Valuing catastrophe bonds involving correlation and CIR interest rate model

Transcription

1 Comp. Appl. Mah : hps://doi.org/ /s Valuing caasrophe bonds involving correlaion and CIR ineres rae model Pior Nowak 1 Maciej Romaniuk 1,2 Received: 16 July 2015 / Revised: 4 November 2015 / Acceped: 25 April 2016 / Published online: 12 May 2016 The Auhors This aricle is published wih open access a Springerlink.com Absrac Naural caasrophes lead o problems of insurance and reinsurance indusry. Classic insurance mechanisms are ofen inadequae for dealing wih consequences of caasrophic evens. Therefore, new financial insrumens, including caasrophe bonds ca bonds, were developed. In his paper we price he caasrophe bonds wih a generalized payoff srucure, assuming ha he bondholder s payoff depends on an underlying asse driven by a sochasic jump-diffusion process. Simulaneously, he risk-free spo ineres rae has also a sochasic form and is described by he muli-facor Cox Ingersoll Ross model. We assume he possibiliy of correlaion beween he Brownian par of he underlying asse and he componens of he ineres rae model. Using sochasic mehods, we prove he valuaion formula, which can be applied o he ca bonds wih various payoff funcions. We use adapive Mone Carlo simulaions o analyze he numerical properies of he obained pricing formula for various seings, including some similar o he pracical cases. Keywords Caasrophe bonds Asse pricing Sochasic models Mone Carlo simulaions CIR model Mahemaics Subjec Classificaion 91B25 60H30 91G60 1 Inroducion Nowadays overwhelming risks caused by naural caasrophes, like hurricanes, floods and earhquakes, lead o severe problems of insurance and reinsurance indusry. For example, Communicaed by Jorge Zubelli. B Pior Nowak pnowak@ibspan.waw.pl 1 Sysems Research Insiue Polish Academy of Sciences, ul. Newelska 6, Warsaw, Poland 2 The John Paul II Caholic Universiy of Lublin, Lublin, Poland

2 366 P. Nowak, M. Romaniuk losses from Hurricane Andrew reached US$30 billion in 1992 and he losses from Hurricane Karina in 2005 are esimaed a $40 60 billion see Muermann Such exreme losses from a single caasrophic even cause problems relaing o reserve adequacy or even lead o bankrupcy of insurers. For example, afer Hurricane Andrew more han 60 insurance companies fell ino insolvency see Muermann The main reason of he menioned problems is relaed o assumpions used in classical insurance mechanisms. In radiional insurance models see, e.g., Borch 1974 risk claims ha are independen and small in relaion o he value of he whole insurance porfolio e.g. caused by car crashes are he norm. Then he classic sraegy of building porfolio he graer he number of risks, he beer qualiy of he whole porfolio is jusified by he law of large numbers and he cenral limi heorem see, e.g., Borch 1974; Ermoliev e al In he case of naural caasrophes, he sources of risks are sricly dependen on ime and locaion. Addiionally, problems wih adverse selecion, moral hazard and he cycles of prices of reinsurer s policies should be noed see, e.g., Ermoliev e al. 2001; Finken and Laux Therefore, new financial derivaives which connec boh he financial markes and he insurance indusry were developed. The main aim of hese insrumens is o securize he caasrophic losses, i.e. o ransfer insurance risks ino financial markes by packaging of risks ino special radable asses caasrophic derivaives see, e.g., Cummins e al. 2002; Freeman and Kunreuher 1997; Froo 2001; Harringon and Niehaus 2003; Nowak 1999; Nowak and Romaniuk 2010b, c, d; Nowak e al One of he mos popular caasrophe-linked securiy is a caasrophe bond known also as a ca bond or an Ac-of-God bond see, e.g., Cox e al. 2000; D Arcy and France 1992; Ermolieva e al. 2007; George 1999; Nowak and Romaniuk 2009a; O Brien 1997; Romaniuk and Ermolieva 2005; Vaugirard In 1993, caasrophe derivaives were inroduced by he Chicago Board of Trade CBoT. These financial derivaives were based on underlying indexes reflecing he insured propery losses due o naural caasrophes repored by insurance and reinsurance companies. Then new approaches o developmen of he ca bonds were applied see, e.g., Kwok 2008; Lee and Yu The payoff received by he ca bondholder is linked o an addiional random variable, which is called riggering poin. This even indemniy rigger, parameric rigger or index rigger is usually relaed o occurrence of specified caasrophe like hurricane in given region and fixed ime inerval or i is conneced wih he value of issuer s acual losses from caasrophic even like flood, losses modeled by special sofware based on he real parameers of a caasrophe, or he whole insurance indusry index, or he real parameers of a caasrophe e.g., earhquake magniude or wind speeds in case of windsorms, or he hybrid index relaed o modeled losses see, e.g., George 1999; Niedzielski 1997; Vaugirard 2003; Walker In he case of some ca bonds, he riggering poin is relaed o he second or even he hird even during a fixed period of ime. Addiionally, he srucure of paymens for he ca bonds depends also on some primary underlying asse e.g. he LIBOR. As noed by many auhors see, e.g., Ermoliev e al. 2001; Finken and Laux 2009; Vaugirard 2003, he ca bonds are imporan ools for insurers and reinsurers. Among oher advanages, hey sressed ha using he ca bonds lowers he coss of reinsurance and reduces he risks caused by moral hazard. The cash flows relaed o he ca bond are usually managed by special ailor-made fund, called a special-purpose vehicle SPV or a special purpose company SPC see, e.g., Lee and Yu 2007; Vaugirard The hedger e.g. insurer or reinsurer pays an insurance premium in exchange for coverage in he case if riggering poin occurs. The invesors purchase an insurance-linked securiy for cash. The menioned premium and cash flows are direced o SPV, which issues he caasrophe bonds. Usually, SPV purchases safe securiies in order

3 Valuing caasrophe bonds involving correlaion o saisfy fuure possible demands. Invesors hold he issued asses whose coupons and/or principal depend on occurrence of he riggering poin, e.g. he caasrophic even. If his even occurs during he specified period, he SPV compensaes he insurer and he cash flows for invesors are changed. Usually, hese flows are lowered, i.e. here is full or parial forgiveness of he repaymen of principal and/or ineres. However, if he riggering poin does no occur, he invesors usually receive he full paymen. In he lieraure concerning he caasrophe bonds and heir pricing many auhors apply sochasic models. Among hem one should menion wo advanced approaches, where sochasic processes wih discree ime are used: Cox and Pedersen 2000 wihin he framework of represenaive agen equilibrium and Reshear 2008, where he payoff funcions depend on caasrophic propery losses and caasrophic moraliy. More auhors apply sochasic models wih coninuous ime. To incorporae various characerisics of he caasrophe process compound Poisson models are used in Baryshnikov e al In his approach, no analyical pricing formula is obained and he problem of change of probabiliy measure in he arbirage mehod is no discussed. However, advanced numerical simulaions are conduced and analyzed. The auhors of Burnecki e al correc he mehod proposed in Baryshnikov e al In urn, he approach from Burneckieal.2003 is applied in Härdle and Lopez 2010 for he ca bonds conneced wih earhquakes in Mexico. In Albrecher e al he doubly sochasic compound Poisson process is used and reporing lags of he occurred claims are incorporaed o he model. The model behavior is analyzed wih applicaion of QMC algorihms. The arbirage mehod for ca bonds pricing is used by Vaugirard He addresses he problem of non compleeness of he marke, caused by caasrophic risk, and non-raded insurance-linked underlyings in he Meron s manner see Meron In he approach proposed in Lin e al he Markov-modulaed Poisson process is applied for descripion of he arrival rae of naural caasrophes. Jarrow in Jarrow 2010 obained an analyically closed ca bond valuaion formula, considering he LIBOR erm srucure of ineres raes. In Nowak and Romaniuk 2013a we applied he approach similar o he one proposed in he Vaugirard s paper. We proved a generalized caasrophe bond pricing formula, assuming he one-facor sochasic diffusion form of he risk-free ineres rae process. In conradisincion o he Vaugirard s approach, where caasrophe bonds payoffs were dependen on risk indexes, we considered he ca bond payoffs dependen only on he cumulaed caasrophic losses. Moreover, we conduced Mone Carlo simulaions o analyze he behavior of he valuaion formula. The menioned paper summarized and generalized our earlier resuls from Nowak and Romaniuk 2010b, c, d, Nowak e al Shorly afer our publicaion, resuls similar o ours were obained in Ma andma 2013, where he auhors assumed he one-facor Cox Ingersoll Ross CIR model of he risk-free ineres rae. In Nowak and Romaniuk 2009b, 2013b we considered he problem of ca bond pricing in fuzzy framework, incorporaing uncerain financial marke parameers o he model. Similar approach was also applied by us in Nowak and Romaniuk 2010a, 2013c, 2014, where he sochasic analysis, including he Jacod Grigelionis characerisics see, e.g., Shiryaev 1999; Nowak 2002, and he fuzzy ses heory were employed o find he European opion pricing formulas. In his paper we coninue our consideraions concerning valuaion of he caasrophe bonds. We assume no arbirage on he marke and he possibiliy of replicaion of ineres rae changes by financial insrumens exising on he marke. We use he maringale mehod of pricing. We apply he d-dimensional Brownian moion wih d 1 for descripion of he risk-free spo ineres rae and he one-dimensional Brownian moion and he compound Poisson process o model an underlying asse I, conneced wih he cumulaive caasrophic

4 368 P. Nowak, M. Romaniuk losses. We assume ha I is similar o a synheic insurance indusry asse. Our conribuion is hreefold. Firs, we consider he caasrophe bonds wih a generalized in comparison o Vaugirard 2003; Nowak and Romaniuk 2013a payoffsrucure, dependingon I. Second, in conradisincion o our previous approaches, where he one-facor spo ineres rae models were applied, he ineres rae behavior is described by he muli-facor CIR model. Third, we assume he possibiliy of correlaion beween he Brownian par of he process I,used in he caasrophe bond payoff, and he componens of he model of he ineres rae. To our bes knowledge such he approach assuming correlaion srucure has no been considered in he pricing lieraure. There are several essenial differences beween he approach presened in his paper and he model of Vaugirard We apply he muli-facor Cox Ingersol Ross model of he risk-free spo ineres rae, whereas in Vaugirard 2003 he one-facor Vasicek ineres rae model is used. In Vaugirard s approach he caasrophe bond payoff depends on a physical risk index driven by he Poisson jump-diffusion process. Is Brownian par models he unanicipaed insananeous index change, reflecing causes ha have marginal impac on he gauge. In urn, jumps, connecing wih caasrophic evens, increase he value of he risk index. In our approach we also use he jump-diffusion process o model an underlying asse, similar o a synheic insurance indusry asse. However, in conradisincion o Vaugirard 2003, is Brownian par plays a more imporan role, modelling, similarly as in Meron 1976, vibraions in price caused by emporary imbalance beween supply and demand on he marke. Jumps, conneced wih occurrences of caasrophic losses, decrease he value of an underlying insrumen. For echnical reasons we use a ransformaion of he process of an underlying asse for descripion of he bondholder payoff. The payoff srucure in our model is much more general hen he one considered in Vaugirard In paricular, i is possible o use a wide class of funcions for descripion of dependence beween he bondholder payoff and he ransformed underlying asse process. Finally, as we have menioned above, in conradisincion o he Vaugirard s model and models of oher auhors, our approach enables aking ino accoun he possibiliy of correlaion beween he Brownian par of he underlying asse and he Brownian moions modelling he ineres rae behavior. Since we use sochasic models of he spo ineres rae and he underlying asse, sochasic analysis mehods play he key role in derivaion and proof of he ca bond pricing formula. In paricular, he Girsanov heorem and Lévy s characerizaion of he Brownian moion is used. Furhermore, he correcness of he applied mehod of change of probabiliy measure is proved in deailed way. The proposed by us payoff srucure enables o use a wide class of funcions describing dependence beween he values of bondholder s payoff and he asse I, including sepwise, piecewise linear and piecewise quadraic one. Apar from heoreical consideraions, we conduc simulaions o compare behavior of hese models for differen payoff srucures. In numerical experimens we find he prices of he caasrophe bonds applying linear and quadraic payoff funcions. To analyze he behavior of he obained prices, we aler some parameers of he appropriae ineres rae model and he model of value of caasrophic losses. This paper is organized as follows: Sec. 2 conains necessary noaions and definiions concerning sochasic noions and processes used in he paper. Moreover, assumpions concerning he financial marke are formulaed. Secion 3 conains generalized definiion of he caasrophe bond payoff srucure as well as descripion of he muli-facor CIR ineres rae model. In Sec. 4 he caasrophe bond pricing formula is inroduced and proved. Since he menioned above risk-free ineres rae is modeled by he muli-facor affine process, he underlying asse is defined by he sochasic jump-diffusion and he Brownian pars of boh processes can be correlaed, he derivaion and proof of he valuaion formula required

5 Valuing caasrophe bonds involving correlaion applicaion of sochasic analysis. Apar from Theorem 3, which is he main heorem of he paper, Lemma 2 is formulaed o describe he caasrophe bond price a he momen zero and simplify compuaions of he expeced bondholder s payoff. In Sec. 5 adapive Mone Carlo simulaions are conduced for he inroduced formula of ca bond pricing. Firs, some necessary numerical algorihms are considered. Then he ca bond prices are esimaed and analyzed for various seings, including he se of parameers similar o he pracical case. Special aenion is paid o he influence of he parameers of he underlying asse like correlaion coefficiens which are imporan properies of he model considered in his paper on he numerically evaluaed price. 2 Sochasic and financial preliminaries In his secion we inroduce some necessary noaions, definiions and assumpions concerning sochasic models of caasrophe losses and financial marke. We denoe by. he Euclidean norm in R d, i.e. for each vecor x R d of he form x = x 1, x 2,...,x d x = x x = d x i 2. Here denoes ransposiion so ha x is a column vecor. Moreover, we will use he symbol. o denoe he Euclidean norm in R d+1. R d d denoes he space of d d marices of real numbers. In he furher par of he paper we will use he noion of quadraic covariance, which is generally defined for semimaringales for deails we refer he reader o Shiryaev Le T [0, be a ime inerval. Definiion 1 We call a sochasic process X = X T a semimaringale if i is represenable as a sum i=1 X = X 0 + A + M, T, where A is a process of bounded variaion over each finie inerval [0, ], M is a local maringale, boh defined on a filered probabiliy space, saisfying he usual condiions. Definiion 2 For wo semimaringales X and Y, on a filered probabiliy space, he quadraic covariance process is he process [X, Y ] = [X, Y ] T defined on he same filered probabiliy space, such ha [X, Y ] = X Y X s dy s Y s dx s X 0 Y 0, where 0 X s dy s and 0 Y s dx s are sochasic inegrals wih respec o Y and X, respecively. For descripion of losses caused by naural caasrophes and behavior of he risk-free spo ineres rae on he marke we apply sochasic processes wih coninuous ime. In he paper we consider hree differen probabiliy measures. For a probabiliy measure M we denoe by E M he expeced value wih respec o his measure. In paricular, all he sochasic processes and random variables inroduced in his secion are defined wih respec o a probabiliy P.

6 370 P. Nowak, M. Romaniuk All he economic aciviy will be assumed o ake place on a finie horizon [ 0, T ],where T is a posiive consan. Le for a posiive ineger d W X = W 1, W 2,...,W d [0,T ] be he sandard d-dimensional Brownian moion. Tha is, each W i is he one-dimensional Brownian moion and he differen componens W 1, W 2,...,W d are independen. The process W X will be used for descripion of he ineres rae on he marke. We addiionally consider he one-dimensional Brownian moion W I [0,T ],usedinhefurherparof he paper for descripion of an underlying asse. We assume ha for i = 1, 2,...,d W I and W i can be correlaed wih a correlaion coefficien ρ i, i.e. he quadraic covariaions have he form [ ] W I, W i = ρ i, i = 1, 2,...,d, [ 0, T ], and he sequence ρ i i=1 d saisfies he inequaliy ρ < 1, where ρ = ρ 1,ρ 2,...,ρ d. We inroduce a sequence U i i=1 of independen and idenically disribued non-negaive random variables wih finie expecaion o describe values of losses during caasrophic evens. For each [ 0, T ] cumulaive caasrophe losses unil he momen are modeled by he compound Poisson process N Ñ = U i, [ 0, T ], 2 i=1 where N is he sandard Poisson process wih a consan inensiy κ>0. Momens of jumps of he process N correspond o momens of caasrophic evens. We denoe by N he jump process N = Ñ κe 1,wheree 1 = 1 E P e U i. As we menioned earlier, all he discussed above sochasic processes and random variables are defined on a probabiliy space,f, P. We inroduce he filraion F [0,T ] generaed by W and Ñ. Moreover, he filraion F [0,T ] is augmened o encompass P-null ses from F = F T. W X [0,T ], N [0,T ] and U i i=1 as well as W I [0,T ], N [0,T ] and U i i=1 are independen. Furhermore, he probabiliy space wih filraion,f, F [0,T ], P saisfies he usual assumpions: he σ -algebra F is P-complee, he filraion F [0,T ] is righ coninuous and each F conains all he P-null ses from F. By he symbol B [0,T ] we denoe banking accoun saisfying he sandard sochasic equaion: db = r B d, B 0 = 1, where r = r [0,T ] is he risk-free spo ineres rae, i.e. shor-erm rae for risk-free borrowing or lending a ime over he infiniesimal ime inerval [, + d]. In he paper we assume ha r is modeled by a ime-homogenous d-dimensional Markov diffusion process X = X 1, X 2,...,X d, given by he equaion dx = αx d + σ X dw X, wih he value space S R d. The funcions α : S R d and σ : S R d d are sufficienly regular so ha he above equaion has a unique soluion. We consider a paricular diffusion

7 Valuing caasrophe bonds involving correlaion model, i.e. he muli-facor Cox Ingersoll Ross model, where he spo risk-free ineres rae process r [0,T ] has he form r = rx, [ 0, T ], for an affine funcion r : S R and X, defined in deail in Sec For all [ 0, T ] he banking accoun process B has he form B = exp r s ds. 0 We assume ha zero-coupon bonds are raded on he marke and by he symbol B, T we denoe he price a he ime, [ 0, T ], of he zero-coupon bond wih he face value equal o 1 and he mauriy dae T T. Similarly as in Vaugirard 2003, we ake ino accoun he possibiliy of caasrophic evens, using he jump-diffusion process I = I 0 exp μ + σ I W I N 3 wih μ = μ 0 σ 2 I 2, μ 0 R, σ I > 0, for descripion of an underlying asse I [0,T ].Since our approach is general, we do no characerize precisely he insrumen I. However, i can be inerpreed as an insrumen similar o a synheic insurance indusry underlying asse. Moreover, we inroduce he sochasic process I 0 Ī = sup, [ 0, T ], 4 s [0,] I s which will be used in definiion of he caasrophe bond payoff funcion in Sec We make he following assumpions concerning financial marke: here is no possibiliy of arbirage; here are no resricions for borrowing and shor selling; rading on he marke akes place coninuously in ime; here are no ransacion coss; lending and borrowing raes are equal and changes in he ineres rae r can be replicaed by exising financial insrumens. 3 Descripion of he caasrophe bond 3.1 Payoff srucure The payoff srucure of he caasrophe bond is described by classes W, and K defined below. We fix a posiive ineger n 1, a face value of he caasrophe bond Fv > 0anda mauriy dae of he ca bond T [0, T ]. The class of sequences w = w 1,w 2,...,w n, where 0 w 1,w 2,...,w n and n i=1 w i 1, is denoed by W. The parial sums of w W are denoed by w 0 = 0, w k = k w i, k = 1, 2,...,n. i=1 is he class of sequences of funcions ϕ = ϕ 1,ϕ 2,...,ϕ n fulfilling he following condiions:

8 372 P. Nowak, M. Romaniuk i ϕ i : [0, 1] [0, 1], i = 1,...,n; ii ϕ i C [0, 1] and is non-decreasing for each i = 1, 2,...,n; iii ϕ i 0 = 0, i = 1,...,n. and In paricular, we consider he following subclasses of : 0 = {ϕ : ϕ i 0 for i = 1, 2,...,n} ; 1 = {ϕ : ϕ i 1 = 1 for i = 1, 2,...,n} ; 1,l = {ϕ : ϕ i x = x for x [0, 1] and i = 1, 2,...,n} ; 1,q = { ϕ : ϕ i x = x 2 for x [0, 1] and i = 1, 2,...,n }. Clearly, 1,l and 1,q are subclasses of 1. Finally, K is he class of increasing sequences K = K 0, K 1, K 2,...,K n, where 1 K 0 < K 1 < < K n. We proceed o define he caasrophe bond payoff funcion. Le w W,ϕ and K W. We inroduce an auxiliary funcion [ f w,ϕ,k :[0, Fv 1 w n ], Fv saisfying he following assumpions: i f w,ϕ,k [0,K0 ] Fv; ii f w,ϕ,k x Ki 1,K i ] = Fv 1 w i 1 ϕ x Ki 1 i iii f Kn, Fv 1 w n. K i K i 1 w i, i = 1, 2,...,n; Definiion 3 Le w W, ϕ and K W. WedenoebyIBw, ϕ, K he caasrophe bond wih he face value Fv, he mauriy and he payoff dae T if is payoff funcion is he random variable ν w,ϕ,k given by he equaliy ν w,ϕ,k = f w,ϕ,k ĪT. The payoff funcion ν w,ϕ,k of IBw, ϕ, K will be called sepwise piecewise linear or piecewise quadraic if ϕ 0 ϕ 1,l or ϕ 1,q. The following remark shows basic facs concerning he ca bond defined above. The presened formulas are obained by sraighforward compuaions. Remark 1 The caasrophe bond IBw, ϕ, K has he following properies: 1. The general formula describing he payoff as a funcion of Ī T can be wrien in he form [ ν w,ϕ,k =Fv 1 n ] ĪT K i Ī T K i 1 n ϕ i w i 1 ϕ i 1w i I { } Ī i=1 K i K T >K i. i 1 i=1 5

9 Valuing caasrophe bonds involving correlaion In paricular, Fv 1 n i=1 w i I { } Ī T >K i [ ν w,ϕ,k = Fv 1 ] n Ī T K i Ī T K i 1 i=1 K i K i 1 w i [ Fv 1 ] n ĪT K i Ī T K 2 i 1 i=1 K i K wi i 1 for ϕ 0 ; for ϕ 1,l ; for ϕ 1,q. 2. If Ī T is relaively small i.e., Ī T K 0, he bondholder receives he payoff equal o is face value Fv. 3. If Ī T > K n, he bondholder receives he payoff equal o Fv1 w n. 4. If K i 1 < Ī T K i for i = 1, 2,...,n, he bondholder receives he payoff equal o Fv 1 w i 1 ϕ i ĪT K i 1 K i K i 1 w i. In case of he sepwise payoff funcion his payoff is consan and equal o Fv 1 w i 1 when Ī T belongs o inerval K i 1, K i ].Forϕ 1,l ϕ 1,q he payoff decreases linearly quadraically from value Fv 1 w i 1 o value Fv 1 w i as he funcion of Ī T in he inerval K i 1, K i ]. 3.2 The muli-facor Cox Ingersoll Ross ineres rae model Muli-facor affine ineres rae models were inroduced by Duffie and Kan Their paper see Duffie and Kan 1996 is regarded as a cornersone in he ineres raes erm srucure heory. Dai and Singelon 2000 provided classificaion of he muli-facor affine ineres rae models and reasoning on heir srucure. The populariy of he menioned models follows from heir racabiliy for bond prices and bond opion prices. A muli-facor affine model of he ineres rae is described by a ime homogeneous diffusion model given by dx = ϕ κ X d + Ɣ V X dw X, 6 where κ and Ɣ are consan d d marices, ϕ = ϕ 1,ϕ 2,...,ϕ d is a consan vecor, υ 1 + ν 1 x υ 2 + ν 2 V x = x , υ d + ν 1 x for i = 1, 2,...,d υ i are consans and ν i = ν i1,ν i2,...,ν id R d, i = 1, 2,...,d, are consan vecors, We also assume ha here is a real consan ξ 0 and a consan vecor ξ= ξ 1,ξ 2,...,ξ d such ha r x = ξ 0 + ξ x. 8 As we menioned earlier, in his paper we consider he muli-facor Cox Ingersoll Ross model, which is an affine ineres rae model of he form 6 wih ϕ i > 0, ξ i = 1and υ i = 0 for i = 1, 2,...,d as well as { 1 for i = j ν ij = 0 for i = j for i, j = 1, 2,...,d. Moreover, we assume ha Ɣ i := Ɣ ii > 0, κ i := κ ii = 0for i = 1, 2,...,d, whereas Ɣ ij = κ ij = 0fori = j, i, j = 1, 2,...,d.

10 374 P. Nowak, M. Romaniuk 4 Caasrophe bond pricing formula Our aim in his secion was o prove he caasrophe bond pricing formula. We will apply he following version of he Levy heorems see, e.g., Ikeda and Waanabe 1989; Shreve Theorem 1 Le M be a maringale relaive o a filraion G [0,T ].LeM 0 = 0, Mbe coninuous and [M, M] = for all [ 0, T ]. Then M is he Brownian moion. Theorem 2 Le M 1,M 2 be maringales relaive o a filraion G 0. Assume ha for i = 1, 2, M i = 0, M i is coninuous and [M i, M i ] = for all [ 0, T ]. If, in addiion, [M 1, M 2 ] = 0 for all [ 0, T ],henm 1,M 2 are he independen Brownian moions. Lemma 1 Le Z 1, Z 2,...,Z k+1,k 1, be he Brownian moions such ha [ Z i, Z j ] = δ ij, i, j {1, 2,...,k}, [ 0, T ], and [ Z k+1, Z i ] = ρ i, i {1, 2,...,k}, [ 0, T ]. Le ρ 2 = k i=1 ρi 2 < 1. Then here exiss he Brownian moion Z k+1 independen of Z 1, Z 2,...,Z k, such ha k Z k+1 = 1 ρ 2 Z k+1 + ρ i Z i. Proof Le [ 0, T ]. The process Z k+1 1 = 1 ρ Z k+1 1 ki=1 ρ i Z 1 ρ i is a 2 2 coninuous maringale saring from 0. [ [ k ] Z k+1, Z k+1] = 1 k 1 ρ 2 2 ρ 2 + ρ i dz i, ρ i dz i 1 = ρ 2 1 ρ 2 =. Theorem 1 implies ha Z k+1 is he Brownian moion. Moreover, for i {1, 2,...,k}, i=1 i=1 [ ] Z k+1, Z i = 1 [ ] Z k+1, Z i 1 1 ρ 2 1 ρ 2 = 1 ρ 1 i ρ i = 0. 1 ρ 2 1 ρ 2 k j=1 i=1 ρ j [ Z j, Z i ] Therefore, from Theorem 2 i follows ha Z k+1 and Z i are independen. In Vaugirard 2003 he auhor considered a simple form of he caasrophe bond payoff funcion. The riggering poin was defined as he firs passage ime hrough a level of losses K of a naural risk index I. He assumed ha if he riggering poin does no occur, he bondholder is paid he face value Fv; and if he riggering poin occurs, he payoff is equal o he face value minus a coefficien in percenage w, i.e. Fv1 w. Bondholders were regarded o be in a shor posiion on a one-ouch up-and-in digial opion on I and, similarly as in case of opions, he maringale mehod was used o find he caasrophe bonds valuaion expression. In our approach we also use he condiional expecaion wih respec o equivalen risk-neural measure o obain he analyical form of he ca bond pricing formula. According o our earlier definiions, he model considered by us has he more general payoff srucure, he underlying asse I is conneced wih he insurance indusry and here is he possibiliy of correlaion beween he coninuous pars of he processes describing r and I. Now we formulae and prove he main heorem concerning caasrophe bond pricing.

11 Valuing caasrophe bonds involving correlaion Theorem 3 Le r [0,T ] be a risk-free spo ineres rae process given by he muli-facor CIR model r = rx, [ 0, T ],wherex is he vecor process wih parameers described in he previous secion. Le w W, ϕ,k K, T T and le I B w,ϕ,k 0 be he price a ime 0 of I B w, ϕ, K. Then here exiss a probabiliy measure Q F, equivalen o P, such ha IB w,ϕ,k 0 = B 0, T E Q F νw,ϕ,k, 9 where i wih B, T = e at d i=1 b i T X i, [0, T ], 10 b i τ = a τ = ξ 0 τ e γ i τ 1 ˆκ i +γ i 2 e γ i τ 1 + γ i, i = 1, 2,...,d, d 2ϕ i ln i=1 Ɣ 2 i γ i ˆκ i +γ i 2 e γ i τ 1 + γ i ˆκi + γ i τ +, 2 γ i = ˆκ i 2 + 2Ɣi 2, ˆκ i = κ i + Ɣ i λ i, i = 1, 2,...,d, and he consan vecor λ = λ 1, λ 2... λ d used in he definiion of he marke price of risk according o formula 16; ii ν w,ϕ,k = f w,ϕ,k ĪT and he price of he underlying asse wih respec o he probabiliy measure Q F has he form I = I 0 exp 0 rx s σ 2 I 2 + σ I ρ σ T s where he vecor process X is described by he sochasic equaion ds + σ I W I N, 11 dx = ϕ κ X d + Ɣ V X d W X, 12 wih he marix κ of he form { κi +Ɣ κ ij = i λ i + Ɣi 2b i T for i = j 0 for i = j and σ T = σ T X, = σ1 T X,,σ2 T X,,...,σd T X,, σi T x, = x i Ɣ i b i T, i = 1, 2,...,d, x = x 1, x 2,...,x d R d. In formulas 11 and 12 W X = W 1, W 2,..., W d is he d-dimensional and W I is he one-dimensional Q F -Brownian moion, respecively. Moreover, [ W i, W I ] = ρ i, i = 1, 2,...,d, [0, T ]. 13

12 376 P. Nowak, M. Romaniuk Proof From he heory of asses pricing i follows ha IB w,ϕ,k = E Q e T r u du ν w,ϕ,k F, [0, T ] for risk-neural equivalen probabiliy measure Q.LeW d+1 be he Brownian moion obained by Lemma 1 for k = d, Z i = W i, 1 i d, andz d+1 = W I.ThenW d+1 saisfies he equaliy: W d+1 1 = W I ρ W X ρ 2 The change of probabiliy measure is described by he Radon-Nikodym derivaive dq dp = Z T P-a.s., where Z = e 0 λ X s dw X s d+1 0 λ s dws d λ X s 2 + λ d+1 s 2 ds [ 0, T ], 15 for he marke price of risk processes λ X and λ d+1 of he form λ X = λ 1, λ 2,..., λ d = V X λ. 16 In formula 16 λ = λ 1, λ 2,..., λ d is a consan vecor and for λ I = μ 0 r σ I, [0, T ]. Ifweareableoproveha λ d+1 = λ I ρ λ X 1 ρ 2 E P Z T = 1, 17 hen Z [0,T ] is a maringale wih respec o P and Q is a probabiliy measure equivalen o P. Le us assume ha he equaliy 17 is saisfied. Then from he Girsanov heorem see, e.g., Karazas andshreve1988, Chaper 3.5.A. i follows ha here exis he wo independen Q- Brownian moions: d-dimensional W X = W 1, W 2,..., W d and one-dimensional W d+1, saisfying he equaliies: d W X = dw X + λ X d; dd W d+1 = dw d+1 + λ d+1 d, [ 0, T ]. Le for [0, T ] W I = 1 ρ 2 W d+1 + ρ W X. 18 Since W I is a coninuous maringale saring from 0 and [ W I, W I ] =, by Theorem 1, i is he Q-Brownian moion. Moreover, for [0, T ] and 1 i d, [ W I, W i ] = ρ i. From 14 i follows ha dw I = 1 ρ 2 dw d+1 + ρ dw X = 1 ρ 2 d W d+1 λ d+1 d + ρ d W X λ X d = d W I λ I d. 19

13 Valuing caasrophe bonds involving correlaion Under Q he vecor process X is given by he equaion where dx = ϕ ˆκ X d + Ɣ V X d W X, 20 ˆκ ij = { κi +Ɣ i λ i for i = j 0 for i = j for each i, j {1, 2,...,d}. Formula 19 implies I = I 0 exp μ σ I λ I s ds + σ I W I N 0 0 and afer reformulaion I = I 0 exp r s σ I 2 ds + σ I W I N I remains o prove 17. To his end, we apply an idea similar o he one used in Cheridio e al Le ˆX be he soluion of he equaion [0,T ] wih respec o P. Le d ˆX = ϕ ˆκ ˆX d + Ɣ λ X = λ X V ˆX dw X, [ 0, T ], 22, λ d+1, [ 0, T ]. Le n 1. We inroduce he following sopping imes: The process τ n = inf { > 0 : λ X n} T, { } ˆτ n = inf > 0 : λ ˆX n T. λ n = λ X I { τn } = λ n,x, λ n,d+1, [ 0, T ], where λ n,x and λ n,d+1 corresponds, respecively, o λ X and λ d+1, saisfies he Novikov condiion see, e.g., Karazas and Shreve 1988, Chaper 3.5.D., i.e. E exp P 1 T λ n 2 2 d e n2 T 2 <. 0 Therefore, he probabiliy measure Q n, defined by he Radon-Nikodym derivaive dqn dp = Z n T,where Z n = e 0 λ n,x dw X s s n,d+1 0 λ s dws d λ n,x s 2 + λ n,d+1 s 2 ds, [ 0, T ], 23

14 378 P. Nowak, M. Romaniuk is a probabiliy measure equivalen o P and he process W n,x = W n,1, W n,2,..., W n,d, saisfying he equaliy d W n,x = dw X + λ n,x d, is he Q n - Brownian moion. Furhermore, he process τn τn X τn = ϕ ˆκ X s ds + Ɣ V X s d W s n,x, [ 0, T ], 0 0 wih respec o Q n has he same disribuion as he process ˆX ˆτn wih respec o [0,T ] P. Therefore, Q n τ n = T = P ˆτ n = T. 24 Moreover, one can check ha and lim n P τ n = T = lim n P ˆτ n = T = 1 lim n T I {τ n =T } = Z T P-a.s. Applying he monoone convergence heorem see, e.g., Billingsley 1986, Theorem 16.2, we obain he equaliy E P Z T = lim Z T I {τn =T n }. 25 Since, for each n 1, E P Z T I {τn =T } = Q n τ n = T, he equaliies 24, 25imply17. For = 0 he zero-coupon bond price has he form B 0, T = E Q e T 0 r udu. Moreover, from Munk 2011 i follows ha B, T, [0, T ], saisfies he equaion db, T B, T = r d + σ T d W X, where σ T = σ T X, = σ1 T X,,σ2 T X,,...,σd T X,, σi T x, = x i Ɣ i b i T, i = 1, 2,...,d, x = x 1, x 2,...,x d R d, and is soluion has he form 10, which finishes he proof of he asserion i. We inroduce he nex probabiliy measure Q F, equivalen o Q, given by he following Radon Nikodym derivaive: dq F dq T = e 0 r d = e 1 2 B 0, T T 0 σ T 2 T d+ 0 σ T d W X Q-a.s.

15 Valuing caasrophe bonds involving correlaion The Girsanov heorem implies ha W d+1 d W X,where and W X = W 1, W 2,..., W d = d W X are he independen Q F Brownian moions. Since W I = 1 ρ 2 W d+1 σ T d, 26 + ρ W X 27 [ is a coninuous maringale saring from 0 and W I, W ] I = for [0, T ], Theorem 1 [ implies ha W I is he Q F -Brownian moion. Moreover, for i = 1, 2,...,d and [0, T ], W I, W ] i = ρ i. Equaliies 18, 26and27 imply he equaliy d W I = d W I ρ σ T d. Therefore, he processes 20and21 ake he following form wih respec o Q F : dx = ϕ κ X d + Ɣ V X d W X, I = I 0 exp rx s σ I σ I ρ σs T ds + σ I W I N, where { κi +Ɣ κ ij = i λ i + Ɣi 2b i T for i = j 0 for i = j. This finishes he proof of he asserion ii. IB w,ϕ,k 0 = E Q e T 0 rudu ν w,ϕ,k = B 0, T E Q B 0, T 1 B 1 = B 0, T E Q dq F dq ν w,ϕ,k Clearly, dq E Q F dq ν w,ϕ,k = E Q F νw,ϕ,k and applicaion of formula 29 o28gives9. T ν w,ϕ,k. 28 By sraighforward compuaions, applying Theorem 3, we obain he following lemma concerning a deailed form of he caasrophe bond price a he momen 0. Lemma 2 The price a ime 0 of I B w, ϕ, K can be expressed in he following form: IB w,ϕ,k 0 = B 0, T E Q F νw,ϕ,k, where B 0, T is described by 10, { } n n+1 E Q F ν w,ϕ,k = Fv ψ 0 w i e i + 1 w i 1 ψ i ψ i 1, 30 i=1 i=1 29 ψ i = Q F Ī T K i, i = 0, 1, 2,...,n and ψ n+1 = 1, 31

16 380 P. Nowak, M. Romaniuk { } e i = E Q F ĪT K i 1 ϕ i I { } K K i K i 1 <Ī T K i, i = 1, 2,...,n. 32 i 1 In paricular, n+1 E Q F ν w,ϕ,k = Fv ψ w i 1 ψ i ψ i 1 i=1 for ϕ 0. The above lemma simplifies he numerical compuaions of he caasrophe bond price. 5 Numerical simulaions In order o analyze he behavior of prices of he ca bonds, he Mone Carlo simulaions are conduced in his secion. To uilize he obained general pricing formula 9 proved in Theorem 3, ieraive schemes of simulaions for he process I given by 11 and he process X given by 12 are applied wih fixed ime sep. Because he final esimaor of he price especially depends on supremum of he generaed rajecory of he process Ī defined by 4, he adapive approach wih adjusmen of he lengh of was inroduced. The necessary algorihms are considered in Sec In he following we illusrae he possibiliy of pricing ca bonds in various parameric seings via numerical compuaions, despie he complex naure of he formulas considered in Theorem 3 and Lemma 2. We sar our consideraions from he simplified, synheic seup discussed in Sec In some pars his firs seing is similar in naure o he one considered in Vaugirard 2003 or oher financial papers see, e.g., Nowak and Romaniuk 2010a. The following discussion enables us o rack down he mos imporan deails of behavior of prices of he considered ypes of he caasrophe bonds. Then in Sec. 5.3 we urn o oher approach, which is more complex, real-life seup, because i is parially based on he model of ineres raes and he model of caasrophic evens adaped from Chen and Sco 2003 and Chernobai e al. 2006, respecively. In hese wo papers he real-life daa were considered and he parameers of he relaed saisical models were esimaed for his daa. Therefore, we show ha even for more complex seing which is closer o he pracical cases, he evaluaion and analysis of he prices of he considered ca bonds is possible and leads o imporan conclusions for praciioners. Also he srucure of paymens of he ca bond is saisically analyzed in his case. In he following, one- or mulifacor CIR models of ineres raes are discussed. We also assume ha he generaed losses U i are of a caasrophic naure hey are rare, bu each loss has a high value. Therefore, he quaniy of losses is modeled by HPP homogeneous Poisson process and he value of each loss is given by a random variable wih a relaively high expeced value and variance i.e., high risk wih high variabiliy. We limi our consideraions o he case when he value of each loss is modeled by lognormal disribuion. This disribuion is commonly used in simulaions of risk evens in insurance indusry. However, oher disribuions, e.g. Weibull, gammma, GEV see Chernobai e al. 2005; Furman 2008; Hewi and Lefkowiz 1979; Hogg and Klugman 1983; Melnick and Tenenbein 2000; Papush and Parik 2001; Rioux and Klugman 2006, or simulaions based on hisorical records see Ermolieva and Ermoliev 2005; Pekárová e al are possible and hey can be easily incorporaed ino he approach presened in his paper.

17 Valuing caasrophe bonds involving correlaion We assume ha he face value of he bond in each numerical experimen is se o 1 i.e. one moneary uni assumpion is used and he rading horizon of he caasrophe bond is se o 1 year. The saring value I 0 of he process 3 is equal o 1. In each experimen we generae N = 1,000,000 simulaions. The se of oher necessary parameers of he caasrophe bond, he model of ineres raes and he model of caasrophe evens are described in deails for each analysis. In our consideraions we focus on he piecewise linear paymen funcion 1,l of he ca bond as defined in Remark 1. As previously noed, he price of he ca bond for his ype of he paymen funcion is direcly relaed o supremum evaluaed for he process Ī. Therefore, he ime sep applied in he ieraive Mone Carlo scheme is adaped according o his value. Usually, = long = 0.02 is se which is close o 1-week cycle for assumed T = 1. Bu if he value of he process Ī is inside he inerval of he values close o he firs riggering poin K 0 or he las one K n, he ime sep is shorened o = shor = Therefore, he obained esimaor has beer qualiy and he whole numerical procedure is more flexible. Of course, addiional momens relaed o he jumps caused by he caasrophic evens U i are also aken ino accoun in our approach as explained furher. 5.1 Algorihms As indicaed by Theorem 3 and Lemma 2, he evaluaed ca bond price IBw, ϕ, K depends on hree processes: he jump process Ñ defined by 2 or is ransformaion N, equivalenly, he price of he underlying asse I wih respec o he probabiliy measure Q F described by 11 and he ineres rae model X given by 12. The processes I and X are correlaed via he Brownian moions W X and W I as defined by 13. Therefore, o apply he pricing formula 9, wo ypes of simulaions should be used. During he firs one Algorihm 1, he rajecory of Ñ is generaed. During he second one Algorihm 2, boh of he rajecories I and X are generaed joinly using some inpu from he firs ype of simulaion. Then, based on all of he rajecories, he expeced value E Q F ν w,ϕ,k given by 30 is esimaed via Mone Carlo approach Algorihm 3. Addiionally, he discouning facor B 0, T given by 10is evaluaed. Evenually, he main formula 9 could be used, merging hese wo oupus. We sar from descripion of he algorihm which is used o generae he process Ñ. Because HPP is applied, hen he inervals beween he consecuive jumps are given by iid random variables from exponenial disribuion wih he parameer κ see, e.g., Romaniuk and Nowak To generae he jumps U i, some fixed random disribuion should be used. In he case of he lognormal disribuion considered in his paper he relevan algorihms are widely sudied in he lieraure see, e.g., Romaniuk and Nowak Then we have he following seps which generae single rajecory of he jump process: Algorihm 1 Inpu Parameers of: he Poisson process κ, he disribuion of losses e.g. μ LN,σ LN for lognormal disribuion. Sep 1 Se Ñ 0 = 0, 0 = 0, j = 0. Sep 2 Generae s from he exponenial disribuion wih he parameer κ. Sep 3 If j + s > T, hen reurn he sored values Ñ 1, Ñ 2,...and he sequence jumps = 1, 2,... Sep 4 Generae U from he disribuion of losses. Se j = j + 1, Ñ j = Ñ j 1 + U, j = j 1 + s. Sore Ñ j, pu j ino he sequence jumps in increased order. Sep 5 Reurn o he sep 2. Oupu The rajecory of Ñ and he jump momens jumps.

18 382 P. Nowak, M. Romaniuk The ransformaion of Ñ ino he process N is sraighforward if he expeced value E P e U for he disribuion of he jump U is a leas numerically known. In he model considered in his paper here is embedded dependency beween he processes I and X.From13, i is relaed o he correlaion marix of d + 1 dimensional normal disribuion given by ρ ρ ρ 1 ρ 2 ρ Using Cholesky decomposiion he relevan rajecories of W 1, W 2,..., W d and W I could be simulaed see, e.g., Romaniuk and Nowak 2015 for he given se of imes. Because of he special form of 33, W I is generaed as a linear combinaion of independen normal random variables used in simulaion of W 1, W 2,..., W d and one addiional normal sample. In order o simulae he rajecories I and X, we use he ieraive scheme for 0 = 0 < 1 <. Firs of all, he ime sep = j+1 j depends on he process Ī in which supremum of I is used. If for some he value of I is such ha I 0 I is inside he inerval [K 0 ε, K n +ε] for he fixed parameer ε>0, hen he ime sep is shorened o shor > 0; Oherwise, i is se o long > shor. Such approach improves he efficiency of numerical simulaions. Addiionally, ino he se of imes for which I and X are generaed, he sequence of momens of jumps jumps from Algorihm 1 should be also incorporaed. In he considered seup Euler schemes are hen used. From 12 and he assumpions abou 7 inroduced in Sec. 3.2 we ge X i, j = ϕ i κ i +Ɣ i λ i + Ɣi 2 b i T j X i, j + Ɣ i X i, j W i, 34 where i = 1,...,d and W 1,...,W d are iid sandard normal variables. From 11 andhe assumpions inroduced in Secion 2 we have I j = exp rx j σ I 2 d 2 + σ I ρ i X i, j Ɣ i b i T j + σ I W I j N j. i=1 35 In he above formula, W I j is incremen of he Brownian moion W I generaed using he menioned earlier linear combinaion of W 1,...,W d wih one addiional normal variable W d+1, e.g. if d = 2henwehaveρ 1 W 1 + ρ 2 W ρ1 2 ρ2 2 W 3. Moreover, N j = Ñ j κe 1, 36 where Ñ j is incremen of he jump process Ñ obained using Algorihm 1. These consideraions lead us o he following seps:

19 Valuing caasrophe bonds involving correlaion Algorihm 2 Inpu Parameers of: he muli-facor CIR model, he Poisson process κ, he disribuion of losses, he caasrophe bond K 1,...,K n, he Brownian moions; he saring value I 0, he accuracy rule ε. Sep 1 Run Algorihm 1 o obain he rajecory of Ñ and is jump momens jumps. Sep 2 Se all = jumps, I 0, X 0, j = 0, 0 = 0. Sep 3 If I 0 I [K 0 ε, K n + ε], hense = shor, oherwise = long. j Sep 4 Le j+1 = j + and pu j+1 ino he sequence of momens all in increased order. Sep 5 Remove he earlies momen j+1 from he sequence all.if j+1 > T,hen j+1 = T.Le = j+1 j. Sep 6 Find N j using 36 and daa from he sep 1. Sep 7 Evaluae X i, j for i = 1,...,d using 34. Le X i, j+1 = X i, j + X i, j for i = 1,...,d. Sore X j+1. Sep 8 Evaluae I j using 35. Le I j+1 = I j I j.sorei j+1. Sep 9 If j+1 = T, hen reurn he obained rajecories I and X. Sep 10 Le j = j+1, j = j + 1. Reurn o he sep 3. Oupu The rajecory of I and X. Then he sampled rajecory of I is ransformed ino Ī. I allows us o obain values necessary for evaluaion of 30 e.g. I { K i 1 <Ī T K } i. The las phase consiss of approximaion of he expeced value E Q F νw,ϕ,k, considered in Lemma 2 via crude Mone Carlo esimaor see, e.g., Romaniuk and Nowak 2015, and applicaion of he main formula 9. Algorihm 3 Inpu Parameers of: he muli-facor CIR model, he Poisson process κ, he disribuion of losses, he caasrophe bond, he Brownian moions; he saring value I 0, he accuracy rule ε, he number of simulaions n. Sep 1 Run n imes Algorihm 1 and Algorihm 2 o sample rajecories Ī 1 n T,...,Ī T. Sep 2 Find esimaors of ψ i see 31 and e i see 32 using relevan averages based on he sample Ī 1 n T,...,Ī T. Sep 3 Evaluae B 0, T given by 10 and approximae E Q F ν w,ϕ,k see 30 using he esimaors found in he sep 2. Sep 4 Find he ca bond price wih he pricing formula 9. Oupu The ca bond price IB w,ϕ,k Simplified seup In Vaugirard 2003 he caasrophe derivaives wrien on he caasrophe index were considered. As he model of ineres raes, Vasicek model was used. The inensiy of caasrophic evens was modeled by HPP, and he lognormal disribuion described he value of single caasrophic loss. Then he simplified seup wih inuiive parameers for he menioned models was discussed o analyze he behavior of he ca bond prices. In his paper, he process I for he insrumen similar o a synheic insurance indusry underlying asse wih is ransformaion Ī defined by 4 is considered. Therefore, we sar our numerical analysis also from he simplified seup, which is close in is naure o Vaugirard 2003, o emphasize he mos imporan feaures in he evaluaion of he ca bond prices. Model I: The relevan parameers of his pricing model are enumeraed in Table 1. Similarly o Vaugirard 2003, in his simplified seup he inuiive values for he ineres rae

20 384 P. Nowak, M. Romaniuk Table 1 Parameers of Model I Parameers CIR model one-facor ϕ = 0.1, κ = 0.1, Ɣ = 0.03,ξ 0 = 0.1 Brownian moions ρ 1 = 0.5,σ I = 0.2 Inensiy of HPP κ HPP = 1 Lognormal disribuion μ LN = 0.1,σ LN = 0.2 Triggering poins K 0 = 5, K 1 = 10 Values of losses coefficiens w 1 = 0.9 model are used. In his case we consider he one-facor CIR model insead of Vasicek model as in Vaugirard The parameers of he lognormal disribuion and he inensiy of HPP are he same as in some of he analysis in Vaugirard Also he simple paymen funcion wih only wo riggering poins is considered. The value of reducion coefficien of he payoff w 1 = 0.9 isveryhighoemphasizehereducionof paymen of he ca bond if he riggering poin occurs. Such value is also similar o he one used in Vaugirard For illusraive purposes, wo sraighforward values of he riggering poins K 0 and K 1 are also se see Table 1. The parameers of he Brownian componen of he process I are also presened in Table 1. To ake ino accoun he dependency beween he behavior of he rajecory of he underlying insrumen and he ineres raes as described by Theorem 3, he correlaion coefficien ρ 1 beween he Brownian moions of hese processes is se o 0.5. The variabiliy σ I = 0.2 is relaively high comparing o he volailiy of he ineres raes Ɣ and he parameers of he process of he caasrophic evens. Then he esimaed price of he caasrophe bond in his case is equal o Model I, Analysis I The inensiy of HPP is imporan parameer of he model of caasrophic evens. Therefore, in Vaugirard 2003 he dependency beween he price of he caasrophe bond and he inensiy κ HPP is analyzed for a few simple cases. We also adop he similar approach bu conduc he relevan numerical simulaions for he whole inerval κ HPP [0.4, 1.6] insead of a few values as in Vaugirard 2003 seefig.1for he graph of he obained ca bond prices. The oher parameers in his analysis are he same as in Table 1. As i could be seen, he ca bond price is sricly decreasing funcion of κ HPP. Model I, Analysis II In he model of he process I, especially wo parameers are imporan comparing o approaches considered in oher papers: he volailiy of he Brownian moion of he underlying asse σ I and he correlaion coefficien ρ 1 beween he Brownian moion W I and he behavior of he model of he ineres rae. Therefore, he influence of hese parameers on he obained esimaor of he ca bond price should be analyzed. The ca bond prices obained for he wide range of he menioned parameers, i.e. σ I [0.1, 0.9] and ρ 1 [0.1, 0.9] may be found in Fig. 2. The oher parameers are he same as in Table 1. As easily seen, he price is he decreasing funcion of σ I. However, he influence of ρ 1 is no so sraighforwardly noiceable from Fig. 2. Only using he single cu of he relevan surface from Fig. 2 for he fixed value σ I = 0.6 wih new, appropriae scale he dependency beween he parameer ρ 1 and he ca bond price is easier o found see Fig. 3. Then he price is also decreasing funcion of ρ 1, bu he influence of σ I on he obained esimaor is more significan in his seing. Model I, Analysis III Usually, he esimaion of he disribuion of he single caasrophic even is based on hisorical daa. I is possible ha he fuure jumps in he relevan process will follow oher paerns or here could be some error in esimaion procedure. Therefore,

21 Valuing caasrophe bonds involving correlaion Price Inensiy Fig. 1 Model I, Analysis I: price of he bond as he funcion of κ HPP Price rho_1 sigma_i Fig. 2 Model I, Analysis II: price of he bond as he funcion of σ I and ρ 1 as in Vaugirard 2003, he influence of he parameers of he considered disribuion on he ca bond prices should be analyzed. Using numerical simulaions, he ca bond prices for he parameers of he lognormal disribuion from he wide inervals μ LN [0.05, 0.3] and σ LN [0.05, 0.25] are calculaed see Fig. 4. The oher parameers are he same as in Table 1. As i may be seen, he ca bond price is decreasing funcion of boh μ LN and σ LN. Model I, Analysis IV The enerprise which issues he caasrophe bond may be also ineresed in he dependency beween he price of such insrumen and he parameers of is paymen funcion. For example, behavior of he ca bond price for various values of he coefficien w 1 may be analyzed. Example of he oupu of he relaed simulaions can be

22 386 P. Nowak, M. Romaniuk Price rho_1 Fig. 3 Model I, Analysis II: price of he bond as he funcion ρ 1 for σ I = Price sigma_LN mu_ln Fig. 4 Model I, Analysis III: price of he bond as he funcion of μ LN and σ LN found a Fig. 5. As i may be seen, in he considered seup given by he parameers from Table 1 he ca bond price is almos linearly decreasing funcion of w 1. Model I, Analysis V As i may be seen from he formula 11, here is imporan relaion beween he processes I and X. Therefore, he parameers of he model of ineres rae also affecs he final price of he caasrophe bond. For example, such influence may be seen if he price is numerically evaluaed for various values of Ɣ see Fig. 6. The oher parameers are he same as in Table 1. Then for he given inerval Ɣ [0.1, 1.0] he esimaed price is explicily non-linear convex funcion.

23 Valuing caasrophe bonds involving correlaion Price w_1 Fig. 5 Model I, Analysis IV: price of he bond as he funcion of w 1 Price Gamma Fig. 6 Model I, Analysis V: price of he bond as he funcion of Ɣ 5.3 Complex seup The ca bond pricing is also possible for he more complex seup which is closer o he reallife cases. Therefore, he wo-facor CIR model of ineres raes is applied. The parameers of his model were esimaed in Chen and Sco 2003 based on monhly daa of he Treasury bond marke using Kalman filer. Also he parameers of he Poisson process and he applied disribuion of he value of he single loss are based on real-life daa in his seing. These parameers are adaped from Chernobai e al. 2006, where he informaion of caasrophe losses in he Unied Saes provided by he Propery Claim Services PCS of he ISO Insurance Service Office Inc. and he relevan esimaion procedure for his daa are considered.