On Pricing Kernels and Dynamic Portfolios

Transcription

1 On Pricing Kernels and Dynamic Porfolios By Philippe Henroe Groupe HEC, Déparemen Finance e Economie Jouy-en-Josas Cede, France henroe@hec.fr April 2002 Absrac We invesigae he srucure of he pricing kernels in a general dynamic invesmen seing by making use of heir dualiy wih he self financing porfolios. We generalize he variance bound on he ineremporal marginal rae of subsiuion inroduced in Hansen and Jagannahan (1991 along wo dimensions, firs by looking a he variance of he pricing kernels over several rading periods, and second by sudying he resricions imposed by he marke prices of a se of securiies. he variance bound is he square of he opimal Sharpe raio which can be achieved hrough a dynamic self financing sraegy. his Sharpe raio may be furher enhanced by invesing dynamically in some addiional securiies. We ehibi he kernel which yields he smalles possible increase in opimal dynamic Sharpe raio while agreeing wih he curren marke quoes of he addiional insrumens. Keywords: Hedging. Pricing Kernel, Sharpe Raio, Self Financing Porfolio, Variance-Opimal

2 1 Inroducion he dualiy beween pricing kernels and porfolio payoffs is he key o many fundamenal resuls in asse pricing heory. In a one-period seing, a pricing kernel is a random variable m +1 which saisfies he equaliy (1 E m +1 w +1 = R f,+1 E m +1 w for every porfolio wih payoff w +1 a ime ( + 1 and value w a ime, where R f,+1 and E denoe respecively he (gross risk free rae from o ( + 1 and he condiional epecaion operaor corresponding o he informaion available a ime. Harrison and Kreps (1979 show ha he eisence of a pricing kernel is equivalen o he law of one price while he absence of arbirage corresponds o he eisence of a posiive pricing kernel. If we know he prices oday and he payoffs omorrow of a se of securiies, hen a posiive pricing kernel m +1 consisen wih hese securiies provides an efficien mehod o produce coningen claim prices in an arbirage free framework. he kernel m +1 yields an arbirage free price F oday for a payoff F +1 omorrow hrough he equaion R f,+1 E m +1 F = E m +1 F +1. his echnique is especially useful when he marke is incomplee and he claim F +1 canno be obained as he payoff of a porfolio based on he primiive securiies. Every posiive pricing kernel yields however a differen arbirage free price sysem, and in many siuaions he resuling range of coningen claim prices is so wide as o be of lile pracical use. I is hen naural o seek a raionale o reduce he se of admissible pricing kernels, and in urn he range of corresponding prices. he ques for such a raionale is a cenral heme in asse pricing heory. Bernardo and Ledoi (2000 show for insance ha seing upper and lower bounds o a pricing kernel in every sae of he world conrols he maimum gain-loss raio of every invesmen sraegy. Balduzzi and Kallal (1997 consider he resricions imposed by he risk premia assigned by he pricing kernels on some arbirary sources of risk. he variance bound on he pricing kernels inroduced in Hansen and Jagannahan (1991 is anoher imporan consequence of he dualiy beween kernels and porfolios. he square of he Sharpe raio of every porfolio is smaller han he variance of every pricing kernel, once properly normalized, and equaliy obains for a unique porfolio whose payoff is also 2

3 iself a pricing kernel. his resul is useful in wo ways. On he one hand, he variance of every pricing kernel yields an upper bound o he Sharpe raios which porfolio managers may epec o obain in he marke. On he oher hand he Sharpe raio of any porfolio is a lower bound o he variance of he pricing kernels, and his allows o rejec he asse pricing heories for which he discoun facor does no display enough variaion across he saes of naure. Bekaer and Liu (2001 give an eensive accoun of he growing use of hese bounds in financial economics. In view of his resul, Cochrane and Saá-Requejo (2000 reduce he se of admissible pricing kernels by rejecing candidaes wih large variance on he ground ha hey may give rise o abnormal good deals in he form of invesmen opporuniies wih large Sharpe raios. hey reason ha alhough posiive pricing kernels wih large variance do no creae arbirage opporuniies, hey are neverheless suspicious and should be discarded. Cochrane and Saá-Requejo (2000 compue he upper and lower bounds for he price of a coningen claim when a variance bound is imposed on he kernels. Our conribuion is o eend he invesigaion of he dualiy beween invesmen sraegies and pricing kernels from a single period o several consecuive rading periods. A pricing kernel from ime o horizon is a random variable m which saisfy he equaliy (2 E s m w = R f s E s m w s, for every inermediae period s beween and and for every self financing porfolio whose value varies from w s o w beween ime s and horizon. We denoe here Rs f he risk free rae from s o. he ime dimension of his dualiy has so far been limied o he descripion of he informaion se implici in he condiional epecaion of Equaion 1. We generalize he variance bound of Hansen and Jagannahan (1991 o a muliperiod seing by showing ha he sandard deviaion of he ineremporal marginal rae of subsiuion over a span of rading periods is larger han he opimal Sharpe raio available over he corresponding invesmen horizon hrough dynamic self financing sraegies. Every invesmen span gives rise o a differen variance bound, and i is legiimae o epec a sharper resricion on he pricing kernels han he one which resuls from a single rading period. he asse pricing resuls in he lieraure which follow from resricions on he pricing kernels are derived hrough a repeaed use of a single period analysis. his is for insance he case in boh Bernardo and Ledoi (2000 and Cochrane and Saá-Requejo (2000 who 3

4 compue coningen claim price bounds recursively. hey canno deal wih a consrain on he kernels which is defined over several periods and which canno be wrien as a succession of consrains on he one period ineremporal marginal raes of subsiuion. An imporan eample of such a consrain is he observaion of he curren marke prices of a se of new securiies on op of he original ones. hese quoes may be he only informaion available abou he price process of he new securiies, and i is logical o resric he kernels o he ones which agree wih hem. If he payoffs of he new securiies span several rading periods, his consrain canno be wrien in a convenien ime separable way. Our muliperiod analysis handles hese consrains and allows us o ehibi he sharper variance bounds which hey generae. We propose a heory of pricing kernels in a general dynamic invesmen environmen. We describe he srucure of he pricing kernels which are consisen wih he sochasic evoluion of a finie number of securiies. Equaion 2 shows ha he se of pricing kernels is he dual of he se of he self financing porfolios which inves in hese securiies. We show ha he pricing kernel wih minimum condiional variance over a span of rading periods is he unique kernel which is also he final value of a self financing porfolio. his invesmen sraegy happens o be dynamically mean-variance efficien. he analysis of his dualiy yields a number of resuls, boh on he pricing kernels and on he dynamic invesmen sraegies. As eplained above, posiive pricing kernels allow o derive he price dynamics of new insrumens in an arbirage free framework. his echnique is also ofen described as he choice of a risk neural probabiliy disribuion in which discouned securiy prices are maringales. he new insrumens can for insance be derivaives wrien on he original securiies. We ake a parial equilibrium poin of view and we assume ha he new securiies have no effec on he dynamics of he original ones. he inroducion of addiional insrumens may herefore only enhance he efficien fronier available hrough dynamic rading. his increase in efficiency depends on he price dynamics of he new insrumens. We show ha if he price process followed by he new insrumens is derived from a pricing kernel consisen wih he original securiies, hen he increase in he opimal dynamic Sharpe raio is a funcion of he een o which he new insrumens help dynamically replicae he kernel. he maimum gain in efficiency is obained once he kernel is perfecly replicaed wih boh he original and he addiional securiies so ha i becomes he final 4

5 value of a self financing sraegy. he maimum dynamic Sharpe raio is hen he sandard deviaion of he pricing kernel. his also proves ha he sandard deviaion of a given pricing kernel is an upper bound o he dynamic Sharpe raio which can be reached hrough dynamic self financing sraegies which inves in a arbirarily large number of insrumens, provided ha he price process of hese insrumens is derived from he given kernel. Once he pricing kernel is perfecly replicaed, no more mean-variance efficiency gain may be epeced from he inroducion of new securiies and he sraegy which replicaes he kernel belongs o he enhanced efficien fronier. If we use a pricing kernel which is already he final value of a self financing sraegy based on he original securiies in he firs place, hen no efficiency gain is possible righ from he sar. his means ha every new insrumen is priced by his kernel in such a way as o be useless for he consrucion of a dynamically mean-variance efficien sraegy. he pricing kernel wih minimum variance is he only kernel enjoying his propery. his special kernel corresponds herefore o a min-ma in erms of dynamic Sharpe raio. Cochrane and Saá-Requejo (2000 have proposed o eliminae dynamics which creae good deals, where hey define a good deal as an invesmen sraegy wih a large insananeous Sharpe raio. he minimum-variance kernel eends his mehodology o an ineremporal Sharpe raio. I generaes conservaive dynamics which do no allow any increase in Sharpe raio, hereby eliminaing good deals in a dynamic sense. Besides is inerpreaion in erms of porfolio managemen, he minimum-variance pricing kernel has received aenion in he finance lieraure for anoher relaed issue: he variance-opimal hedging of a coningen claim. Schweizer (1995 derives he price of a coningen claim from he cos of is opimal replicaion by means of self financing sraegies. Opimaliy is measured by a quadraic loss funcion. his price happens o be idenical o he one derived from he minimum-variance pricing kernel. he imporance of he varianceopimal hedging sraegy is highlighed by he fac ha every pricing kernel can be wrien as he variance opimal hedge residual of a coningen claim. We prove ha he cos of he variance-opimal hedge of a securiy does no change as new hedging insrumens are inroduced, as long as hese insrumens are hemselves priced according o he cos of heir variance-opimal hedge, ha is if heir price dynamics is derived from he minimum-variance pricing kernel. We ne invesigae he siuaion where, on op of he original securiies, he curren marke prices of a se of addiional securiies are available. hese new insrumens could 5

6 ypically be a se of acively raded calls and pus wrien on he original securiies. In line wih he opion pricing lieraure, we shall someimes refer o he collecion of hese prices as a smile. We illusrae he significance of his siuaion by considering wo dynamic invesmen problems, he dynamic managemen of a porfolio on he one hand, and he pricing and hedging of a coningen claim on he oher hand. We consider firs a fund manager who rades in a finie number of securiies and who considers invesing in derivaive insrumens wrien on hem. Markes are fricionless and perfecly compeiive and we assume ha he manager knows he price dynamics of he underlying securiies. Alhough she observes he prices of all raded securiies every period, she does no know he fuure price dynamics of he derivaive insrumens. he manager could for insance be an equiy porfolio manager who is considering invesing in converible bonds wrien on he shares in which she is rading. he manager faces several inerconneced quesions. Which derivaives should she selec? Which price dynamics will hey follow? How should she opimally manage her porfolio wih he new insrumens? Which performance gain can she epec from epanding her invesmen scope? Consider now an invesmen banker who is seeking o price and hedge an eoic derivaive insrumen wrien on some underlying securiies. he banker knows he price process followed by he underlying securiies, and he observes he marke quoes of a se of acively raded derivaives wrien on hem, for insance vanilla calls and pus, bu he does no know heir price dynamics. he eoic derivaive is no acively raded and no marke price is readily available. he banker seeks o use he raded derivaives, ogeher wih he underlying securiies, in order o hedge he eoic insrumen. He is confroned wih several quesions, echoing he quesions raised by he fund manager. Which price dynamics will follow he raded derivaives? A which price should he deal in he eoic insrumen? Which is he bes replicaion sraegy, using boh he underlying securiies and he raded derivaives? In a complee marke seing, he quesions raised by boh he fund manager and he invesmen banker find immediae answers. For every derivaive insrumen, only one price dynamics is consisen wih absence of arbirage, and i is given by he value process of is eac replicaion sraegy. No performance gain can be epeced in he managemen of a porfolio by he inroducion of new securiies since he opporuniy se is no changed by he addiion of redundan securiies. here is no need eiher for he banker o hedge he eoic insrumen wih he raded derivaives since i is already perfecly replicaed wih 6

7 he underlying securiies. In an incomplee marke seing however, eac replicaion is ypically no possible and many price dynamics for he new insrumens may be consisen wih he observed marke quoes and he principle of absence of arbirage. An imporan quesion arises as o which raionale allows o reduce he choice among admissible price dynamics. We offer a raionale which answers he concerns of boh he fund manager and he invesmen banker. Following again he logic of limiing good deals in a dynamic sense, we characerize he kernel which yields a minimum increase in opimum Sharpe raio while agreeing wih he prices of he insrumens for which marke quoes are available. Drawing on he dualiy wih he dynamic porfolios, we describe he efficien invesmen sraegies which corresponds o his kernel. hey solve a ma-min problem in erms of dynamic Sharpe raio. hese sraegies keep a fied quaniy of every quoed insrumen, on op of an invesmen in he L 2 minimum porfolio for he original securiies. he consrain of maching he smile reduces he se of admissible pricing kernels and leads o a higher variance bound on he kernels. We describe his se and we show ha he increase in he variance bound is given by he disance, in he meric of he varianceopimal hedge residuals, beween he observed marke quoes of he insrumens and he cos of heir variance-opimal hedge. We show ha he pricing kernel which limis dynamic good deals while agreeing wih he smile is also opimal in erms of variance-opimal hedge for wo reasons. Firs i prices a coningen claim as close as possible o he cos of is variance-opimal hedge. Second his price is he iniial value of a consrained opimal replicaion sraegy. In boh cases, he consrained opimaliy corresponds o a min-ma where we consider he wors possible coningen claim. We show finally ha he coningen claim price derived from his kernel is equal o he value of he variance-opimal hedge of he claim, when he dynamic hedging sraegy uses boh he original securiies and he insrumens of he smile. he paper is organized as follows. Secions 2 o 4 describe he self financing porfolios and heir mean-variance properies. hey draw heavily on Henroe (2001 which provides an eensive accoun of he srucure of hese dynamic invesmen sraegies. Secion 5 sudies he srucure of he pricing kernels and generalizes he Hansen and Jagannahan (1991 variance bound o a muliperiod seing. Secion 6 eplains how o price addiional securiies in an incomplee marke seing while avoiding mean-variance good deals in a dynamic sense. I relaes he increase in he slope of he efficien fronier wih he een o which 7

8 he addiional securiies help replicae he kernel. Secion 7 sudies he pricing kernels and he price dynamics which are consisen wih he consrain of maching he marke quoes of a given se of securiies. We derive a lower bound o he variance of hese kernels and we describe he minimum increase in he opimum dynamic Sharpe raio implied by his consrain. his lower bound and his minimum are reached for a pricing kernel and an efficien dynamic sraegy which we describe in Secion 8. We propose his dynamics as a soluion o our wo invesmen problems in incomplee markes, he mean-variance managemen of a porfolio and he opimal replicaion of a coningen claim. 2 Dynamic Porfolios 2.1 Iniial Marke Srucure We consider a finie number n of underlying securiies raded in a fricionless and compeiive marke over a se of discree imes wih finie horizon. We inde he rading daes by he inegers beween 0 and a final horizon. Informaion is described by a filraion F def. = {F } 0 over a probabiliy space (Ω, F, P. hroughou he aricle, equaliies and inequaliies beween random variables are undersood o hold P almos surely. We denoe respecively EF and E F he epeced value and he condiional epecaion wih respec o F of a random variable F in L 1 (P. We le L 2 (P and L 2 (P ; RI n be respecively he space of random variables and random vecors in RI n which are boh measurable wih respec o F and in L 2 (P. If f is posiive and measurable wih respec o F, we define L 2 (P, f as he se of random variables F such ha f F belongs o L 2 (P. We define in he same way L 2 (P, f ; RI n for random vecors in RI n. We close his lis of echnical noaions by leing y denoe he usual scalar produc of wo vecors and y in RI n. An unspecified numeraire is fied every period and we le p be he vecor of prices of he n securiies in his numeraire a ime. We le d be he numeraire dividend disribued by he securiies a ime. he owner of one uni of securiy i a ime is eniled o receive he dividend d i +1 in numeraire he ne period. We le φ def. = (p + d be he cum-dividend price vecor of he securiies a ime. he vecor processes {p } 0, {d } 0, and {φ } 0 are adaped o he filraion F. We do no rule ou ha some securiy migh be redundan a some rading period and in 8

9 some sae of he world bu we do impose ha he law of one price holds. For he remainder of he aricle, we shall assume ha he following wo assumpions are saisfied. Assumpion 1 Prices and reurns of he securiies do no vanish. For every period beween 0 and and for every period s beween 1 and he price vecors p and φ s are P almos surely differen from he null vecor. Assumpion 2 Law of one price. For every period beween 0 and ( 1, and for every random vecors X and Y in RI n measurable wih respec o F, he equaliy φ +1 X = φ +1 Y implies p X = p Y. 2.2 Self Financing Porfolios A dynamic porfolio X saring a ime is a process in RI n adaped o F and indeed by ime s wih s ( 1, where X i s represens he number of unis of securiy i held in porfolio X a ime s. We le w(x be he value process of porfolio X, naurally defined by w s (X def. = p sx s for s ( 1 and we le w (X = φ X 1. We say ha a dynamic porfolio X saring a ime is self financing a ime s whenever w s (X = φ sx s 1 and ha i is self financing whenever i is self financing from ( + 1 o. We remark ha he definiion of he final value of he sraegy implies ha a dynamic porfolio is always self financing a ime. I is easily checked ha he law of one price implies ha wo self financing porfolios wih idenical final values a ime share he same value process. his propery will allow us laer o idenify wo such dynamic porfolios. Henroe (2001 characerizes he se of self financing dynamic porfolios saring a ime wih he propery ha heir final wealh a ime is in L 2 (P. Saving on noaion, we denoe X his se wih no eplici reference o since which we shall keep his final horizon consan hroughou our analysis. We also le w (X def. = {w (X ; X X } be he se in L 2 (P of erminal wealhs of porfolios in X. Besides he self financing condiion, no resricion is imposed on he value process of he porfolios a periods prior o he final horizon. Henroe (2001 builds a posiive process h by backward inducion from he final value h = 1 a ime. his process plays a cenral role in he descripion of he srucure of X, and more generally in he mean-variance analysis. I is closely linked o he noion of dynamic Sharpe raio and i can be inerpreed as a correcion lens for myopic invesors. 9

10 We denoe N + he Moore-Penrose generalized inverse of a symmeric mari N in RI n RI n. he mari N + is iself symmeric, commues wih N, and saisfies 1 (3 (4 NN + N = N, N + NN + = N +. If N is a random mari measurable wih respec o F, hen N + (ω is defined for every ω in Ω and N + is also measurable wih respec o F. Proposiion 1 he adaped process h defined by h equaion (5 ( def. h = p N + 1 p = 1 a ime and he backward def. wih N = E h+1 φ +1 φ +1 for 0 ( 1, is well defined, P almos surely posiive, and saisfies φ L 2 (P, h ; RI n for every period beween 0 and as soon as he following wo condiions are me: (a. φ L 2 (P ; RI n ; (b. d L 2 (P, h ; RI n for every period wih 0 ( 1. he following properies hen hold. (i. For every dynamic porfolio X X he process {h s w s (X 2 } s is a submaringale, ha is, for every period s wih s ( 1 we have h s w s (X 2 E s h s+1 w s+1 (X 2 E s h w (X 2. (ii. he se X is he se of self financing dynamic porfolios saring a ime such ha w s (X L 2 s(p, h s for every period s. (iii. he se w (X is closed in L 2 (P. Condiion (b of Proposiion 1 involves he variable h which is derived recursively hrough Equaion 5. he following lemma provides a sufficien condiion independen of h. 1 see heil (1983 for a general descripion of he Moore-Penrose inverse. 10

11 Lemma 1 If φ is an elemen of L 2 (P ; RI n (Condiion (a of Proposiion 1, hen d belongs o L 2 (P, h ; RI n for every period from 0 o ( 1 (Condiion (b of Proposiion 1 if one securiy, say Securiy k, pays no dividend and is such ha (p k /pk d is an elemen of L 2 (P ; RI n for every period from 0 o ( 1. For he remainder of he aricle, we assume ha Condiions (a and (b of Proposiion 1 are saisfied so ha he resuls of his proposiion apply. Assumpion 3 Condiions (a and (b of Proposiion 1 are saisfied. wo equaions will prove useful. For every period beween 0 and ( 1, (6 φ +1 = N N + φ +1, and he law of one price implies hen ha (7 p = N N + p. he process h acs as a weigh which regularizes he prices and he values of he self financing porfolios in X every period. Once we muliply hose processes by he square roo of h, hey all have finie second momens every period. Henroe (2001 shows ha he process h is he larges process wih value h = 1 a horizon having his regularizaion propery. 3 Opimal Replicaion his secion invesigaes he replicaion properies of he self financing dynamic porfolios. We firs show how o consruc a dynamic sraegy which bes replicaes a payoff F a horizon, saring from a wealh w a ime. he loss funcion which we choose a horizon is he norm of L 2 (P, which is well defined for he porfolios in X. We hen sudy he cos and qualiy of he opimal replicaion and we show ha he value process of he opimal soluion is unique. When he final payoff F is zero, we obain as a special case he minimum L 2 porfolio which is he hedging numeraire used by Gouriérou e al. (1998. We show ha are analysis can be eended o deal wih he opimal replicaion of securiies described by a sequence of coningen cash flows insead of a single final payoff. We inroduce ineres raes by mean of defaul free zero coupon bonds and we relae our work wih he concep of variance-opimal signed maringale measure inroduced in Schweizer (

12 3.1 Consrucion of an Opimal Replicaion he opimal L 2 replicaion of a coningen claim involves a miure of forward and backward equaions. We derive firs he cos of he opimal replicaion every period in a backward way, and we hen use his process in order o consruc he opimal replicaing sraegy hrough a forward equaion. Proposiion 2 For every period such ha 0 ( 1, for every funcion w in L 2 (P, h, and for every funcion F in L 2 (P : essinf E (F w (X 2 ( 2 = E F w (X,w,F = h (F w 2 + g. (8 X X s.. w (X = w def. F and g are defined by backward inducion by g = 0 and for s ( 1, F s g s def. = p sn s + E s h s+1 F s+1 φ s+1, def. = E s g s+1 + E s h s+1 Fs+1 2 E s hs+1 F s+1 φ s+1 N + s E s h s+1 F s+1 φ s+1. For every period s beween ime and ( 1, F s L 2 s(p, h s, g s is measurable wih respec o F s, g s L 1 (P, and g s 0. he dynamic porfolio X,w,F is defined recursively by (9 (10 X,w,F X,w,F s def. = h (w F N + p + N + E h +1 F +1 φ +1, def. = h s (φ sx,w,f s 1 F s N + s p s + N + s E s h s+1 F s+1 φ s+1, for ( + 1 s ( 1. he dynamic porfolio X,w,F wih iniial wealh w, and saisfies (11 ( 2 ( 2 E s F w (X,w,F = h s F s w s (X,w,F + gs for every period s beween and. belongs o X, sars a ime I is easily checked ha if F, F a, and F b are in L2 (P, if w, w a, and w b are in L 2 (P, h, and if γ is measurable wih respec o F wih γ F in L 2 (P and γ w in L 2 (P, h, hen (12 X,wa,F a + X,w b,f b = X,wa +wb,f a+f b, (13 γx,w,f = X,γw,γF. 12

13 I is clear from Proposiion 2 ha he opimizaion program essinf E (F w (X 2 is solved in X,F,F s.. X X wih g as opimal value. he iniial value F can be seen as he iniial cos of he bes replicaion sraegy, while g describes he qualiy of his replicaion. We remark ha he consrucion of boh F s and g s from F in Proposiion 2 is respecively linear and quadraic and does no depend on he saring ime as long as s. his allows us o consruc boh a linear operaor Q and a quadraic operaor G for every period beween 0 and from he space of random variables in L 2 (P o he space of random variables measurable wih respec o F such ha Q (F def. = F and G (F def. = g as defined recursively in Proposiion 2. his Proposiion shows ha Q (F belongs o L 2 (P, h while G (F is an elemen of L 1 (P. A ime, he operaors Q and G are rivially respecively he ideniy and he null operaor. We sill denoe G he bilinear operaor obained hrough polarizaion of G and defined by G (F a, F b = 1 2 ( G (F a + F b G (F a G (F b for F a and F b in L2 (P. Equaion 11 of Proposiion 2 wries ( 2 ( 2 G s (F = E s F w (X,w,F h s Q s (F w s (X,w,F for any iniial wealh level w in L 2 (P, h. In paricular for s = and w = Q (F we obain (14 so ha, by polarizaion, (15 ( 2 G (F = E F w (X,Q(F,F. G (F a, F b = E ( F a w (X,Q (F a,f a ( F b w (X,Q (F b,f b. he following lemma liss some properies of hese operaors which will be used hroughou our analysis. Lemma 2 Le s and be wo periods such ha s, le F be a random variable in L 2 (P, and le w be a random variable in L 2 (P, h. (i. For every dynamic porfolio X in X, Q s (w (X = w s (X. 13

14 (ii. G (F = 0 if and only if F w (X. For every dynamic porfolio X in X, G (w (X, F = 0. (iii. h s w s (X,w,0 Q s (F = E s w (X,w,0 F. (iv. For every dynamic porfolio X in X s, ( h s (Q s (F w s (X,w,F w s (X = E s F w (X,w,F w (X. 3.2 Uniqueness of he Opimal Replicaion he ne resul shows ha he opimizaion problem 8 of Proposiion 2 has a unique soluion, a leas in erms of value a ime, and herefore also in erms of value process. We recall ha we canno epec o obain a unique porfolio because we do no rule ou redundancy beween he securiies. Lemma 3 We consider a period beween 0 and ( 1, an iniial wealh w in L 2 (P, h, and a payoff F in L 2 (P. For every dynamic porfolio Y in X, he equaliy w (Y = w (X,w,F holds P almos surely on he se A (Y in F defined by A (Y def. = { ω Ω such ha: E (F w (Y 2 = h (Q (F w 2 + G (F and w (Y = w }. 3.3 L 2 Minimum Porfolio One special choice of final payoff F and iniial wealh w a ime will prove imporan, i is obained for F = 0 and w = 1/ h. Noice ha for his choice of iniial wealh h w = 1, which belongs o L 2 (P. We inroduce he simplified noaion X def. = X,1/ h,0 and ws def. = w s (X for s. Equaions 9 and 10 of Proposiion 2 show ha he self financing sraegy X is obained by invesing every period s beween and ( 1 he value w s in he porfolio h s N + s p s whose value a ime s is h s p sn + s p s = 1 and X s = h s w sn + s p s. he self financing condiion implies ha w s+1 = φ s+1 X s so ha (16 w s+1 = w sh s φ s+1n + s p s. 14

15 Proposiion 2 proves ha his sraegy yields a dynamic porfolio in X which saisfies (17 essinf E w (X 2 = E (w 2 = h (w 2 = 1. X X s.. w (X = 1/ h (18 Saemen (iii of Lemma 2 wih s = and w = 1/ h implies ha Q (F = 1 h E w F, which shows ha Q is a posiive operaor whenever w is iself posiive. If w s does no vanish a ime s beween and, we also have (19 Q s (F = 1 h s ws E s w F. 3.4 Replicaion of a Sequence of Cash Flows We generalize our analysis from a single payoff a horizon o a sequence of coningen cash flows every period up o. We consider a period beween 0 and ( 1 and we le f = {f s } +1 s be a sequence of cash flows from ( + 1 up o adaped o F. We say ha a dynamic porfolio X saring a ime finances he cash flow f s a ime s wih s ( 1 whenever w s (X = φ sx s 1 f s and ha i finances he sequence of cash flow f whenever i finances he cash flows f s from ( + 1 o ( 1. A he las period, we recall ha we have defined he final value of a dynamic porfolio X by he equaion w (X = φ X 1. We creae a one o one operaor θ f on he se of dynamic porfolios saring a ime as follows. For every dynamic porfolio X saring a ime, we le Y = θ f (X be he dynamic porfolio saring a ime defined by Y = X a ime and s (20 Y s = X s f u hu ws u h s N s + p s u=+1 for ( + 1 s ( 1. he following lemma yields some firs properies of his operaor. Lemma 4 Le X and Y be wo dynamic porfolios saring a ime such ha Y = θ f (X. (i. he porfolio X is self financing if and only if he porfolio Y finances he sequence of cash flows f. 15

16 (ii. w (Y = w (X and (f w (Y = (F w (X wih F = f s hs w s, where we le w = 1. s=+1 We remark ha he payoff F is obained a ime by invesing every cash flow of he sequence f in he L 2 minimum sraegy up o ime. We le X (f be he se of dynamic porfolios saring a ime which finance f and which end up a horizon wih a value in L 2 (P. he following proposiion proves he equivalence beween he variance-opimal replicaion of F hrough self financing porfolios in X and he L 2 opimal replicaion of he sequence f by means of dynamic sraegies in X (f. Some inegrabiliy condiion on he sequence f are needed for his resul. Proposiion 3 Le f = {f s } +1 s be a sequence of cash flows such ha f s belongs o L 2 s(p, h s for every period s from ( + 1 o. he payoff F = s=+1 f s hs w s is in L 2 (P and he mapping θ f is one o one from X o X (f. For every iniial wealh w in L 2 (P, h, we have essinf E (f w (Y 2 = essinf E (F w (X 2 Y X (f X X s.. s.. w (Y = w w (X = w and he firs program is solved in Y = θ f ( X,w,F. he opimal replicaion sraegies for he wo equivalen opimizaion programs of Proposiion 3 sar wih an idenical iniial wealh a ime equal o Q (F and lead o he same replicaion error described by G (F. he ne lemma eplains how boh he opimal hedging cos Q (F and he opimal hedging qualiy G (F can be direcly compued from he sequence f. Lemma 5 Le f = {f s } +1 s be a sequence of cash flows such ha f s is in L 2 s(p, h s for every period s from ( + 1 o and le F = s=+1 f s hs w s. We define he processes f = { f s } s and ḡ = {ḡ s } s from he sequence f by backward inducion as follows. 16

17 We le f def. = ḡ = 0, and f s ḡ s def. = p sn s + E s hs+1 ( f s+1 + f s+1 φ s+1, def. = E s ḡ s+1 + E s h s+1 ( f s+1 + f s+1 2 E s hs+1 ( f s+1 + f s+1 φ s+1 N + s E s hs+1 ( f s+1 + f s+1 φ s+1, for s ( 1. For every period s beween and we have s Q s (F = f u hu ws u + f s, u=+1 G s (F = ḡ s, wih he convenion ha s u=+1 f u hu ws u = 0 when s = so ha Q (F = f. 3.5 Ineres Raes We inroduce from now on a money marke. For he res of he aricle we assume ha Securiy 1 is a risk free zero coupon bond paying a unique dividend of one uni of numeraire a mauriy. Assumpion 4 For every period beween 0 and ( 1 he price p 1 of he zero coupon bond is posiive. R f Since he price of he bond is assumed posiive, we define R f def. = 1/p 1 for 0 ( 1. is he nominal risk free reurn from invesing in he zero coupon bond from ime up o horizon, for our choice of numeraire every period. his buy and hold sraegy belongs o X, we denoe i 1. We remark ha w s (1 = Q s (1 = p 1 s = 1/Rs f for s and we learn from Saemen (i of Proposiion 1 ha h (p 1 2 E h+1 (p We define def. H = h /(R f 2 so ha, wih his normalizaion, his las inequaliy wries H E H +1 1 and he normalized process H is a posiive submaringale. Noice ha we have essinf E w (X 2 = H, X X s.. w (X = p 1 = 1/R f 17

18 which is reached in H X. We derive from Saemen (iii of Lemma 2 wih F = 1 and w = 1/ h, (21 (22 E s w E w = h sw s = Rs f h R f for s, = H. We remark ha he sufficien condiion of Lemma 1 which requires ha (p k /pk d be in L 2 (P ; RI n holds wih k = 1 as soon as R f d is in L 2 (P ; RI n for every period beween 0 and ( 1. his is he case for insance if R f is bounded and d belongs o L 2 (P ; RI n. 3.6 Variance-Opimal Maringale Measure We have seen ha he operaor Q is posiive as soon as he final wealh w = w (X of he L 2 minimum porfolio X is iself posiive. We show ha when his happens, he cos Q s (F a ime s beween and ( 1 of he opimal replicaion of a payoff F in L 2 (P can be epressed as he discouned condiional epecaion of F in a probabiliy disribuion differen from he original probabiliy P. his new probabiliy disribuion is called he minimum variance probabiliy disribuion or he variance-opimal maringale probabiliy. We firs noice from Equaion 21 ha if w is posiive, hen he value w s of he sraegy X a ime s is also posiive. Saemen (iii of Lemma 2, ogeher wih Equaion 21, yields he following resul (23 Q s (F = 1 E s w F Rs f E s w. If f is a posiive random variable in L 1 (P, we denoe P f and E f he probabiliy disribuion and is corresponding epecaion operaor obained from he original probabiliy P by means of he posiive Radon-Nikodym derivaive f/ef. For every random variable F such ha ff is in L 1 (P we have E f F = EfF /Ef and E f F = E ff /E f. We use his consruc here wih f = w, which is in L1 (P. We obain Q s (F = 1 Rs f E w s F, which shows ha Q s (F can indeed be wrien as a discouned epecaion in he modified probabiliy disribuion P w. 18

19 One can usually no epec w o be posiive when he cum-dividend prices assume unbounded values. his fac has been noed in Schweizer (1995. When his happens, he minimum variance probabiliy becomes he variance-opimal signed maringale measure and he operaor Q, alhough sill well defined, is no posiive. In a coninuous ime seing, Gouriérou e al. (1998 shows ha w is always posiive as long as prices follow coninuous semimaringales wih no dividend disribuion. hey assume a no arbirage condiion which is more sric han he law of one price. 4 Mean-Variance Porfolio Selecion We summarize in his secion he mean-variance properies of self financing dynamic porfolios. We consider in his secion a ime period beween 0 and ( 1 and we sudy he noions of dynamic Sharpe raio and efficien fronier condiioned on he informaion a dae. For every dynamic porfolio X in X, we denoe SR (X he Sharpe raio condiioned on he informaion a ime, which resuls from following he self financing invesmen sraegy X from ime o horizon. We le SR (X def. = E w (X R f w (X Var w (X when Var w (X is non zero and we se SR (X def. = 0 whenever Var w (X = 0. We denoe R (X def. = w (X/w (X he oal reurn from period o horizon of a dynamic porfolio X in X wih non vanishing wealh w (X a dae. In paricular we have R f = R (1 when X = 1 is he sraegy which invess wihou rebalancing in he defaul free zero coupon bond wih mauriy from ime on. If w (X and Var w (X are P almos surely differen from zero, we also have SR (X = E R (X R f Var R (X, he usual definiion of a Sharpe raio. Our definiion of reurn is no innocuous. he choice of non annualized simple oal reurn allows us o bring ogeher in an common framework he heories of dynamic replicaion and of dynamic mean-variance analysis. his nice convergence may no hold for oher specificaions of he reurns. 19

20 We le he dynamic mean-variance efficien fronier a ime wih horizon, which we denoe EF, be he se of porfolios in X which are soluion o he opimizaion program essinf Var R (X X X s.. w (X = w E R (X = R for some epeced reurn arge R measurable wih respec o F and some posiive iniial wealh w in L 2 (P, h. Henroe (2001 shows ha he opimum dynamic Sharpe raio from ime o horizon, condiioned on he informaion available a ime, wries ( def. 1 SR = 1 H and esssup SR (X 2 = SR (X 2 = (SR 2. s.. X X he opimum dynamic Sharpe raio obains for he porfolios on he efficien fronier EF. Under some regulariy condiion, every efficien porfolio on EF can be idenified wih a combinaion of he sraegy X and he zero-coupon bond wih mauriy. In paricular he sraegy X belongs o he efficien fronier EF. 5 Pricing Kernels We le PK be he se of pricing kernels corresponding o he dynamics of he underlying securiies from period unil he horizon. I is defined as he se of random variables m in L 2 (P such ha (24 E s m w (X = R f s E s m w s (X, for every period s beween and and for every dynamic porfolio X in X s. We do no require any a priori posiiviy condiion on he pricing kernels and PK is a vecor subspace of L 2 def. (P. For every period s beween and we denoe m s = E s m he condiional epecaion a ime s of a pricing kernel m in PK and we define PK 0 def. = {m PK such ha m = 0}. 20

21 he following lemma proves ha a variable m in L 2 (P is a pricing kernel if and only if i prices correcly he n securiies from one rading period o he ne. Lemma 6 A random variable m in L 2 (P is a pricing kernel in PK if and only if for every period s beween and ( 1. E s R f s+1 m s+1φ s+1 = R f s m s p s 5.1 Srucure of Pricing Kernels he ne proposiion describes he srucure of he se PK of pricing kernels. he noion of condiional orhogonaliy will be useful. We say ha wo random variables f and g respecively in L 2 (P and L 2 (P are condiionally orhogonal a ime if and only if E fg = 0. If A is a subse of L 2 (P, we le A be he se of random variables in L 2 (P condiionally orhogonal a ime wih every random variable in A. Proposiion 4 For every period and s such ha 0 s ( 1, (i. w PK and PK is herefore no reduced o zero. (ii. PK 0 = w (X. (iii. Every pricing kernel m in PK saisfies m s Hs = h s Q s (m so ha m s belongs o L 2 s(p, 1/ H s. (iv. PK is he se of random variables m in L 2 (P which can be wrien m = m 0 + ξ w for some random variables m 0 in PK0 and ξ in L 2 (P. In oher words ( PK = w (X + L 2 (P w where L 2 (P w def. = { ξ w wih ξ L 2 (P }. (v. PK w (X = L 2 (P w. 21

22 {( (vi. PK = PK 0 = F w (X,w,F {( F w (X,Q (F,F wih F L 2 (P and w L 2 (P, } h } wih F L 2 (P (vii. he wo ses PK 0 and PK are closed in L 2 (P and w (X =. and ( PK 0. Saemen (i proves ha w is a pricing kernel and Saemen (iv shows ha every oher pricing kernel in PK can be decomposed as he sum of his pricing kernel and a pricing kernel condiionally orhogonal o he final values of he dynamic porfolios. Saemen (v proves ha he final value of he L 2 minimum porfolio X, possibly normalized by an iniial value ξ in L 2 (P, is he only pricing kernel which is also he final value of a dynamic porfolio in X. Saemen (vi proves ha a pricing kernels is he minimum-variance hedge residual of some coningen claim. F = 0 and w = 1/h. he L 2 minimum porfolio obains for insance for he ne lemma describes how a pricing kernel in PK evaluaes a payoff in L 2 (P possibly ouside w (X. Lemma 7 We consider wo periods and s such ha 0 s, a pricing kernel m in PK and a payoff F in L 2 (P, we have (25 E s m F = R f s m s Q s (F + G s (m, F. We remark ha when F is in w (X and wries w (X, Equaion 25 is Equaion 24, since Q s (w (X = w s (X and G s (m, w (X = 0, according o Saemens (i and (ii of Lemma Variance Bounds on Pricing Kernels We have seen in Secion 4 ha he dynamic porfolio X is mean-variance opimal wihin X. he ne proposiion shows ha w is also L2 opimal wihin PK. his resuls yields ineremporal bounds on he variance of he pricing kernels. Proposiion 5 We consider wo periods and s such ha 0 s. Every pricing kernel m in PK saisfies he equaliies (26 E s m 2 = m2 s + G s (m, H s (27 Var s m = m 2 s(sr s 2 + G s (m. 22

23 In paricular every pricing kernel m in PK saisfies he inequaliies (28 (29 m 2 s E s m 2, H s m 2 s(sr s 2 Var s m. Inequaliies 28 and 29 become equaliies if m = ξ w wih ξ in L 2 (P, and, for s =, if and only if m = ξ w wih ξ in L 2 (P. I resuls from Proposiion 5 ha if ξ is a random variable in L 2 (P, hen he pricing kernel ξ w solves essinf E m 2 = ξ 2. (30 m PK s.. m = H ξ We also derive ha if m is a random variable in L 2 (P, 1/ H, hen he pricing kernel ( m / H w solves essinf Var m = m 2 SR 2. (31 m PK s.. m = m Inequaliy 29 provides a series of variance bounds on pricing kernels wihin PK for every inermediae period s beween and as a funcion of he opimal dynamic Sharpe raio beween s and horizon. Equaion 27 idenifies he disance o he bound o he qualiy of he replicaion of he kernel, as measured by G s (m. Ecess variance of a pricing kernel is due o is componen which is condiionally orhogonal o he space w (X of payoffs which can be reached hrough self financing sraegies. When he kernel is in w (X, as i is he case for w, he replicaion is perfec and he inequaliy becomes an equaliy. 6 Eension of he Invesmen Scope he analysis of he self financing porfolios and heir pricing kernels which we developed so far will help us now ackle a cenral issue in incomplee markes. We sudy he implicaions of selecing a price process for some derivaive insrumens in a way which is consisen wih he dynamic behavior of heir underlying securiies. We focus on wo relaed invesmen problems, he dynamic managemen of a porfolio on he one hand, and he opimal hedging 23

24 of a coningen claim on he oher hand. We deal in his secion wih basic issues and we pospone unil he ne one he analysis of he addiional consrain imposed by a smile. In addiion o he original n securiies, we consider n new securiies which disribue some numeraire dividends every period described by he vecor process {d } 1 ( 1. For every period beween 0 and ( 2, he owner of one uni of securiy j a ime, receives he ne period he quaniy d,j +1 of numeraires as dividend. A ime ( 1, one uni of securiy j gives righ o he final payoff φ a ime. One can hink of φ as he sum of a dividend and a residual value. We assume ha he dividend process {d } 1 ( 1 and he final payoff φ are given and known. We furher assume ha he dividend process is adaped o F, ha he final payoff φ is a random vecor in L2 (P ; RI n, and ha for every period beween 1 and ( 1 and for every inde j he dividend d,j belongs o L 2 (P, h. hese new insrumens may be for insance derivaives wrien on he original securiies, in which case he dividends and he final payoff are funcions of he prices of he original securiies. We do no however limi ourselves o his special siuaion. We consider a period beween 0 and ( 1 and we le he vecor processes in RI n {p s} s ( 1 and {φ s} s ( 1 be respecively he e and cum dividend price dynamics of he new securiies beween and ( 1. We say ha his price dynamics saring a ime is admissible if i is adaped o F, if φ s = (p s + d s every period, and if i saisfies he law of one price ogeher wih he prices of he original n securiies. In line wih Assumpion 2, his las requiremen means ha for every period s beween and ( 1 and for every vecor (u, v in RI n RI n measurable wih respec o F s, he equaliy φ s+1 u + (φ s+1 v = 0 implies p su+(p s v = 0. I is a weak noion of absence of arbirage, he minimum srucure which we need in order o apply our dynamic mean-variance analysis. We shall limi our invesigaions o admissible price dynamics for he new securiies. Le us remark ha our approach is purely parial equilibrium. We do no sudy for insance how he inroducion of he new securiies changes he price dynamics of he original ones, a comple and fascinaing quesion. he denominaion original and new securiies is herefore somewha misleading, i is only a convenien way o describe he eension of he invesmen scope. 24

25 6.1 Admissible Price Dynamics A firs quesion is he eisence and he consrucion of an admissible price dynamics for he new securiies. he following lemma shows ha an admissible price dynamics may be derived from a posiive pricing kernel for he original securiies. I is well known ha such a posiive kernel prevens he eisence of arbirage opporuniies, as would resul from any violaion o he law of one price. he proof of his lemma, lef in he Appendi, is a sraighforward applicaion of Lemma 6. Lemma 8 Le m be a posiive pricing kernel in PK. he processes {p s, φ s} s ( 1 defined by he backward equaions (32 p s = E s R f s+1 m s+1φ s+1 φ s = p s + d s, form an admissible price dynamics for he new securiies. / (R s f m s, Even when no posiive kernel is available, and in paricular even if we do no know if w is posiive, i is possible o creae an admissible price dynamics for he new securiies. Lemma 9 he processes { p s, φ s} s ( 1 in RI n n by he backward equaions p,j s = p sn s + E s h φ,j s+1 s+1 φ s+1, φ,j s = p,j s + d,j s, form an admissible price dynamics for he new securiies. defined for every inde j from one o We remark ha if we describe he dividends saring from ( + 1 and he final payoff of securiy j as a sequence of cash flows wih f = {f s } +1 s defined by f s = d,j s for s beween ( + 1 and ( 1 and f = φ,j, hen he process { p,j s } s ( 1 coincides wih he process { f s } s ( 1 defined in Secion 3.4. In paricular if we le (33 F def. = 1 s=+1 hs w s d s + φ, hen we learn from Lemma 5 ha p,j = Q (F,j. his means ha p,j s represens he cos a ime s of he opimal replicaion of he sequence of cash flows generaed by new securiy j from (s + 1 up o horizon. 25

26 We also remark ha for every pricing kernel m in PK we have (34 E m F = E 6.2 Eended Asse Srucure 1 s=+1 Rs f m s d s + m φ. For an admissible price dynamics {p s, φ s} s ( 1, we consider he eended asse srucure beween ime and horizon which consiss in he n original securiies ogeher wih he n payoffs priced according o he dynamics {p s, φ s} s ( 1. We denoe p e s and φ e s he corresponding e and cum dividend prices a ime s. he firs n componens of he vecors p e s and φ e s are respecively p s and φ s while heir las n componens are respecively p s and φ s. he eended asse srucure saisfies boh Assumpions 1 and 2 and Condiions (a and (b of Proposiion 1. he zero coupon bond is a securiy of he original asse srucure and i remains raded in he eended one. he resuls of Secions 2 o 5 can herefore be brough o bear, wih period corresponding o he iniial rading period 0 in hese secions. We se h e = h = 1 and for s ( 1, we le h e s, H e s, X e s, Q e s, G e s, PK e s, X s,e, w s,e, SR e s be he counerpars o h s, H s, X s, Q s, G s, PK s, X s, w s, SR s for he eended asse srucure. Noice ha for every period s beween and ( 1 we have H e s and SR e s def. = ( 1 H e s 1. def. = h e s/(rs f 2 I is clear ha PK e s is a subse of PK s. he ne lemma, anoher direc applicaion of Lemma 6, shows ha a necessary and sufficien condiion for a pricing kernel in PK o belong o PK e is o price correcly he new securiies. Lemma 10 Le m be a pricing kernel m in PK. he following hree saemens are equivalen. (i. m belongs o PK e. (ii. For every period s beween and ( 1, E s R f s+1 m s+1φ s+1 = Rs f m s p s. 26

27 (iii. For every period s beween and ( 1, Rs f m s p s = E s 1 u=s+1 Rum f u d u + m φ. If m is a pricing kernel in PK e for he eended asse srucure hen we derive from Saemen (iii of Lemma 10 and Equaion 34 ha (35 E m F = R f m p. 6.3 Sharpe Raio Improvemen he opimum dynamic Sharpe raio may only increase as a resul of he eension of he invesmen se, which means ha for every period s beween and ( 1 we have SR s SR e s and H e s H s. he following resul quanifies his increase in erms of pricing kernels, i is a direc applicaion of Equaion 27 o he eended asse srucure. We recall ha w,e is he value a ime of he L 2 minimum porfolio X,e in he se of self financing sraegies X e for he eended asse srucure. Resul 1 For every pricing kernel m in PK e and for every period s beween and ( 1, m 2 s (SR e s 2 (SR s 2 = G s (m G e s(m and in paricular (SR e 2 (SR 2 = G (w,e H e. Resul 1 ells us ha he opimum dynamic Sharpe raio increases inasmuch as he pricing kernels for he eended asse srucure are beer replicaed wih he help of he new securiies. he increase in he square of he Sharpe raio is also direcly relaed o he disance beween he L 2 minimum porfolio for he eended asse srucure and he final values of he self financing sraegies based on he iniial securiies, as measured by G (w,e. If, as in Lemma 8, a posiive pricing kernel m is used in order o generae he price dynamics of an increasing number of new insrumens, hen he Sharpe raio increases as 27

28 long as G e s(m decreases and he new insrumens help replicae he kernel. Once enough insrumens have been inroduced so ha m is perfecly replicaed, he opimum dynamic Sharpe raio ceases o increase as new insrumens are added. he opimum dynamic Sharpe raio from s o reaches hen is maimum possible value given by (SR e s 2 = Var s m /m s. Wih no clear indicaion on which pricing kernel o choose, a fund manager runs he risk of picking a kernel wih oo large a variance, leading o large poenial increases in performance for some carefully seleced new insrumens. We ne invesigae he conservaive siuaion which corresponds o a min-ma in erms of dynamic Sharpe raio. We sudy he admissible price dynamics which yields he lowes possible increase in Sharpe raio for he corresponding opimal dynamic sraegy. Wihou any smile consrain, i is possible o avoid any mean-variance good deal alogeher. 6.4 Absence of Good Deal We consider an admissible price dynamics {p s, φ s} s ( 1 and he eended asse srucure which i generaes from ime up o horizon. he following proposiions characerize he siuaion where no gain in dynamic Sharpe raio may be epeced from rading in he new securiies. Proposiion 6 SR e = SR if and only if he following equivalen condiions hold. (i. w,e = w. (ii. w belongs o PKe. When his happens, p = p. If no good deal is available from o, i seems inuiive ha no good deal should eis beween a laer rading dae s and. We only prove his fac for he periods s such ha he value w s of he L 2 minimum sraegy does no vanish. Proposiion 7 If SR e = SR, hen a every period s beween and ( 1 such ha w s does no vanish we have SR e s = SR s. 28