A Discrete Time Benchmark Approach for Finance and Insurance Hans Buhlmann 1 and Eckhard Platen 2 March 26, 2002 Abstract. This paper proposes an inte

Transcription

1 A Dscrete Tme Benchmar Approach for Fnance and Insurance Hans Buhlmann 1 and chard Platen March 6, 00 Abstract. Ths paper proposes an ntegrated approach to dscrete tme modellng n nance and nsurance. Ths approach s based on the exstence of a specc benchmar portfolo, nown as the growth optmal portfolo. When used as numerare, ths portfolo ensures that all benchmared prce processes are supermartngales. A far prce s characterzed n terms of the type of maxmum that the growth rate of the perturbed growth optmal portfolo attans. In general, arbtrage amounts arse due to the supermartngale property of benchmared traded prces. No measure transformaton s needed for the far prcng of nsurance polces and dervatves Mathematcs Subject Classcaton: prmary 90A1 secondary 60G30, 6P0. JL Classcaton: G10, G13 Key words and phrases: nancal and nsurance maret model, benchmar approach, growth optmal portfolo, numerare portfolo, arbtrage amount, far prcng, unt lned nsurance. 1 Department of Mathematcs, dgenosssche Technsche Hochschule, 809 Zurch, Swtzerland Unversty oftechnology Sydney, School of Fnance & conomcs and Department of Mathematcal Scences, PO Box 13, Broadway, NSW, 007, Australa

2 1 Introducton There exsts a stream of lterature that explots the concept of a growth optmal portfolo (GOP), orgnally developed by Kelly (1956) and later extended and dscussed, for nstance, n Long (1990), Artzner (1997), Bajeux-Besnanou & Portat (1997), Karatzas & Shreve (1998), Kramov &Schachermayer (1999) and Goll & Kallsen (00). Under certan assumptons the GOP concdes wth the numerare portfolo, whch maes prces, when expressed n unts of ths partcular portfolo, nto martngales under the gven probablty measure. In Kramov & Schachermayer (1999) and Platen (001) t was demonstrated that prces when benchmared by the GOP can become supermartngales wthout assumng the exstence of an equvalent martngale measure. The noton of a numerare portfolo was recently extended by Becherer (001), tang nto account benchmared prces that are supermartngales when an equvalent local martngale measure exsts. In standard cases wth an equvalent martngale measure the numerare portfolo has been shown to concde wth the nverse of the deator or state prce densty, see Constatndes (199), Due (1996) or Rogers (1997). Furthermore, n Buhlmann (199, 1995) and Buhlmann, Delbaen, mbrechts & Shryaev (1998) the deator has been suggested for the modellng of nancal and nsurance marets. Smlarly, n Platen (001) a nancal maret has been constructed by characterzaton of the GOP as benchmar portfolo. Wthn ths paper we follow a dscrete tme benchmar approach, where we characterze ey features of a nancal and nsurance maret va the GOP wthout assumng the exstence of any equvalent local martngale measure as n most of the above mentoned lterature. In partcular, far prces of dervatves and nsurance polces are obtaned n a consstent manner va condtonal expectatons wth respect to the real world probablty measure. Ths provdes a bass for the equvalence prncple that s wdely used by actuares. An example of a dscrete tme maret, sutable for nance and nsurance applcatons, wll llustrate some ey features of the benchmar approach. Dscrete Tme Maret Let us consder a dscrete tme maret on a gven probablty space ( A P). Asset prces are assumed to change ther values only at the gven dscrete tmes 0 t 0 <t 1 <:::<t n < 1 for n f0 1 :::g. The nformaton structure n ths maret s descrbed by the ltraton A = (A t ) f0 1 ::: ng wth A t0 beng the -algebra consstng of all null sets and ther complements. In ths paper we consder d + 1 prmary assets, d f1 :::g, whch generate nterest, dvdend, coupon or other payments as ncome or loss, ncurred from holdng the respectve asset. We denote by S (j)

3 the nonnegatve value at tme t of a prmary securty account. Ths account holds only unts of the jth asset n addton to the earnngs from holdng the jth asset. That s, all ncome s renvested nto ths account. Thus the jth prmary securty expresses the tme value of the jth prmary asset. The 0th prmary securty account s the domestc savngs account. For smplcty, we assume at tme t 0, that the jth prmary securty account conssts of one unt of the jth prmary asset. Accordng to the above descrpton, the domestc savngs account S (0) s then a roll-over short term bond account, where the nterest payments are renvested at each tme step. If the jth prmary asset s a share, then S (j) s the value at tme t of such shares ncludng accrued dvdends. The quantty S (j) then represents the jth cum-dvdend share prceattmet. We assume that a.s. for all j f0 1 ::: dg. S (j) 0 > 0 (.1) Now, we ntroduce the growth rato h (j) +1 of the jth prmary securty account at tme t +1 n the form h (j) +1 = 8 < : S (j) +1 S (j) for 0 otherwse S (j) > 0 (.) for f0 1 ::: n; 1g and j f0 1 ::: dg. Note that the return of S (j) at tme t +1 equals h (j) +1 ;1. We assume that h(j) s A +1 t +1 -measurable and a.s. nte. The growth rate of the prmary securty account S (0) for the domestc currency shall be strctly postve, that s h (0) +1 > 0 (.3) a.s. for all f0 1 ::: n; 1g wth S (0) 0 = 1. We can then express the prce of the jth prmary securty account at tme t,say the jth cum-dvdend share prce S (j), n the form S (j) = S (j) 0 Y `=1 h (j) ` (.4) for f0 1 ::: ng and j f0 1 ::: dg. Note that due to (.4) and (.3) we have for all f0 1 ::: ng. S (0) > 0 (.5) In the gven dscrete tme maret t s possble to form self-nancng portfolos contanng the above prmary securty accounts. For the characterzaton of a self-nancng portfolo at tme t t s sucent to descrbe the proporton (j) 3

4 (;1 1) of ts value that at ths tme s nvested n the jth prmary securty account, j f0 1 ::: dg. Obvously, the proportons add to one, that s dx j=0 (j) =1 (.6) for all f0 1 ::: dg. The vector process = f = ( (0) (1) ::: (d) ) f0 1 ::: ngg denotes the correspondng process of proportons. We assume that s A t -measurable, whch means that all proportons at a gven tme do not depend on any future events. The value of the correspondng self-nancng portfolo at tme t s denoted by We obtan the growth rato ` and we wrte = f of ths portfolo process at tme t` n the form ` = dx j=0 f0 1 ::: ngg. (j) `;1 h(j) ` (.7) for ` f1 ::: ng, where ts value at tme t satses the expresson for f0 1 ::: ng. = 0 Y `=1 ` (.8) 3 Dscrete Tme Maret of Fnte Growth In the gven dscrete tme maret let us denote by V the set of all self-nancng, strctly postve portfolo processes Ths means, for a portfolo process V t holds +1 (0 1) a.s. for all f0 1 ::: n; 1g. Due to (.3) - (.5), S (0) (t) s always strctly postve. Consequently, V s not empty. We dene for a gven portfolo process V wth correspondng process of proportons ts growth rate g () at tme t by the followng condtonal expectaton g () = log( +1 ) At (3.1) for all f0 1 ::: n; 1g. Ths allows us to ntroduce the optmal growth rate g at tme t as the supremum for all f0 1 ::: n; 1g. g = sup g () (3.) V 4

5 If the optmal growth rate could reach any arbtrarly large value at some tme, then some self-nancng portfolo would have unlmted growth wth strctly postve probablty. Ths can be nterpreted as some form of arbtrage. We exclude such arbtrage opportunty by ntroducng the followng notons. Denton 3.1 We say that a dscrete tme maret s of nte growth f max g < 1 (3.3) f0 1 ::: n;1g a.s. Denton 3. If there exsts n a dscrete tme maret of nte growth a portfolo V wth a correspondng process of proportons such that 0 =1 (3.4) g () = g (3.5) and h() A t < 1 (3.6) for all f0 1 ::: n; 1g and V, then we call growth optmal and the maret ntegrable. From the vewpont ofannvestor, a growth optmal portfolo, GOP, can be nterpreted as a best benchmar portfolo because there s no other strctly postve, self-nancng portfolo that can outperform ts growth rate. In what follows we call prces, whch are expressed nuntsofagop, benchmared prces and ther growth ratos benchmared growth ratos. The condton (3.6) guarantees the ntegrablty of benchmared growth ratos. We can now formulate the followng result. Theorem 3.3 In an ntegrable maret a portfolo process V s growth optmal f and only f all portfolos V, when expressed n unts of ths portfolo, are (A P)-supermartngales, that s for all f0 1 ::: n; 1g. h() A t 1 (3.7) 5

6 The proof of ths theorem s gven n Appendx A. Under the assumpton of the exstence of an equvalent local martngale measure, Becherer (001) proved a smlar result for semmartngales. Let us consder two portfolos that are both growth optmal n an nvertble maret. Accordng to Theorem 3.3 the rst portfolo, when expressed n unts of the second, must be a supermartngale. Addtonally, by the same argument the second, expressed n unts of the rst, must be also a supermartngale. Ths can only be true f both processes are dentcal, whch yelds the followng result. Corollary 3.4 In an ntegrable maret the GOP s unque. Note that the stated unqueness of the GOP does not mply that ts proportons have to be unque. 4 Far Maret The benchmared prce ^S() by the relaton at tme t of a self-nancng portfolo s dened for all f0 1 ::: ng. ^ = S() (4.1) By Theorem 3.3, n an ntegrable maret the benchmared prce of a strctly postve, self-nancng portfolo V s a supermartngale and t follows ^ ^S() for all f0 1 ::: ng and f0 1 ::: g. At (4.) One can nterpret the last observed benchmared prce of a self-nancng, strctly postve portfolo as ts benchmared traded prce. In the sense of relaton (4.) t turns out that ths benchmared traded prce s never below the best forecast of ts future benchmared values. Denton 4.1 If n an ntegrable maret the benchmared prce ^S() at tme t of a self-nancng portfolo process equals the best forecast of ts future benchmared values, that s ^ = ^S() +1 At for all f0 1 ::: n; 1g, then we call a far prce process. 6 (4.3)

7 quaton (4.3) means that ^S() s an (A P)-martngale. Note, for a far prce process we have always ^ = ^S() n At (4.4) for all f0 1 ::: ng, whch means that we can nterpret ^S() as the best forecast process of ts nal value. Furthermore, equvalently to(4.3) we have for a far prce process that h() A t =1 (4.5) for all f0 1 ::: n; 1g. If all self-nancng portfolo processes n an ntegrable maret are far, then we call t a far maret. Snce n a far maret all benchmared, self-nancng portfolo processes are martngales one can show that such a maret s arbtrage free n the sense of Harrson & Kreps (1979) and Harrson & Plsa (1981) and an equvalent rs neutral measure exsts. To see whether a portfolo n an ntegrable maret s far t s essental to dentfy the type of maxmum that s obtaned by the growth rate of the GOP f t would be perturbed by ths portfolo. For (0 1 )ands() Vwe construct a perturbed GOP V Vwth growth rato +1 = V +1 h g V at tme t +1 and correspondng growth rate = log = +1 +(1; ) +1 (4.6) h +1 At (4.7) at tme t for f0 1 ::: n; 1g. Ths allows us to dene the dervatve of the growth rate of the perturbed GOP n the drecton of the portfolo V at tme t as the lmt for f0 1 ::: =0 = lm 0+ Now we can formulate the followng dentty. 1 g ; g () (4.8) Theorem 4. In an ntegrable maret for a portfolo V and f0 1 ::: n; 1g the followng =0 = h() A t ; 1: (4.9)

8 The proof of ths theorem follows drectly from the proof of Theorem 3.3 that wll be gven n Appendx A. From Theorem 4. and (4.5) we obtan mmedately the followng characterzaton of a far portfolo. Corollary 4.3 In an ntegrable maret a portfolo V s far f and only f for all f0 1 ::: n; 1g =0 =0: (4.10) Intutvely, Corollary 4.3 expresses the fact that a portfolo s far f the maxmum that the growth rate of the perturbed GOP attans, s a genune maxmum that satses the usual rst order condton n the drecton of the portfolo. It must not be a maxmum that arses, for nstance, at the boundary of V because of the constrant that the GOP has to reman strctly postve. 5 Far Contngent Clam Prcng We consder a contngent clam H,whchsanA t -measurable, possbly negatve payo, expressed n unts of the domestc currency, that has to be pad at a maturty datet, f0 1 ::: ng. Note that the clam H s not only contngent on the nformaton provded by the observed prmary assets S (j) ` up untl tme t, j f0 1 ::: dg, ` f0 1 ::: g, but as well on addtonal nformaton contaned n A t as, for nstance, the occurrence of defaults or nsured events. If we dene the far prce U (H ) at tme t for the contngent clam H by the relaton U (H ) = H A t (5.1) for f0 1 ::: g, then we obtan n a natural way a consstent system of prces. All far prces of nstruments ncludng prmary securtes, dervatve products and contngent clams have then a correspondng benchmared prce of the type ^U (H ) = U (H) (5.) (H for all f0 1 ::: g, f0 1 ::: ng. Obvously, the process ^U ) = (H f ^U ) f0 1 ::: gg forms an (A P)-martngale. The argument can be easly extended to sums of contngent clams wth A-measurable maturty dates. 8

9 In nance, the prcng formula (5.1) s typcally used for projectng future cash- ows nto present values, that s for t t. Formally, one can use (5.1) also for assessng the present value for cashows that occurred n the past, that s for t >t. Then we obtan U (H ) = H (5.3) for f0 1 ::: g, f0 1 ::: ng. In (5.3) we express the benchmared accumulated value of the payment H at tme t n terms of tme t value. Ths nterpretaton s mportant for nsurance accountng as we wll dscuss below. For the prcng of an nsurance polcy the actuaral tas s the valuaton of a sequence of cashows X 0 X 1 ::: X n, whch are pad at the tmes t 0 t 1 ::: t n, respectvely. After each payment, ts value s nvested by the nsurance company n a strctly postve, self-nancng portfolo, characterzed by a process of proportons. The benchmared far prce ^Q 0 at tme t 0 for the above sequence of cashows s accordng to (5.) gven by the expresson ^Q 0 = mx =0 X A t 0 : (5.4) The benchmared far prce ^Q at tme t for f0 1 ::: n; 1g of ths sequence of cashows becomes ^Q = ^C + ^R (5.5) for f0 1 ::: ng. Here we choose an arbtrary process of proportons, representng the nvestment portfolo of the nsurance company, to obtan ^C = 1 X =0 X ;1 Y `= `+1 (5.6) whch then expresses the benchmared far value of the already accumulated payments. Furthermore, ^R = nx =+1 X A t (5.7) s the benchmared far prce at tme t for the remanng payments, whch s called the prospectve reserve. It s easy to chec that the process ^Q = f ^Q f0 1 ::: ngg forms an (A P)-martngale for all choces of. When expressed n unts of the domestc currency, we have at tme t for the above sequence of cashows the far value Q = ^Q (5.8) 9

10 for all f0 1 ::: ng. The above result s mportant, for nstance, for the far prcng of lfe nsurance polces. ach nsurance carrer can choose ts own process of proportons to nvest the payments that arse. However, the GOP, that s needed to value the prospectve reserve, must be the same for all nsurance companes n the same maret. Above we clared the role of the GOP for prcng the prospectve reserve. We pont out that the above analyss says nothng about the performance and rsness of derent nvestment strateges that the nsurance carrer can choose. The growth rate for the nvestment portfolo becomes optmal, f the proportons of the GOP are used. If the nsurance company ams to maxmze the growth rate of ts wealth, then the prcng of an nsurance polcy and the optmzaton of the nvestment portfolo both nvolve the GOP. Note that we dd not use any measure transformaton to obtan a far prce system. In the case of a far maret wth contnuous securty prces one can equvalently derve the resultng far prces by the use of the mnmal equvalent martngale measure P ~, whch s related to local rs mnmzaton, as descrbed n Follmer & Sondermann (1986), Follmer & Schwezer (1991) or Hofmann, Platen & Schwezer (199). The correspondng Radon-Nodym dervatve process for the mnmal equvalent martngale measure s then d P ~ = S(0), whch n ths case dp s an (A P)-martngale. 6 Arbtrage Amount Typcally, arbtrage free marets have been studed n a rs neutral settng n the lterature. In our termnology these are far marets. As we wll see, t appears to be realstc to allow some form of arbtrage n a gven maret. In an ntegrable maret a partcular form of arbtrage becomes vsble n the postve derence between a traded benchmared prce of a portfolo V and ts expected benchmared future prce, see (4.). We call ths derence the benchmared arbtrage amount ^A () = ^S() ; ^S() +1 At (6.1) at tme t, f0 1 ::: n; 1g. As has been shown for contnuous tme n Heath & Platen (00), one can model and quantfy arbtrage amounts, see also our example n Secton 8. In a developed maret t s reasonable to expect that arbtrage amounts are typcally small. However, strctly postve arbtrage amounts may exst. In an ntegrable maret one could, n prncple, nterpret any nonnegatve contngent clam as prmary asset. Its benchmared traded prce forms the correspondng benchmared prmary securty account, whch s a supermartngale. If 10

11 the correspondng contngent clam prces are addtonally chosen to be far, that means they have zero arbtrage amounts, then they are equvalent to those obtaned n (5.1). In general, however, demand and supply prmarly determne the securty prce evoluton and extreme demand or supply can lead to some strctly postve arbtrage amounts. Under the above descrbed benchmar framewor, prces are allowed to be derent from far prces. Note, n an ntegrable maret the mnmal possble prce for a nonnegatve contngent clam s always gven by the far prce. Ths s a consequence of the supermartngale property of benchmared prces. 7 Unt Lned Insurance Contracts In the nsurance context we loo agan at the cash ows X 0 X 1 ::: X n but assume a specc form for these random varables. Intutvely, they stand now for unt lned clams and premums. Hence they can be of ether sgn. The cashow at tme t s of the form X = D (7.1) for f1 ::: ng. The payments are lned to some self-nancng, strctly postve reference portfolo V wth gven proportons. The nsurance contract speces the reference portfolo and the random varables D, whch are contngent on the occurrence of nsured events durng the perod (t ;1 t ], for nstance, death, dsablement or accdents. The standard actuaral technque treats such contracts by usng the reference portfolo process as numerare and then deals wth the unt lned random varables D 0 D 1 ::: D n at nterest rate zero. It s reasonable to assume that these random varables are A-adapted and ndependent of the reference portfolo process. As n Secton 5, let us nowlooatthevalue W () of the payment stream at tme t. It s determned by the accumulated payments C () and the lablty or prospectve reserve r. Let us follow the standard actuaral methodology, assumng that the nsurer nvests all accumulated payments n the reference portfolo. Then we obtan for W (), when expressed n unts of the domestc currency, the expresson W () = C () + r (7.) wth accumulated payments C () = X =1 D (7.3) 11

12 and the lablty or prospectve reserve r = nx =+1 D At (7.4) for f0 1 ::: ng. In an ntegrable maret the benchmared value ^W () = W () at tme t for the cashows of ths unt lned nsurance contract s then by (7.) of the form ^W () = C() + r (7.5) for f0 1 ::: ng. On the other hand, the benchmared far value at tme t of the cashows of ths contract s accordng to (5.4) - (5.8) gven by the expresson ^Q () ^Q () = C() wth benchmared far prospectve reserve R = nx =+1 + R D A t (7.6) (7.7) for f0 1 ::: ng. It s mportant to note that under qute natural condtons one can prove that the benchmared far prospectve reserve s less or equal the actuaral prospectve reserve. The proof of ths nequalty reles on the supermartngale property of, f0 1 ::: ng shown n Theorem 3.3 and the nequaltes nx =+1 D A t n;1 nx =+1 D ^(n;1) ^(n;1) A t n;1 f (D n ja tn;1) 0, nx =+1 D ^(n;1) ^(n;1) A t n; nx =+1 D ^(n;) ^(n;) A t n; f (D n + D n;1 ja tn;) 0,. 1

13 nx =+1 D ^(+1) ^(+1) A t nx =+1 D ^ ^ A t = r f (D n +D n;1 +:::+D +1 ja t ) 0, for f0 1 ::: n;1g. Tang condtonal expectaton wth respect to A t, the nequaltes above become a chan, whose rst member equals R r, and the last member becomes. We formulate the above result as a lemma. Lemma 7.1 If for all m f0 1 ::: n; 1g, then for all f0 1 ::: m; 1g. As by (7.4) nx =m+1 D At m r m = m nx 0 (7.8) R r (7.9) =m+1 D At m the condton (7.8) of the lemma means that the nsurance contract denes a cashow whose actuaral prospectve reserve never becomes negatve. Ths s usually observed as practcal rule, snce products that allow for negatve reserves have many defects. For nstance, they permt antselecton. From (7.5) and (7.6) we mmedately have under condton (7.8) ^Q () ^W () for f0 1 ::: ng. However, we observe that nether ^Q() = f ^Q() () () f0 1 ::: ngg nor ^W = f ^W f0 1 ::: ngg s n general a supermartngale, even under condton (7.8). Revertng to property (7.9) we observe that there s, n general, a nonnegatve derence A = r ; R 0 between the actuaral and the far prospectve reserve. Ths arbtrage amountsa consequence of the classcal actuaral prce calculaton leadng to the prospectve reserve r n (7.4). Of course, the actuaral and the far prospectve reserve concde f one uses the GOP as reference portfolo. 13

14 8 A Lognormal xample To llustrate ey features of the dscrete tme benchmar approach, let us dscuss a smple example of a maret wth two prmary assets, whch s a dscrete tme verson of the Blac-Scholes model, see also Becherer (001). Other examples, whch also demonstrate a supermartngale property for benchmared prces, can be found n Kramov & Schachermayer (1999) and Heath & Platen (00). The two prmary assets are the domestc currency, whch s assumed to paynonterest, and a stoc thatpays also no dvdends. The prmary securty account at tme t for the domestc currency s then smply the constant S (0) domestc currency for f0 1 ::: ng. The stoc prce S (1) by the product S (1) = S (1) 0 Y `=1 =1of one unt of the at tme t s gven h (1) ` (8.1) for f0 1 ::: ng, where we assume the growth rato at tme t` to be an ndependent, lognormal dstrbuted random varable, that s Y` =log h (1) ` N( ) (8.) wth mean, varance > 0 and some gven parameter > 0. The GOP proportons (0) and (1) must be such that (0) =1; (1) (8.3) for all f0 1 ::: ng. Furthermore, snce the GOP has to be always strctly postve we must have g () (1) [0 1] (8.4) for all f0 1 ::: ng. Obvously, the set V of strctly postve, self-nancng portfolos s here characterzed by those processes of proportons for whch (1) [0 1] for all f0 1 ::: ng. The growth rate at tme t for a portfolo V s then gven by the expresson = log (8.5) 1+ (1) (exp(y +1 ) ; 1) At for all f0 1 ::: n; 1g. Its rst dervatve wth respect to (1) = and ts second dervatve has the g () = (1) 0 exp(y +1 ) ; 1 1+ (1) (exp(y +1 ) ; 1) (exp(y +1 ) ; 1) 1+ (1) (exp(y +1 ) ; 1) A t 1 A t C s (8.6) A (8.7) 14

15 for f0 1 ::: n; 1g. We note that the second dervatve s always negatve, whch ndcates that the growth rate has at most one maxmum for some proporton (1) [0 1]. We also observe that the rst dervatve of the growth rate for = 0has the value (1) = exp (1) =0 + ; 1 (8.8) and s for (1) = 1 gven by the (1) for f0 1 ::: n; 1g. =1; exp (1) =1 ; + (8.9) 1. To show that we have a far maret, we rst clarfy whether the (1) can become zero for (1) can be only the case for jj show that the optmal proporton (1) [0 1]. Tang (8.6) - (8.9) nto account, ths. In ths case t s then straghtforward to for the GOP converges to the value lm (1) = as 0 for f0 1 ::: n; 1g. Asymptotcally for 0 we have a far maret for ths parameter choce because lm 0 h(0) A t =1 and lm 0 h(1) A t =1: Obvously, by Corollary 4.3 and (6.1) wehave then for all Vand f0 1 ::: n; 1g a vanshng arbtrage amount =0 =0. The Radon-Nodym dervatve process ^S(0) = S(0) becomes here for 0 a martngale and the standard rs neutral approach can be appled.. In the case <; we obtan from (8.6) - (8.7) the optmal proporton (1) =0 15

16 for all f0 1 ::: n; 1g. Ths requres to hold for the GOP all nvestments n domestc currency. Here we get h(1) A t = exp + whch shows that the benchmared stoc prce process < 1 ^S (1) = S(1) strct supermartngale and not a martngale. Obvously, ^S(0) = S(0) =1s a martngale and would be the canddate for the Radon-Nodym dervatve process for an equvalent rs neutral measure. However, snce ^S(1) s not a s a martngale we have not a far maret and the standard rs neutral prcng approach does not apply. For =( (0) (1) )=(0 1) we have the arbtrage amount ^A () =1; exp + > =0 = exp + ; 1 < 0: 3. For > t follows by (8.6) - (8.7) the optmal proporton (1) =1 for f0 1 ::: n ; 1g. Ths means for sucently large mean of the logarthm of the growth rato of the stoc one has to hold n the GOP all nvestments n the stoc. In ths case we get h(0) A t = exp ; + < 1 whch says that the benchmared domestc savngs account ^S(0) = S(0) s a strct supermartngale and not a martngale. Ths means that ^S (0), whch s the canddate for the Radon-Nodym dervatve process of an equvalent rs neutral measure, s not a martngale. Ths means, the maret s not far and the standard rs neutral approach does not apply. However, note that ^S(1) = S(1) = 1 s a martngale. For = ( (0) (1) ) =(1 0) we have then the arbtrage amount ^A () =1; exp ; + > =0 = exp ; 16 + ; 1 < 0:

17 Ths example demonstrates that benchmared prces are not always martngales and strctly postve arbtrage amounts may exst. However, these benchmared prces become martngales f the correspondng rst dervatves of the growth rate of the perturbed GOP n the drecton of these securtes are zero. In all above cases the benchmar approach provdes a unque prce system, whereas the standard rs neutral approach s not applcable n the last two cases. Concluson We have shown that the growth optmal portfolo plays a central role n the understandng of ey propertes of a nancal and nsurance maret. If the rst dervatves of ts growth rate vansh, then ths turns out to be equvalenttohavng a far maret. It s well nown that ths portfolo also plays a ey role n portfolo optmzaton, equvalent to the mutual fund. The proposed benchmar approach provdes not only an ntegrated framewor to man applcatons n nance and nsurance but fund management, also. Acnowledgement The second author les to express hs grattude to TH Zurch for ts nd hosptalty durng a vst n 001. Both authors than Freddy Delbaen, Walter Schachermayer and Albert Shryaev for valuable suggestons on mprovements of ths wor. A Appendx Proof of Theorem For (0 1 ) and S() Vwe consder the perturbed GOP V V, that s wth growth rato h +1 > 0 (A.1) gven n (4.6) for f0 1 ::: n; 1g. One can then show, usng log(x) x ; 1and(4.6), that = h() G = 1 log h 1 h +1 ; ; 1 (A.) +1 17

18 and G +1 ;1 +1 ; 1 = h() h ; +1 : (A.3) h +1 We obtan n (A.3) for ; +1 h() +1 0 because of h +1 > 0thenequalty G +1 0 (A.4) and for +1 ; +1 < 0 from (A.3) because of (0 1 h +1 > 0 that ), h() +1 > 0, and ;h() G h +1 = ; 1 1 ; + h() ; 1 1 ; ;: (A.5) Summarzng (A.) - (A.5) we have for f0 1 ::: n; 1g and V the upper and lower bounds ; G +1 h() ; 1 (A.6) where by (3.6) h() +1 < 1 (A.7) +1 f s growth optmal. Then by usng (A.6) and (A.7) t follows by the Domnated Convergence Theorem that 0 lm G +1 A t 0+ = lm G A log h +1 = h() A t =0 A t ; 1 (A.8) for f0 1 ::: n; 1g and V. Ths shows us that the benchmared portfolo process ^ s then a supermartngale. 18

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