ON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT


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1 Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT UDC A. V. MELNIKOV AND M. L. NECHAEV Absrac. The paper deals wih he problem of pricing an equiylinked insurance conrac based on sock prices. The sock prices are supposed o follow a sochasic exponen model wih respec o a given Gaussian maringale. The model gives a possibiliy o obain unified formulas for mean variance hedging and he corresponding premium for boh naural cases: Black Scholes and Gaussian discree ime models. 1. Inroducion Suppose ha an insurance company has a porfolio of l insurance conracs. Any conrac is associaed wih a random ime τ i, i = 1,...,l, which indicaes he ime of inciden occurrence. The corresponding premiums should be disribued beween financial asses o guaranee he bes correspondence beween liabiliies of he company and is capial V π. As a crierion of he qualiy of a financial porfolio π we shall use hemeanvariancedisance E (VT π f T ) 2, where f T represens he claim ha should be paid by he company a he erminal ime T. The insurance conrac based on he marke s price of a given asse S is called an equiylinked life insurance conrac. An appropriae descripion of such a conrac was given, for insance, in 3, in he framework of he Black Scholes model for S. Thispaper is devoed o he pricing of such insurance conracs and in a more general case of a financial marke driven by Gaussian maringale. 2. Financial marke and insurance porfolio Suppose he company invess in he (B,S)marke wih wo raded asses: bank accoun B 1andsockS, { S = S exp Y 1 } 2 Y, where Y is he righ coninuous Gaussian maringale (see 2) on a given sochasic basis (Ω 1, F 1,P 1,F 1 ) wih filraion F 1 generaed by Y. 2 Mahemaics Subjec Classificaion. Primary 6H3; Secondary 91B28, 91B3. Key words and phrases. Hedging, (B,S)marke, coningen claim, insurance liabiliies, insurance premiums and claims. This paper was parially suppored by grans of NSERC and SSHRC. 15 c 25 American Mahemaical Sociey
2 16 A. V. MELNIKOV AND M. L. NECHAEV Any selffinancing rading sraegy π =(β,γ) can be deermined by is capial V π = V π + γ u ds u, where γ is he number of socks in he porfolio a ime (see 4). The number of bank accoun unis β can be idenified from he balance equaion V = γ S + β. Using he explici form of S and he Kolmogorov Iô formula 2, we can derive he following expression for he capial V π (1) V π = V π + : γ u S u dy u + u γ u S u ( { exp Y u 1 } ) 2 Y u 1 Y u. The mauriy ime of his conrac will be denoed by T. We also assume a pure echnical condiion: Y T =. The corresponding coningen claim has he form f T = f(s T ), where he Borel funcion f saisfies he following condiion: ( f(x) c 1+x p1) x p2, c, p 1, p 2, x. In his case (see 1) we can define he funcion { ( 1 f(z) ln(x/z)+ 1 2 F (u, x) = exp ( Y T u)) } 2 dz. 2π Y T u R z 2( Y + T u) I is easy o prove ha his funcion is wice coninuously differeniable wih respec o boh argumen u and x and admis he represenaion (2) F ( Y,S )=E f(s T ) F 1 for any <T. Using he maringale convergence heorem 2 we can conclude ha F ( Y T,S T )=f(s T ). The Kolmogorov Iô formula gives direcly ha (3) F ( Y,S )=F (,S ) u F x ( Y u,s u )S u dy u F u ( Y u,s u )S u d Y u F xx ( Y u,s u )Su 2 d Y c u {,S u ),S u ) F x ( Y u,s u )S u Y u F u ( Y u,s u )S u Y u }, where F x, F u,andf xx are he corresponding derivaives. Taking ino accoun ha he process F ( Y,S ) should be a maringale, we can reduce (3) o he equaliy {u } F ( Y,S )=F (,S )+ F x ( Y u,s u )S u dy u (4) + {,S u ),S u ) F x ( Y u,s u )S u Y u }. The insurance porfolio of l conracs can be characerized by random imes τ i, i =1,...,l,
3 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS 17 of inciden occurrence and claim paymen values. We shall assume ha τ i are i.i.d. random variables on some probabiliy space (Ω 2, F 2,P 2 ). The paymen funcion g( ) for he conrac indicaes he value g(s T ) which should be paid if no inciden occurs during he insured period. Suppose ha he disribuion of τ i admis he following represenaion: { } P 2 (τ ) =1 exp µ s ds, where µ is called a force of moraliy. Denoe I k = I {τk }, andn = I I. l The couning process N indicaes he number of incidens on,. We shall equip (Ω 2, F 2,P 2 ) wih a filraion F 2 =(F 2 ) generaed by (N ). The process is a compensaor of N and M = N is a maringale wih respec o F 2. (l N s )µ s ds (l N s )µ s ds 3. Main resuls and examples I is quie naural o hink ha he financial marke and he lives of insured are independen. Hence he general probabiliy space for he model can be defined as a produc of (Ω 1, F 1,F 1,P 1 )and(ω 2, F 2,F 2,P 2 ) wih he general filraion F generaed by F 1 and F 2 : Ω=Ω 1 Ω 2, F = F 1 F 2, P = P 1 P 2, F = F 1 F 2 = { F = F 1 } F2. The paymen funcion of such a conrac has he form f T = g(s T )(l N T ), where he Borel funcion g( ) saisfies he condiions menioned above, g(x) c(1 + x p1 )x p2, c, p 1, p 2, x >. To opimize is liabiliies he company should choose an iniial capial ˆv and a selffinancing rading sraegy ˆπ such ha E (VT ˆπ (ˆv) f T ) 2 E (VT π (v) f T ) 2 for any v and any selffinancing sraegy π. Denoe by V = Ef T F =Eg(S T )(l N T ) F he socalled racking process V ; using he independence of S and N we ge V = E 1 g(s T ) F 1 E 2 (l N T ) F 2. I is easy o check ha (5) E 2 (l N T ) F 2 l = E 2 (1 I {τi T }) F 2 =(l N ) p T, i=1 where p T = E 2 1 I {<τi T } F =exp { T µ s ds } is he probabiliy of inciden occurrence afer expiraion dae T. Using he represenaion (2) we have V = F ( Y,S )(l N ) p T.
4 18 A. V. MELNIKOV AND M. L. NECHAEV we obain he inegral repre Applying he Kolmogorov Iô formula o he process V senaion (6) V = V + + u (l N u ) u p T d,s u )+,S u )(l N u ) u p T µ u du,s u ) u p T dn u,s u )(l N u ) u p T,S u )(l N u ) u p T Theequaliy(6)canberewrienas (7) V = V + + u (l N u ) u p T,S u )+,S u ) u p T N u. (l N u ) u p T d,s u ),S u ) u p T dm u,s u )(l N u ) u p T,S u )(l N u ) u p T where (l N u ) u p T,S u )+,S u ) u p T N u, M = N (l N u )µ u du. Taking ino accoun he represenaion (4) for F ( Y,S )wehavefrom(7)ha V = V + (l N u ) u p T F x ( Y u,s u )S u dy u,s u ) u p T dm u +,S u )(l N u ) u p T,S u )(l N u ) u p T u +,S u ) u p T N u (l N u ) u p T F x ( Y u,s u ) Y u. Noe ha any capial V π conrolled by a selffinancing sraegy can be represened in he form (1). Consider he mean variance disance beween VT π and V T = f T : R(π, v) =E (VT π VT ) 2. Because of he maringale properies of V π and V, i is clear ha he iniial capial ˆv for he opimal rading sraegy should be equal o V : ˆv = le 1 g(s T )p(,t)=lf(,s ) P({τ >T}). Le he iniial capial v of he selffinancing sraegy π be equal o ˆv; hen ( R(π, v) =E ((l N u ) u p T F x ( Y u,s u ) γ u )S u dy u (8) + u,s u ) u p T dm u (l N u ) u p T,S u ) (l N u ) u p T F x ( Y u,s u )S u Y u γ u S u + γ u S u Y u ) 2.
5 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS 19 Since he difference M = V V π is a maringale wih respec o F, here is a unique represenaion of he form M c + M d, where M c and M d are respecively he purely coninuous and disconinuous pars of M, which are orhogonal. Consequenly E M 2 = E (M c ) 2 + E (M d ) 2. Le us rewrie M c in he form M c ( ) = (l Nu ) u p T F x ( Y u,s u ) γ u Su dyu c,s u ) u p T dmu. c Since Y u and M u are independen, we ge E (M c ) 2 ( ) 2S = E (l 2 Nu ) u p T F x ( Y u,s u ) γ u u d Y c u + E (,S u )p T ) 2 d M c u. In he case of he purely disconinuous par M d, we have he formula M d = u u p T,S u ) N u +(l N u ) u p T,S u ) γ u S u, and herefore E (M d ) 2 = E ( u u p T,S u ) N u +(l N u ) u p T,S u ) γ u S u ) 2. I is well known (see 2) ha he imes of he jumps of he Gaussian maringale are deerminisic. Denoe he corresponding se by A. Define he processes γ and γ by he following formulas: γ s = γ s χ A, γ s = γ s χ A, where χ A is he indicaor funcion of he se A. Take γ s = γ s + γ s. Since A is a counable se, E (M c ) 2 = E ((l N u ) u p T F x ( Y u,s u ) γ u ) 2 Su 2 d Y c u (9) + E (,S u ) u p T ) 2 d M c u.
6 11 A. V. MELNIKOV AND M. L. NECHAEV On he oher hand i is clear ha E ( (M d ) 2 = E u p T,S u ) N u (1) = E u A ( + E u A ( +(l N u ) u p T,S u ) γ u S u u p T, S u ) N u u/ A u/ A ) 2 u p T,S u ) N u +(l N u ) u p T,S u ) γ u S u ) 2 ) 2 u p T,S u ) N u. Equaions (9) and (1) give us he explici forms of γ and γ: (11) γ =(l N ) p T F x ( Y,S )χ A and (12) γ = p T (l N ) E F ( Y,S ) S F E, (S ) 2 F where / is supposed o be equal o. So, we ge he following main resul of he paper: Theorem 1. For he (B,S)marke conrolled by a Gaussian maringale and a porfolio of l homogeneous unilinked pure endowmen insurance conracs, here is an opimal meanvariance hedging sraegy ˆγ = γ + γ, where he coninuous par γ and he purely disconinuous par γ are defined by (11) and (12), respecively. The iniial capial ˆv of he sraegy can be calculaed by ˆv = l Eg(S T ) P(τ >T), where g(s ) is a paymen funcion for one insurance conrac, T is a erminal ime, and τ is a random ime wih disribuion of he inciden occurrence. Example 1. The model invesigaed above includes he model considered by T. Møller (see 3) when Y is Brownian moion. Taking, in our seing, γ, ˆγ = γ =(l N ) P T F x ( Y,S ) gives he model and he resul presened in 3. Example 2. Anoher ineresing example is he discree model { S n = S exp M n 1 } 2 M n, B n 1, where M n = h h n, h i = σɛ i, ɛ i N(, 1), and ɛ i are i.i.d. random variables on he probabiliy space (Ω 1, F 1,P 1 ) wih filraion F 1 = { F n}, 1 F 1 n = σ{ɛ i,i n}. I is clear ha M n = σ 2 n and } S n = S exp {h h n σ2 2 n.
7 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS 111 Using he independence of ɛ i,wehave ES n F n 1 =S n 1 E exp { h n σ 2 /2 } F n 1 = Sn 1. We can embed he discree model o our general model by he following sandard way:,, 1), Y = M n, n, n +1),n<N, M N, N,N + δ), N+ δ = T, F 1 = F 1, F 1 = {F 1 }. I is clear ha Y is a Gaussian maringale on he sandard sochasic basis ( Ω 1, F 1,F 1,P 1) saisfying he echnical condiion Y T =. Inviewofheheorem γ n = because here is no coninuous par of Y. Regarding he oher par of a hedging sraegy γ n we have E F ( Y n,s n ) S n F n 1 =E(F ( Y n,s n ) F ( Y n 1,S n 1 )) S n F n 1 = EF ( Y n,s n ) S n F n 1 =Eg(S T S n ) F n 1. Hence we ge where S T = S N. γ = γ n = n p T (l N n 1 ) E g(s T S n ) F n 1 E ( S n ) 2, F n 1 Bibliography 1. O. Gloni and Z. Khechinashvili, Financial (B, S)marke wih Gaussian maringale mean square opimal hedging sraegies, Proc. A. Razmadze Mah. Ins. 115 (1997), MR16391 (99j:91) 2. R. Lipser and A. N. Shiryaev, Theory of Maringales, Nauka, (Russian) MR (88h:691) 3. T. Møller, Riskminimizing hedging sraegies for unilinked life insurance conracs, Asin Bullein 28 (1998), no A. N. Shiryaev, Essenials of Sochasic Financial Mahemaics, World Sci., River Edge, NJ, MR (2e:9185) Seklov Mahemaical Insiue, Gubkina 8, Moscow , Russia Curren address: Deparmen of Mahemaical and Saisical Sciences, Universiy of Albera, Edmonon, Albera T6G 2G1, Canada Seklov Mahemaical Insiue, Gubkina 8, Moscow , Russia Received 12/MAR/23 Originally published in English
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