ANALYTICAL FORMULAS FOR OPTIONS EMBEDDED IN LIFE INSURANCE POLICIES

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1 ANALYTICAL FORMULAS FOR OPTIONS EMBEDDED IN LIFE INSURANCE POLICIES Vnnucci Emnuele Dip.to di Sttitic e Mtemtic Applict ll Economi, Univerità di Pi (Preenting nd correponding uthor) Vi C. Ridolfi, Pi e.vnnucci@ec.unipi.it Vnnucci Luigi Dip.to di Mtemtic per le Deciioni, Univerità di Firenze Vi C. Lomroo, 6/7-54 Firenze luigi.vnnucci@dmd.unifi.it ABSTRACT Anlyticl formul for evluting the integer moment of the preent vlue of gurnteed enefi t emedded in Unit Linked life inurnce policie nd their exercie proilitie re found. Such gurntee re given in the ce of inured deth or in the ce of urrender, with the urrender condition contrctully ettled. Some reult propoed in Crr (998) re generlized in everl direction nd in prticulr the ence of ritrge opportunity hyp othei i rem oved. KEYWORDS Unit Linked life inurnce policie, urrender nd deth gurntee, tochtic mturity option.

2 . INTRODUCTION While for Blck-Schole Europen option formul the tndrd norml ditriution tultion i necery, it i well-known, urpriing reult tht thepriceofeuropenndamericnoptionigivenycloednlyticlformul, without the neceity of ny tultion, if uch option hve exponentil tochtic mturity in Blck-Schole cenrio with the ence of ritrge opportunity. Such formul re derived in Crr (998) which h the im to otin qui nlyticl formul for pricing clicl Americn option (e.g with fixed determinitic mturity) y men of recurive method thnk to the reult for tochtic (exponentil) mturity. Crr method i ed on equence of Erlng ditriution tht converge to degenerte ditriution with zero vrince nd with expected vlue equl to the determinitic mturity. Other uthor invetigte the prolem of pricing Americn option with tochtic mturity, ut not for cturil purpoe: ee El Kroui, Mrtellini (), Dorontu, Pontier (6). A urvey of ppliction of the tochtic mturity hypothei i in Berrd (5), in which it i decried how to otin the optiml exercie oundry nd the expected vlue, for rel option nd employee tock option with tochtic (uniform, tringulr) mturity. We generlize Crr reult in the light of n cturil point of view: in life cturil frmework conidering the tochtic mturity of the option the tochtic deth time i quite nturl. The exponentil mturity implie contnt deth intenity, which i reonle hypothei for firt pproximtion. The content of thi note hve to e et in the re of the vlution of option emedded in life inurnce policie, widely conidered in cturil literture oth for Europen type option, in Brennn, Schwrtz (976) nd Bcinello, Ortu (993), nd for Americn type option, in Groen, Jorgenen (997 e ), Bcinello ( e 4) nd Vnnucci E. (999, 3 e 3). Here we find nlyticl formul, which do not need ny tultion, for evluting (proility ditriution, proilitie of exercie, moment of ny order...) tochtic enefit of Unit Linked life policie with gurntee, which re given in ce of deth or in ce of urrender. The urrender condition re contrctully ettled nd they force the holder to urrender hi policy with gurntee of minimum return rte, when the current vlue of the underlying et hit fixed rrier for the firt time. Alterntively, if the inured die efore the rrier i hitted, he receive t let fixed mount when the underlying etvlueinotufficiently ried. From n opertive point of view, the nlyticl reult llow to uild policie which comply with the cutomer requirement out level nd cot of gurntee. An importnt feture i to hve contrctully ettled the urrender condition. If thee condition re left to the free deciion of policy-holder the repective rik i hrd to evlute: for n ttempt of modelling thi pect ee Romgnoli (7).

3 By the wy the nlyi of innovtion in life inurnce policie with flexile return gurntee h een the oject of MIUR (Itly) reerch (3): for ome contriution ee Vnnucci E. (5), Vnnucci L. (5). The pper i tructured follow. In ection we decrie the finncil cenrio nd the tructure of the policy. In ection 3 we otin the nlitycl formul for evluting the option emedded in the policy. In ection 4 we ccommodte our reult with the exiting one, reintroducing f.i. the ence of the ritrge opportunity hypothei. Some numericl exmple re propoed in ection 5. In ection 6 we conider firt demogrphic improvement, which cn ring to cloed nlyticl formul once more. The pper end with rief concluion nd two technicl ppendixe.. POLICY STRUCTURE We conider Unit Linked type policy iued t time. We ume tht the underlying et vlue S t,witht nd initil vlue S, follow Geometric-Brownin motion, typicl of the Blck-Schole cenrio; uch policy provide two kind of gurntee, one in ce of deth nd one in ce of urrender. In detil policy enefit re: t the exponentil ditriuted deth time, T, deth-pyment i mde: thi pyment i equl to the mximum etween the rndom underlying vlue, S T,ndfixed gurnteed mount k ; lterntively, if t the rndom time U, with U<T, the underlying unit vlue hit for the firt time the rrier, with<, then the policy mut e urrendered nd the holder receive the pyment k exp (gu), with k nd g i the gurntee interet intenity on the cpitl k. A nturl hypothei it i to conider T, relted to the demogrphic rik, nd U, relted to the finncil rik, independent rndom vrile. Firt, uch kind of policy would protect the holder (or more preciely hi fmily) from the negtive economic conequence of deth event nd, in econd order, from too lrge loe in the underlying invetment: hence, it my e fir to conider k much igger thn, not only of k, ut lo of. We remrk tht the cot of deth gurntee without the poiility of urrender i equivlent to n Europen option with rndom (exponentil) mturity nd with fixed trike price k. Beide, if we ume g, k k nd the ence of ritrge opportunity hypothei then we hve, prticulr ce of our model, Crr model for the Americn option with tochtic (exponentil) mturity. Letting C e the rndom cot of deth gurntee nd C the rndom cot of urrender gurntee. The im of thi pper i to otin the nlyticl olution for the proility of exercie of ech gurntee: P (C > ) nd P (C > ) 3

4 the expected vlue nd the vrince, ut lo ny integer moment, of the rndom cot of ech gurntee: E C h nd E C h for h,,... For the totl rndom cot C C, conidering h i tht the event C > nd C > re mutully excluive o tht E CC i j when i nd j re poitive integer, we hve P (C C > ) P (C > ) P (C > ) nd eide for h,,... h E (C C ) hi E C h E C h Oviouly, y men of the h moment, for h,,..., of generic rndom vrile X other ueful informtion cn e otined, uch vrince σ [X] E X E [X] kewne γ [X] E X 3 3E X E [X]E 3 [X] µqe [X ] E [X] 3 kurtoi κ [X] E X 4 4E X 3 E [X]6E X E [X] 3E 4 [X] ³p E [X ] E [X] 4 We remrk tht the four prmeter, k, k nd g cn e freely ued to comply with cutomer requirement (nd lo thoe of the inurnce compny!) out level nd cot of gurntee. Other prmeter of the model decrie the Blck-Schole cenrio, relted to the rndom motion of the underlying et vlue, nd they re µ the drift of the Brownin proce of the rndom return of the underlying et µ (it vlue, S t, i lognorml ditriuted), viz. the drift of the proce St ln,witht [, ) nd with S σ> the voltility of the me Brownin proce r the vlution rte in flt, here umed, interet rte tructure with dicount fctor exp ( rt) for t [, ) If we ume the ence of ritrge opportunity hypothei then it will follow for thee prmeter r µ σ Conidering n cturil frmework, µ nd σ decrie the invetment kill of the inurnce compny nd r the free rik opportunity of the policy-holder; o tht here we ume 4

5 r µ σ ut our reult will work in ny ce, o lo if r>µ σ. Finlly, let e the contnt inured deth intenity with P (T >t)exp( t) From n opertive point of view we cn conider the invere of the expected further life of the policy-holder firt pproximtion of the vlue of. Oviouly, with exponentil umption for T, deth time cn e een the firt topping event of Poion proce of prmeter. In ection 6 of thi note we will propoe relxtion of the inured deth contnt denity hypothei, tht cn ring to cloed nlyticl formul once more. 3. FORMULAS FOR EMBEDDED OPTIONS For the cturil purpoe of thi pper, we prefer to tudy the poed prolem with clicl method, which llow immedite interprettion nd very helpful informtion, in comprion to thoe gettle y tochtic differentil eqution nd Ito clculu, generlly limited to the expected vlue of the rndom vrile of interet. To egin with we hve to conider the two following proilitie. Firt, the proility of the event U [t, t dt) nd T > t(dt infiniteiml), viz. P (U [t, t dt),t >t) m (t) dt Ã! z σ πt exp (z µt) 3 σ exp ( t) dt t with t (, ) nd where z ln repreent the negtive return, pplied to the initil vlue of the underlying et, with which the rrier (whtever t) i hitted. A ove-mentioned, t the firt rrier hitting the holder mut urrender hi policy. µ St Secondly, the proility of the event T [t, t dt) nd ln [x, x dx) nd U (t, ] (dt nd dx infiniteiml), viz. µ µ St P T [t, t dt), ln [x, x dx),u (t, ] n (t, x) dt dx exp ( t) dt Ã! Ã! (x µt) exp σ t µ exp (x z µt) σ z µ σ t exp πt σ σ πt dx 5

6 with t (, ) nd x µln,. The firt reult, reltive to m (t) dt, i well-known for Brownin motion: ee e.g. Bhttchry, Wymire (99). For completene we remrk tht µ P (U (t, )) z ³ µz Z exp z vσ exp µ v σ π σ σ dv v 3 nd jut differentiting thi function with repect to t we cn otin the denity proility function m (t). Note tht the vlue of P (U (t, )) i le thn for ech t [, ) if either µ>, z < or µ<, z > : in thee ce P (U (, )) give the proility tht the underlying et vlue, t time, never hit the rrier, viz.p (U ). The econd reult, reltive to n (t, x) dt dx, ilecommon:itcnefound y the Reflection Principle for the trjectorie of Brownin motion (ee for n hint the Appendix A). The following fctor in n (t, x) dt dx à exp σ πt (x µt) σ t! µ exp z µ exp σ t à (x z µt) σ t σ πt! dx µ µ St repreent the proility P ln [x, x dx),u (t, ] for ech given t (, ): note tht if U then the rrier h never een hitted. Anlyzing the formul of thi proility, we cn point out how the event U (t, ], or equivlently S τ >for ech τ [,t], "twit" the uul (when no rrier i conidered) µ norml denity (with prmeter µt nd σ t)of St the rndom vrile ln. It follow y integrtion tht P (U (t, ]) Ã! Ã! (x µt) Z exp σ t µ exp (x z µt) σ z µ σ t exp πt σ σ πt dx µ ln nd oviouly P (U (t, ]) P (U (t, )), eing P (U ) P (U (t, ]) P (U (t, )) 6

7 With the formul of thee two proilitie it i quite evident how to olve the poed prolem. In detil, we hve to clculte the following integrl: for C with h,,... P (C > ) Z ln k Z z n (t, x) dx dt E Z C h ln k Z z nd for C with h,,... Z P (C > ) m (t) dt E Z C h n (t, x) (k exp x) h exp ( hrt) dx dt ³ m (t) (k exp (gt) ) h exp ( hrt) dt All thee integrl cn e clculted y men of the reult reported in Appendix B of thi pper. Conidering the enefit C, y uing the reult of Appendix B, where we define G h p µ σ ( hr) for h,,,..., it i necery to clculte integrl of the following type with h,,,... nd i,,,..., h ψ j (h, i) ln k Z z µ exp G h µ µ G h exp x σ G h x z σ ³ µx exp σ exp G h ³ µx σ eix dx etting j if k nd j if k >. Clculting thee integrl i ey. In prticulr, with k it follow (x i negtive in the integrtion intervl) µ µg h k σ i µ µg h σ i ψ (h, i) µ G h σ G h i µ µ G h k µ G σ i µ µ G h σ i h σ µ G h σ i 7

8 while, with k >it follow (x chnge ign in the integrtion intervl) µ µ G h k σ i µ µg h σ i ψ (h, i) µ G h µ G h σ G h i σ i µ µ G h k µ G σ i µ µ G h σ i h σ µ G h i Given the two function ψ j (h, i), wherej or j, for the rndom enefit C we otin P (C > ) ψ j (, ) E [C ]k ψ j (, ) ψ j (, ) E C k ψ j (, ) ψ j (, ) k ψ j (, ) nd more generlly for ech h,,... E hx µ h C h k h i ( ) i ψ i j (h, i) i For enefit C the clculu i eier. The introduction of the function (ee Appendix B) ϕ (β) µ µ µ σ (β) σ llow to otin immeditely P (C > ) ϕ () E [C ]k ϕ (r g) ϕ (r) E C k ϕ (r g) ϕ (r) k ϕ (r g) nd more generlly for ech h,,... E hx µ h C h k h i ( ) i ϕ (hr (h i) g) i i 4. PARTICULAR CASES To find ome well-known reult concerning Americn type option, e.g. Crr formul for E [C C ], trting from our, we mut conider g the ence of ritrge opportunity hypothei σ 8

9 With thee two umption we hve relevnt implifiction for the nlyticl olution of E [C C ]. Indeed, eing G G µ G µ G σ σ r, µ σ thi difference i with the ence of ritrge opportunity hypothei. With thee two umption, our reult llow to otin for k imple formul for E [C C ] tht point out the role of prmeter in the emedded option vlution. Note tht the term G G µ G µ G σ σ i reported for completene ut it could e cncelled in the following expnion of E [C C ] E [C C ]ω () k k G µ µg k G σ µ G µ G σ σ G G µ G µ G σ σ µ G µ σ k µ G k σ k G G µ G µ G σ σ µ µg k k σ k G G µ G µ G σ σ Anlogouly, for k >(gin the ove-mentioned term could e cncelled) we hve E [C C ]ω () k k G µ µg k G σ µ G µ G σ σ G G µ G µ G σ σ 9

10 k G µ G σ µ k µ G σ µ µ G k σ µ G σ k G µ G σ µ G G σ G k µ G σ µ k µ G σ µ G σ µ G σ With the foreid umption (g nd no ritrge opportunity hypothei) the mximiztion of ω () or of ω () ring to the me exct optiml rrier vlue k (µ G ) ott k k σ µ G µ G µ G σ σ G µ if ott (, min (k,k )), otherwie the optiml olution i corner olution: ott or ott min(k,k ). The condition ott (, min (k,k )) i verified if k k in Crr model, ut it my not hold for k much igger thn k : ee the exmple reltive to T. in the ection 5. The ove-mentioned vlue of ott nd thoe of ω ( ott ) nd ω ( ott ) when k k,wellg nd no ritrge opportunity hypothei, coincide with thoe of Crr: he derive hi formul trting from the theory of tochtic differentil eqution nd pplying Ito Lemm, without giving the detiled reult which we hve here otined pplying more tndrd method. We remrk tht, with ritrge opportunity hypothei, viz. r 6 µ σ, nd with g, the mximiztion of E [C C ] ring to n eqution in, which doen t hve exct explicit olution, ut only numericl pproximtion of them. In the mximiztion (of E [C C ])prolem,toerch ott i equivlent to mximize repect to function A (C ) D E with the rel prmeter A, D, E poitive nd E>A. The rel prmeter C i "mll" in olute vlue, with the ign equl to the ign of µ σ r. Note tht it i C if nd only if we ume the no ritrge opportunity hypothei: in thi ce, with g, the equivlent function to mximize ecome A D E, which llow exct nlyticl tretment of the mximiztion prolem. Finlly, with ritrge opportunity hypothei, viz. r 6 µ σ,ndwith g>, the ove-poed prolem ecome even more difficult: the mximiztion of E [C C ] i in generl equivlent to mximize repect to function

11 F A (C H) D E with the rel prmeter A, D, E, F, H poitive, E>A>Fnd with C ove-mentioned. Ltly, we remrk tht if then no urrender gurntee i ctive: in thi ce the option emedded i of Europen type with tochtic (exponentil) mturity. If then no deth gurntee i ctive: in thi ce the option emedded i of Ruin type with infinity mturity. In oth ce the correponding formul would e coniderly reduced. 5. NUMERICAL EVIDENCE All the reult given in thi ection re otined conidering policy-holder with n expected further life of 4 yer, o it i fir to ume.5. The other prmeter of the model re fixed follow:, k 4, k 95, g., µ.5, σ., r.3. InT.wehowthevlueof () P (C > ), ()E [C ],(3)σ [C ],(4)P (C > ), (5)E [C ],(6)σ [C ],(7) P (C C > ), (8)E [C C ],(9)σ [C C ] for ome increing vlue of T., k 4, k 95, g., µ.5, σ., r.3 () () (3) (4) (5) (6) (7) (8) (9) Some remrk on the reult propoed in T.. For itnce, the figure of the row correponding to 75tell tht the policy-holder py premium of l l (l i the loding) with.7 proility he die receiving pyment of 4, prtilly covered y the vlue of the underlying et with.43 proility the policy i urrendered (efore deth) with.7.99 proility he die receiving pyment of more thn 4, fully covered y the vlue of the underlying et.

12 We remrk lo tht the vlue of E [C C ] i "tle" from 5to 45. InthecereltivetoT.,k i much igger thn k o tht the mximum vlue for E [C C ] could e (or ner to) the corner vlue, ott :igvlue of in the rnge [, 95] coniderly reduce E [C ], ee (), nd the increingdecreing movement of E [C ], ee (5) with mximum vlue "ner" 75, doen t chnge the decree of E [C C ]. To find the expected vlue of the gurntee cot t current vlue we cn conider our formul with r :for 75we otin (me interprettion of T. ) T., k 4, k 95, g., µ.5, σ., r () () (3) (4) (5) (6) (7) (8) (9) We cn deduce y T. (row 75) nd T. tht with.7 proility the pyment of 4 i covered y the vlue of the underlying et, in verge / , nd y the deth gurntee enefit, in verge 56.65/ , with expected preent vlue / : n etimtion of the pyment time of the deth gurntee enefit i o otinle y the olution of the eqution exp (.3t), viz.t T 8.96 with.43 proility the policy i urrendered (efore deth) with n verge pyment of 75.5/ with expected preent vlue / : n etimtion of the pyment time of the urrender gurntee enefiti o otinle ythe olution of exp (.3t), viz. t U.83 For completene, we how in T. 3 (without comment) for 75ome vlue of kewne nd of kurtoi: () γ [C ],()κ [C ],(3)γ [C ],(4)κ [C ], (5) γ [C C ],(6)κ [C C ] T. 3, k 4, k 95, g., µ.5, σ., r.3 () () (3) (4) (5) (6) With the me vlue for the prmeter of the model, except now k 4 nd k, we how in the following T. 4 the vlue of () P (C > ), () E [C ],(3)σ [C ],(4)P (C > ), (5)E [C ],(6)σ [C ],(7)P (C C > ), (8) E [C C ],(9)σ [C C ] for ome increing vlue of, in T.,

13 T. 4, k 4, k, g., µ.5, σ., r.3 () () (3) (4) (5) (6) (7) (8) (9) In thi ce k nd k re "ufficiently" ner o tht ott (for the mximiztion of E [C C ])iinteriorto(, ). The mximum vlue of E [C C ] in T. 4 i otined with 45. We remrk only tht the vlue of E [C C ] i here "tle" from 5to A DEMOGRAPHIC IMPROVEMENT We re wre tht our reult re ed on pecil umption, nmely the Geometric-Brownin motion of the underlying et vlue nd the contnt deth intenity nd tht oth ought to hold for very lrge period of time conidering the cturil frmework. Of coure, nturl generliztion could e opportune nd here we conider firt ttempt for deth time T.IntheplceofP (T >t)e t we cn conider for t [, ) nx P (T >t) α i exp ( γ i t) i with γ i > nd proper other prmeter α i for i,,..., n. For itnce, with n we cn ume F T (t) P (T >t)ce t ( c) e t with nd poitive rel prmeter with <nd c deth intenity i F T (t) F T (t) ce t ( c)e t ce t ( c)e t h i,.inthice Thi intenity vrie from c ( c) for t to for t : it decree ³ if c [, i ); it i contnt if c nd it incree (more reliticlly) if c,. Note tht the firt three moment of relitic T could e wellmtched even in thi imple ce with only three prmeter,, c. It i immedite to convince ourelve tht nlyticl type formul, tht we hve found with contnt deth intenity, could e eily generlized to thi 3

14 ce. Oviouly, without contnt deth intenity the optiml urrender rrier doen t correpond to the optiml exercie frontier of the emedded Americn option, even if g nd no ritrge opportunity hypothei re umed. CONCLUSION The nlyticl formul we hve derived eily llow ome vlule enitivity nlye in order to clirte the premium tht hve to e pid for gurnteed enefit. Beide, thee formul cn e ued to control the efficiency of imultion method. When they re teted with the hypothee here umed, they mut give vlution ner to the exct one! Finlly, the uthor togheter hve dicued ll the pect of the work. Both hve verified nlyticl procedure nd the properne of reult. If diviion of the work mut e mde, then E. h een more engged in the formultion of cturil model nd L. in the proilitic feture. APPENDIX A µ St In thi ppendix we will outline how to otin the proility denity of ln conditioning µ to the event "S t doen t hittherrier efore t" or equivlently St "ln doen t hittherrierz efore t". For the Bernoullin rndom wlk, trting in, with independent increment nd, with the up proility p, ndwithrrierin B with B fixed poitive integer, y pplying the Reflexion Principle, we hve (S k i the poition of the rndom wlk t time k with k,,..., n nd n h even) P (S n h S k > B for k,,..., n) µ µ n n n h n h p nh ( p) n h B Bing on the Fundmentl Correpondence Limit Theorem, to otin the nlogou reult in t for Wiener proce with prmeter µ nd σ>, we divide the time intervl [,t] in ntc uintervl,echoflenght n,ndwefix the increment n > nd n nd the up proility p n follow Ã! p n µ p nσ µ p nσ µ n n With thi prmetriztion if H ntc i the poition t time t of the recled wlk, with generic determintion h ntc, it follow E H ntc n (p n ) ntc µt 4

15 σ H ntc 4 n p n ( p n ) ntc σ t From (note z < in the model) h ntc 'ntc (p n ) x pp n ( p n ) ntc B n ' z p nc σ the expected reult cn e otined y the following limit p lim pn ( p n ) ntc n µ ntc µ ntc ntch ntc ntc h ntc ntc h ntc ntc h ntc p n ( p n ) B n Ã! Ã! (x µt) exp σ t µ exp (x z µt) σ z µ σ t exp πt σ σ πt APPENDIX B To otin nliticl olution for the moment of C it i necery to olve improper integrl of the type (A nd B re two poitive rel contnt) Z (ince t 3 exp µ Z r A B Z r A exp (AB) B r A B Z Z µa t B dt (given the utitution t B t A w ) µ µ exp AB µw w dw Z Ã Ã exp AB µ w!! dw w,withw w z it i w z ± z 4 ) Z exp (AB) Z exp ABz d z z 4 exp ABz d z z 4 r Z A exp (AB) exp ABz d z B 4 5

16 (given z cohx nd o z 4inhx) r Z A ³ ³ 4 exp (AB) exp 4AB (coh x) d inh x B 4 r A exp (AB) B (finlly, given inh x r A 4 B Z y AB ) Z exp ( AB) AB ³ ³ ³ exp 4AB (inh x) d inh x µ exp µ z dz r r A B π π exp ( AB) AB exp ( AB) B In our pecific cez i negtive nd if we conider generic rel vlue β (with β or β r or β r g or β r or β r g or β r g or...) it i Z à z σ πt 3 exp à z σ π exp ³ z µ σ (z µt) σ t Z!! exp ( t) exp ( βt) dt µ t 3 exp µ z σ t à z µ z p! µ exp σ ( β) σ µ µ σ ( β) σ t dt µ µ p µ σ ( β) σ ϕ (β) To otin nliticl olution for the moment of C it i necery to olve improper integrl of the type (A nd B re two poitive rel contnt) Z µ exp t Z u 3 exp µ µa t B dt (y the utitution t u ) t µb u A π du exp ( AB) u A ince thi integrl i tken ck, y the foreid utitution, to the previou one. 6

17 In thi pecific ce, given G h p µ σ ( hr) with h,,,..., Ã! (x µt) Z exp σ t exp ( t) σ t π exp ( hrt) dt µ p µ σ ( hr) exp x µ exp x ³ µx G h σ G h exp σ p µ σ σ ( hr) ³ µx exp σ nd Ã! Z µ exp (x z µt) z µ σ t exp ( t) exp σ σ t π exp ( hrt) dt µ p µ σ ( hr) exp µ exp x z G h σ G h x z σ ³ µx exp σ p ³ µx µ σ ( hr) exp σ REFERENCES Bcinello A.R., Ortu F. (993); Pricing equity-linked life inurnce with endogenou minimum-gurntee; Inurnce: Mthemtic nd Economic,, pp Bcinello A.R. (); Pricing Gurnteed Life Inurnce Prtecipting Policie with Periodicl Premium nd Surrender Option; Quderni del Diprtimento di Mtemtic Applict lle Scienze Economiche Sttitiche e Atturili Bruno de Finetti, Univerità degli Studi di Triete, n./. Bcinello A.R. (4); Modelling the urrender condition in equity-linked life inurnce; Quderni del Diprtimento di Mtemtic Applict lle Scienze Economiche Sttitiche e Atturili Bruno de Finetti, Univerità degli Studi di Triete, n./4. Berrd T. (5); Americn Contingent Clim with Stochtic Mturity: Vlution nd Appliction; Numericl Method in Finnce; Ed. H. Ben Ameur nd M. Breton, Springer, pp Bhttchry R.N., Wymire E.C.; Stochtic Procee With Appliction; Wiley (99). Brennn M.J., Schwrtz E.S. (976); The pricing of equity-linked life inurnce policie with n et vlue gurntee; Journl of Finncil Economic, 3, pp

18 Brodie M., Detemple J. (996); Americn option vlution: new ound, pproximtion nd comprion of exiting method; Review of Finncil Studie, 9, pp. -5. Brodie M., Detemple J. (4); Option Pricing: Vlution Model nd Appliction; Mngement Science, Vol. 5, N. 9, pp Crr P. (998); Rndomiztion nd the Americn Put; The Review of Finncil Studie, N. 3, pp Dorontu D., Pontier M.(6); Riky Det nd Optiml Coupon Policy nd Other Optiml Strtegie; Stochtic procee nd ppliction to mthemticl finnce Proceeding of the 6th Ritumeikn Interntionl Sympoium, ed. J. Akhori, S. Ogw & S. Wtne, World Scientific Pulihing Co. Pte. Ltd., pp El Kroui N., Mrtellini L. (); Dynmic Aet Pricing Theory with Uncertin Time-Horizon; WorkingPper,MrhllSchoolofBuine,USC. Groen A., Jorgenen P.L. (997); Vlution of erly exercile interet rte gurntee; Journl of Rik nd Inurnce, 64, N. 3, pp Groen A., Jorgenen P.L. (); Fir vlution of life inurnce liilitie: the impct of interet rte gurntee, urrender option, nd onu policie; Inurnce: Mthemtic nd Economic, 6, pp Romgnoli A. (7); Le copule per getire i fenomeni di dipendenz nelle icurzioni vit; Tei di dottorto in cienze Atturili, XIX Ciclo, Univerità di Rom. Vnnucci E. (999); Un modello di vlutzione del coto di grnzie di rendimenti minimi eigiili nel continuo e condizionte ll oprvvivenz; Giornle dell Itituto Itlino degli Atturi, LXII, pp Vnnucci E. (3); The vlution of unit linked policie with miniml return gurntee under ymmetric nd ymmetric informtion hypothee; Diprtimento di Sttitic e Mtemtic Applict ll Economi dell Univerità di Pi, Report n.37. Vnnucci E. (3); An evlution of the rikine of Unit Linked policie with miniml return gurntee; Proceeding of the VI Spnih-Itlin Meeting on Finncil Mthemtic, Triete, pp Vnnucci E. (5); Polizze Unit Linked con pretzioni minime grntite che i delineno nel coro dell durt contrttule; Anlii dei richi ed ottimlità delle grnzie nei prodotti icurtivi vit con protezione; Ricerc Interuniveritri Prin 3 cofinnzit dl Miur, ed. Bellieri dei Bellier A. & Mzzoleni P., pp Vnnucci L. (5); Suun opzione generlizznte quell di Mrgre; Anlii dei richi ed ottimlità delle grnzie nei prodotti icurtivi vit con protezione; Ricerc Interuniveritri Prin 3 cofinnzit dl Miur, ed. Bellieri dei Bellier A. & Mzzoleni P., pp