Catastrophe Insurance Products in Markov Jump Diffusion Model

Transcription

1 Catastrophe Insurance Products in Markov Jup Diffusion Model (Topic of paper: Risk anageent of an insurance enterprise) in Shih-Kuei Assistant Professor Departent of Finance National University of Kaohsiung No. 7 Kaohsiung University Rd. Nan Tzu Dist. 811 Kaohsiung Taiwan R.O.C. E-ail: square@nuk.edu.tw Tel: Fax: Chang Chia-Chien Ph.D Student Departent of Finance National Sun Yat-sen University No. 7 ien-hai Rd 84 Kaohsiung Taiwan R.O.C. E-ail: u75147@yahoo.co.tw Tel: Fax: Abstract For catastrophic events the assuption that catastrophe clais occur in ters of the Poisson process is inadequate as it has constant intensity. This article proposes Markov Modulated Poisson process (MMPP) to odel the arrival process for catastrophic events. Under this process the underlying state is governed by a hoogenous Markov chain and it is the generalization of Cuins and Gean (1993) Chang Chang and Yu (1996) and Gean and Yor (1997). We apply Markov jup diffusion odel to derive pricing forulas and hedging forulas for catastrophe insurance products included futures call option catastrophe PCS call spread and catastrophe bond. The nuerical analysis shows how the catastrophe insurance products prices are related to jup rate of catastrophe events standard deviation of jup size and ean of jup size. Keywords: Markov odulated Poisson process Markov jup diffusion odel futures call option catastrophe PCS call spread catastrophe bond. 1

2 I. Introduction Insurance copanies traditionally often liit their liabilities given the large capital requireents required to cover high-loss-severity catastrophe risk by reinsurance. However the reinsurance industry is also liit in size relative to the agnitude of these daages creating large fluctuations in the price and availability of reinsurance during years as ultiple catastrophes occur. During the last decade the high level of worldwide catastrophe losses in ters of frequency and severity had a arked effect on the reinsurance arket. The catastrophes such as Stor Daria (Europe 199) Hurricane Andrew (USA 199) Northridge earthquake (USA 1994) and the Kobe earthquake (Japan 1995) have ipacted the profitability and capital bases of reinsurance copanies. Therefore because of worldwide capacity shortage and the increent nuber of natural catastrophes in recent years the illustration of a viable substitute to reinsurance is viewed by the insurance industry to be both tiely and desirable. The Chicago Board of Trade (CBOT) launched catastrophe insurance futures contract in 199 options on catastrophe insurance futures in These contract values link to the loss ratios coplied by the Insurance Service Office (ISO). Due to the low trading volue in these derivatives trading was given up in They are replaced by a new generation of options called Property Clai Services (PCS) options which is introduced at the CBOT in Septeber PCS Options are based on catastrophe loss indices provided daily by PCS a US industry authority which estiates catastrophic property daage since Unlike reinsurance that is costly lengthy and irreversible these catastrophe insurance derivatives have the advantages of reversibility transactional efficiency and anonyify because they are standardized and exchange-traded. These innovative catastrophe derivatives provide the insurance industry access to financial arkets to transfer catastrophe exposure to contract counterparties willing to bear the risk. This access allows the insurance industry to stabilize losses and increases overall capacity in the insurance or reinsurance arket. Besides underwriters ainstrea institutional investors ay also enjoy the diversification effect of these derivatives since catastrophic losses display no correlation with the price oveents of stocks or bonds. Another innovation product for catastrophe risk anageent is the catastrophe bond (CAT bond) (or naed as Act of God bond or insurance-liked bond ) which is a liability hedged instruent for insurance copanies. CAT bond provisions have debt-forgiveness triggers whose generic design grants the interest payent and/or the return of principal

3 forgiveness. Meanwhile the debt forgiveness can be triggered by the insurer s actual losses or a coposite index of insurer s losses during the specific period. Under this structure the insurance copany could transfer the catastrophe risk to increase the ability to provide insurance protection. The first successful CAT bond was issued in 1997 by Swiss Re. In 1999 the first CAT bond by a non-financial fir was issued to cover earthquake losses for in the Tokyo region for Oriental and Copany td. Guy Carpenter & Copany show that the CAT bond arket recorded total issuance of $1.99 billion in 5 a 74 percent increase over the $1.14 billion issuance in 4 and 15 percent higher than the previous record of $1.73 billion issued in 3. During the period Guy Carpenter and MMC Securities Corporation reported that 69 catastrophe bonds have been issued with total risk liits of $1.65 billion whose predoinant sponsors were insurers and reinsurers. Certain articles focus on the pricing of catastrophe-linked securities. For exaple Cox and Schwebach (199) Cuins and Gean (1993) Cuins and Gean (1995) and Chang Chang and Yu (1996) price the CAT futures and CAT call spreads under deterinistic interest rate and PCS loss process. itzenberger Beaglehole and Reynolds (1996) price a zero-coupon CAT bond with hypothetical catastrophe loss distribution. Zajdenweber (1998) follows itzenberger Beaglehole and Reynolds (1996) yet change the catastrophe loss distribution to evy distribution. ouberge Kellezi and Gilli (1999) calculate the CAT bond under the assuption that the catastrophe loss stands for a pure Poisson process the loss severity is an independently identical lognoral distribution and the interest rate follows a binoial rando process. ee and Yu () assue that the catastrophe aggregate loss follows the copound Poisson process and then copute risk-free and default-risky CAT under the consideration of default risk basis risk and oral hazard using Monte Carlo ethod. Vaugirard (3a 3b) adapts the jup-diffusion odel of Merton (1976) to develop a valuation fraework that allows for catastrophic events interest rate uncertainty and non-traded underlying state variables and then reports the fairly siulations of insurance-linked securities. Dassios and Jang (3) use the Cox process (or a doubly stochastic Poisson process) to odel the clai arrival process for catastrophic events and then apply the odel to the pricing of stop-loss catastrophe reinsurance contracts and catastrophe insurance derivatives under constant interest rate. Therefore expect for Dassios and Jang (3) ost pricing odels for catastrophe insurance products assue that the loss clai arrival process follows Poisson process. This study intends to contribute to the literature by providing an alternative point process needs to be used to generate the arrival process. We will propose a new odel a Markov jup 3

4 diffusion odel which extends the Poisson process used in the jup diffusion odel to be Markov odulated Poisson process. Markov odulated Poisson process stands for a doubly stochastic Poisson process where the underlying state is driven by a hoogenous Markov chain (cf. ast and Brandt 1995). More precisely instead of constant average jup rate in the years in the jup diffusion odel under Markov odulated Poisson process the arrival rates of new inforation are different fro the abnoral vibrations of the loss dependent on current situation. The Markov jup diffusion odel with two states the so-called switched jup diffusion odel depends on the status of the econoy. In switched jup diffusion odel the jup rates are different in different status ore precisely; the jup rates are large in one state and sall in other status. Figure 1 show PCS loss quarterly in the United State fro 195 first quarterly to 5 first quarterly respectively 1. In general if the arrival process (jup rate) of natural catastrophes stands for Poisson process then it could appear constant average jup rate in the years. However figure 1 sees to reveal the saller jup rate of natural catastrophes before 199s and larger jup rate of natural catastrophes after 199s. Therefore we could infer that the arrival process (jup rate) of natural catastrophes could be different at different status and the Markov odulated Poisson process could be ore fit than Poisson process to capture the arrival rate of natural catastrophes. 5 PCS loss (USD illion) /Q 1954/Q4 1961/Q3 1968/Q 197/Q4 1976/Q1 1979/Q3 1983/Q 1987/Q3 199/Q4 1994/Q 1997/Q3 1/Q 5/Q1 Year Figure 1: PCS loss in the United State during 195 Q1 to 5 Q1 The contributions of this paper are follows: (1) To capture the phenoenon of figure 1 that the arrival process of natural catastrophes could be different at different status we propose a ore general Markov jup diffusion odel to odel loss process and it could be the generalization of Cuins and Gean (1993) Chang Chang and Yu (1996) and Gean 4

5 and Yor (1997). () We apply the Markov jup diffusion odel to evaluate accurately the valuation of catastrophe insurance products including European futures call option catastrophe PCS call spread and CAT bond with the constant interest rate assuption. Meanwhile we propose the hedging strategy for these catastrophe insurance products. We ake use of Merton s (1976) assuption that the risk associated with jups can be diversified away and therefore the jup size distribution and transition rate are not altered by the easure changed. Our pricing forula of European futures call option can also reduce to Poisson su of Black's prices of Chang Chang and Yu (1996) and Black's prices of Cox and Schwebach (199). The reainder of the paper is organized as follows. Section presents the general fraework of the odel. Section 3 derives the pricing forulas and hedging strategy of three catastrophe insurance products: European futures call option catastrophe PCS call spread and CAT bond. Section 4 deonstrates nuerical analysis. Section 5 suarizes the article and gives the conclusions. For ease of exposition ost proofs are in an appendix. II. General Fraework of The Model 1. Markov Modulated Poisson Process (MMPP) The Markov odulated Poisson process Φ (t) stands for a doubly stochastic Poisson process where the underlying state is governed by a hoogenous Markov chain. Since the Markov Chain has a finite nuber of states the Poisson arrival rate takes discrete values corresponding to each state. More precisely we consider a finite state space X = { 1...I } and let { X P i : i X } Ψ( i j) denoted as: where where be a Markov jup process on a state space X with transition rate ν ( i j) i j Ψ ( i j) = ν ( i j) otherwise. j i i j X. Keeping copany with X we consider non-negative nubers λ 1 λ.. λ I λ is the intensity of a doubly stochastic Poisson process Φ (t) if X is at state i. i That is we consider a point process Φ ( t) = ( T n ) which is for all X stochastic Poisson process with σ ( X ) -intensity kernel υ( dt) : = E Φ( ) 1 X _ dt X = λ dt { t< T } X ( t ) i i a { } X P - doubly 5

6 where X T is the point of explosion of X. More precisely Pi ( X ) Φ is Pi -a.s a distribution of a non-hoogeneous Poisson process with intensity function { } λ _. For < z < 1 define n= with P( ) = 1{ = } Dij where D ij = 1 if i = j; otherwise. 1 X t< T X ( t ) * P ( z t) = P( t) z (.1) By using Kologorov's forward equation the derivative of P( t ) becoes d P ( t ) = ( Ψ Λ ) P ( t ) + 1 { 1} Λ P ( 1 t ) dt and the derivative of P * (z t ) becoes d P * t P * t P * t dt it s unique solution can be obtained as (z ) = ( Ψ Λ ) (z ) + z (z ) Λ [ Ψ (1 z) Λ]t * P ( z t) = e (.) where P( t) : = P ( t) denotes the transition probability at jup ties fro ij state X ( ) = i to state X ( t) = j. Ψ : = Ψ( i j) represents transition rate and Λ denotes I I diagonal atrix with diagonal eleents λ i. Finally by using aplace inverse transfor (.1) and the unique solution (.) we obtain the joint distribution of X and Φ (t) at tie t when * P( t) = P ( z t) z=. z! We assue that the jup rate under different status is unknown and then to consider the hidden switch Poisson process. et the transition rate under hidden switch Poisson process is given by α Ψ = α α α 1 1 and the jup rate under hidden switch Poisson process is λ1 Λ =. λ By using the unique solution (.) with z = 1 we have * ( t) P (1 t) = e Ψ. Hence 6

7 -( 1 ) -( 1 ) * 1 α + α t α + α t α + α1e α1 -α1e P (1 t) =. α -( α1 α ) t -( α1 α ) t 1 + α + + α -αe α1 + αe It can be easy to get the liiting distribution as α π1 = li Pi 1(1 t) = t α + α 1 α1 π = li Pi (1 t) = t α + α 1 where π 1 and π denote the initial probability that the jup rate stays in state 1 and state respectively. et Q( t) = ( κ + 1) exp(- Λ κt) P( t) thus * Q ( z t) = Q( t) z = ( κ + 1) exp(- Λκt) P( t) z = = = ( z( κ + 1)) exp(- Λκt) P( t) = * and the unique solution of Q ( z t) is * Ψ ( 1 ( κ + 1) ) Λ Q ( z t) = e e z t Λκt Ψ ( (1 z) ( κ + 1) Λ) t = e. Therefore under the original Markov odulated Poisson process Φ (t) the original transition probability is P( t ) with transition rate Ψ and I I diagonal atrix Λ with diagonal eleents λ i. Through the change of easure the risk neutral transition probability becoes Q( t ) with transition rate Ψ and I I diagonal atrix (κ + 1) Λ with diagonal eleents λ i.. Total Insured oss process Cox and Schwebach (199) assue that the aggregate inured losses follow lognoral distribution and then derive the closed-for solution of catastrophe futures call option. The pricing forula is siilar to Black s (1976) forula for pricing futures options which assues that the underlying futures price stands for a pure diffusion. However option pricing based on a copound Poisson process has been well-developed in insurance literature. Cuins and Gean (1993) odel the increents of the loss index as a geoetric Brownian otion plus a jup process that is assued to be a Poisson process with fixed loss sizes. Chang Chang and Yu (1996) use a randoized operational tie approach to price European 7

8 futures options. The randoized operational tie approach concept in probability theory indicates that a siple change of tie scale will frequently reduce a general nonstationary process to a ore tractable stationary one. More precisely the tie change transfors a general nonstationary process in the usual calendar-tie scale to a stationary counterpart in an operational-tie scale. They assue that parent process of the futures return is a lognoral diffusion and the directing process is a hoogeneous Poisson and thus derive the option pricing forula as a risk-neutral Poisson su of Black's prices. The extent of the usefulness of this pricing equation is deterined only by a careful epirical evaluation using arket data. Unfortunately this approach is today inappropriate in the current situation since the futures contract stopped trading in 1995 and no transaction tie can be derived fro the in order to price options. Gean and Yor (1997) represent the dynaics of is directly odeled as a geoetric Brownian otion plus a Poisson process with constant jup sizes and then derive the PCS call price. Aase ( ) takes a different odeling approach and uses a copound Poisson process with rando jup sizes to describe the dynaics of the loss index. However for catastrophic events the assuption that resulting clais occur in ters of the Poisson process is inadequate as it has deterinistic intensity. Dassios and Jang (3) use the Cox process to odel the arrival process for catastrophic events and price catastrophe insurance derivatives under constant interest rate. On the other hand only a few articles focus on the pricing of CAT bonds. ee and Yu () use an arbitrage-free fraework to price catastrophe bonds and assue that the catastrophe aggregate loss follows the copound Poisson process and then copute risk-free and default-risky CAT under the consideration of default risk basis risk and oral hazard using Monte Carlo ethod. Vaugirard (3 a 3b) extends and adapts the jup diffusion odel of Merton (1976) to develop a CAT bond valuation fraework that allows for catastrophic events interest rate uncertainty and non-traded underlying state variables. Therefore when pricing catastrophe insurance products ost prior studies assue the arrival process of catastrophe event follows Poisson process with constant arrival rate. However figure 1 displays that the arrival rate of catastrophe events could be significant difference at different situation. To capture this phenoenon we extend the odeling fraeworks of Cuins and Gean (1993) Gean and Yor (1997) and Vaugirard (3a 3b) to assue that total insured loss index is driven by a Markov jup diffusion process instead of Poisson jup diffusion process. Under Markov odulated Poisson process the underlying state is governed by a hoogeneous Markov chain. For siplicity we assue the 8

9 interest rate is constant thus under the risk neutral probability easure Q the dynaic process of total insured loss can be written as: Φ( t) 1 Q ( t) = ()exp ( r σ ) t + σ W t + lnyn Λκt (.3) n= 1 where () is the initial total insured loss r is the drift paraeter and σ is the constant volatility of the Brownian coponent of the process. { Q ( ) : } Brownian otion and { : 1... } n W t t > is a standard Y n = are independent for the sequence and identically distributed nonnegative rando variables representing the size of the n - th loss. { ( t) : t } Φ > is a Markov odulated Poisson process under risk-neutral easure with arrival rate of catastrophe events Λ ( κ + 1) where κ E( Y n -1). The last ter Λ κt in equation (.3) is the copensator for the Markov odulated jup process under the risk-neutral easure. We assue that κ < which iplies that the eans of the jup sizes are finite for the total losses of the insured. In addition all three sources of randoness W ( t ) standard Brownian otion Φ (t) Markov odulated Poisson process and Y n the jup size are assued to be independent. Changes in the total insured loss coprise three coponents: the expected instantaneous total insured loss change conditional on no occurrences of catastrophes the unanticipated instantaneous total insured loss change which is the reflection of causes that have a arginal ipact on the gauge and the instantaneous change due to the arrival of the catastrophe. The total insured loss ( t ) follows the geoetric Brownian otion during the tie period ( t ] given that no inforation of catastrophe event arrivals during the tie period. When the inforation of catastrophe event arrivals at tie t the total insured loss changes instantaneously fro ( t ) to Y ( t _). Various statistical distributions are used in actuarial n odels of insurance clais processes to describe the jup size of total insured loss such as lognoral gaa Pareto distribution. This paper follows prior studies such as Chang Chang and Yu (1996) and Vaugirard (3 a 3b) to adapt the lognoral distribution. Note that for valuation purpose we need to know the loss dynaic under the risk neutral probability easure. When the loss process has jups the arket becoes incoplete and then there is no unique pricing easure. Hence we follow Merton (1976) and suppose that investors acknowledge that natural catastrophe shock is idiosyncratic risk when it coes to Y n 9

10 pricing contingent clais. The rationale underlying this stance is that natural catastrophes such as hurricanes and earthquakes are barely correlated to financial stors which is supported for exaple by the epirical study of Hoyt and McCullough (1999). Therefore CAT insurance products provide a valuable tool of diversification for investors because catastrophic losses are zero-beta events in the sense of the Capital Asset Pricing Model as ephasized for exaple by itzenberger Beaglehole and Reynolds (1996) or Canter Cole and Sandor (1997). By assuing that such the jup risk is nonsysteatic and diversifiable attaching a risk preiu to the risk is unnecessary. III. Pricing Catastrophe Insurance Product 1. European Catastrophe Call Futures Option Pricing The structure of European Catastrophe Call Futures Option In the early 199s CBOT launched catastrophe futures contracts and catastrophe futures call option with contract values linked to the loss ratios copiled by PCS. These two catastrophe derivatives provide underwriters and risk anagers an effective alternative to hedge and trade catastrophic losses. Unlike reinsurance negotiations that are costly lengthy and irreversible these two catastrophe derivative contracts have the advantages of reversibility transactional efficiency and anonyity due to the standardized and exchange-traded properties. Besides because catastrophic losses show no correlation with the price oveents of stocks and bonds underwriters and ainstrea institutional investors ay also enjoy the diversification effect of these derivatives These two catastrophe derivatives trade on a quarterly basis: that is Jan-Mar Apr-June July-Sep and Oct-Dec. The CBOT devised a loss ratio index as the underlying instruent for catastrophe insurance futures and options contracts. The loss ratio index is the reported losses incurred in a given quarter and reported by the end of the following quarter divided by one fourth of the preius received in the previous year. Further it is likely that a cap is needed to liit the credit risk in the case of unusually large losses. However to date there has not been an incident where the axiu loss ratio has been reached thus we ignore the axiu loss ratio at %. Then the value of an insurance future at tie T F( T ) is the noinal contract value US$5 ties the loss ratio index. The value of a catastrophe insurance call option on the future of the option C( T ) at tie T is given by C( T) = ax( F( T T) K) (3.1) 1

11 where K denotes the pre-deterined horizon exercise price. Fro the equation (3.1) we know that the catastrophe call futures call option is uch like a regular European futures call option. This right is only exercisable when the catastrophe futures price exceeds a constant critical value during the life tie of the option; otherwise the call value is. Pricing of European Catastrophe Call Futures Option Under the risk-neutral easure Q the value of the catastrophe call futures option can be obtained by discounted expectations. For siplicity we assue the interest rate is constant. Thus the price of the European catastrophe call futures option at tie t C( t ) can be given as follows: ( ) + = - ( - ) ( ) r T t Q C t e E F( T ) K where r represents the risk-free interest rate. To derive the pricing forula of European catastrophe call futures option we assue that the size of the n - th loss variance Y n follows a lognoral distribution with ean θ and δ and then apply the classical artingale ethod under risk neutral easure. A detailed proof is sketched in Appendix A thus the forula of the European catastrophe call futures option can be obtained as Theore 1: Theore 1:The value of European catastrophe call futures option is where F F φ 1 φ (3.) = C( t) = Q( T t)exp( r ( T t)) F '( t) ( d ) K ( d ) ( I I ) atrix d F 1 1 ln [ F( t) K ] ± σ ( T - t) + θ Λκ( T - t) = σ ( T - t) + δ 1 F( t) ' = F( t) exp ( Λκ)( T t) + θ + δ -1 r θ + δ = ( r Λ κ) + ( T t) φ( ) denotes the cuulative distribution function for a standard noral rando variable. This pricing forula can be viewed as the transition probability ultiplied by Black s prices with jup coponent. The jup rate of catastrophe events depends on the different status. For exaple if the jup rate of catastrophe events follows the switched jup 11

12 diffusion odel which is larger in one state and saller in other status. Since a jup process gives rise to fatter tails for the underlying asset return distribution it results in larger transition probability and volatility of jup size and then akes the value of the European catastrophe call futures option increase. Note that when λ1 = λ =...= λ I = λ the Markov odulated Poisson reduces to the Poisson process with intensity λ. Hence the pricing forula (3.) can be rewritten as below: where = - λ ( κ + 1)( T t) ( λ( κ + 1)( T - t) ) e F F C( t) = exp( r ( T t)) F '( t) φ( d1 ) - Kφ( d) (3.3)! d F 1 1 ln [ F( t) K ] ± σ ( T - t) + θ λκ( T - t) = r σ ( T - t) + δ 1 θ + δ = ( r λκ) + ( T t) This iplies that the expected nuber of jups of catastrophe events per tie unit is driven by the Poisson process. Chang Chang and Yu (1996) assue that the catastrophe futures price change follows a jup subordinated process in calendar-tie. A subordinated process is obtained by rando the tie clock of the stationary process called the parent process using a new tie clock called the directing process or subordinator. Therefore by changing the tie scale the subordinated process is transfored to a stationary process. Besides the parent process of the futures return is a lognoral diffusion and the directing process is a hoogeneous Poisson. If no inforation of catastrophe events arrives the futures price stay put. On the other hand if inforation of catastrophe events arrives then futures price jups instantaneously according to the lognoral distribution. Although our odel setting is different with Chang Chang and Yu (1996) our pricing forula is siilar to Chang Chang and Yu (1996) which is the Poisson su of Black s prices. If λ1 = λ =...= λ = and total insured loss follows lognoral distribution then European I catastrophe call futures option price can be given by: where d F 1 F ( ) = φ 1 φ F C t exp ( - r( T - t)) F( t) ( d ) K ( d ) 1 ln [ F( t) K ] ± σ ( T - t) =. σ ( T - t) Hence our pricing forula equation (3.) will reduce to the pricing forula of Cox and Schwebach (199) which is siilar to Black s (1976) forula for pricing futures options. The diffusion assuption ignores the sporadic nature of catastrophes and jup in the nuber of clais..

13 Dynaic Hedging of European catastrophe call futures option When the investors provide European catastrophe call futures option they will siultaneously hedge their position to avoid taking on huge losses. Hence this section will illustrate how to hedge against oderate changes in the total insured loss. In coplete arkets Delta Gaa hedging techniques will be used to easure the sensitivity of the option s price to total insured loss oveents at the first and second order. Hence by partial differential of the equation (3.) we have: F F ( t) = C( t) = Q( T t)exp( r( T t)) φ( d1 ) F( t) = = φ '( d1 F ) Γ ( ) ( ) ( )exp( ( )) F t = C t = Q T t r T t F( t) ( t) σ ( T t). 1 F F ( d1 ) where φ '( d1 ) = exp - is standard noral distribution function. π Since the underlying status follows Markov chain and different arrival rate of catastrophe events depends on different status under MMPP our hedging forula are the su of transition probability ultiplied by hedging forula of Black s price adding jup coponent. If no catastrophe occurs our hedging forulas reduce to hedging forula of Black s price.. European PCS Catastrophe Call Spread Pricing The structure of European PCS Catastrophe Call Spread PCS catastrophe insurance options were introduced at the CBOT in Septeber They are standardized exchange-traded contracts that are based on catastrophe loss indices provided daily by PCS. The PCS indices reflect estiated insured industry losses for catastrophes that occur over a specific period. Only cash options on these indices are available; no physical entity underlies the contracts. They can be traded as calls puts or spreads. Most of the trading activity occurs in call spreads since they essentially work like aggregate excess-of-loss reinsurance agreeents or layers of reinsurance that provide liited risk profiles to both the buyer and seller. These European cash options are also quoted in percentage points and tenths of point but each point equals $ cash value. The CBOT lists PCS options both as sall cap contracts which liit the aount of aggregate industry losses that can be included under the contract to $ billion and as large cap contracts which track losses fro $ billion to $5 billion. Based on the PCS loss index the cash value of the PCS call spread at aturity 13

14 T C ( T ) is: ( K - A) K ( T ) < C ( T ) = ( ( T ) - A) A ( T ) < K ( T ) < A = ( ( T ) A)1 { ( T ) > A} ( ( T ) K)1{ ( T ) > K } where ( T ) presents the relevant PCS loss index K and A represent the cap point and strike point of the call with K = and < A for a sall cap and K = 5 and A 5 for a large cap. 1{} denotes the indicator function. Pricing of European PCS Catastrophe Call Spread Under the risk-neutral easure Q the price of the PCS catastrophe call spread at tie t is the discounted expected value of C ( T ). For siplicity we assue the interest rate is constant. Thus the price of the European PCS catastrophe call spread at tie t C ( t ) can be given as follows: C t = e E T A T K - r ( T - t) Q ( ) ( ( ) - )1 ( ( ) - )1 T A T K where r represents the risk-free interest rate. { ( ) > } { ( ) > } To derive the pricing forula of European PCS catastrophe call spread we follow Chang Chang and Yu (1996) to assue that the size of the n - th loss lognoral distribution with ean θ and variance Y n follows a δ and then apply the classical artingale ethod under risk neutral easure. The pricing process is siilar to the European future call option thus the forula of the European PCS catastrophe call spread can be obtained as Theore : Theore :The value of European PCS catastrophe call spread is = φ 1 φ = C ( t) Q( T t) ( t) ( d ) Aexp( r ( T t)) ( d ) where ( I I ) atrix ( t) φ( d3) K exp( r ( T t)) φ( d4) d 1 1 ln [ ( t) A] + ( r ± σ )( T - t) + θ Λκ( T - t) = σ ( T - t) + δ (3.4) 14

15 ( I I ) atrix d 34 1 ln [ ( t) K ] + ( r ± σ )( T - t) + θ Λκ( T - t) =. σ ( T - t) + δ Note that when λ1 = λ =...= λ I = λ the Markov odulated Poisson reduces to the Poisson process with constant intensity λ. Hence the equation (.3) could reduce to the odel of Cuins and Gean (1993) and Gean and Yor (1997) and the pricing forula equation (3.4) can be rewritten as below: ( λ( κ 1)( T - t) ) - λ( κ + 1)( T t) e = φ 1 φ =! C ( t) + ( t) ( d ) Aexp( r ( T t)) ( d ) (3.5) ( t) φ( d3) K exp( r ( T t)) φ( d4) where d 1 1 ln [ ( t) A] + ( r ± σ ) ( T - t) + θ λκ( T - t) = σ ( T - t) + δ d 34 1 ln [ ( t) K ] + ( r ± σ )( T - t) + θ λκ( T - t) =. σ ( T - t) + δ This pricing forula iplies that the expected nuber of jups of catastrophe events per tie unit is driven by the Poisson process. In addition this pricing forula is also siilar to Chang Chang and Yu (1996) if the underlying is the catastrophe future price. If λ1 = λ =...= λ = then European PCS catastrophe call spread can be given by: I ( φ 1 φ φ 3 φ 4 ) C( t) = ( t) ( d ) Aexp( r( T t)) ( d ) ( t) ( d ) K exp( r( T t)) ( d ). (3.6) Hence the pricing forula equation (3.6) will becoe siilar to traditional Black and Scholes s pricing forula for European call spread. Dynaic Hedging of European PCS catastrophe call spread This section will exaine how to hedge against oderate changes in the total insured loss. In coplete arkets Delta Gaa hedging techniques will be used to easure the sensitivity of the price of European PCS catastrophe call spread to total insured loss oveents at the first and second order. Hence by partial differential of the equation (3.3) we have: 15

16 t C t Q T t φ d1 φ d3 ( t) = ( ) = ( ) = ( ) ( ) ( ) φ '( d1 ) φ '( d3 ) Γ ( t) = C ( t) = Q( T t) ( t) ( t) σ ( T t) ( t) σ ( T t). = Since our odel considers the status follows Markov chain and the arrival rate of catastrophe events depends on status our hedging forulas are the su of transition probability ultiplied by hedging forula of Black and Scholes adding jup coponent. If no catastrophe occurs our hedging forulas reduce to hedging forula of Black and Scholes. 3. Default-Free Catastrophe Bonds The structure of Default-Free Catastrophe Bonds CAT bond is a liability hedged instruent for insurance copanies. This security provisions have debt-forgiveness triggers whose generic design grants the interest payent and/or the return of principal forgiveness. Meanwhile the debt forgiveness can be triggered by the insurer s actual losses or a coposite index of insurer s losses during the specific period. Hence under this structure the insurance copany could transfer the catastrophe risk to increase the ability to provide insurance protection. The CAT bondholder accepts losing interest payents or a fraction of the principal if the loss index or the aggregate loss of issuing fir standing for natural risk or associated insurance clais hits a pre-specified threshold K. More specifically if the loss index or the aggregate loss of issuing fir does not reach the threshold during a risk exposure period the bondholder is paid the face aount. Otherwise he receives the portion of the principal needed to be paid to bondholders. Consider the CAT discount bond whose payoffs PO i T at aturity date T are as follows: PO i T a if i T K = δ a if i T > K where i T is the aggregate loss of issuing fir i at aturity T. K is the trigger level in the CAT bond provisions and δ is the portion of the principal needed to be paid to bondholders if the forgiveness trigger has been pulled. is the face aount of the issuing fir s total debts. a is the ratio of CAT bond s face aount to total outstanding debts. 16

17 Pricing of Default-Free Catastrophe Bonds According to the payoff of CAT bonds the price of discount default-free CAT bonds at tie t Pi ( t ) can be valued as follows: r( T - t) Q Pi ( t) e E a 1 r( T t) Q = { i T K} + e E η a 1 { i T > K } (3.7) This paper follows Vaugirard (3 a 3b) to adapt the lognoral distribution with ean θ and variance δ then equation (3.7) can be rewritten as follows: Theore 3:The value of discount default-free CAT bonds is r( T t) Pi ( t) = e P( T - t) φ( d4) + η 1 P( T - t) φ( d4) a = = (3.8) If λ 1= λ =... λ I = λ then new Markov odulated Poisson process Φ ( t) siplifies to new Poisson process N( t ) i.e. total insured loss index could reduce to the odel of Vaugirard (3a 3b) which is driven by a Poisson jup diffusion process. Thus the equation (3.8) becoes: - λ ( T t ) ( ) - λ ( T t ) r( T t) e λ( T - t) e ( λ( T - t) ) Pi ( t) = e φ( d4) + η 1 φ( d4) a =! =!. If λ 1= λ =... λ I = then the equation (3.8) reduces to r( T - t) Pi ( t) = e a φ( d4 ) + η (1 φ ( d4 )). This pricing forula is siilar to ouberge Kellezi and Gilli (1999) under the assuption that the catastrophe loss index or the aggregate loss of issuing fir stands for geoetric Brownian otion process. Dynaic Hedging of CAT bonds Siilar to European futures call and European PCS catastrophe call spread we also provide the hedging strategy of CAT bonds. In coplete arkets Delta Gaa hedging techniques will be used to estiate the sensitivity of the CAT bond s price to total insured loss oveents at the first and second order. Hence by partial differential of the equation (3.8) we have: 17

18 r( T t) φ d4 φ d4 '( ) '( ) ( t) = C ( t) = e a P( T t) + η ( t) = ( t) σ ( T t) ( t) σ ( T t) r( T t) φ '( d4) φ '( d4 ) Γ ( t) = C ( t) = e a P( T t) η ( t). = ( t) σ ( T t) ( t) σ ( T t) IV. Nuerical Analysis of Catastrophe Insurance Products This section exaines the sensitivity analysis for catastrophe insurance products including European futures call European PCS catastrophe call spread and CAT bonds. The values of catastrophe insurance products depend on several paraeters in a Markov jup diffusion odel when the jup size is assued to follow the lognoral distribution. We will establish a set of paraeters and base value to evaluate catastrophe insurance products in Markov jup diffusion odel and copare it to jup diffusion odel forula. To deonstrate how the European futures call is related to various paraeters we ake soe assuptions as follows: futures price F = 4 ; exercise price K = 8 ; interest rate r =.5 ; catastrophe loss volatility σ =.4 ; option ter T =.5 T =.5 ; truncation n = 8 ; transition rate leaving state 1 and state α 1 = 1 α = 1. Table 1 report the value of European futures calls as a function of jup rate of catastrophe events the standard deviation of jup size and ean of jup size under Markov jup diffusion odel and jup diffusion odel. For the paraeters of jup rate of catastrophe events we consider two jup rate: 1 and 3 to exaine the jup rate effect. When jup rate of catastrophe events increases this causes the volatility of European futures call increases the Poisson probability increases and exercise tie early. Thus higher jup rate of catastrophe events higher value of European futures call. In the other hand copared with MJ(13) we find that Poi(1) underestiates MJ(13) and Poi(3) overestiates MJ(13). Hence if different econoy status has significant different jup arrival intensity in real econoy using jup diffusion odel to evaluate the European futures calls could cause significant ispricing. For the concern of paraeters of the standard deviation of jup size and ean of jup size due to sae reason with jup rate of catastrophe events this table also reveals that higher the standard deviation of jup size and ean of jup size due to catastrophe occurrences result in higher the European Futures Call price. Next we exaine the sensitivity analysis for European PCS catastrophe sall cap thus 18

19 the cap point ( K ) is. We copare nuerically European PCS catastrophe call spread values obtained fro equation (3.4) to equation (3.5) in jup diffusion odel. We assue the PCS loss index ( ) is 4 and the riskless rate ( r ) is.5. Option ter ( T ) is.5 and truncation ( n ) is 8. The ean of jup size ( θ ) is.1 and the strike point ( A ) is 4 and 8 respectively to represent deep-in the-oney at-the-oney and deep-out-of-the-oney respectively. Assue the jup rate of catastrophe loss at state 1 or state are set to be 1 and 3 respectively to reflect the frequencies of catastrophe events per year. The transition rate of these two states is 1 and 1 respectively to capture the leaving length for jup rate at different state. We also report volatility of jup size ranges fro.1 year to. in order to exaine the volatility effect. Table reports the value of European PCS catastrophe call spread as a function of oneyness jup rate and standard deviation of jup size. For out-the-oney and at-the-oney options the results are also consistent with the prediction that the European PCS catastrophe call spread experiences as single call option. Hence when jup rate of catastrophe events increases or standard deviation of jup size increases the value of European PCS catastrophe call spread increases. However for in-the-oney options because catastrophe event happens a lot or akes huge loss when jup rate of catastrophe events increases the probability that PCS loss index exceed cap point higher and thus European PCS catastrophe call spread decreases. By the sae taken for in-the-oney options we could find that higher standard deviation of jup size lower European PCS catastrophe call spread. In addition copared with MJ(13) the results are also consistent with the prediction that no atter what case of oneyness Poi(1) underprices MJ(13) and Poi(3) overprices MJ(13). Finally for the nuerical analysis of CAT bond assue that the total aount of the issuing fir s debts which include CAT bonds has a face value of $1. The strike value ( K ) is set to be 1 and the loss aount ( i T ) is set at 1. The interest rate ( r ) is.5 and the ean of jup size ( θ ) is 1. Truncation ( n ) is set at 8. The portion of principal needed to be repaid ( η ) is set at.5 when debt forgiveness has been triggered. The ratio of the aount of CAT bonds to the insurer s total debt ( a ) is set to be.1. We report standard deviation of jup size ranges fro. to.5 to deonstrate the volatility effect. We also report aturity ranges fro.5 year to 1 year in order to exaine the aturity effect. In Markov jup diffusion odel assue the jup rate of catastrophe loss at state 1 or state are set to be 1 and 3 respectively and the transition rate of these two states is 1 and 1 respectively. 19

20 Table3 reports the CAT bond prices under alternative sets of occurrence jup rate aturity and standard deviation of jup size. As jup rate rises up the probability that loss aount exceed strike value increase then according to the payoff of CAT bond we can obtain CAT bond price decreasing. In the other hand for volatility effect due to the siilar phenoenon with jup rate we also find that higher standard deviation of jup size due to catastrophe occurrences results in lower CAT bond price. For aturity effect longer aturity leads to higher discount rate and thereby lower CAT bond price. V. Conclusions We propose a Markov jup diffusion odel which extends the Poisson process used in the jup diffusion odel to be Markov odulated Poisson process. Markov odulated Poisson process stands for a doubly stochastic Poisson process where the underlying state is governed by a hoogenous Markov chain to odel the arrival process for catastrophic events. It is also the generalization of Cuins and Gean (1993) Chang Chang and Yu (1996) and Gean and Yor (1997). Further we apply Markov jup diffusion odel to derive pricing forulas and hedging strategy of catastrophe insurance products: European futures call option catastrophe PCS call spread and CAT bond. For nuerical analysis we evaluate catastrophe insurance products in Markov jup diffusion odel and copare it to jup diffusion odel forula. Copared with MJ(13) we find that MJ(13) is between Poi(1) and Poi(3). Hence if different econoy status has significant different jup arrival intensity in real econoy using jup diffusion odel to evaluate the European futures calls could cause significant ispricing.

21 Table 1 Value of European Futures Call δ θ Model..5 Poi(1) Poi(3) MJ(13) Poi(1) Poi(3) MJ(13) This table reports the value of European Futures Call as a function of transition rate jup rate and ean of jup size. The paraeters are K = 8 F = 4 r =.5 σ =.4 T =.5 n = 8. Poi(1) and Poi(3) denote the value with annual jup rate 1 and 3 respectively in jup diffusion odel. MJ(13) denote the value with annual jup rate is 1 at state 1 and 3 at state in Markov jup diffusion odel. 1

22 Table Value of European PCS Call Spread / A Model Poi(1) Poi(3) MJ(13) Poi(1) Poi(3) MJ(13) Poi(1) Poi(3) MJ(13) δ This table reports the value of European PCS Call Spread as a function of transition rate of two states jup rate and ean of jup size. The paraeters are K = = 4 r =.5 σ =.4 T =.5 n = 8. Poi(1) and Poi(3) denote the value with annual jup rate 1 and 3 respectively in jup diffusion odel. MJ(13) denote the value with annual jup rate is 1 at state 1 and 3 at state in Markov jup diffusion odel.

23 Table 3 Value of Catastrophe Bonds δ T Model..5 Poi(1) Poi(3) MJ(13) Poi(1) Poi(3) MJ(13) This table reports the value of Catastrophe Bonds as a function of transition rate jup rate and ean of jup size. The paraeters are K = 1 = 1 r =.5 σ =.4 n = 8 = 1 a =.1 η =.5. Poi(1) and Poi(3) denote the value with annual jup rate 1 and 3 respectively in jup diffusion odel. MJ(13) denote the value with annual jup rate is 1 at state 1 and 3 at state in Markov jup diffusion odel. i T 3

24 Endnotes 1 The quarterly data coes fro the Insurance Service Office (ISO). The PCS track insured catastrophe loss estiates on national regional and state basis in the US fro inforation obtained by PCS. The ter natural catastrophe includes hurricanes stors floods waves and earthquakes. The event of natural catastrophe denotes a natural disaster that affects any insurers and when clais are expected to reach a certain dollar threshold. Initially the threshold was set to $5 illion. In 1997 ISO increased its dollar threshold to $5 illion. Appendix A The payoff of call option on a future contract under risk neutral easure Q is ( ) = - ( - ) ( ) r T t Q C t e E F( T ) K + where = ( ( ) ) 1 { ( T ) K } - r( T - t) Q e E T K > ( ( )1 { ( ) } 1 T > K { ( T ) > K} ) - r( T - t) Q Q = e E T E K Q A = E ( T )1 > { ( T ) K} - r( T - t = e ) ( A B) ( T - t) Q Φ 1 Q Λκ ( T - t) = E ( t) exp ( r σ )( T - t) + σ W ( T - t) Yn e 1{ ( T ) > K } n= By Girsanov s theory (A.1) T R Q W W σ t dt = equation (A.1) can be rewritten as: Φ( T - t) R A = F( t) E exp ln Y κ ( T - t) Λ 1 n= { ( ) > } n T K ( n κ ) { ( T ) > K} R = P( T t) F( t) E exp ln Y Λ ( T - t) 1 Φ ( T - t) = (A.) = n= By Radon-Nikody derivative of transition probability and jup size dq( t) = ( κ + 1) exp(- Λ κt) dp( t) the equation (A.) can be obtained: Q Q Q 1 = dq( Y Y... Y ) y f ( y) ( κ + 1) 4

25 Q( T t) F( t) E 1 Φ ( T - t) = (A.3) R 1 R Q Λκ ( T - t ) = ( t)exp σ ( T - t) + σ W ( T - t) Yn e > K n= Assue that Y n stands for lognoral distribution with ean θ and variance δ i.e lny n ~ N( θ δ ) and then 1 θ + δ κ + 1 = E ( Y n) = e. Since Q n dq( Y ) = yfy ( y ) we have: ( κ + 1) Since Q 1 (ln y ( θ + δ )) dq( Yn ) = exp y πδ δ Y n is independent identically distributed nonnegative rando variables the new jup size under Q is Hence we have: Q lnyn = ln Yn ( θ + δ ). n= 1 n= 1 Q( T - t) F( t) E 1 Φ ( T - t) = (A.4) R 1 R Q = F ( t)exp σ ( T - t) + σ W ( T t) + ln yn + ( θ + δ ) Λκ ( T t) > K n= 1 R et W ( T t) = T - tz where Z ~ N(1). By the sae taken set hence equation (A.4) can be described as: Q Q lnyn = δ Z n= 1 R Q K 1 A = Q( T t) F( t) P σ T -tz + δ Z > ln σ ( T t) = F( t) +Λκ ( T t) θ δ Φ ( T - t) = F( t) 1 ln + σ ( T - t) Λ κ( T - t) + θ + δ R K = Q( T t) F( t) P Z < Φ( T t) = = σ ( T - t) + δ δ ( T - t) F( t) 1 1 ln + - Λ κ + ( θ + δ ) ( T - t) ( T - t) + σ + ( T t) R K = Q( T t) F( t) P Z < Φ( T t) = = δ ( σ ) T -t + T - t 5

26 F( t) 1 ln + η ( T - t) + σ ( T - t) R K = Q( T t) F( t) P Z < Φ( T t) = = σ ( T t) = = Q( T - t) F( t) N( d ) where ( I I ) atrix d 1 1 F( t) 1 ln + η ( T t) + σ ( T t) K = σ ( T t) 1 θ + δ η = Λ κ + ( T t) σ = σ + δ. ( T - t ) Siilarly B E Q K = 1{ ( T ) > K} 1 ( t) Q Φ Q Λ κ ( T t) = KP ( t)exp ( r σ )( T - t) + σ W ( T - t) Yn e > K n= rt Q 1 Q = P( T t) Ke P ( t) exp ( r σ )( T - t) + σ W ( T - t) = = + ln Yn Λ κ ( T - t) > K Φ ( T - t) = n= Q K 1 = P( T t) K P σ T tz + δ Z > ln + σ ( T t) + Λκ ( T t) θ Φ( T t) = F( t) 1 ln [ F( t) K ] σ ( T - t) Λκ ( T - t) θ Q P( T t) K P Z = < Φ ( T - t) = (A.5) = σ ( T - t) + δ Λk ( T t) Due to Q( T t) ( κ 1) e P( T t) = + and + 1) = [ E( Y )] 1 κ ( n = exp θ + δ thus equation (A.5) becoes 1 ln [ F( t) K ] + η ( ) σ ( ) exp( )( ) T t T t Λκ T t Q ( ) B = KQ T P Z ( T t) < Φ = = 1 σ ( T t) exp θ + δ 6

27 = exp( Λκ )( T t) KQ( T ) N( d) 1 exp θ + δ = where ( I I ) atrix 1 ln [ F( t) K ] + η ( T t) σ ( T t) d = σ ( T t) δ σ = σ +. ( T - t) Therefore the option price on future contract can be described as: F F C( t) = exp( r ( T t)) Q( T t) F '( t) N( d1 ) KQ( T ) N( d1 ) = =. 7

28 References 1. Aase K Catastrophe Insurance Futures Contracts. Norwegian School of Econoics and Business Adinistration Institute of Finance and Manageent Science Working Paper.. Aase K An Equilibriu Model of Catastrophe Insurance Futures and Spreads. Geneva Papers on Risk and Insurance Theory Canter M. J. B. Cole and R.. Sandor 1997 Insurance derivatives: A New Asset Class for the Capital Markets and A New Hedging Tool for the Insurance Industry. Journal of Applied Corporate Finance 1(3) Chang C. J. Chang and M.-T. Yu 1996 Pricing Catastrophe Insurance Futures Call Spreads: A Randoized Operational Tie Approach Journal of Risk and Insurance Cox S. H. and R. G. Schwebach 199 Insurance Futures and Hedging Insurance Price Risk Journal of Risk and Insurance Cuins J. D. and H. Gean 1993 An Asian Option to the Valuation of Insurance Futures Contracts Wharton School Center for Financial Institutions Working Paper. 7. Cuins J. D. and H. Gean 1995 Pricing Catastrophe Futures and Call Spreads: An arbitrage Approach Journal of Fixed Incoe Dassios A. and J.-W. Jang 3 Pricing of Catastrophe Reinsurance and Derivatives Using the Cox Process with Shot Noise Intensity Financial Stochastic Gean H. and M. Yor 1997 Stochastic tie changes in catastrophe option pricing Insurance: Matheatics and Econoics Hoyt R. E. and K. A. McCullough 1999 Catastrophe Insurance Options: Are They Zero-Beta Assets? The Journal of Insurance Issues () ast G. and A. Brandt 1995 Marked Point Processes on the Real ine: The Dynaic Approach. Springer-Verlag New York. 1. ee J.-P. and M.-T. Yu Pricing Default-Risky CAT Bonds with Moral Hazard and Basis Risk Journal of Risk and Insurance itzenberger R. H. D. R. Beaglehole and C. E. Reynolds 1996 Assessing Catastrophe Reinsurance-inked Securities as a New Asset Class Journal of Portfolio Manageent ouberge H. E. Kellezi and M. Gilli 1999 Using Catastrophe-inked Securities to Diversify Insurance Risk: A Financial Analysis of CAT Bonds Journal of Risk and 8

29 Insurance Merton R. C Option Pricing when Underlying Stock Returns are Discontinuous Journal of Financial Econoics Vaugirard V. 3a Valuing Catastrophe Bonds by Monte Carlo Siulations Applied Matheatical Finance Vaugirard V. 3b Pricing Catastrophe Bonds by an Arbitrage. The Quarterly Review of Econoics and Finance Approach Zajdenweber D The Valuation of Catastrophe-Reinsurance-inked Securities Paper presented at Aerican Risk and Insurance Association Meeting. 9