FLOOR OPTIONS ON STRUCTURED PRODUCTS AND LIFE INSURANCE CONTRACTS

Transcription

1 6 Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 FLOOR OPIONS ON SRUURED PRODUS AND LIFE INSURANE ONRAS Rami Yos Absac W consid an xoic call opion dind on sucud poducs and on wo ps o li insuanc conacs: pu ndowmn insuanc and is insuanc conacs. Upon xcis, s opion conacs pomis ig o i uu valu o invsd und in is- ins a, wic is dind in opion conac o uu valu o und invsd in a bas o is asss. andomnss o ins a is modulad b wo socasic pocsss: Onsin-Ulnbc () pocss and Vasic pocss. In ac cas considd, an xplici xpssion o valu o opions as wll as numical xampls a povidd. K wods: Amican call opion; Euopan call opion; Floo opion; Onsin- Ulnbc pocss; Pu ndowmn insuanc; Ris insuanc; Vasic pocss.. Inoducion ans in Isal av cnl sad oing nw invsmn avnus calld Sucud Poducs (SP). s poducs a dsignd o invsos insd in invsing i mon in an avnu wi pobabili o ig ild o maui, wiou ising i invsd und. s SP av a vai o invsmn pas. mos common pas a os a pomis invsos ig o i uu valu o und invsdinis- ins a o uu valu o und invsd in a bas o is asss. Suc a bas o is asss could includ invsmn o 5% o und in S&P 5, 5% in DJ Euosoxx 5 and maining 5% in KOSPI. On maui da o invsmn in SP, ban undas o pa ig uu valu o i invsd und wi a dminisic ins a as dind in SP conac and uu valu o und invsd in s is asss. Moivad b SP, w suggs an xoic opion dind on SP and li insuanc conacs. W us wo ps o li insuanc conacs: pu ndowmn insuanc and is insuanc. In is cas, w xoic opion is dind on SP andonpu ndowmn insuanc, opion olds bu is opion conac and dposi an amoun o mon, d o as invsd und. invsd und is invsd o a dind da o maui. opion wi opns a cun accoun o invsd undand bgins managmn o accoun b invsing und in ma in a basois divaivs. I opion olds suviv oug xcis da o opion, could xcis opion conac and civ ig o i uu valu o invsd und in is- ins a, wic is dind in opion conac o uu valu o invsd und wic is invsd in bas o is asss. amoun o mon paid b opion wi upon xcis o opion conac is paid a dducion o a commission, wic is a pcnag o dinc bwn uu valuo invsd und in is- ins a and uu valu o invsd undin is asss. No a opion olds will onl xcis opion conac on maui da i valu o undling ass ( invsd und in is asss) is, a a im, wo mo an uu valu o invsd und discound b is- ins a dind in opion conac. Also no a in opion conac suggsd, i opion olds do no suviv oug maui da o opion, valu o opion conac is zo and bniciais o opion olds do no g bac invsd und. Onc pic o opion conac suggsd as bnound, iis simpl o ind valu o opion conac in cass w vn i opion olds do no suviv oug xcis da, invsd und discound b is- ins a vs o bniciais. ollowing a sval poins wo o mnion : a) is opion conac Rami Yos, 6

2 Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 6 could onl b xcisd on maui da o opion. is p o opion conac bavs li a Euopan p o opion and is acuall a loo opion. b) opionwiismoivad o acivaig ildomaui on invsmns, as commission is dind as a pcnag o dinc bwn uu valu o invsd und in is-insaand uuvalu o invsd und wic is invsd in bas o is asss. c) I opion olds dcid o xcis opion on maui da, opion wi bus undling ass om opion olds o psn valu o invsd und, discound b a is- ins a, and pas opion olds invsd und muliplid b ins a andb bas o is asss a a pcnag as bn dducd as commission. usispo opion conac is acuall a vsion o a Euopan call opion. No a si pic o is opion conac is uu valu o invsd und, discound b is- ins a. d) I is obvious w opion olds a moivad o invs inis ind o xoic opion conac insad o in oiginal SP od b bans. answis sam ason o invsing in pu ndowmn insuanc insad o invsing in ban savings. In is p o opion conac, invsos xpc a ig ild o maui an invsmn in oiginal SP. ) Moov, sould opion olds no suviv oug xcis da, valu o opion conac is zo. is is sam as in a pu ndowmn insuanc conac, and mans a vn invsd und dos no v bac o bniciais. u as aomniond, a picing is ind o opion conac, valu o is opion conac can asil b ound, vn i opion olds di duing li o opion, uu valu o invsd und wi is- ins a could v bac o bniciais. In scond cas considd, w din an opion conac on SP and is insuanc conacs. A is insuanc conac mans a i insuds do no suviv oug maui da o polic conac, bniciais o insuds civ sum assud, as dind in insuanc conac, om insuanc compan. In is p o opion conac dind on SP and on is insuanc, commns discussd wi gad o is conac old u in is cas as wll. Opion olds bu opion conac o ixd pic, wic is pic o opion. dposi an amoun o mon in opion wi accoun, wic is n invsd b opion wi in abaso a is asss. I opion olds sould di duing li o opion, i bniciais (wo could b namd in opion conac) could xcis opion conac and g om opion wi uu valu o invsd und discound b ig o is- ins a o ins a acivd on invsmnin bas o is asss a commission as bn dducd. I opion olds suviv oug xcis da, canno xcis opion conac, and valu o is opion conac is zo. us opion olds (i bniciais) los invsd und as wll. Howv, as aomniond, is opion conac can b undwin, suc a vn i opion olds suviv oug xcis da, invsd und discound b is- ins as could v bac. No a is xoic opion is miniscn o an Amican call opion, bo in ms o xcis da and si pic. In wo cass o xoic opion conacs considd, wo ps o socasic pocsss a usd o modula andomnss in ins as: Vasic and Onsin- Ulnbc ( ) pocsss. In ac o s socasic pocsss, pics o s opions a valuad and numical xampls a povidd. Rcn sudis in acuaial liau inga mamaics o inanc wi mamaics o insuanc. Saing wi uni-lind li insuanc, nnan and Scwaz (976) cognizd opion sucu o an unlind-lind li insuanc conac wi a guaan. is and d Vann (994) dal wi bonus opion o polic old and banupc opion o (owns o ) insuanc compan in ms o coningn claims analsis. O cn sudis wic dal wi bonus opion a Milsn and Psson (998) o Gosn and JØgnsn (). O conxs in wic wo o mo socasic pocsss govn li o a pu opion a av bn sudid in liau a picing o pu opions on daulabl bonds o swaps, and picing o Asian xcang a opions und socasic ins as. sud o opions in o conxs, in wic wo o mo socasic pocsss govn li o a

3 6 Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 daulabl bond o swaps as a long iso, bu sminal pap in is ild is mos lil a win b Du and Singlon (997). islss, insananous ins a is adjusd b im issuing bond o swap daul azad, o ild a modl a omall smbls daul- cas and a can b solvd in a simila mann. adjusmn, owv, involvs sum o wo azad-li ms wic implis indpndnc, dspi ac a som p o laionsip lil xiss bwn daul azad and insananous ins a. Similal, Asian opions a win on xcang a in a wo-cunc conom. In valuing s opions, bo socasic nau o oign and domsic zo-coupon bond pics and xcang a pocss a modld. A cn amn o poblm is givn b Nilsn and Sandmann (), in wic wo counis zo-couponbondpic pocsss a assumd o b indpndn gomic-ownian moions, bu xcang a pocss is modld b a socasic dinial quaion a is a gomic ownian moion basd on dinc o so-m ins a pocsss in wo counis. o disc and coninuous-im socasic modls o ins a pocsss av bn psnd in acuaial liau, pimail Gaussian auogssiv pocsss. Panj and llous (98) povid a ooug viw o auogssiv pocsss o od, AR(), and o od, AR(), wi consan volaili (vaianc). Pa (994), and ncs in, discusss modling oc o ins, vsus modling accumulad oc o ins, using a coninuous-im auogssiv pocss o od : pocss wi a supimposd lina nd, and Win pocss wi lina nd. Mo cnl, Milvs and Pomislow () av modld so-a pocss isl as a ox-ingsoll-ross (IR) pocss. IR pocss in an AR() pocss in coninuous im, wi andom volaili a is popoional o squa oo o insananous ins a jus pio o im. O cn sudis wic combin call opions on pnsion annui insuanc plans w conducd b alloa and Habman (3) and Yos, nzion, and Goss (3). s o is pap is sucud as ollows: In Scion w psn loo opion dind on SP and on pu ndowmn insuanc conacs. W ind xplici xpssions o valu o is opion conac in cas o Vasic and pocsss, wic modula andomnss in ins a pocss. In Scion 3 w solv cas o a loo opion dind on SP and on is insuanc conacs in wo cass o socasic pocsss psnd abov. Numics and conclusions a povidd in Scion 4. No a w clal igno xpnss, pois and o adminisaiv cags and us assum a ving is psnd on a n basis.. Floo Opion on SP and Pu Endowmn Insuanc is scion ocuss on valuaing xoic call opion dind on SP and pu ndowmn insuanc as psnd abov, und socasic sucu o ins as. W can wi is loo opion w moali and ins a a socasic b: PE P E, () w im om o nd o polic conac. andom vaiabl dscibing oal liim o an individual. consan is- ins innsi. is- ins a. consan aco dnoing commission o opion wi. consan aco. () andom ins a pocss. valu o invsd und a im. PE dnos pic o is xoic-loo call opion dind on SP and on pu ndowmn insuanc a im.

4 Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 63 No a as aomniond, is p o opion conac is acuall a looopion. A im zo, opion olds pa pic o opion wic is: PE and also dposi an amoun o mon qual o in opion wi accoun. I opion olds suviv oug xcis da, can xcis opion, and poi om xcising opion conac is E. No a is opion conac could onl b xcisd wn opion olds suviv oug xcis da o opion. I do no suviv, valu o opion is zo. In is cas opion olds spnd mon o pucas opion and los invsd und as wll, maning a vn invsd und dos no v bac o i bniciais. In od o valua pic o an opion conac in cas o da o opion old duing li o opion conac and cip o invsd und b bniciais, ollowing quaion mus b solvd: P E PE, wic will b as o valuaion a solving (). W assum a socasic sucu o ins a ollows wo inds o socasic pocsss: Vasic modl and pocss. s wo pocsss av insing bavios. pocss as an advanag a is sampl uncions nd o v o iniial posiion, a pop wic sms appopia o man ins a scnaios. ini dimnsional disibuions a nomal, and pocss as a Maovian pop (s aman & Fulling (99,99)). Vasic modl as a ndnc o lucua aound ixd ins a >, wi an vnuall sabilizing volaili. conncion bwn s wo pocsss and mo abou and Vasic pocsss will b givn in ollowing. Dno b () Vasic pocss wic is dind as a diusion pocss saising socasic dinial quaion: d d d, () w (,, ) > and is sandad ownian moion wi di and vaianc p uni im (s ax and Rnni, 996, p. 53). In ms o socasic ingal, soluion o () is givn b s d s. (3) Accoding o (3), as a di owads o (sa-dpndn) siz,wic is us popoional o disanc om. No a o a spcial cas, w =,wg pocss. Also no a w can psn Vasic pocss b ( ), (4) w () is pocss. Kalin and alo (98, p. 33) Nx,, w /, so Nx,.In is cas, om (4), w can wi L o Vasic pocss,,b: siz Lmma. L b: E x, R. (5) ollow Vasic pocss as dscibd in (), n w can wi E

5 Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 64 x x, w ( ) is cumulaiv disibuion uncion o nomal disibuion, is Laplac ansom (L ) o Vasic pocss wic is givn in (5) a,and w /. Poo. Dno b d cumulaiv disibuion uncion o Vasic pocss a in poin,n o > df E now sinc x N, w can solv is ingal and g: x x x df w. So pic o is loo opion dind on SP and pu ndowmn insuanc und Vasic pocss a im zo, Vasic PE SP,can b win b

6 Vasic SPPE P Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 65 w x x is L o Vasic pocss wic is givn in (5) a poin, > and w /. Now b ling =, w g pic o loo opion dind on SP and pu ndowmn insuanc und pocss. 3. Floo Opion on SP and Ris Insuanc In is scion w xamin picing valu o an xoic call opion dind on SP and on is insuanc und socasic sucu o ins as. As pviousl mniond, in is ind o opion conac, opion olds acqui opion o an amoun o mon and mus dposi an invsd und in opion wi accoun. opion wi sas invsing is und in a bas o is asss suc as in soc xcang oo is divaivs in od o aciv bs ild o maui on invsmns. In cas o da o opion olds duing li o opion conac, bniciais av opion o xcising opion and civing om opion wi ig o i invsd und discound b is- ins a o discoun valu o invsd und wi al ins a acivd b opion wi oug maui da, a a commission as bn dducd. is commission is dind as a pcnag o dinc bwn is- ins a and al ins a acivd b opion wi on invsmns, muliplid b invsd und. Fumo, i opion olds suviv oug xcis da, wo o is opion is zo. us opion wi is moivad o maximiz is dinc and aciv bs possibl ild on invsmns. No a da o opion olds could, o cous, b an im unil xcis da, maning a opion could b xcisd b bniciais a an im oug xcis da. is p o opion is acuall a loo opion a acs li an Amican call opion in ms o xcis da and si pic. I is loo opion sinc in cas o da o opion olds duing li o opion, bniciais will, a v las, av invsd und discound b is ins a. I is also considd a call opion, bcaus i opion conac is xcisd, bniciais bu invsd und om opion wi b paing pic o uu valu o invsdundinis- ins as, and inunciv om opion wi uu valu o invsd und wi al ins a acivd on i invsmns. Now valu o is loo opion a im zo can b win b: SPRI () E E, (7) w,,,,,,, a as dind in abov scion, and w SP RI() is valu o is loo opion a im und s paams and vaiabls. No a a im zo opion olds pa pic o opion ) and dposi an amoun o mon qual o in opion wi accoun. I opion olds do no suviv oug xcis da, i bniciais could xcis opion wi a gain o: RI PE (, (6)

7 Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 66 E. Also no a sould opion olds suviv oug xcis da o opion, valu o opion is zo. In is cas, opion olds spnd mon o pucas opion and los i invsd und. o valua pic o an opion conac, w vn i opion old suvivs oug xcisdawillg bac iinvsdund, ollowing quaion mus b solvd: () RI SP E E,wic will b as o valuaion a calculaing (7). Now calculaion o opion conac psnd in (7) und Vasic pocss wic is dnod b Vasic RI SP, is givn in ollowing Lmma. Lmma. L ollow Vasic pocss as dscibd in (). n o >,w can wi (7) b: Vasic RI SP df x df x (8) w df is cumulaiv disibuion uncion o, is L o, ( ) is cumulaiv disibuion uncion o nomal disibuion, and is L o Vasic pocss wic is givn in (). Poo. L, min.n o >, Vasic RI SP E E E. Now siz o condiional xpcaion wi spc o ( ), could b win b: F d E so ollowing lmma w g:

8 Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 67 x x Now sinc = o <, and sinc valu o is opion is o >, aing xpcaion wi spc o =, wg quid sul o (7), wic is givnin lmma. As aomniond, i w wan o ind valu o is loo opion und pocss, SP RI, all a is ndd is o plac =. 4. Numics and onclusions W will consid wo cass o andom vaiabl o oal liim o an individual (). is cas is an xponnial liim, w P(> )= o posiiv consans,. scond cas is Gompz low: µ ag =wc ag o posiiv consans w and c. In is cas w can wi suvival liim as: P w ag c c ln c, w paam ag s o Vasic psn ag o insud. In ac cas w will ind pics o s loo opions SP PE, Vasic Vasic PE, p, SP RI and RI und sval assumpions o paams. W will compa s suls o picso s loo opions, w pobabili o insud o suviv oug xcis da is, i.. P(> )=. Suppos consan paams o pocsss a: =.5, =., =., =., =.7, x =.5, ban commission is = %, and =$. suls o som scnaio o is- ins a,and o som posiiv, and paam ag will b givn in ollowing abls ( is givninas). Also suppos a consan paams o Gompz low a: w = 4,c =..abl psns valus o loo opions dind on SP and on pu ndowmn insuanc conacs, and abl psns suls o loo opions dind on SP and on is insuanc conacs. abl oulins pics o loo opions dind on SP and on pu ndowmn insuanc. As mniond in Scion, a im zo opion olds pa pic o opion PE() and dposi an amoun o mon qual o: o opion wi accoun. I opion olds suviv oug xcis da, can xcis opion and i poi om xcising opion conac is PE. E No a all o numics a o =$. I, o xampl, w xamin cas w consans a =.3, =., =5 (5 as), ag =3, pics o opion conac in cas o and Vasic pocsss is $.887 in xponnial cas. is mans a opion olds will onl av a gain om xcising is opion conac i annual ins a acivd b opion wi on invsmns xcds.77%.

9 68 Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 Pics o loo opions on SP and on pu ndowmn insuanc (,,,ag) PE P Vasic PE PE Vasic PE abl 4 ag.. ln. PE Vasic PE (.3,.,5, 3 4 ) (.3,.5,5, 3 4 ) (.3,.,5, 3 4 ) (.3,.5,5, 3 4 ) (.3,.,3, 3 4 ) (.3,.5,3, 3 4 ) (.5,.,5, 3 4 ) (.5,.5,5, 3 4 ) (.5,.,5, 3 4 ) (.5,.5,5, 3 4 ) (.5,.,3, 3 4 ) (.5,.5,3, 3 4 ) (,,,ag) Pics o loo opions on SP and on is insuanc RI P abl 4 ag.. ln. Vasic SP RI RI Vasic SP RI RI (.3,.,5, 3 4 ) (.3,.5,5, 3 4 ) (.3,.,5, 3 4 ) Vasic SP RI

10 Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 69 (,,,ag) RI P abl (conionuous) 4 ag.. ln. Vasic SP RI RI Vasic SP RI RI (.3,.5,5, 3 4 ) (.3,.,3, 3 4 ) (.3,.5,3, 3 4 ) (.5,.,5, 3 4 ) (.5,.5,5, 3 4 ) (.5,.,5, 3 4 ) (.5,.5,5, 3 4 ) (.5,.,3, 3 4 ) (.5,.5,3, 3 4 ) Vasic SP RI abl indicas pics o loo opion dind on SP and on is insuanc conacs. I w a, o xampl, s consan paams =.5, =.5, =3, w can s a pics o is opion conac wi pocss in cas o a cain liim is $.34 andincaso Vasic pocss is $.68. pics o is opion in xponnial cas is $783 in pocss and $.78 wi Vasic pocss. Addiionall no a som dincs xis bwn opion pics und Gompz low o moali and xponnial cas, pdominanl du o lac o mmo wi gad o ag o insud pais aibud o xponnial liim. 5. onclusions In is pap w inoduc a loo opion dind on SP and on li insuanc conacs. s opion conacs could lad insuanc companis, i w in o m as opion wis, o b mo involvd in capial ma, an objciv impoan o all pais involvd, paiculal in a coun as small as Isal. Fumo, in Isal, all insuanc policis wi a componn o savings on svs o policis, suc as pu ndowmn insuanc, ndowmn insuanc and annui insuanc conacs, av o invsapcnag o i poolio in is asss. bni om a commission wic is a pcnag o gain o i invsmns, bu also njo a ixd commission vn i av bad invsmns. us ind o conacs suggsd is also v impoan o insud pais insd in buing an insuanc conac wi a savings lmn suc as pu ndowmn insuanc and would li o poc i invsd und om bad invsmns o insuanc companis. s conacs will also accla compiion bwn insuanc companis o maximiz poi om i invsmns.

11 7 Invsmn Managmn and Financial Innovaions, Volum 3, Issu 3, 6 Rncs. alloa, L., Habman, S., (3). Valuaion o Guaand Annui onvsion Opions. Insuanc; Mamaics and Economics, 33, pp ax, M., Rnni, A., (996). Financial alculus, ambidg Univsi Pss. 3. aman, J.A., Fulling,.P., (99). Ins and Moali Randomnss in Som Annuiis. Insuanc: Mamaics and Economics, 9, pp man, J.A., Fulling,.P., (99). Exa Randomnss in ain Annui Modls. Insuanc; Mamaics and Economics,, pp nnan, M.J., Scwaz, E.S., (976). Picing o Equi-Lind Li Insuanc Policis wi an Ass Valu Guaan. Jounal o Financial Economics, 3, pp is, E., D Vann, F., (994). Li Insuanc in a oningn laim Famwo: Picing and Rgulao Implicaions. Gnva Paps o Ris an Insuanc o, 9, pp Gosn, A., JØgnsn, P.L., (). Fai Valuaion o Li Insuanc Liabiliis: Impac o Ins Ra Guaans, Sund Opions, and onus Policis. Insuanc: Mamaics and Economics, 6, pp Du, D., Singlon, K., (997). An Economic Modl o m Sucu o Ins Ra Swap Yilds. Jounal o Financ, 5, 4, pp Kalin,S., alo, H.M., (98). A Scond ous in Socasic Pocsss, nd. Ed. Acadmic Pss, Inc. Nw-Yo.. Milvs, M.A., Pomislow, D.A., (). Moali Divaivs and Opion o Annuiis. Insuanc: Mamaics and Economics, 9, pp Milsn, K., Psson, S.A., (998). Guaand Invsmn onacs: Disibud Excss Run. cnical Rpo. Dpamn o Financ, Univsi o Odns.. Nilsn, A.J., Sandmann, K., (). Picing o Asian Excang Ra Opions Und Socasic Ins Ras as a Sum o Opions. Financ and Socasics, 6, pp Panj, H.H., and llous, D.R., (98). Socasic Modlling o Ins Ras wi Applicaions o Li oningncis. Jounal o Ris and Insuanc, 47, pp Pa, G. (994). wo Socasic Appoacs o Discouning Acuaial Funcions. Asin ulin, 4,, pp Yos, R., U. nzion, and S.. Goss, 4, Picing o a Euopan all Opion on Pnsion Annui Insuanc, Jounal o Insuanc Issu, 7,, pp