PRICING OF REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS ABSTRACT KEYWORDS

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1 PRICING OF REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS BY GARETH G. HASLIP AND LADIMIR K. KAISHE ABSTRACT A mhodology for pricing of rinsuranc conracs in h prsnc of a caasroph bond is dvlopd. An imporan advanag of his approach is ha i allows for h pricing of rinsuranc conracs consisn wih h obsrvd mark prics of caasroph bonds on h sam undrlying risk procss. Wihin h proposd mhodology, an appropria financial pricing formula is drivd, undr a mark implid risk nural probabiliy masur for boh a caasroph bond and an aggrga xcss of loss rinsuranc conrac, using a gnralisd Fourir ransform. Efficin numrical mhods for h valuaion of his formula, such as h Fas Fourir ransform and Fracional Fas Fourir ransform, ar considrd. Th mhodology is illusrad on svral xampls including Paro and Gamma claim svriis. KEYWORDS Rinsuranc, caasroph bonds, risk nural valuaion, scuriisaion, Fas Fourir Transform, Fracional Fas Fourir Transform. 1. INTRODUCTION Th naur of h rinsuranc indusry is rapidly changing. Ovr h las dcad larg insiuions and h financial marks hav dvlopd a rang of nw financial producs ha provid dirc xposur o h risks ha prviously had bn h sol inrs of h insuranc indusry. Ths includ caasroph bonds, indusry loss warranis, indusry loss fuurs and a rang of ohr insuranc linkd scuriis, drivd from spcial purpos vhicls, such as sidcars. Th convrgnc of h insuranc and capial marks has bn acclraing ovr rcn yars. Considring h mark for caasroph (ca) bonds alon, h oal ousanding issuanc a nd of 007 was $1.8 billion up 6 prcn ovr h nd of 006. In fac $7 billion of publicly disclosd ca bonds wr issud in 007, compard o $4.7 billion in 006 and $ billion in 005 (s Asin Bullin 40(1), doi: 10.14/AST by Asin Bullin. All righs rsrvd.

2 08 G.G. HASLIP AND.K. KAISHE McGh al. (007)). Th lvl of issuanc in 008 and 009 has subsanially rducd du o h impac of h crdi crisis, wih $5.8 billion nw issuanc in 008 and only $1.7 billion for h firs half of 009, as dscribd by Aon Bnfild Scuriis (009). Howvr, h rcn dclin in populariy mayb aribud o h impac of h collaps of Lhman Brohrs, who acd as a oal rurn swap counrpary for a numbr of ca bonds and h gnral cauion of invsors o non-sandard ass classs during h crisis. Thr ar svral rasons why scuriisd insuranc insrumns had bn incrasing in populariy prior o h crdi crisis. Th acclrad issuanc of ca bonds was in rspons o limid caasroph capaciy in h rinsuranc indusry, following Hurrican Karina, Ria and Wilma. During h arly issus of ca bonds, prics wr rgardd as bing high, compard o radiional rinsuranc and h bonds acd o fill gaps in h mark, whr capaciy was limid. Ovr im, as h financial marks bcam comforabl wih h concp of insuranc linkd scuriis, dmand for hm has incrasd. Many larg invsmn niis, such as hdg funds, ar consanly looking for invsmn vhicls ha provid divrsificaion from h rs of hir porfolio. As mos arly ca bonds wr wrin o covr high lvls of losss, h majoriy did no riggr and providd ras of rurn wll abov LIBOR. This incrasd dmand furhr and h insuranc indusry racd by incrasing issuanc of ca bonds and ohr rlad scuriis. Mor rcnly, as dscribd by h Insuranc Journal (00), invsmn banks hav movd ino h rinsuranc mark and ar now boh capialising and sing up rinsuranc companis. Th aim of his iniiaiv is wo-fold. Firsly i provids hm wih a dirc xposur o h hisorically high lvls of profiabiliy in h rinsuranc scor. Scondly, i allows hm o xploi pric diffrnials bwn h dirc rinsuranc mark and insuranc linkd scuriis in h capial marks. This can b achivd by issuing insuranc linkd scuriis on h risks insurd by h rinsuranc companis s up by invsmn banks. Typically, h dsign of h scuriisaion follows ha of ass backd scuriis by forming a muli-ranch forma of varying risk. This approach hlps o mak h scuriisd insuranc insrumn araciv o h wids rang of invsors and hnc achiv grar prics for a givn lvl of issuanc. As h rnd of moving owards h capial marks for procion agains caasroph risk incrass, an imporan qusion ariss. Dos h currn mark pric of insuranc linkd scuriis imply somhing abou how much rinsurrs should b charging o insur similar risks? If profis can b mad by xploiing pric diffrnials bwn rinsuranc prmiums and h pric invsors ar prpard o pay for insuranc linkd scuriis, hn h answr is a dfini ys. From h prspciv of a dirc insurr, i is imporan o b abl o assss whhr purchasing insuranc linkd scuriis or rinsuranc covr provids br valu for mony. Similarly, a rinsurr should b abl o judg whhr

3 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS 09 h prics i is offring ar highr or lowr han ha implid by h capial marks. Th aim of his papr is o s up a framwork in which h obsrvd mark prics of insuranc linkd scuriis can b usd o assss a mark consisn pric for radiional rinsuranc. Rahr han looking a h nir univrs of insuranc linkd scuriis and rinsuranc conracs, w will considr only h ca bond and h aggrga xcss of loss conrac. Howvr, h approach akn could b xndd o considr a widr rang of scuriisaion vhicls and rinsuranc policis. Th papr is organisd as follows: In Scion w dscrib a gnral approach for pricing rinsuranc in h prsnc of a mark for ca bonds. This provids h foundaions upon which w build a consisn pricing mhodology ha is applid o boh ca bonds and rinsuranc conracs. Scions and 4 implmn his pricing framwork building upon h risknural pricing approach originally dvlopd for insuranc by Dlban and Handonck (1989). W apply h financial pricing approach firs dscribd for rinsuranc by Sondrmann (1991) and lar for ca bonds by Baryshnikov al. (001). Following his approach lads o a pricing formula basd around calculaing h xpcd discound valu of h rinsuranc or ca bond conrac payoff undr a risk-nural probabiliy masur. Mos hory, rlaing o valuaing h risk-nural pricing formula, is drivd from incompl mark drivaiv pricing hory in financ. Th financial world has dvlopd many chniqus for daling wih pricing drivaiv conracs whr h undrlying sock pric follows a Lévy procss. For xampl of such rsarch w rfr h inrsd radr o Con and Tankov (004). Th ida of applying risk-nural valuaion is no nw o h acuarial profssion. For xampl, Holan (004) dscribs applying opion pricing chniqus o insuranc conracs. Furhr, Murmann (00) applis a Fourir ransform basd approach for pricing caasroph drivaivs and rinsuranc basd on a classic opion pricing chniqu, dscribd by Carr and Madan (1999). W will follow h approach of risk-nural valuaion bu will uilis a mor rcn pricing chniqu, dscribd by Lwis (001): h gnralisd Fourir ransformaion mhod. This will provid us wih an lgan mchanism for valuaing h horical prics of ca bonds and rinsuranc conracs. I will lad us o an ingral xprssion for h conrac pric in rms of h gnralisd Fourir ransform of h payoff funcion and h aggrga loss characrisic funcion. Gnral invrsion formula which can b usd for insuranc and opion pricing basd on h Parsval s horm, hav bn rcnly obaind by Dufrsn al. (006). Addiionally, a Fourir spac im-spping mhodology, applid o h problm of pricing caasroph quiy pu opions is prsnd by Jaimungal al. (007).

4 10 G.G. HASLIP AND.K. KAISHE In Scions 5-7 w considr diffrn numrical mhods of valuaing h gnral pricing formula for ca bonds and rinsuranc conracs obaind in Scions and 4. W bgin by implmning h Fas Fourir Transform o provid an fficin numrical compuaion of rinsuranc conrac and ca bond prics, as suggsd by Murmann (00) as an xnsion of his work. W hn follow h approach of Chourdakis (005) from drivaiv pricing hory o dmonsra how h Fracional Fas Fourir Transform can b usd as an fficin numrical algorihm o valua ca bonds and rinsuranc prics. Scion 8 considrs wo pracical xampls of applying h horical and numrical pricing chniqus. Th firs xampl dmonsras ha h pricing mhod works succssfully in h cas of a Gamma disribuion for claim svriy. For h scond xampl, w considr pricing conracs undr h Paro yp II svriy disribuion. I is dmonsrad ha h pricing mhod is succssful for ca bonds bu fails for an aggrga xcss of loss rinsuranc conrac du o h bhaviour of h Paro s characrisic funcion. In ordr o rsolv his issu w driv a pu-call pariy rlaionship for aggrga rinsuranc conracs. This lads us o rcovr h pric of h rinsuranc conrac in rms of h pric of a pu opion on h aggrga claims procss. This approach is a succssful dmonsraion of h pracicaliy of h gnralisd Fourir ransform pricing mhod. In Scion 9 w compar h accuracy of h pricing mhod o ha of Mon-Carlo simulaion and conclud ha for calibraion purposs, h analyical formula ar prfrabl, sinc for a givn lvl of accuracy, Mon Carlo simulaion is significanly slowr han h analyical mhod. Finally in Scion 10 w provid som commns and conclusions.. GENERAL APPROACH In his papr w uilis risk nural valuaion for pricing boh ca bonds and rinsuranc conracs. This diffrs o h sandard acuarial approach of applying ral-world prmium principls o conracs involving insuranc risk. W rfr h rad o Holan (004) for a comparison of h wo mhodologis and o Baxr and Rnni (1996) for a daild inroducion o risk nural financial pricing. Following Embrchs (1996), w dscrib h insuranc mark as a filrd probabiliy spac ( W, F, ( F ) 0, P ), whr ( F ) is an incrasing family of s-algbras, ha rprsn all h informaion prsn in h hisory of h insuranc risk procss. W dno by ( S ) 0 T h accumulad losss a im from an undrlying insuranc risk procss. Throughou his papr w will assum ha S can b adqualy modlld as a compound Poisson procss. Using a ( risk-nural ) quivaln maringal probabiliy masur Q o h ral-world probabiliy masur P, h arbirag fr pric,, of a coningn

5 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS 11 claim wih payoff c( S T ) a im T is givn by h fundamnal horm of ass pricing as -rt ( -) = E $ c( S ) F., (1) Q T whr r is h coninuously compoundd risk-fr inrs ra. For mor informaion abou h fundamnal horm of ass pricing, w rfr h radr o Dlban and Schachrmayr (1994). Insuranc risk modlling is usually carrid ou using incompl mark modls, which mans ha hr is no uniqu quivaln maringal masur. Insad, h insuranc mark uss a wid rang of risk-nural masurs ha corrspond o h many diffrn acuarial prmium principls. Dlban and Handonck (1989) characris h s of quivaln maringal masurs Q, undr which h srucur of h insuranc risk procss rmains a compound Poisson procss undr h ral-world masur P. Thy also idnify diffrn risk-nural probabiliy masurs ha corrspond o som of h acuarial prmium principls. Rcnly, Murmann (00) characriss h mark pric of risk implid by diffrn prmium principls and dscribs a pricing chniqu for rinsuranc conracs wih a Europan payoff ( ha is no pah dpndn ) undr h associad risk-nural probabiliy masur. In Murmann (00) h auhor idnifis h implid risk-nural masur associad wih diffrn invsor prfrncs whn pricing caasroph drivaivs. In his mos rcn papr, Murmann (006) invsigas whhr h mark pric of caasroph risk can b calculad by comparing h mark pric of rinsuranc and caasroph drivaivs. H provids a mhod of calculaing h mark pric of caasroph risk undr h rsricion ha a singl caasroph vn will b sufficin o bring an ou-of-h-mony caasroph drivaiv ino h mony. In his papr, rahr han choosing an quivaln maringal masur ha corrsponds o a paricular prmium principl, w will follow Murmann (006) and adop a mark implid approach. Tha is, w will pric rinsuranc conracs using h ( risk-nural ) quivaln maringal probabiliy masur, implid by h obsrvd mark prics of ca bonds. This is analogous o how xoic sock opions ar usually pricd using h risk-nural masur implid by h obsrvd mark prics of vanilla Europan sock opions, as dscribd by Schouns (00). W no ha in h currn scondary mark for ca bonds, hr is no sufficin liquidiy o radily obain mark prics for variy of ca bonds on a spcific rgion and pril. Howvr, w bliv ha givn h coninuing growh in h insuranc linkd scuriy mark, i is only a mar of im bfor such a mark dvlops. W will hrfor procd on h basis ha h liquidiy of h scuriisaion mark will improv wih im and w assum ha prics ar radily availabl for ca bonds a a rang of xpiraion das and riggr lvls on h insurd risk.

6 1 G.G. HASLIP AND.K. KAISHE W will assum ha undr h mark implid risk-nural probabiliy masur, h undrlying loss procss follows a compound Poisson disribuion. This assumpion is suppord by Dlban and Handonck (1989), who show ha if h loss procss is compound Poisson undr h ral-world probabiliy masur P, hn i will rmain compound Poisson undr any quivaln risknural masur Q, providd ha insuranc prmiums on h undrlying risk ar linar wih rspc o im. Th rquirmn for linariy of insuranc prmiums can b xplaind as follows: h insuranc company s liabiliis a im ar assumd o b of h form X + p, whr X is h accumulad claims up o im and p is a ( prdicabl ) prmium for ransfrring h risk for h rmaining priod of covr [, T ] o a hird pary. If h insuranc prmium is linar hn w rquir ha p = p( T ) whr p is h prmium dnsiy. Th assumpion of prmium linariy is fasibl o assum sinc i holds in many cass in non-lif insuranc. Howvr i should b nod ha hr ar xcpions, for xampl, whn h undrlying insuranc risk procss xhibis sasonaliy ffcs. W will no ackl any of h issus surrounding calibraion chniqus in his papr. Insad, w will focus on h acual pracicaliis of applying risk-nural valuaion o insuranc procsss. This is a ncssary prrquisi o h calibraion procss. As such, w rurn o h calibraion problm in a fuur papr.. PRICING CAT BONDS W will bgin by providing som background informaion on how ca bonds opra. Ca bonds ar a rlaivly nw yp of bond ha provids a sris of coupon paymns and rurn of capial o an invsor, coningn on a riggr vn no occurring. Th riggr vn is dfind o b whr a masurabl quaniy rlad o an undrlying insurd risk xcds a prdrmind lvl. Thr ar many diffrn variis of ca bond, whr h riggr vn could b basd on modlld losss, indusry losss or h svriy of a naural disasr xcding a spcifid limi. In his papr w will only b considring ca bonds wih an indmniy basd riggr. Th indmniy vn is riggrd whn h acual losss of h bond issur xcd a hrshold riggr lvl ( so h issur is ffcivly fully indmnifid agains losss in a layr saring a h riggr lvl and xnding o h riggr lvl plus h n prsn valu of ousanding coupons and h rdmpion paymn ). I will b assumd ha undr a risk-nural probabiliy masur Q, h aggrga loss procss S follows a compound Poisson procss and S = / X j, wih h convnion ha S = 0 if N = 0, () N j = 1 whr N is h numbr of claims ha hav occurrd by im and X j is a random variabl rprsning h svriy of h j-h claim.

7 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS 1 W assum ha N is a Poisson procss wih arrival ra l and h X j ( j = 1,, N ) ar indpndn idnically disribud absoluly coninuous random variabls wih probabiliy dnsiy funcion f ( x ), which ar also indpndn of N. I is assumd also ha h ca bond is of h indmniy yp wih riggr lvl D, D > 0. Th bond is assumd o maur a im T and coupons ar paid a ra C j a ims < 1 < < < n = T ( n 1 ). W apply h financial pricing formula ( 1) o assr ha h pric, ca, pr $1 nominal, of a ca bond a im can b wrin as ca ca_ coup ca_ cap = + n / -r ( j -) -r( T - ) = C E % 1 F/ + E $ 1 F., j = 1 j Q { S < D} Q { ST< D} j () whr 1 {S < D} is h indicaor funcion and ca_coup, ca_cap dno h coupon and h capial pars of h bond pric rspcivly. For furhr dails rlad o h drivaion of his approach, w rfr o Baryshnikov al. (001). In ordr o calcula h xpcaion in ( ), w will us h gnralisd Fourir ransform mhod. This was inroducd o financial mahmaics by Lwis (001) who dmonsrad is us for opion pricing undr a Lévy procss. W dfin h gnralisd Fourir ransform F of a funcion w : R " R as i x - F{ w( x)} = w ( ) = # w( x) dx, whr = u+ iv and u, v! R Th invrs gnralisd Fourir ransform is dfind as F iv ix { w ( )} = wx ( ) = # wd ( ), - whr ingraion is prformd along a sraigh lin paralll o h ral axis, along which says wihin a srip of rgulariy. Th ida bhind h gnralisd Fourir ransform pricing mhod is o uilis h abiliy o swich h ordr of xpcaion and ingraion, whn applying a conscuiv Fourir and Fourir invrs ransformaion o h indicaor procss 1 {S < D}. Swiching h ordr of xpcaion and ingraion is quivaln o swiching h ordr of ingraion undr a doubl ingral. This is prmissibl undr Fubini s horm, providd ha h ral and imaginary pars of h ingrand ar boh L 1 ( R ) funcions ( s.g. Wir (197) ). W will procd undr h assumpion ha his is saisfid. In pracic, his will normally b h cas, sinc h usual choics of svriy disribuions such as h Gamma, Paro and Log-Normal ar wll bhavd.

8 14 G.G. HASLIP AND.K. KAISHE I is no difficul o s ha h Fourir ransform of h indicaor procss is givn by F 1{ S T < D} # (-, D) T - ist i { } = 1 ( S ) dst = - for Im( ) < 0. id L us now rurn o h pricing formula ( ) and xprss h indicaor procss in rms of is Fourir ransform. W bgin by simplifying h capial par, ca_cap, of h bond pric as follows ca_ cap -rt ( -) = Q ) E id i F - 1c- m F = -rt ( -) E Q 1 * iv+ # -ist -i id d F 4 (4) i = - -rt ( -) iv+ # id -ist E ( F ) d. Q Now, in ordr o valua h ingral givn in ( 4 ) w will firs nd o simplify h xprssion E Q ( is T F ). Following dfiniion ( ) of h compound Poisson modl h accumulad losss a h conrac xpiry im T can b xprssd in rms of h accumulad losss a im ( i.. using h informaion givn by h filraion F ) as S NT = S + / X = S + D T j, T j= N + 1, whr D, T = NT / X. Hnc w hav j= N + 1 j -ist -is -idt, -is -idt, Q` = Q` = Q` E Fj E Fj E Fj. Now, E Q ( id,t F ) is acually h momn gnraing funcion ( MGF ), M D,, of D, T valuad a = i. A sandard approach o valua h MGF yilds M N -id F MD = EQ T, ^ h = MN ln M (),, T - N # X - whr T - N is h MGF of h Poisson numbr of claims N T N in h im inrval (,T ) and w assum ha X is such ha M X ( ) xiss for imaginary.

9 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS 15 This shows ha -is -is E ( F ) = M { ln M (-i)} Q = -is M NT - N NT - N { ln f (-)} -is = xp_ l ( T -)( f (-)-1) i, X X X (5) whr f X ( ) is h characrisic funcion of h claim svriy disribuion. Applying h sam approach o simplify h coupon paymn par, ca_coup, of h pricing formula, w arriv a h following gnral pricing formula for ca bonds ca n j i =-/ C # j j = 1 i - -r( -) iv+ i( D - S) + l( j-)[ fx (-)-1] -r( T-) iv+ i( D - S) + l( T -)[ fx (-)-1] # d, d (6) whr Im( ) < 0. Th ingral in (6 ) will gnrally b compud numrically along a srip blow and paralll o h ral axis in h complx plan. Th choic of svriy disribuion will impos rsricion on h xac srip chosn, h criria bing o avoid passing hrough any poins of singulariy. I is unforuna ha ralisic choics of h svriy disribuion will prvn h us of rsidu calculus o valua h ingral xplicily du o h ingrand no dcaying fas nough as ". Howvr, mos mahmaical packags provid faciliis o valua his yp of ingral and w will look a fficin algorihms for is compuaion lar in h papr. In ordr o us his pricing formula on nds simas of l and f X (.). To obain such simas on can us ( 6 ) and a sampl of obsrvd mark prics of ca bonds. Th calibraion procss for ca bonds undr his yp of modl is dscribd by Burncki (005) and w ar no going o considr i hr. Assuming ha w hav calibrad h compound Poisson modl succssfully ( and hrfor hav dscribd h loss procss undr h risk nural probabiliy masur ), w will now procd o look a mark consisn rinsuranc pricing. 4. CAT BOND CONSISTENT REINSURANCE PRICING Thr ar many variis of rinsuranc conracs. For simpliciy, w will only considr an aggrga xcss of loss ( aggrga XL ). This is somims rfrrd o as a sop-loss conrac in h acuarial liraur.

10 16 G.G. HASLIP AND.K. KAISHE I is assumd ha h aggrga XL conrac will xpir a im T and all claims ar sld a h nd of h conrac. Undr his assumpion, h payoff from h conrac a im T will b X T T = max( S - K, 0) = ( S -K ) +, whr K > 0, is h prioriy of h conrac. Ohr rinsuranc conracs ha dpnd only on h oal loss a im T can b xprssd in a similar way. Th analogis bwn h payoff funcion of his rinsuranc conrac and a Europan opion ar obvious and ar a clar moivaion for applying financial pricing mhods. W rfr h radr o Embrchs (1996) and Holan (004) for a simulaing discussion of h conncions bwn h acuarial and financial filds in his conx. Furhr insigh ino his dualiy can b gaind from Dufrsn al. (006). As dscribd in dail arlir in Scion, w ar adoping a mark implid approach o drmining h risk-nural paramrs of h appropria risk procss, S, from h obsrvd prics of ca bonds. Undr assumpions of h framwork of Dlban and Handonck (1989), i holds ha h insuranc loss procss follows a compound Poisson procss undr h implid risk-nural probabiliy masur. W apply h fundamnal horm of ass pricing, o assr ha h valu of h rinsuranc conrac ( on h sam undrlying insuranc risk ) a im is XL + = E * xpf-# r ds p( S -K) F 4, (7) or in h cas of drminisic inrs ras XL Q T s T -rt ( ) = E $ ( S - K) F.. (8) Q - + T I is worh noing ha w can build mor faurs ino h aggrga XL conrac by combining svral simpl conracs, in an analogous way o craing sprads using a porfolio of opions. For insanc, if w wish h XL conrac o aach a lvl K 1 and hav an uppr limi of K, hn w cra h insuranc quivaln of a Bull Sprad. Tha is, w ffcivly buy an aggrga XL conrac wih prioriy K 1 and sll an aggrga XL conrac wih prioriy K. Thus h pric a im of h aggrga XL conrac ha aachs a K 1 wih limi K is -rt ( -) + -rt ( -) + ST - 1) F - Q ST - ) F E $ ( K. E $ ( K.. Q In ordr o valua hs xprssions, w will again us h Fourir ransform mhod ha was inroducd in Scion. W bgin by noing ha h Fourir ransform of h payoff funcion ( S T K ) + is asily sn o b

11 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS 17 ik + F $ ( S T - K ). =-, whr Im( ) > 0. From ( 8) w hrfor hav ha i XL -rt ( -) 1 = EQ * F - - K o F 4 -rt ( -) 1 = EQ * # -rt ( -) =- -rt ( -) =- iv+ # iv+ E ` -ist -ist - F j d F 4 d iv+ i( K -S) + l( T -)[ fx( -)-1] # Q ik ik d, (9) whr h ingraion mus b compud along a srip abov and paralll o h ral axis. Again, his is prformd numrically and will nd o b wihin a srip of rgulariy ( drmind by h choic of svriy funcion ). 5. HOW TO EALUATE THE INTEGRALS Whil h financial pricing approach is araciv from a horical sandpoin, i is of lil valu unlss h complx ingrals can acually b valuad in a pracical mannr. In his scion w will considr diffrn ways of compuing h ingrals in ( 6 ) for h pric of a ca bond. Wihou loss of gnraliy, w will only considr a ro coupon ca bond. This simplifis h problm o calculaing a singl ingral, rahr han on pr coupon paymn. Th pric of h coupon paying ca bond can hn b calculad as a linar combinaion of ro coupon bonds of varying duraion. Th numrical chniqus dvlopd ar also applid in compuing h ingral in ( 9 ) and pricing aggrga XL conracs ( S Scion 6 (14) ). Th problm w ar aiming o solv is o find a mhod of compuing h following ingral ca -rt ( -) iv + i( D S ) ( T )[ (-) 1] X i = l - f - # d, (10) whr ca dnos h pric of a ro coupon ca bond.

12 18 G.G. HASLIP AND.K. KAISHE Th simpls way o valua his yp of ingral is o rprsn i as a Rimann sum and hn ihr compu i dircly or apply an fficin numrical ingraion algorihm. For insanc, adapiv Gauss-Kronrod ingraion dscribd by Calvi al. (000) can b applid by making a subsiuion o rmov h complx limis. Alrnaivly, chniqus such as h Fas Fourir Transform ( FFT ) can b applid if fficin valuaion of h ingral is rquird a a rang of riggr lvls. For insanc, during h calibraion procss, on will amp o rplica obsrvd ca bond pric for a variy of diffrn riggr lvls by rpad r-valuaion a diffrn paramr valus. I is imporan ha a ach iraion of h calibraion procss h ca bond prics can b valuad quickly. Th firs sp in dvloping h numrical compuaion of ca bond prics is o simplify h ingral xprssion. I currnly has complx limis ha prvn i bing rprsnd as a summaion, w hrfor mak h subsiuion = iv, which yilds ca -rt- ) i( + iv) ( D - S ) + l( T - )[ f (- - iv) - 1] ( X i = - # iv d, + (11) - whr h ral and imaginary pars of h ingrand funcion ar assumd o b in L 1 ( R ). In ( 11), w runca h limis of ingraion a A/, A/ and w hav ca. -i A -r ( T- ) i( + iv)( D-S) + l( T-)[ fx (--iv)-1] # A - + iv d A -r ( T-) -v( D- S ) + l( T-)[ fx (- -iv)-1] i i( S) = - D- # + iv A - d. Convrgnc of his ingral is guarand following h propry ha ingrals ar coninuous funcions of hir limis. Finally, w dfin A D =, N - 1 A =- + m D, m whr N is h numbr of sps in h numrical approximaion, 0 < 1 <... < N dfin a uniform pariion of h inrval [ A/, A/] and D is h widh of

13 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS 19 h pariion. This is a classical Rimann approximaion of h ingral ( 10 ), which yilds -rt ( - ) N - 1 A - vd ( - S) + l( T-)[ f X(-m- iv) -1] ca i i (- + md)( D -S). - / m + iv m = 0 i =-D ia -rt ( -)-( + v)( D-S) N 1 imd( D - S) + l( T -)[ fx( -m- iv) -1] - / m = 0 m + iv. D (1) Clarly, as A and N " his will convrg o h rquird ingral ( 10 ). Th convrgnc of his approximaion could b vry asily improvd by using a br numrical ingraion mhod such as a sandard quadraur rul or a mor advancd mhod such as h aformniond adapiv Gauss-Kronrod approach. Th pricing formula can hrfor b valuad numrically using a simpl compur program. Howvr, as mniond abov, hr is a mor fficin way of calculaing his yp of ingral, whn h pric is rquird for a rang of riggr lvls D: compuaion using h FFT or h Fracional Fas Fourir Transform ( FFFT ). 6. THE FAST FOURIER TRANSFORM Th FFT is an fficin algorihm ha can b usd o numrically valua h Discr Fourir Transform ( DFT ). W rfr h radr o Carr and Madan (1999) for a daild discussion of h FFT for h applicaion of opion pricing. Rurning o quaion ( 1 ), w can r-xprss i as a DFT by making h following subsiuion n Dn = S+, ND as an approximaion o D for a suiabl choic of n. W hn approxima h pric of h ca bond as ia -rt ( -)-( + v)( Dn-S) 1 ca i N-,D n. -D m = 0 / f N, (1) m inm whr f m = l m ( T- ) [f X (- -iv ) -1 ] m + iv. This is in prcisly h form of h DFT which mans ha w can us h Fas Fourir Transform o prform h rquird numrical compuaion. Th FFT

14 0 G.G. HASLIP AND.K. KAISHE ca N - 1 will rurn us wih an array, { D, } n= 0, ach lmn of which n n, ca rprsns N h pric of a ca bond wih riggr D n = S + ND, whr losss o da ar S. Thus, w ar quickly abl o pric an nir rang of ca bonds wih a rang N of riggr lvls D n = S o S + A, in a singl applicaion of h FFT. No ha any ca bonds for which D is blow S will hav alrady bn riggrd. Similar compuaions yild h following approximaion formula for h XL pric,, of h aggrga XL conrac in h arlir xampl, considrd in Scion 4. ia -rt ( -)-( + v)( Kn -S) 1 XL i N-,K n = -D m = 0 / f N, (14) m inm whr f m = l m ( T- ) [fx (- -iv ) -1 ] ( m + iv) and K n = S + n ND. Th FFT applid o ( 14 ) will yild aggrga XL rinsuranc prics for a rang N of prioriy lvls K n = S o S + A. 7. THE FRACTIONAL FAST FOURIER TRANSFORM Th Fas Fourir Transformaion was shown in h prvious scion o provid a good mhod for valuaing h conrac pric for a rang of riggrs / prioriy lvls. Howvr, is main disadvanag is ha h riggr / prioriy lvls all n N - 1 li on h msh dfind by { ND } n = 0. This mans ha o pric a paricular poins, inrpolaion mus b usd, which inroducs addiional rror. In ordr o ovrcom his sourc of rror h Fracional Fas Fourir Transform mhod suggsd by Baily (1990) and implmnd for opion pricing by Chourdakis (005) can b usful. As an xampl of applying h FFFT o pricing conracs, w considr h ca bond xampl from Scion 6 on h FFT. Suppos w wish o pric ca bonds using (1) for a rang of riggr lvls D = D L o D U. W dfin D U = DL + D D n for n N n, - - L = 0, 1,, - 1 N 1 and subsiu ino (1) yilding h following xprssion for ca bond prics, ca D, n i. -D i =-D ia -rt ( -)-( + v)( Dn -S) ia -rt ( -)-( + v)( Dn -S) N - 1 / m = 0 N - 1 DU - DL l( T-)[ fx(-m- iv) -1] imd( DL + n - S ) N - 1 m + iv imng / (15) f, m = 0 m

15 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS 1 lt[ f (- -iv)-1] X m D DU - DL imd( DL-S whr g = p ` N - 1 j and f ) m = m + iv. Exprssion ( 15 ) can b compud for h xac rang of rquird riggr lvls in singl applicaion of h FFFT. 8. EXAMPLE W will now look a an xampl of pricing h ca bond and h aggrga xcss of loss conrac undr a paricular choics of svriy disribuion o dmonsra h mhod is fasibl from a pracical prspciv Gamma Svriy I is assumd ha S follows a compound Poisson disribuion wih Poisson paramr l and individual loss amoun follows a Gamma disribuion wih shap paramr a and scal paramr b. For simpliciy, w assum ha h ca bond pays no coupons. Th characrisic funcion of h Gamma disribuion is -a i b fx ( ) = d1 - =. b n d b- i n Applying formula ( 6 ) w hav h valu of a ro coupon ca bond a im a ca - - ) iv+ b i( D S ) ( T )[( ) a - 1]. rt ( i i = l - b+ # d (16) To valua his numrically, w rquir Im( ) < 0 and w mus avoid h irrgulariy a = bi ( which will mak b + i = 0 ). W hrfor should ingra along a sraigh lin ha lis bnah h ral axis. No ha sinc h poin = bi lis abov h ral axis, i dos no concrn us. Nx, w considr h pricing formula ( 9 ) for h aggrga XL conrac. Subsiuing in h formula for h characrisic funcion, w immdialy s XL =- -rt ( -) iv+ b i( K - S ) ( T )[( ) a + l - - 1] b+ i d, # (17) whr ingraion is carrid ou on a sraigh lin ha lis abov h ral axis and blow h poin = bi. 8.. Paro Svriy On of h mor popular svriy disribuions in pracical applicaions is h Paro disribuion. W will considr h wo paramr vrsion of h Paro disribuion and driv h pricing formula for h wo conracs.

16 G.G. HASLIP AND.K. KAISHE Th wo paramr Paro disribuion wih paramrs k and a dos no hav a momn gnraing funcion, w only rquir h xisnc of is characrisic funcion, which can b shown o b f () = k (-ia) (-k, -ia), (18) X whr G( a, ) is h incompl uppr Gamma funcion dfind as k G a -1 -y G ( a, ) = # y dy. Th drivaion of h characrisic funcion is sandard bu is rahr lnghy and hnc is omid. For an xampl of similar calculaions w rfr o Boai (007) ( s also Dufrsn al. (006) ). This rprsnaion of h characrisic funcion is only dfind for in h uppr half of h complx plan, ha is, whr Im( ) 0. This rsricion will hav a srious ffc on our abiliy o pric crain conracs whr h pricing formula involvs calculaion of h characrisic funcion in h ngaiv half of h complx plan. No ha hr is a numrically fficin mhod of valuaing h incompl gamma funcion dscribd in Numrical Rcips (1988), which is asily adapd o h complx paramr cas. Again, applying formula ( 6 ) w hav h valu of a ro coupon ca bond a im o b ca -rt ( - ) iv + k i( D- S) + ( T -)[ k(ia) G( -k, ia) -1] i = - l # d. (19) To valua his numrically, w simply rquir Im( ) < 0. Sinc h Paro characrisic funcion is bing valuad a, his will nsur ha w ar only valuaing h characrisic funcion in h uppr half of h complx plan. W now coninu o look a h cas of h aggrga XL conrac. Using h sam approach w hav ha h valu of aggrga XL conrac is XL =- - r( T- iv+ k ) i( K - S) + l( T -)[ k(ia) G( -k, ia) -1] # d, (0) whr ingraion is carrid ou on a sraigh lin ha lis abov h ral axis. This mans ha w nd o valua h characrisic funcion in h lowr half of h complx plan. Howvr, as prviously nod, h numrical form of h characrisic funcion is divrgn in his rgion, which mans ha his pricing mhod fails.

17 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS W nd o xplor an alrnaiv approach of valuaing h aggrga XL conrac pric in h cas of Paro svriy. Forunaly, w can draw inspiraion from h financ world onc mor. Th ky conncion bwn h financial and acuarial filds xploid in his papr is h form of h aggrga XL payoff funcion, max( S T K, 0 ), which is idnical o h payoff of a Europan opion conrac. An imporan formula in opion pricing hory is h pu-call pariy rlaionship c + K ) = p + s, (1) -rt ( - whr c, p ar h prics a im of a call and pu opion xpiring a im T on sock s, wih srik pric K. W can driv a similar rlaionship in insuranc. W bgin by dfining h insuranc quivaln of a pu opion as dscribd by Wack (1997). This will b a conrac ha provids payoff max( K S T, 0 ) () a im T. W can consruc a pu-call pariy rlaionship for insuranc by considring no-arbirag argumns undr h assumpion ha h noional pu conrac xiss. Considr h following porfolios. Porfolio 1: A currn im, purchas a pu conrac a cos P and addiionally purchas an insuranc policy on h aggrga claims procss for pric. By im T, if h aggrga claims procss S T has xcdd prioriy K, hn h pu conrac will provid ro payoff. If h aggrga claims do no xcd K hn h pu conrac will provid payoff K S T. In boh cass h insuranc policy will provid payoff S T. Thus h ovrall payoff from his porfolio a im T is ST, if ST $ K * K, if S < K. In a similar fashion, w s up a scond porfolio blow. Porfolio : A currn im, purchas an aggrga XL conrac a cos XL and addiionally invs amoun K r( T ) in risk-fr cash. By im T, if h aggrga claims procss S T has xcdd prioriy K, hn h aggrga XL conrac will provid payoff S T K. If h aggrga claims do no xcd K hn h conrac will provid ro payoff. Th cash will maur o amoun K. Thus h ovrall payoff from his porfolio a im T is T ST, * K, if S T $ K if S < K, T

18 4 G.G. HASLIP AND.K. KAISHE which is prcisly h sam as Porfolio 1. Applying h principl of no-arbirag, w assr ha boh porfolios hav qual valu a im as wll. Ohrwis i would b possibl o mak risk-fr profi by slling on porfolio shor and aking a long posiion in h ohr. W hrfor hav h following pu-call rlaionship for insuranc policis XL rt ( K - - ) + = + P. Th prmium paid for h insuranc policy on h aggrga claims procss can b pricd undr a risk-nural probabiliy masur as follows = E ( S F ) = E ( S F ). () -rt ( - ) ( -rt- ) Q T Q T W can calcula his xpcaion using a similar approach and noaion o ha usd in ( 5 ) for calculaing h characrisic funcion of h aggrga claims procss wih rspc o h filraion a im. W hav E( S F ) = E( S + D F ) T, T NT = + f / j = N + 1 S E X p = S + E( X) E( N -N F ) T = S + le( X)( T-), j whr in h las qualiy w hav usd h fac ha h cnrd Poisson procss ( N T lt ) is a maringal wih rspc o h filraion F. This provids us wih pu-call pariy rlaionship for insuranc procsss undr h aggrga claims compound Poisson modl XL ( ( + -rt- ) -rt- ) K = le( X)( T- ) + P. (4) W now rurn o h problm of valuaing h pric of h aggrga XL conrac in ( 0 ). Using h pu-call pariy rlaionship w can r-xprss XL as XL = P - + E( X)( T -). K -rt ( - ) -rt ( - ) l So, providd w can succssfully calcula h valu of h pu conrac undr h Paro claims svriy disribuion, w will b abl o work h pric of h aggrga XL conrac. Th gnralisd Fourir ransform of h pu payoff ( ) is asily calculad o b

19 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS 5 ik + F $ ( K - ST ). =-, (5) whr Im( ) < 0. This is virually h sam as h Fourir ransform of h aggrga XL payoff funcion, xcp is now rsricd o h lowr half of h complx plan. Th pu conrac can hrfor b valud as P =- - r( T- i + k ) v i( K - S) + l( T -)[ k(ia) G( -k, ia) -1] # d, (6) whr ingraion is carrid ou along a sraigh lin ha lis bnah h ral axis. This mans ha w will nd o valua h characrisic funcion in h uppr half of h complx plan, which is wihin h rgion of rgulariy. W hrfor hav succssfully found a mans of pricing aggrga XL conracs in h cas of a Paro svriy disribuion. L us no ha an alrnaiv xprssion for E(( S T K ) + F ) has bn obaind by Dufrsn al. (006) using Parsval s horm. 8.. Numrical Compuaion As an illusraion of h pracicaliy of h FFFT approach, h pric of ca bonds and aggrga XL conracs wr compud using h proposd mhodology and h paramrisaion from h xampl in Scion 8.1, for a rang of riggrs / prioriy lvls and duraions bwn 0 and 1. A D plo of h aggrga XL prics is shown in Figur 1 and h corrsponding plo for h ca bond prics is shown in Figur. I is noicabl ha a h boundary poins whr h riggr / prioriy lvl is clos o ro, h ingral approximaion ( 15) dos no convrg wll. This is a wll known problm in h applicaion of h DFT o opion pricing problms in financ. Howvr, i is no a significan issu, sinc no ca bonds Pric Tim o Expiry Prioriy FIGURE 1: Aggrga XL prics undr Gamma(, ) svriy and l = claims frquncy.

20 6 G.G. HASLIP AND.K. KAISHE Pric Tim o Expiry Prioriy FIGURE : Caasroph bond prics undr gamma(, ) svriy and l = claims frquncy. or rinsuranc conracs ar issud wih a riggr / prioriy lvl clos o ro. This would bcom a mor srious problm for pricing dirc insuranc. For normal conracs arising in h rinsuranc mark h ingral approximaion ( 15 ) convrgs quickly and accuraly. 9. COMPARISON WITH MONTE CARLO SIMULATION To vrify ha h analyical formula for h ca bond and rinsuranc prics in Scions and 4 drivd using h gnralisd Fourir ransform mhod provid h anicipad rsuls, hy will b valuad using h Rimann summaion approximaion and compard o ha achivd using Mon Carlo simulaion for a varying numbr of simulaions. For his purpos, w will s up a simpl modl in which losss ar gnrad for a on yar priod according o a compound Poisson disribuion wih ra l = and svriy disribuion Gamma wih paramrs a = b = 1. Th gnrad losss ar aggrgad and hn h rcovris ar valuad for boh an aggrga XL conrac and a ca bond. Th aggrga XL conrac has aachmn poin a 4.75 and has no limi of rinsamns or uppr limi on rcovris. Th ca bond is a simpl ro coupon riggr basd bond wih prioriy lvl W will compar h pric of hs conracs using h analyical formula ( 16 ) and ( 17 ) o ha achivd hrough simulaion. Calculaing h analyical formula numrically, undr h Rimann summaion approximaion o 7 dcimal placs of prcision, w find h prics for h aggrga XL and ca bond conracs ar and rspcivly, assuming h risk fr ra of inrs is Undr Mon-Carlo simulaion, h pricing rsuls for diffrn simulaion sis ar shown in h abl blow. I is inrsing o obsrv ha convrgnc undr Mon-Carlo is qui slow and rquirs around wo million rials o achiv an aggrga XL pric wihin 0.1% of h analyical pric. This suggss ha for pricing applicaions h analyical mhods of compuing prics ar mor fficin han Mon

21 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS 7 Trials XL Man ( MC ) Error % Ca Bond Man ( MC ) Error % % % % % % % % % % % % % Carlo simulaion chniqus. In paricular, his maks h analyical mhod suiabl for calibraing h modl o obsrvd mark prics. This would no b possibl o achiv in a rasonabl im priod using Mon-Carlo, sinc h calibraion procss usually involvs an opimising rouin rcalculaing modlld prics rpadly using diffrn paramr valus. 10. COMMENTS AND CONCLUSIONS In his papr w hav providd a framwork for pricing rinsuranc conracs in a way ha is consisn wih h prics of ca bonds on h sam undrlying loss procss. W hav uilisd xising work in his ara by applying an opion pricing chniqu dvlopd by Lwis (001) ha applis h gnralisd Fourir ransform o pric drivaiv conracs on an undrlying Lévy procss. W hn dmonsrad pricing an aggrga xcss of loss conrac and ca bond undr his framwork, for boh h Gamma and Paro Typ II svriy disribuion. Whil h mahmaics involvd undr his approach is mor complicad han radiional acuarial pricing mhods, w hav shown ha i is rlaivly asy o compu h pricing formula drivd using fficin numrical mhods. In paricular w hav dmonsrad ha h Fracional Fas Fourir Transform ( FFFT ) can provid a usful rol in acuarial scinc. Using h FFFT w hav shown how insuranc conracs can b pricd in a singl calculaion for a rang of prioriy / riggr lvls. This provids a clar advanag ovr Mon- Carlo basd mhods, as i mans ha h modlld ca bond pric can b compud a all rquird riggr lvls in around on scond ( on a Gh Inl CPU ). Th sourc cod implmning h mhodology proposd in h papr is availabl upon rqus o h auhors. Undoubdly, h mos difficul par of applying his approach will b calibraing h compound Poisson disribuion o h mark prics of ca bonds and hisorical loss frquncy / svriy daa. Howvr, his is crainly achivabl and is an xnsion of xising rsarch ha has focusd on calibraing agains hisorical daa. Som work has alrady bn carrid ou in his ara by Burncki (005), who dscribs h calibraion procss for pricing ca bonds on h Propry Claims Srvics ( PCS ) indx in h Unid Sas.

22 8 G.G. HASLIP AND.K. KAISHE W bliv ha h FFFT could provid an fficin mhod of calibraing h modl by mans of an opimisaion algorihm. In paricular w suggs o follow h calibraion mhods normally usd in opion pricing modls. For xampl, an xhausiv algorihm such as adapd simulad annaling or a gnic algorihm could b applid o find h modl paramrs ha minimis h oal squard rror bwn obsrvd ca bond prics and modlld prics. Finally, w no ha h mhodology prsnd in h papr can b asily gnralisd o h cas of sochasic inrs ras undr h assumpion of indpndnc bwn h insuranc and inrs ra procsss. ACKNOWLEDGEMENTS W ar vry graful o Alan Lwis for his hlpful suggsions for h numrical implmnaion of h gnralisd Fourir ransform mhod. W ar also graful o an anonymous rfr for h consruciv commns and suggsd improvmns. REFERENCES AON BENFIELD SECURITIES ( 009 ) Insuranc-Linkd Scuriis Adaping o an Evolving Mark 009. Aon Bnfild Scuriis Limid. BAILEY, D.H. and SWARZTRAUBER, P.N. ( 1990 ) Th Fracional Fourir Transform and Applicaions, NAS Sysm Division. NASA Ams Rsarch Cnr. BARYSHNIKO, Y., MAYO, A. and TAYLOR, D.R. ( 001 ) Pricing of CAT Bonds. Working papr. BAXTER, M.W. and RENNIE, A.J.O. ( 1996 ) Financial Calculus: An Inroducion o Drivaiv Pricing. Prss Syndica of h Univrsiy of Cambridg. BOTTAZZI, G. ( 007 ) On h Paro Typ III disribuion. Laboraory of Economics and Managmn ( LEM ) San Anna School of Advancd Sudis, Pisa, Ialy. BURNECKI, K., KUKLA, G. and TAYLOR, D. ( 005 ) Pricing of caasroph bonds. Saisical Tools for Financ and Insuranc, Springr, Brlin. BURNECKI, K. ( 005 ) Pricing caasroph bonds in a compound non-homognous Poisson modl wih lf-runcad loss disribuions. Mahmaics in Financ Confrnc, Brg-n- Dal. CALETTI, D., GOLUB, G.H., GRAGG, W.B. and REICHEL, L. ( 000 ) Compuaion of Gauss-Kronrod Quadraur Ruls. Mah. Compu. 69, CARR P. and MADAN D. ( 1999 ) Opion aluaion Using Fas Fourir Transform. Journal of Compuaional Financ ( 4 ), CHOURDAKIS, K. ( 005 ) Opion Pricing Using h Fracional FFT. Journal of Compuaional Financ 8( ), CONT, R. and TANKO, P. ( 004 ) Financial Modlling Wih Jump Procsss. Chapman and Hall/ CRC Financial Mahmaics Sris. DELBAEN, F. and HAEZENDONCK, J. ( 1989 ) A Maringal Approach o Prmium Calculaion Principls in an Arbirag Fr Mark. Insuranc: Mahmaics and Economics 8, DELBAEN, F. and SCHACHERMAYER, W. ( 1994 ) A gnral vrsion of h fundamnal horm of ass pricing. Mahmaisch Annaln 00, DUFRESNE, D., GARRIDO, J. and MORALES, M. ( 006 ) Fourir Invrsion Formula Opion Pricing and Insuranc. Mhodology and Compuing In Applid Probabiliy, 11( ), EMBRECHTS, P. ( 1996 ) Acuarial vrsus financial pricing of insuranc. Risk Financ 1( 4 ), HOLTAN, J. ( 004 ) Pragmaic Insuranc Pricing, XXXh ASTIN Colloquium. THE INSURANCE INSIDER ( 007 ), Excuiv Brifing Auumn 007,

23 PRICING REINSURANCE CONTRACTS IN THE PRESENCE OF CATASTROPHE BONDS 9 INSURANCE JOURNAL PROPERTY AND CASUALTY MAGAZINE ( 00 ) Mrrill Lynch Forms Brmuda Rinsuranc Co., Wlls Publishing, hp:// 00/06/06/168.hm. JAIMUNGAL, S., JACKSON, K.R., and SURKO,. ( 007 ) Opion Pricing wih Rgim Swiching Lévy Procsss using Fourir Spac Tim-spping. Procding of h Fourh IASTED Inrnaional Confrnc on Financial Enginring and Applicaions, 9-97, 007. LEWIS, A.L. ( 001 ) A simpl opion formula for gnral jump-diffusion and ohr xponnial Lévy procsss. Envision Financial Sysms and Opion Ciy.n. MCGHEE, C., CLARKE, R., FUGIT, J. and HATHAWAY, J. ( 007 ) Th Caasroph Bond Mark a Yar-End 007: Th Mark Gos Mainsram. Invsmn Banking Spcialiy Pracic, MMC Scuriis Corp. MCGHEE, C., CLARKE, R., and COLLURA, J. ( 006 ) Th Caasroph Bond Mark a Yar-End 006. Invsmn Banking Spcialiy Pracic, MMC Scuriis Corp. MUERMANN, A. ( 00 ) Acuarially Consisn aluaion in an Ingrad Mark. Working Papr, Financial Insiuions Cnr, Th Wharon School, Univrsiy of Pnnsylvania. MUERMANN, A. ( 00 ) Acuarially Consisn aluaion of Caasroph Drivaivs. Working Papr, Financial Insiuions Cnr, Th Wharon School, Univrsiy of Pnnsylvania. MUERMANN, A. ( 006 ) Mark Pric of Insuranc Risk Implid by Caasroph Drivaivs. Working Papr, Financial Insiuions Cnr, Th Wharon School, Univrsiy of Pnnsylvania. PRESS, W.H., FLANNERY, P.F., TEUKOLSKY, S.A. and ETTERLING, W.T. ( 1988 ) Numrical Rcips in C: Th Ar of Scinific Compuing. Prss Syndica of h Univrsiy of Cambridg. SCHOUTENS, W. ( 00 ) Lvy Procsss in Financ: Pricing Financial Drivaivs, Wily, Nw York, 00. SONDERMANN, D. ( 1991 ) Rinsuranc in arbirag-fr marks. Insuranc: Mahmaics and Economics 10, WACEK, M.G. ( 1997 ) Applicaion of h Opion Mark Paradigm o h Soluion of Insuranc Problms. Procdings of h Casualy Acuarial Sociy LXXXI, WEIR, A.J. ( 197 ) Lbsgu Ingraion and Masur. Prss Syndica of h Univrsiy of Cambridg. YOUNG,.R. ( 004 ) Encyclopdia of Acuarial Scinc. Wily. GARETH HASLIP Sir John Cass Businss School, Ciy Univrsiy, 106 Bunhill Road, London EC1Y 8TZ, Tl.: +44 ( 0 ) garh@haslip.co.uk LADIMIR K. KAISHE Sir John Cass Businss School, Ciy Univrsiy, 106 Bunhill Road, London EC1Y 8TZ, Tl.: +44 ( 0 ) v.kaishv@ciy.ac.uk