Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate


 Ami Griffin
 3 years ago
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1 Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology Sockholm
2 Acknowledgmens During he progress of he work on his hesis we have received a grea deal of help, suppor and guidance from a large number of people. Special hanks o Professor Boualem Djehiche as our supervisor a he deparmen of mahemaics, KH for valuable suggesions. Furhermore, we would like o express our graiude o our families. Finally, we would like o hank our girlfriends, Xin Chen and Liu Jiaxin for heir suppors.
3 Absrac We use Black and Scholes opions heory o obain marke valuaion of ypical life insurance conracs. he conracs are specified as in rosen & Loche Jorgensen (001), bu wih a sligh exension where he guaraneed rae is simulaed from marke models of shor ineres rae such as Vasicek, CoxIngesollRoss and HoLee models. Anoher exension is ha we assume he guaraneed rae is equal o he risk free ineres rae. Excep he above exensions, we keep oher facors of model similar as rosen & Loche Jorgensen (001). Firs, he liabiliy holders have prior claim for company asse han equiyholders. Second, a regulaory mechanism is added ino he model in order o reduce insolvency risk. Finally, we derive valuaion formulas and give numerical examples for iniial fair conracs and marke values of conracs a differen ime poins.
4 Conens 1 Inroducion 1 Basic Model 1 3 Conrac Specificaions 3.1 he liabiliyholders Claim 3. he Equiyholder s Claim 7 4 Valuaions 9 5 Numerical Examples and Comparison Fair Conracs Componens of Fair conracs Conrac componen prices 0 Appendix 31
5 1 Inroducion he fair valuaion of life insurance liabiliies has been discussed in he pas decades. One reason is how o give a suiable srucure of liabiliies conracs, which is also he main purpose of his paper. For example, mos insurance companies will promise o give policy holders an explici ineres rae added o heir accoun, named as guaraneed ineres rae. An unsuiable guaraneed rae will cerainly bring insolvency risk o insurance company. he purpose of his paper is o discuss he effecs of he variabiliy of he guaraneed rae on he insolvency risk, and we will make some exension in such poin. In our model, we assume ha he guaraneed rae is a deerminisic funcion, which means we can use Black and Scholes opion heory o derive price formulas. hen, we use shor rae models, such as Vasicek, CoxIngesollRoss, and HoLee models, o simulaed he above guaraneed rae and obain discree ime poin valuaions. Finally, we insered hem ino he formulas and obain corresponding valuaion resuls for his simulaed guaraneed rae. Furhermore, we will presen a reasonable connecion beween i and risk free rae. Anoher mechanism we will inroduce, which was also inroduced in he lieraures, is regulaory resricion. However, he difference here is ha he regulaory line is a random process, raher han deerminisic exponenial funcion as in rosen and Jorgensen (001). Finally, we provide a conclusion abou he valuaions of policyholders claims under he following ineres rae models: Vasicek model, CoxIngesollRoss model and HoLee model. Basic Model In his secion, we will presen a model o deal wih he problem discussed in he inroducion. he basic framework is based on Briys and de Varenne(1997), bu we es he shor rae and guaranee rae by using sochasic process. A =0, policyholders and equiyholders form a muual company, he life insurance company. he oal Asse is A 0, a fracion L 0 = αa 0 (0 α 1) is invesed by policyholders and E 0 = (1 α)a 0 is invesed by equiyholders. he parameer α is called he wealh disribuion coefficien. 1
6 able 1 Asses Liabiliies A 0 L 0 αa 0 E 0 (1 α)a 0 hen he agens acquire a claim for a payoff on or before he mauriy dae. herefore, hese claims are very similar o financial derivaives wih he company s asses as he welldefined underlying asse. Hence, firs, we will give a precise descripion of claims and hen use he powerful apparaus of coningen claims valuaion o price hem. Second, we will choose suiable parameers in he valuaion formula in order o form a fair conrac a =0, which means ha iniial invesmens equal o he ime zero marke value of he claims. 3 Conrac Specificaions In his secion we describe he deails of he sakeholders claims on he company s asses, as suggesed in Briys & Varenne(1997) and Loche & Jorgensen(001). We begin wih a specificaion of he liabiliyholders claim. 3.1 he liabiliyholders Claim As menioned in he inroducion, he life insurance company promises he liabiliyholders a coninuously compounded reurn on he iniial marke value of he liabiliies of r during he life of he conrac: L = L 0 e 0 r (s)ds a ime. Furhermore, anoher imporan assumpion in our model is ha he risk free rae equals o ineres rae guaranee: r r. In Figure1, r is simulaed from he Vasicek shor rae models: dr = a (br ) d+σ r dw, (1) where we fix he parameers a he following values a=0.5, b=0.06, σ r = 0.0, r0=0.04. hese values imply a posiively sloped curve rising from 4% owards
7 approximaely 6% as ime o mauriy ends o infiniy. Figure 1 his ranslaes ino a guaraneed final paymen of L = L 0 e 0 r (s)ds. When here are no any exernal guaranors, he company s promise can be realized only if i urns ou ha A > L a ime. In he opposie case, A L, and in he even ha he company has no been declared insolvey by regulaors, he policyholders receive A and he equiyholders receive nohing. Besides, in addiion o he promised mauriy paymen L, liabiliyholders are also generally given bonus if he marke value of he asses evolves sufficienly favorable. δ[αa L ] () From () i is clear ha in final saes where he policyholders share of oal value A exceeds he promised paymen of L hey will receive a fracion, δ, of his surplus. he parameer δ is called as he paricipaion coefficien and should be in he inerval [0 1]. 3
8 o sum up, he oal mauriy payoff o policyholders, Ψ L (A ), can be described as or A, A < L, Ψ L (A ) = L, L <A < L L +δ[αa L ], A > L α, α, (3) Ψ L (A )= δ[αa L ] + +L +[L A ] +. (4) he firs par of he righhand side has already been named as he bonus opion. he wo remaining erms are fixed mauriy paymen and shored pu opion respecively. Figure is a graphical illusraion of (4), where r is given by he Vasicek model (1). Figure 4
9 In he nex sep, we will impose a regulaory resricion. echnically, suppose ha in he framework above, he asse mus evolve above a cerain level: A > λl 0 e 0 r (s)ds B (5) he curve{b()} 0, will be called he regulaory boundary. he purpose of adding he regulaory boundary is o deermine an absolue lower bound of he asses, under which he company is declared insolvency: L 0 e 0 r (s)ds he policyholders iniial deposi compounded wih he guaraneed rae of reurn up o ime. herefore, only if ha he oal asses A a all imes have been sufficien o cover L 0 e 0 r (s)ds opions keep evolving. muliplied by some prerequisie consan, λ, will he sakeholders In he opposie even, asses will a some poin in ime,τ, equal o B τ, A τ = B τ. In his siuaion regulaory auhoriies close down he company immediaely and disribue he wealh o sakeholders. A his poin we noe ha here are wo ineresing cases. For λ 1 and in he even τ of a boundary hi, liabiliyholders receive L 0 e 0 r (s)ds. A he same ime τ equiyholders receive (λ 1)L 0 e 0 r (s)ds. Conversely, for λ < 1 and in he even of τ a boundary hi, liabiliyholders receive λl 0 e 0 r (s)ds. A he same ime equiyholders receive nohing. he various siuaions are illusraed in he figures below. is 5
10 Figure 3a For λ < 1 Figure 3b For λ 1 6
11 As described above here will be a rebae o liabiliyholders in he even of premaure closure a ime τ. his rebae, Θ L (τ), is given as τ L 0 e 0 r (s)ds, λ 1, Θ L (τ)= (6) τ λl 0 e 0 r (s)ds, λ < 1, or τ Θ L (τ)= (λ 1) L 0 e 0 r (s)ds. (7) Before considering he valuaion of claim, (Ψ L (A ),Θ L (τ)), we describe he deails regarding he equiyholders claim. 3. he Equiyholder s claim As residual claimans, equiyholders will receive a payoff a he mauriy dae condiional on no premaure closure as follows: 0, A < L, Ψ E A = A L, L <A < L A L δ[αa L ], A > L α. α, (8) or Ψ E A = [A L ] + δ[αa L ] +. (9) he above payoff funcion is depiced in Figure 4 below. 7
12 Figure 4 As described above here will be a rebae o equiyholders in he even of echnical insolvency a ime τ. his rebae, Θ E (τ), is given as τ (λ 1)L 0 e 0 r (s)ds, λ 1, Θ E (τ) = (10) 0, λ < 1. or τ Θ E (τ)= [(λ 1) 0] L 0 e 0 r (s)ds. (11) 8
13 4 Valuaions o hese conracs we apply he basic framework of Black and Scholes (1973) where all aciviy occurs on a probabiliy space (Ω, F, F, P). Under such assumpion, he dynamic evoluion of he asses is described by he sochasic differenial equaion: da =μa d + σa dw p (1) where μ and σ are consans and {W p } is a sandard Brownian moion under P. iven he riskless ineres rae, r, we have da = r A d + σa dw Q where {W Q } is a sandard Brownian moion under he riskneural probabiliy measure Q. Here,V i A,, i = L, E, denoe he ime value of he liabiliy and equiyholders claims respecively. We can wrie (see e.g. Ingersoll (1987), p ) (13) V i A, = E Q [e r (s)ds Ψ i A 1 τ + e τ r s ds Θ i (τ)1 τ< F ]= e r (s)ds Ψ i (A ) 0 f(a, ; A, )da + e τ r (s)ds Θ i (τ) g(τ; A, )dτ (14) where we assume r in he above formula a deerminisic funcion and he following heorems are also based on he assumpion. Moreover, f(a, ; A, ) denoes he riskneural densiy for he value of he asses a ime wih an absorbing barrier B 0. Similarly, g(τ; A, ) denoes he riskneural firs hiing ime densiy of asse hrough he absorbing barrier condiional on he posiion A a ime. he explanaion of hese wo densiies is deferred o he Appendix. 9
14 heorem 1. We have he following value of he Liabiliyholders Claim Ψ L (A ): A bonus (call) opion elemen, a fixed paymen, and a shored pu opion elemen. If a premaure barrier is hi here will be a rebae paymen specified by Θ L (τ) in (6). V L A, is he sum of he ime value of hese four elemens as given below in pars IIV where we used he noaion: d γ (x, )= ln x + (r(s) γ 1 σ )ds, σ ( ) I: he ime value of he bonus (call) opion elemen is δα{a N d o + ( A ) [ B + N(d B A o A X, L α e B A X, ) L r(s)ds α e r s ds + N(d o N(d 0 ( A X, )) B A X, )]}. where, X L (λ 1 α ). II: he ime value of he condiional fixed paymen elemen is L e r(s)ds N d r A B, A B N d r B A,. III: he ime value of he shored pu opion elemen is 1 λ<1 L e r(s)ds A N( d 0 L, N d r A B, A N( d 0 + A L, N d + A r, ] B L e A B r(s)ds B N( d 0 A L, N d r B A, + B N( d A 0 A L, B N d + B r, ]}}. A IV: he ime value of he rebae is (λ 1) + A A λ N d r, A 1 + B N d B B r,. A he proof relies on Black & Scholes formula and is presened in he Appendix. 10
15 heorem. We have he following value of he Equiyholders Claim Ψ E (A ): One residual claim call opion and anoher shored bonus opion. if a premaure barrier is hi here will be a rebae paymen specified by Θ L (τ) in (10). V L A, is he sum of he ime value of hese hree elemens as given below. I. he ime value of he long residual claim (call opion) elemen is A B A N d o + B A N d o + A Y, L α e B A Y, L e r(s)ds N(d 0 ( A Y, )) r s ds N d o B A Y,, where, Y = (B L ). II: he ime value of he shored bonus opion is given in par I of heorem 1. III he ime value of he equiy holders rebae is ((λ 1) 0) + A A λ N d r, B A B 1 + B N d r,. A Proof of heorem. Par I is esablished by seing δ = α = 1 in Par I of heorem 1. Par II is precisely Par I of heorem 1 wih he sign reversed. Par III ress on calculaions similar o hose ha esablished Par IV of heorem 1. 5 Numerical Examples and Comparisons 5.1 Fair Conracs he formulas for he values of he liabiliy and equiy claims are closed formulas ha can be calculaed once he relevan parameers are given. Jus as discussed in he inroducion par, i is clear ha no every choice of parameers will represen fair conracs. So he firs quesion o ask is which 11
16 combinaions of parameers will represen fair conracs. A fair conrac should saisfy he iniial equaion: L 0 = αa 0 = V L (A 0, 0; α, δ, λ, σ, r, r ), he equaion is obvious because he liabiliy holders iniial conribuion o he oal asses L 0, should equal o he iniial marke value of coningen claim. Anoher equaion is E 0 = 1 α A 0 = V E A 0, 0; α, δ, λ, σ,, r, r. Nex, we provide some seleced represenaive plos o illusrae some ypical relaions beween parameers of iniially fair conracs, where r=r are simulaed from differen shor ineres rae models such as Vasicek, CIR and HoLee. his exends he work by rosen & Jorgensen(001), who considered he case where r and r are held consan Using Vasicek model Figures 5 illusraes he relaion beween fair values of he paricipaion coefficien, δ, and he wealh disribuion coefficien, α, for some fixed and represenaive values of he remaining parameers. I is noed ha all hese graphs are negaively sloped as a higher wealh disribuion coefficien will be associaed wih a lower paricipaion coefficien in order for he conrac o be fair o boh sides (noe r and r are simulaed from random process and valuaions are averages in hese examples). Figure 5a 1
17 Figure 5b 5.1. Using CIR model Figure 6 and Figure 7 illusrae similar comparison under CIR ineres rae model and HoLee model. We find he resuls are lile differen when we use hree differen ineres rae models: Cox_ingersoll_Ross shor rae model: dr = a(br)d+σ r rdw Here we se parameers as a=0.5, b=0.05, σ r = 0.0 and r 0 =0.1. Under such a seing, he shor rae will sar from r 0 =10% oward o 5% when ime is infinie. he purpose of such an iniial parameer seing is o make shor rae under his model is much more differen o Vasicek model. However, afer careful comparison, we find ha he difference of comparison among hree models is negligible. Anoher hing we wan o menioned here is ha we will keep he above parameer seing in he CIR model when we compare he difference among models in secion 5. and secion
18 Figure 6a Figure 6b 14
19 5.1.3 Using HoLee model HoLee ineres rae model: dr=θ d+σ r dw, Here we se parameers as: Θ = 0.05e, σ r =0.0 and r 0 = 0.1 Under such a seing, he shor rae will sar from r 0 =10% oward o 5% when ime is infinie. he purpose of such an iniial parameer seing is o make shor rae under his model is much more differen o Vasicek model bu similar as CoxIngersollRoss model. We wan o menion is ha we will keep he above parameer seing in he HoLee model when we compare he difference among models in secion 5. and secion 5.3 Figure 7a 15
20 Figure 7b 5. Componens of Fair conracs In Secion 4 we derived value formulas for each of he componens. he following able shows some examples of how he oal conrac value decomposes ino he separae elemens. We will find ha he resuls are lile differen when we use hree differen ineres rae models Using Vasicek model Le r=r be he values simulaed from he following Vasicek shor rae model: dr = a (br)d+σ r dw, r=r. able : Decomposiions of Conrac Values a=0.5, b=0.06, σ r = 0.0, r0=0.04 A 0 = 100, r = r, α = 0.8, = 0 σ δ BO SP CFP RL V L (A 0, 0) RC SBO RE V E (A 0, 0) 16
21 λ λ = λ = λ = λ = λ = λ =1.5 All All From he able several ineresing observaions can be made. For example: If volailiy is increased, δ mus be increased o mainain a fair value disribuion A larger volailiy ends o increase he value of he rebae elemen and o decrease he value of he condiional fixed paymen. his is, of course, explained by he fac ha generally a larger volailiy is associaed wih a larger probabiliy of an early barrier hi. As expeced, he value of he equiy rebae elemen is nil when λ 1 and posiive when λ > 1. 17
22 5.. Using CIR model Here, Le r=r be he values simulaed from he following Cox_ingersoll_Ross shor rae model: dr = a(br)d+σ r rdw able 3: a=0.5, b=0.05, σ r = 0.0 and r 0 =0.1 A 0 = 100, r = r, α = 0.8, = 0 σ δ BO SP CFP RL V L (A 0, 0) RC SBO RE V E (A 0, 0) λ λ = λ = λ = λ = λ = λ =1.5 All All
23 When Lambda is small he volailiy is large, when Sigma is small he volailiy is large Using HoLee model Here, Le r=r be he values simulaed from he following HoLee shor rae model: dr=θ d+σ r dw able 4: Θ = 0.05e, σ r =0.0 and r 0 = 0.1 A 0 = 100, r = r, α = 0.8, = 0 σ δ BO SP CFP RL V L (A 0, 0) RC SBO RE V E (A 0, 0) λ λ = λ = λ = λ =
24 λ = λ =1.5 All All When Lambda is small he volailiy is large, when Sigma is small he volailiy is large HoLee model has a higher volailiy 5.3 Conrac componen prices I is an imporan propery of our model ha i can idenify fair conracs for a given se of iniial condiions. However, i is equally imporan ha he model can price conracs and heir consiuing elemens a any given poin in ime given he iniially specified erms. Now, we presen he following numerical examples in which he values of he componens of he liabiliy holders conracs as a funcion of he sae variable, A, a differen imes during he life of he conrac. he conrac parameers have been se so ha he conrac was fair a =0 and he conrac elemens values are ploed for =0.1 (righ afer incepion), =10, =19 and =0(see rosen & Jorgensen (001)). We will find ha he resuls are lile differen when we use hree differen ineres rae models Using he Vasicek model he corresponding r : 0
25 Figure 8a oal Liabiliy Value Bonus call opion Condiional Fixed Paymen Shored pu opion Rebae 1
26 Figure 8b Figure 8c
27 Figure 8d Figure 8e 3
28 For each ime =0.1,10,19,0, we assume A sar from corresponding B o 450. For differen simulaed values of r, he corresponding prices are differen a ime =0.1,10,19.0. o obain fair prices a differen ime, we should simulae r sufficien many imes and choose averages Using he Cox_Ingersoll_Ross model he corresponding r : Figure 9a oal Liabiliy Value Bonus call opion Condiional Fixed Paymen Shored pu opion Rebae 4
29 Figure 9b Figure 9c 5
30 Figure 9d Figure 9e 6
31 5.3.3 Using he HoLee model he corresponding r : Figure 10a oal Liabiliy Value Bonus call opion Condiional Fixed Paymen Shored pu opion Rebae 7
32 Figure 10b Figure 10c 8
33 Figure 10d Figure 10e 9
34 Reference [1] Briys, E. and F. de Varenne (1994): Life Insurance in a Coningen Claim Framework: Pricing and Regulaory Implicaions, he eneva Papers on Risk and Insurance heory, 19(1):53 7 [] Briys, E. and F. de Varenne (1997): On he Risk of Life Insurance Liabiliies: Debunking Some Common Pifalls, Journal of Risk and Insurance, 64(4): [3] rosen, A. and P. L. Jørgensen (1997): Valuaion of Early Exercisable Ineres Rae uaranees, Journal of Risk and Insurance, 64(3): [4] Han, L.M.,. C. Lai, and R. C. Wi (1997): A FinancialEconomic Evaluaion of Insurance uarany Fund Sysem: An Agency Cos Perspecive, Journal of Banking and Finance,1(8): [5] Bauer, D., Kiesel, R., Kling, A., Ruß, J., (006). Riskneural valuaion of paricipaing life insurance conracs. Insurance: Mahemaics and Economics 39, [6] Cox, J.C., Ingersoll, J.E., Ross, S.A., (1985). A heory of he erm srucure of ineres raes. Economerica 53, [7] Hull, J.C., (006). Opions, Fuures, and oher Derivaives, 6h ed. PreniceHall, Englewood Cliffs, NJ. [8] Vasicek, O., (1977). An equilibrium characerizaion of he erm srucure. Journal of Financial Economics 5, [9] Bjork,., (004): Arbirage heory in Coninuous ime, nd (Oxford Universiy Press). 30
35 Appendix he process of he liabiliy holders claim valuaion In his appendix we presen a series of corollaries each of which can lead o he heorem which gives he closedformula for fair value of every componen of he liabiliy holders claim in main conex. We begin o research on he ime valuaion of he mauriy paymen elemens. Here mos mehods are based on rosen and Jørgensen (00). Firs refer o rosen and Jørgensen (00) who show he derivaion of he defecive ransiion densiy for Brownian moion wih drif and an absorbing barrier a he origin in appendix A. Here we use heir resuls direcly. Namely: f(z, ; z, ) = 1 σ ( ) {N(Z z μ( ) )e z μ σ σ ( ) N( Z+z μ( ) )} (C.1) σ ( ) Noe ha he above densiy funcion is for Brownian moion of he form like zu ( u ) ( Wu W ) z Recall ha we have A u and he exponenial barrier: u 1 rs ds u Wu W A e, u u u u r ds r ds r ds r ds 0 0 u 0 0 B B e B e L e L e, u, (A.1) hrough he whole aricle we assume ha a he valuaion dae he barrier has no ye been reached i.e A B.Furher we le ha r r are deerminisic funcion. Corollary 1 he bonus (call) opion elemen: E e A L 1 r ds rs ds Q e E Q A L s = 1 rs ds =, ;, e A L f A A da B = e rs ds L ( A L ) f ( A, ; A, ) da (A.) = { AN ( B ) X A X 1 ( ) ln rs ds ( ) 31
36 A 1 ln rs ds ( ) L rs ds e N X ( ) B / 1 / A B A 1 ln ( ) A s ln s ( ) r sds B r ds r ds X L N e N X B A ( ) ( ) where f A, ; A, is he densiy of A wih an absorbing barrier Bu as given in (A.1). } Proof of Corollary 1: We firs perform a change of variables in order o be able o use he resul (C.1). Noe firs ha A B u u u 1 u rs ds uwu W r ds B e A e 1 B r ds u W W r ds u u s u ln A, since we choose r r 1 u W W A ln 0 u B Hence, passage of A() hrough B() is equivalen o passage of he Brownian moion 1 A A Zu u Wu W ln hrough zero. Define now z ln and noe ha: B B L A ( B ) X z rs dsz A e X X z ln z r sds A X z ln r sds q B herefore we can wrie (A.) as 3
37 αe r s d s q B e z + r s d s L α f(z, ; z, ) d z where f ( z, ; z, ) is he defecive densiy of Brownian moion wih drif and absorbing barrier a zero. Subsiuing (C.1) for his, he res is jus edious calculaions. Noe ha 1 he drif is in our formula. So we ge: αe r s d s q B e z + r s d s L α f(z, ; z, ) rs ds z rs ds L = e ( B e ) q 1 1 Z z ( ) ( ) 1 Z z 1 z n e n ( ) ( ) ( ) ( ) d z 1 ( ) 1z z ( ) rs ds rs ds z 1 { q = e B e e dz ( ) 1 ( ) 1z z ( ) 1 L e q ( ) dz 1 ( ) 1z z ( ) rsds z 1 q z e B e e dz ( ) 1 ( ) 1z z ( ) 1 L z e e dz q ( ) 1 q z ( ) rsds rsds = e { Ae N ( ) 1 q z ( ) L N ( ) 1 q z ( ) z B rs ds e e N A ( ) } 33
38 1 q z ( ) L e z N } ( ) A 1 ln rs ds ( ) = { AN X ( ) A 1 ln rs ds ( ) L rs ds e N X ( ) B / 1 / A B A 1 ln ( ) A s ln s ( ) r sds B r ds r ds X L N e N X B A ( ) ( ) } Corollary he condiional fixed paymen elemen As a par of he mauriy payoff, he liabiliy holders receive L on he condiion ha he knockou barrier has no been hi. A ime wih A B and r r are deerminisic his payoff elemen is valued as follows: rs ds EQ e L 1 rs ds = e E L 1 Q B rs ds =, ;, L e f A A da A 1 A 1 ln ( ) ln ( ) rs ds B A B = L e N N ( ) B ( ) Proof of Corollary : Changing variables as in Corollary 1 and using (C.1) yields f A, ; A, da B =, ;, 0 f z z dz 34
39 1 1 z ( ) ( ) 1 z z 1 z = z n e n dz 0 ( ) ( ) ( ) ( ) A 1 A 1 ln ( ) ln ( ) B A B = N N ( ) B ( ) which esablishes he resul. _ Corollary 3 he pu opion elemen Noe firs ha his payoff elemen can only be sricly posiive when 1, i.e. when he u r ds 0 barrier lies below he curve Le 0, u [0, ]. he ime value of he pu opion elemen is given as follows: rs ds 1 1 EQ e L A 1 r ds 1 e E L A 1 s = 1 Q rs ds = 1 1 e L A f ( A, ; A, ) da B s L = r ds 1 1 e L A f ( A, ; A, ) da (A.3) = 1 1 { B L 1 B 1 ln ( ) ln ( ) r sds rs ds A A L e N N ( ) ( ) L 1 B 1 ln rs ds ( ) ln ( ) A A A A N N ( ) ( ) B L A 1 A 1 ln ( ) ln ( ) r sds rs ds B B [ L e N N / ( ) ( ) 35
40 L A 1 A 1 ln ( ) ln ( ) r sds B B B N N ]} A ( ) ( ) Proof of Corollary 3: Change variables as before. Firs A B implies ha z 0.Also we should have A L z rs dsz Ae L L z ln z r sds A z L ln B r ds L seing q ln r sds B we can rewrie (A.3) as = s rs ds q s z r dsz 1 1e L (, ;, ) 0 A e f z z dz rs ds q s z r ds 1 e L 0 Be 1 ( ) 1 1 Z z ( ) ( ) 1 Z z 1 z n e n ( ) ( ) ( ) ( ) =. = 1 1 ( ) 1z z ( ) rs ds 1 q 1 e { L e dz ( ) 1 ( ) 1z z q ( ) z rsds 1 0 B e e dz ( ) 36
41 1 ( ) 1z z ( ) q z 1 e ( L 0 e dz ( ) 0 q 1 ( ) 1z z ( ) rsds z 1 Be e dz ( ) 1 = 1 e rs ds L 1 B 1 ln ( ) ln ( ) r sds rs ds A A { L e N N ( ) ( ) 1 1 ( ) ( ) r sds q z z A e N N ( ) ( ) 1 1 q z ( ) z ( ) z e ( L N N ( ) ( ) 1 1 ( ) ( ) rs ds B q z z e N N )} A ( ) ( ) L 1 B 1 ln ( ) ln ( ) r sds rs ds A A 1 { L e N N ( ) ( ) L 1 B 1 ln rs ds ( ) ln ( ) A A A A N N ( ) ( ) B L A 1 A 1 ln ( ) ln ( ) r sds rs ds B B [ L e N N ( ) ( ) = 1 )} 37
42 L A 1 A 1 ln ( ) ln ( ) r sds B B B N N ]} A ( ) ( ) Corollary 4 (he rebae) he ime value of he rebae paymen o he liabiliy holders assuming A B is rs ds EQ e 1 L 1 e EQ 1 L 1 rs ds = 0 = s r ds rds 1 0 ( ;, ) (A.4) e L e g z d A 1 A 1 ln ( ) ln ( ) 1 B B = A N B N ( ) ( ) where g( ; z, ) is he firs hiing ime densiy which will be derived below. Proof of Corollary 4 Firs we would like o presen he firs passage ime densiy of geomeric Brownian moion hrough an exponenial barrier: Since A u and he exponenial barrier: u 1 rs ds u Wu W A e, u u u u r ds r ds r ds r ds 0 0 u 0 0 B B e B e L e L e, u, A 1 Le denoe he smalles u such ha Au Bu.Se ln z and.do B exacly he same calculaion as rosen and Jørgensen (00) did in appendix B. We can esablish ha z z ( ) g( ; z, ) n (C.) ( ) 3 Now subsiuing for g( ; z, ) in (A.4) we ge 38
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