The Valuation of the Catastrophe Equity Puts with Jump Risks

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1 The Valuatio of the Catastrophe Equity Puts with Jump Risks Shih-Kuei Li Natioal Uiversity of Kaohsiug Joit work with Chia-Chie Chag

2 Outlie Catastrophe Isurace Products Literatures ad Motivatios Jump Risk Models Markov modulated Poisso processes (MMPP) Doubly stochastic Poisso processes (DSPP) Marked poit processes (MPP) The Valuatio of the Catastrophe Equity Puts Numerical ad Empirical Experimet Coclusios ad Future Researches

3 Catastrophe Isurace Products Catastrophe (CAT) A evet resultig i the great loss Ma-made hazards ad ature hazards Ma-made hazards are arise by huma beigs, such as Wars, 9 Terrorist attack Nature hazards are atural disasters, such as Earthquakes, Storms, Hurricaes Defiitio of Property Claim Service (PCS) idex A ature CAT is a evet which causes i excess of 5 millio US dollars i isured damages 3

4 Types of Isurace Istrumets Those that trasfer CAT risk are Reisurace Exchage-traded derivatives Swaps CAT bods Those that provide cotiget capital are Cotiget surplus otes CAT equity puts (we focus o Catastrophe equity puts ) 4

5 Catastrophe Equity Puts (CatEPut) () The reisurace compay (writer) issues the CatEPut to isurace compay (buyer). () If the actual CAT loss L(T)-L(t 0 ) exceed the specified losses L ad S(T) is lower tha K, the isurace compay has the right to sell a specified amout of its stock to the writer at K Isurace compay () () Reisurace compay The payoff of CatEPut ( ) ( ) ( ) ( 0) ( ) ( ) ( ) K S T, if S T < K ad L T L t > L, P( T) = 0, if S T K or L T L t0 L, () 5

6 Literatures Cox et al., (004) propose the pure Poisso process to model the price per share of the isurace compay s equity S( t) = S(0)exp µ ( t) σ S( ) ( ) S t+ σ S W t AN t () where S(0) is the iitial price, {W S (t): t>0} a stadard Browia motio, ad {N(t): t>0} is a Poisso process with a costat parameter λ. If a large claim occurs at time t, the price chages istataeously from S t- to S t = e -A S t-.(a is called by jump size.) The model icludes two assumptios with The costat arrival rates of CAT The costat Jump size. The large losses are ot equal whe these hurricaes hit the US east coast. 6

7 Literatures Jaimugal ad Wag (006) propose the compoud Poisso process to model the dyamics process of the stock price (3) S( t) = S(0)exp µ ( t) σ S( ) ( ) S t+ σ S W t αl t The loss process of the isured is assumed as the compoud Poisso L( t ) where {Y, =,, } are i.i.d radom variables represetig the size of the i-th loss with p.d.f f(y) α deotes the percetage drop i the share value price per uit of loss. Jaimugal ad Wag (006) exteds the Cox s model (004) for radom jump sizes. The model icludes two assumptios with The costat arrival rates of CAT evets The costat percetage drops. = N ( t ) = Y (4) 7

8 Poisso Process (PP) The arrival rates are equal i a Poisso process. However, the arrival rates are ot equal i the Uited State durig 950 to 004 i Figure provided by ISO (Isurace Service Office). 60 Numbers of CAT λ Year Figure : Number of CAT i the Uited State durig 950 to 004 8

9 Markov Modulated Poisso Process (MMPP) More precisely, we could assume the umbers of CAT comes from two depedet arrival rates i MMPP. Oe is the small arrival rate, the other oe is larger. The smaller jump rate stay for a log time, the the smaller jump rate chage ito the larger jump rate ad it stays for a log time i Fig.. Numbers of CAT λ Smaller jump rate larger jump rate λ Year Figure : Number of CAT i the Uited State durig 950 to 004 9

10 Doubly Stochastic Poisso Processes (DSPP) Geerally, we assume the umbers of CAT come from a DSPP with determiistic rates λ( t) or with stochastic rates. The arrival rates are determiistic i the Uited State durig 950 to 004 i Figure 3 provided by ISO λ λ Numbers of CAT λ ( t ) 0 λ Year Figure 3: Number of CAT i the Uited State durig 950 to 004 0

11 Marked Poit Processes (MPP) More geerally, we ca assume the arrival rate follows a MPP by a sequece ( τ, Y ), =,,.... τ 60 { } The iterprets the times of potetial CAT evets ad satisfies τ< τ <... τ<...,supτ= I this paper, we propose a geeral stochastic itesity (MPP) depedet o some evirometal variable to model the arrival rates of CAT evets λ( dy, t) Numbers of CAT { τ Y = } (, ),,, Figure 4: Number of CAT i the Year Uited State durig 950 to 004

12 Motivatio Jaimugal ad Wag (006) propose the stock process t S( t) = S(0)exp µ ( s) ds σst+ σs WS ( t) αl( t) 0 The loss-lik-price process of the isured is assumed as follows N ( t ) α L( t ) = =.Assume the arrival rate of CAT follows a Marked Poit process N ( t) M ( t) αy αy (7) = =. Assume the geeral fuctio H ( Y, τ ) > 0 with the percetage drop ad loss distributio, because ot all CAT evets has same effect o stock price. αy (5) (6) M ( t) M ( t) αy H ( Y, τ ) = = (8)

13 Jump Risk Models We propose the stock process with a Marked poit process t S( t) = S(0)exp µ ( s) ds σst+ σs WS ( t) L( t) 0 Three special cases for Marked poit processes are Poisso Process (PP) t (0) S( t) = S(0)exp µ ( s) ds σst+ σs WS ( t) L ( t) 0 Markov modulated Poisso process (MMPP) (9) = M ( t) L( t) H ( Y, τ ) = = N ( t ) L ( t) H ( Y, τ ) = t S( t) = S(0)exp µ ( s) ds σst+ σs WS ( t) L ( t) 0 Doubly stochastic Poisso process (DSPP) t S( t) = S(0)exp µ ( s) ds σst+ σs WS ( t) L3 ( t) 0 () () 3 Φ ( t ) = L ( t) H ( Y, τ ) = = D( t) L ( t) H ( Y, τ ) = 3

14 Some Assumptios Assume the iterest rate follows Vasicek model, d r( t) = k [ θ r( t)] dt+ σ dw ( t) () The iterest rate ad the stock price with Browia motio are correlated. It is atural to assume that the iterest rate process is stochastically idepedet of the loss measure. As Cox et al (004), we also assume a liquid market for CatEPut exists, the stadard derivative pricig theory implies a equivalet probability measure Q exists. r r 4

15 Catastrophe Equity Put (CatEPut) d The payoff of CatEPut P ( T ) = { ( ) ( )} ( K S ( T )) L T > L + L t0 { S ( T ) < K } Theorem : The pricig formula of CatEPut with the MPP M, Q M PM ( t; t0 ) = E K B( t, T ) ( d { ( ) ( ) } ) L T L t > L N T M L S( t)exp L( T ) L( t) h( y, s) λ( dy, ds) ( d ) Ft t 0 N T l ( S( t) K B( t, T )) ± σ ( t, T ) + L( T ) L( t) h( y, s) λ( dy, ds) t 0 = σ ( t, T ) (3) (4) kρσ rσ S + σ r σ r = σ S + [ ] L = L+ L( t ) L( t) σ ( t, T ) ( T t) ( T t) U ( t, T) U ( t, T) k k 0 5

16 Catastrophe Equity Put (CatEPut) The formulas ca be computed i MMPP ad DSPP The price of the CatEPut with a DSPP m T T exp λ ( ds) λ( ds) t t Q DS DS ( 0) = { ( ) ( ) } 0 m! L T L t > L N m= P t; t E K B( t, T) ( d ) T DS L S( t)exp L( T) L( t) h( y, s) λ( s) f ( y) dyds ( d ) Φ T t = m Ft f ( λ( s)) dλ( s) t 0 N (5) d DS, T l ( S( t) K B( t, T )) ± σ ( t, T ) + L( T ) L( t) h( y, s) λ( s) f ( y) dy ds t 0 = σ ( t, T ) 6

17 Catastrophe Equity Put (CatEPut) The formulas ca be computed i MMPP ad DSPP The price of the CatEPut with a MMPP MM MM ( 0 ) = Q { L( T ) L( t) > L } N m= P t; t P(m,T-t) E K B( t, T ) ( d ) (6) T MM L S( t)exp L( T) L( t) h( y, s) f ( y) dyds Λ ( d ) Ψ T-t= m Ft t 0 N d MM, = T l ( S( t) K B( t, T )) ± σ ( t, T ) + L( T ) L( t) h( y, s) f ( y) dy ds Λ t 0 σ ( t, T ) 7

18 Catastrophe Equity Put (CatEPut) If λ = λ =... λ I = λ (or λ = λ ), ad L( t) α Y, the equatio (5) ad t (6) reduce to the pricig formula of Jaimugal ad Wag (006) N t = = m exp ( λ( T t) )( λ( T t) ) Q PP PP ( 0 ) = { ( ) ( ) } m! L T L t > L N m= P t; t E K B( t, T ) ( d ) PP L S( t)exp α ( ( L( T ) L( t)) λκ( T t) ) N( d ) Ψ T t= m Ft (7) d PP, l ( ) (, ) (, ) ( ) ( ) ( ) = σ ( t, T) ( S t K B t T ) ± σ t T + α [ L T L t λκ T t ] 8

19 Catastrophe Equity Put (CatEPut) If λ = λ =... λ I = λ (or λ t = λ ), L( t) = α { N ( t ) > }, ad the iterest rate is determiistic, the equatio (5) ad (6) reduce to the result of Cox et al. (004) ( ) - λ( T- t) m e [ λ( T - t) ] ( ) N α( λ ) N PN t; t Ke ( d ) S( t)exp m k( T t) ( d ) r T t PP PP 0 = m= m! (8) d PP, l ( ) ( S )( ) ( ) = σ ( T t) ( S t K ) + r ± σ T t α [ m λκ T t ] S e α κ = λ ( ) 9

20 Numerical ad Empirical Experimet I our umerical ad empirical experimet, some parameters are give, ad other parameters are estimated with the data. The the true CatEput value is compared with the price of these jump risk models. Which jump risk model has the smallest error? Some parameters are give by Jaimugal ad Wag (006) Iitial stock price: S(0)=5 Volatility i equity: σ = 0. Iitial iterest rate: r(0)=0.0 Parameters i the iterest rate: Exercise price: K=80 Trigger level: L=0.5 Optio term: T=4 S k= 0.3 θ= 5% ρ= 0. σ = 5% r 0

21 Numerical ad Empirical Experimet Data source comes from ISO We use the umber of hurricae i the PCS idex durig 950 to 004 Parameters are estimated for the arrival rate of the differet models PP : The costat arrival rate is 3.93 overall years MMPP : The smaller arrival rate is.095 before 970 ad the larger arrival rate is after 970. The trasitio probability is (0.0) from state () with smaller arrival rate to state () with larger arrival rate DSPP Assume the arrival rate is determiistic as follows λ ( t) =.63 0 (9) Jump size : A logormal distributio is estimated with locatio parameter=0.0, scale parameter= t -6 4

22 Numerical ad Empirical Experimet The four global fit measures for the differece betwee true value ad model value with MMPP, DSPP, ad PP. Average percetage error (APE) N PR PT () Average absolute error (RMSE) Average relative percetage error (ARPE) APE= E( P ) AAE R = N = = P N ARPE= N = Relative measure square error (RMSE) R N P N P R T P P R T () (3) ( PR PT ) RMSE= N = (4) where P R deotes the CatEPut value uder the real umber of CAT, P T presets the CatEPut value uder the model, E(P R ) is the mea of the CatEPut value P R. N

23 Results of Experimet Table shows four measuremet errors of the CaTEPut value uder differet jump risk models, icludig DSPP, MMPP, ad PP. Uder DSPP, the four measuremet errors are smallest tha other risk models. 3

24 Coclusios Model We exted Jaimugal ad Wag s (006) model by stochastic itesity ad stochastic loss-lik-price effect to value the catastrophe equity puts uder a stochastic iterest rate model. Numerical ad empirical experimets Based o the umerical ad empirical experimets, the value of the catastrophe equity put uder the DSPP model is closed to the true value of the catastrophe equity put. 4

25 Future Researches Early excise features Because CatEPut are quite illiquid, the early exercise features will be cosidered. Some other structured risk maagemet products This model ca be applied to other structured risk maagemet products such as double-trigger products. 5

26 The Ed Thaks for your listeig 6

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