VARIABLE STRIKE OPTIONS IN LIFE INSURANCE GUARANTEES

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1 Opions in life insurance guaranees VARIABLE SRIKE OPIONS IN LIFE INSURANCE GUARANEES Piera MAZZOLENI Caholic Universiy Largo Gemelli,, (03) Milan, Ialy piera.mazzoleni(a)unica.i Absrac Variable srike opions offer a flexible hedging ool in finance and hey give new models for guaranees in life insurance policies. An example is given by he -quanile opions: indeed, he srike is sricly relaed wih VaR and we are led o guaranees ha are sraigh measures of he risk level. Usually, eiher only one risk is considered or is final impac on he managemen is summarized. Bu he risk appears under several caegories and we need consider mulivariae quaniles. Since hese ools are very cumbersome and we are able o measure he relaive weigh of he differen kinds of risks, a direcional quanile is inroduced. herefore, we can apply he classical VaR heory also o muliple risks and offer wider guaranees o life insurance policies. Moreover, according o he need of a deailed analysis of he several kinds of risks, ha is promoed by Solvency II, we inroduce an easy-o-pracice ool. Key Words Opions, Value-a-Risk, Insurance guaranees

2 Opions in life insurance guaranees. Variable srike opions he increasing developmen of derivaive financial producs is linked wih speculaion and hedging sraegies, overcoming he radiional risk profile. he innovaion process leads o new kinds of exoic opions and i modifies he ypical elemens such as he underlying, he srike price, he mauriy. In his noe, we are especially ineresed wih he developmen of he srike price role owards guaranees in life insurance policies. Among he wide variey of exoic opions we recognize wo main families, he pah dependen opions and he correlaion opions, while leaving he remaining ones in a hird general class. he payoff a mauriy of a sandard opion depends only on he difference beween he price of he underlying and he srike price, wih no reference o he ime evoluion. Bu he process leading o he final value has o be aken explicily ino accoun: hen we face he family of he pah dependen opions. A flexible family of pah dependen opions is he one of lookback, i.e. no regre opions: he payoff depends no only on he final price S, bu also on he minimum (maximum) price ha has been regisered in an assigned ime period. he srike price may be fixed, bu i is more realisic o adjus he buying (selling) price of he underlying o he minimum (maximum) price observed in he given period. he variable srike price of a lookback opion depends on he frequency of regisraion for he underlying price. Anoher family of flexible pah dependen opions is offered by he clique opions, ha are also called reseing srike opions. Indeed, he srike price is updaed on a se of inermediae daes, in order o beer represen he marke behaviour. Consider a call opion wih a one year mauriy and a srike price assigned a he beginning of each period, according o he marke value: he holder akes advanage of he corresponding quarer performance of he underlying asse. his remark shows ha a clique opion is a srip of one beginning opion and hen forward opions: he srike is given as a spo curren price and hen as he forward marke prices. We menion he clique opions ha adjus he srike price a given daes and he ladder opions, assigning some reference levels o be compared wih he previous srike price. A an exreme side, we find he shou opions, for which he holder can declare he srike a any level and a any dae he judges convenien. I can be proved ha he final value can be spli ino wo componens, he firs one linked wih he difference of he underlying values S S and he oher one wih he difference beween he curren value S and he original srike price { S S } ( S K) max 0, +. S K, ha is In he previous cases he srike price is absoluely random. I akes a funcional form in he Asian opions. hese opions are pah dependen, because he srike price akes he form of a suiable mean over he preassigned se of observaions. For he geomeric mean a closed form expression for he value is available, while we use he approximaed relaion beween he geomeric and he arihmeic means. We can disinguish:

3 Opions in life insurance guaranees - he average price opions, when he srike price is given, while he final price of he underlying asse is a suiable mean of he observed prices; - he average srike opions, ha apply a srike price calculaed as he chosen mean. Denoe M any mean. hen he payoff a mauriy of an average price opion is given by max 0, ϕ M ϕk, ϕ being an indicaor funcion, ϕ = for he call opions, ϕ = { } for he pu opions. Alernaively, he final payoff for he average srike opions is, max{ 0, ϕs ϕm}. Le us assume ha an arihmeic mean is suiably approximaed by he half sum of he iniial and final prices of he underlying asse, A ( S 0 + S ). hen he expeced value of an Asian opion a mauriy is equal o he half expeced value of a sandard opion issued a he money, E S A E S { } = { } max 0, max 0, S 0. Saring from he classical pu-call pariy, some symmery resuls have been proved for opions wih he same srike price. Baes (99) proves a relaion for European pu and call opions wih differen srike prices, when he underlying follows an exponenial Brownian moion. Delbaen-Yor (999) give he equivalence beween a passpor opion and a fixed srike lookback opion. In his secion we give explici menion of a symmery resul for exoic opions wih fixed and variable srike. Indeed, le us exchange he roles of he ineres rae r and of he dividend δ. hen an Asian call opion wih variable srike exhibis he same price as a pu opion wih fixed srike, provided ha K = λs0, { } { } r cf ( S0, λ, r, δ,0, ) = e E max 0, λs A = δ = px( K, S0, δ, r,0, ) = e E max 0, K A An analogous symmery relaion holds for a call opion wih fixed srike and a pu opion wih variable srike. Correlaion opions are devoed o enlarge he number of he underlying asses, ha characerize he opion price. We can disinguish a firs order correlaion, if i direcly influences opions payoffs, and a second order correlaion, if i merely modifies he opions payoffs. Spread opions are buil on he difference among indexes, prices and raes: hen he correlaion is of firs order. An ou-performance opion can use boh he correlaion orders. hink for insance of an ouperformance opion on wo differen indexes A and B, denominaed in currency C: hen we have o consider no only he direc covariance beween A and B, as a firs order effec, bu also he covariances beween A and C, B and C, hus using a second order effec. Among correlaion opions we recognize baske opions for porfolios of underlying asses, for insance several currencies. 3

4 Opions in life insurance guaranees Exchange opions, ha have been proposed by Margrabe (978) allow us o exchange he underlying asse wih anoher one a mauriy. Consider for insance a U.S. invesor who buys yen in exchange for Ausralian dollars, ha is, he exchanges one foreign currency for anoher one. hese opions exhibi a random srike: indeed, an exchange opion can be considered as: - boh a call opion on a firs underlying asse, wih srike price represened by he fuure price of he second asse a mauriy; - and as a pu opion on he second underlying asse, wih srike price given by he fuure price of he firs asse a mauriy. Exchange opions can be used o model several oher ypes of exoic opions: for insance, he rainbow opions are wrien eiher on he maximum or on he minimum of wo underlying asses. Ineres rae spread opions are coningen claims on he difference of wo ineres raes, ypically a shor rae and a long rae, ha allow us o conrol he risk due o changes in he shape of he yield curve. An example of pricing is given in Fu (996).. Random guaranees and he insurer risk he Value-a-Risk measuremen is one of he main issues in life insurance boh on he side of he insurer and on he side of he insured. ake for insance he Equiy Linked policies. he benefi is relaed wih eiher he marke value of muual funds or suiable indices and i is subjec o a high level risk. In he Ialian marke of Uni Linked policies he whole risk is charged o he insured. In he Equiy Linked policies a suiable guaranee is added o weaken he insured s risk (CIA, 00). he mos common guaranee models are: - Guaraneed Minimum Deah Benefi, GMDB he benefi a he insured s deah is he maximum beween he value of he fund quoas and he already paid premia: someimes a roll up mechanism is added and he premia are revalued according o he lengh of period beween he sipulaion of he conrac and he insured s deah. - Guaraneed Minimum Mauriy Benefi, GMMB A suiable quoa from 75% up o 00% of he already paid premia is guaraneed: someimes a revaluaion mechanism is added according o eiher a suiable law (roll up) or o he marke behaviour (rache). - Guaraneed Minimum Accumulaion Benefi, GMAB his guaranee refers o he level of he already paid premia as in he GMMB, bu i adds some daes when he benefi is consolidaed. - Guaraneed Minimum Surrender Benefi, GMSB In his case he GMMB is recognized also o he surrender. he guaranee s price is found under he opion model. 4

5 Opions in life insurance guaranees While in he acuarial approach he guaranee is se a a fixed value and i can be covered by a suiable porfolio of risk-free bonds, in he financial approach he insurer has o build a suiable porfolio replicaing he European pu opions payoff. Le us inroduce he following noaions: F is he marke value a ime of he invesmen fund relaed wih he conrac. Managemen fees are deduced by he fund a he beginning of each monh. he noaions F and F + indicae he fund value a he end of monh before and afer he subracion of he managemen fees and represen he cos of he guaranees; S denoes he value a ime of he invesmen underlying he invesmen fund, wih S 0 = ; G gives he guaranee level per uni of invesed capial; m is he managenen fees amoun, ha is deduced every monh; m C gives he margin offse, ha is he quoa of he managemen fees devoed o he guaranee s funding; M represens he ime gain ha is obained from he margin offse applied o he fund value; C represens he cash flow of he insured s gains and losses from he guaranee. If he fund dynamics is valued saring from he origin, i is described by he following expression S F F ( ) + = 0 m ; S0 and he revenue is given by S M F m m F ( ) = C = C 0 m. S0 hese expressions allow us o give a mahemaical formula for gains and losses of he guaranees: for he GMMB guaranee, we are analysing, he cash flow is given by C τ = p M, = 0,,..., n x τ + C = p ( G F ), = n, n n x n+ where we assume ha he negaive flows C represen an income, he posiive flows, giving he guaranee s paymen, represen a cos. he Black-Scholes formula is applied, when he underlying is he invesmen fund, reduced of he annual charge m. he GMMB opion price is 5

6 Opions in life insurance guaranees ( ) + ( ) r r P0 = e E Q G F = e E Q G S( m ) +. hen, if S0( m) denoes he guaranee s price a = 0 afer he managemen fees, we obain wih ( ) ( ) r P ( ) 0 = Ge Φ d S0 m Φ d { log ( ) / ( σ / ) } d = S0 m G + r+ σ d = d σ. Wihdrawals due o deah and surrender are added as P0 p τ x. he risks affecing he insurer due o he Equiy Linked policies are differen from he classical insurance risks, ha can be eliminaed by applying he muualiy and diversificaion principles. Indeed, he financial side of risk canno be eliminaed and we look for a suiable measure. Denoe L 0 0 he presen value of he financial gains and losses flow n = ( ) L = C + i he quanile risk measure. { : Pr ( 0 ) } V = inf V L V, represens he smalles amoun V o be invesed in risk free asses, so ha he insurer has enough money o pay he guaranee G wih probabiliy a leas. Le us confine our aenion o a GMMB guaranee. Suppose he asses reurn exhibis a lognormal disribuion ( ( µ ln ( )), σ ) LN n + m n, hen he explici expression of V becomes { { ( ( ))}} exp ln V G F z n n m e rn = 0 σ + µ +, wih ( ) z =Φ and Φ he sandard normal disribuion funcion. Such a measure is no coheren and we need consider he Expeced Shorfall, ES, ( ) [ : ] ES L = E L L > V, ha is, he expeced loss condiioned o he upper ail ( ) of he disribuion. 6

7 Opions in life insurance guaranees he explici formula for a GMMB guaranee is rn rn ( n) : n ( ) ES = E G F e F < G V e. Le us consider an Equiy Linked policy wih underlying he EUROSOXX 50. he subscriber is 50 yeas old, he mauriy is 0 years and he risk free rae is i = 3,5%. A 00% GMMB guaranee leads o a 0% probabiliy of loss. Quanile opions and guaranees Up o now we have followed he insurer s poin of view. Bu he flexibiliy of he guaranee can be linked o he VaR measure also for he insured s side, by modelling he guaranee iself as a quanile opion. An inermediae way beween he Uni Linked policies wih he whole risk charged o he insured and he Equiy Linked policies is offered by he inroducion of guaranees using he Value-a-Risk (Baione-Menziei, 005). Le us suppose ha he guaranee is relaed wih he risk aversion degree of he insured and i is modelled as a quanile of he underlying disribuion a mauriy. Denoe S() he underlying, eiher an index or a baske of indices, whose evoluion is described by a geomeric Brownian moion, () µ ( ) σ ( ) ( ) ds = S d+ S dz. herefore he reference disribuion is he normal one and he -quanile has an explici expression N ( ) ( ) ( ) ( ) { } ( ) ( ) S = S 0expN = S 0 f, µ, σ,, being he inverse normal disribuion funcion: i resuls o be funcion of he level, of he disribuion parameers µ, σ and of he mauriy. Under he classical assumpions of he Black-Scholes model, he price for he - quanile call and pu opions are r ( ( 0, ) ( ), ) = ( 0) ( ) ( ) ( ) C S S S N d S e N d r ( ( 0, ) ( ), ) ( ) ( ) ( 0) ( ) P S S = S e N d S N d. For insance, by seing = 5% he srike price for 5 = 0 S ( ) 0,6096. = = is S ( ) 0,6650 = and for his approach allows us o value he exercise price no longer as an exogeneous parameer, bu as a funcion boh of he marke and he insured parameers. 7

8 Opions in life insurance guaranees 3. Modern developmens in he quanile heory he quanile funcion is a fundamenal ool in he modern heory of risk: indeed, a family of quanile risk measures is widely used in finance and insurance, under he classical assumpion of normal disribuion seing. No ye concluded is he discussion on he definiion of a suiable quanile funcion for mulivariae analysis. Several quesions concern he comparison among he differen proposals and he adherence wih he probabilisic naure. Le us remind only some problems. If we proceed from he univariae o he mulivariae seing, i is worh oriening o he cener of he underlying disribuion, inended in median sense and defining quaniles as boundaries demarking inner and ouer regions, having specified probabiliies. Le us sar wih he univariae case. Le F ( ) denoe he usual -quanile of a univariae cumulaive disribuion funcion F. he median M is given by F ( ) and for 0< <, he values F, F represen boundary poins demarking lower and upper ail regions of equal probabiliies oaling. A corresponding median oriened -quanile inner region having probabiliy is given by he closed inerval Q(, ) = F, Q(, ) = F For = i represens an inerquanile region, for = 0 i reduces o he mean. (, ) Sill considering he univariae case a median oriened quanile can be se asq u for u =± denoing direcion from M, and Q( u,0) = M he main properies o be preserved by he mulivariae exension are: - probabilisic inerpreaion - direcional monooniciy - suiable se heoreic inerpreaion. d Le M be he mulidimensional median. S ( M) { :0 } A A γ γ denoes he se of direcions u. = < < represens a family of regions nesed abou M wih A { M} monoonic, ( 0,). A γ A for 0 γ γ ' γ ' < <. Moreover, se inf : P( A γ ) 0 = and { } γ = γ > for hen, Q( u, ) is given by he boundary poin of A γ in he direcion u from M and denoes a -quanile inner region. he boundaries A γ, which we may call conours, have he effecive inerpreaion as median-oriened quanile funcions. 8

9 Opions in life insurance guaranees Differen noions of median M and differen possible shapes for regions A γ lead o differen versions of he quanile funcions. Among he several proposals given in he lieraure, le us remind he generalized quanile approach by Einhmal-Mason (99), ( ) { ( ) ( ) } U = inf λ C : P C, 0 < <. A kind of bivariae analysis is he one reaing randomness and ime evoluion simulaneously, hus leading o a quanile heory for sochasic processes. Le { B, 0} be a one-dimensional Brownian moion saring from 0. Le σ R +, µ R and define X = σ B + µ a Brownian moion wih drif. In Dassios (995) he -quanile of X has been inroduced in funcional form. If we consider level of he quanile funcion as a parameer, while ime plays he role of variable, he -quanile of X is defined as (, ) = inf : ( ) { 0 s } M x I X x ds>. I should be noiced ha 0 Le us se ( ) = { }, lim M (, ) sup { X :0 s } lim M, inf X :0 s s = { }, Y = { X s } Y sup Xs :0 s inf s :0 =. for 0< <, and Y independen random variables. hen he following equaliy in law does hold, Y (, ) M = Y + Y. I is well known ha he normal disribuion does no describe he behaviour of asse reurns in a very realisic way. hen, i is worh overcoming he classical Wiener process in he geomeric Brownian moion: ake for insance some general Lévy process. Le us denoe L a general exponenial Lévy process and consider is () represenaion in he finie variaion case L () = γ () + βw() + Ls ( ), 0, 0 s ha is sum of a drif erm γ (), a Gaussian componen W ( ) erms of L. Le s β and a pure jump par in r R be he riskless ineres rae and σ = ( σ ij ) he variance-covariance marix, d b R a suiable vecor such ha each sock has he desired appreciaion rae. he prices follow he dynamic evoluion i () () dp P i () = bd+ dlˆ, i i 9

10 Opions in life insurance guaranees d where ˆi is such ha exp σ ˆ ijlj = e Li and e denoes he sochasic exponenial j= of he process. L () ( ) d π () R is an admissible porfolio process and i ( ) π wealh X (), which is invesed in asse i ; (,..., )' ( ) 0 π he wealh process X () π () ( ) dx = π + π + π π X π represens he fracion of he s = is he summaion vecor. hen follows he dynamic behaviour (( ' s) r ' b) d ' dl ˆ (), wih 0, X π ( 0) > = x. In he Emmer-Kluppelberg (003) he capial a risk, CaR, is used as a risk measure of he porfolio sraegy o be opimized, r (, π, ) = ( exp( π '( ) )) CaR x xe z b rs he calculaion of CaR requires he knowledge of he quanile z of he sochasic exponenial e( π L ˆ ( )) of process π ˆL ( ) a mauriy : indeed, we have { ( ( ) ) } π π ( ) {( ) } VaR = inf z R: P X z = xz exp ' b rs + r Bu his is a quie complicaed objec according o he menioned reference. 4. Direcional quanile he operaional difficuly o calculae quaniles for sochasic processes and he parameer independence on ime suggess o lead back o a one dimensional definiion. Suppose ha ime and uncerainy have differen weighs in he sochasic analysis. I is worh applying a direcional analysis. hen pair ( ω,) is considered as a unified eniy and he sochasic process (, ω) = ( + ξ ) = ( ξ) X X u d X 0 is read as a random variable wih respec o ξ, along direcion saemen Prob X ( ( ξ ) x) = means ha we look for a pair * * ( * * u = u0 + ξ d = ω, ) d. In correspondence,, such ha he probabiliy no exceeding x is equal o parameer. ha is, ime and randomness are se simulaneously. More generally, expression { :Pr ( ( ) ) } Q = inf u ob X ξ x > defines he lowes even ω * * and he firs ime such ha he probabiliy level is blocked. 0

11 Opions in life insurance guaranees Bu he approach is even more relevan, when we face several risks. Indeed, denoe X = X X he risk vecor. Since we know he weigh of each variable, we can (,..., n ) develop a unidimensional research along ray ( ξ ) 0 X 0 0 (,..., 0 X n ) X = X + ξ D, where = X and D represens he direcion induced by he acual risk weighs. hen we obain he direcional VaR { ξ ( 0 ξ ) } VaR = inf : P X + D z and a corresponding coheren direcional measure ( ξ) : ( ξ) ES = E X X > VaR. he quie deailed approach o risk promoed by Solvency II allows us o recognize he quoas of he several kinds of risks. herefore he VaR measure should be no longer applied o an unidenified SURPLUS, bu raher o he direcion of risks, induced by heir variey. he direcional approach allows us o give an immediae and easy-opracice approach o unify he risks owards Solvency II. References Baione, F. Menziei, M. (005), Forme assicuraive index linked con prezzo di esercizio - quanile, in Bellieri dei Belliera Mazzoleni (eds.) Analisi dei rischi ed oimalià delle garanzie nei prodoi assicuraivi via con proezione. Florence Universiy Press Baes, D.B. (99), he crash of 87: was i expeced? he evidence from Opion marke, Journal of Finance, march 99 Bellieri dei Belliera, A. P.Mazzoleni (eds.) (005). Analisi dei rischi ed oimalià delle garanzie nei prodoi assicuraivi via con proezione. Florence Universiy Press CIA. (00), Research Paper on Use of sochasic echniques o value acuarial liabiliies under Canadian GAAP, Canadian Insiue of Acuaries Dassios, A. (995). he disribuion of he quanile of a Brownian moion wih drif and he pricing of relaed pah dependen opions. he Annals of Applied Probabiliy. 5, Delbaen, F. Yor, M., (999), Passpor opions, echnical Repor, EH, Zurich Drees H. (998). On smooh saisical ail funcionals. Scandinavian Journal of Saisics. 5, 87-0 Einmahl, J.K.J. Mason, D.M. (00). Generalized quanile processes. Annals of Saisics. 0, Emmer, S. Kluppelberg, C. (004), Opimal porfolios when sock prices follow an exponenial Lévy process, Finance and Sochasics, 8, 7-44 Fu, Q. (996) On he valuaion of an opion o exchange one ineres rae for anoher, he Journal of Banking and Finance,0,

12 Opions in life insurance guaranees Margrabe, W., (978) he value of an opion o exchange one asse for anoher, Journal of Finance, 33, Rolski. Schmidli H. Schmid V. eugels J. (999). Sochasic Processes for Insurance and Finance. J.Wiley Serfling R. (00). Quanile funcions for mulivariae analysis: approaches and applicaions. Saisica Neerlandica 56, 4-3

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