink of m as opion wrirs o b mor involvd in capial mark an objciv a is vr imporan o all paris involvd paricularl in a counr as small as Isral REFERENCES Balloa L. and S. Habrman 3 Valuaion of Guarand Annui Convrsion Opions Insuranc; Mamaics and Economics 33: Baxr M. and A. Rnni 996 Financial Calculus Cambridg Univrsi Prss. Bkaman J. A. and C. P. Fulling 99 Inrs and Morali Randomnss in Som Annuiis Insuranc: Mamaics and Economics 9: Bkaman J. A. and C. P. Fulling 99 Exra Randomnss in Crain Annui Modls Insuranc; Mamaics and Economics : Brnnan M. J. and E. S. Scwarz 976 Pricing of Equi-Linkd Lif Insuranc Policis wi an Ass Valu Guaran Journal of Financial Economics 3: 95-3.

15 Bickslr J. L. and A. H. Cn 985 Ingraion of Insuranc and axs in Corpora Pnsion Srag Journal of Financ Bris E. and F. D Varnn 994 Lif Insuranc in a Coningn Claim Framwork: Pricing and Rgulaor Implicaions Gnva Paprs of Risk an Insuranc or 9: Duffi D. and K. Singlon 997 An Economric Modl of rm Srucur of Inrs Ra Swap Yilds Journal of Financ 54: Grosn A. and P. L. Jørgnsn Fair Valuaion of Lif Insuranc Liabiliis: Impac of Inrs Ra Guarans Surrndr Opions and Bonus Policis Insuranc: Mamaics and Economics 6: Karlin S. and H. M. alor 98 A Scond Cours in Socasic Procsss nd. d. Nw York: Acadmic Prss Inc.. Milvsk M. A. and D. A. Promislow Morali Drivaivs and Opion o Annuiis Insuranc: Mamaics and Economics 9: Milrsn K. and S. A. Prsson 998 Guarand Invsmn Conracs: Disribud Excss Rurn. cnical Rpor Dparmn of Financ Univrsi of Odns. Nilsn A. J. and K. Sandmann Pricing of Asian Excang Ra Opions Undr Socasic Inrs Ras as a Sum of Opions Financ and Socasics 6: Panjr H. H. and D. R. Bllous 98 Socasic Modlling of Inrs Ras wi Applicaions o Lif Coningncis Journal of Risk and Insuranc 47: 9-. Parkr G. 994 wo Socasic Approacs for Discouning Acuarial Funcions Asin Bullin 4: Sarp W. F. 976 Corpora Pnsion Funding Polic Journal of Financial Economics Yosf R. U. Bnzion and S.. Gross 4 Pricing of a Europan Call Opion on Pnsion Annui Insuranc Journal of Insuranc Issu 7: 66-8.