The numéraire portfolio, asymmetric information and entropy

Transcription

1 The numéraire porfolio, asymmeric informaion and enropy Peer Imkeller Insiu für Mahemaik Humbold-Universiä zu Berlin Uner den Linden Berlin Germany Evangelia Perou Ab. Wahrscheinlichkeisheorie und Mah. Saisik Universiä Bonn Endenicher Allee Bonn Germany 15 November, 211 Absrac We sudy he relaion beween differen forms of non-exisence of arbirage and he characerisics of he sochasic basis under he differen filraions. This is achieved hrough he analysis of he properies of he numéraire porfolio. Furhermore, we focus on he problem of calculaing he addiional logarihmic uiliy of he beer informed invesor in erms of he Shannon enropy of his addiional informaion. We show ha he expeced logarihmic uiliy incremen due o beer informaion equals is Shannon enropy also in case of a pure jump basis wih jumps ha are quadraically hedgeable, and so exend a similar resul known for bases consising of coninuous semimaringales. 21 AMS subjec classificaions: primary 6 H 3, 91 G 1; secondary 6 G 48, 94 A 17. Key words and phrases: free lunch; arbirage; equivalen maringale measure; numéraire porfolio; special semimaringale; insider rading; asymmeric informaion; enlargemen of filraions; logarihmic uiliy; Shannon enropy. 1 Inroducion One of he fundamenal quesions in mahemaical finance is he exisence of arbirage in he marke. In markes generaed by semimaringales, he mos common no arbirage concep, no free lunch wih vanishing risk NFLVR), is shown, in [7 and [8, o be equivalen wih he exisence of an equivalen sigma maringale measure. Under his measure he dynamics of he marke asses S discouned by a risk free bond B are seen o be sigma maringales. A number of less resricive conceps of arbirage have since been inroduced, wih he laes being he noion of no unbounded profi wih bounded risk NUPBR), see [11. From a mahemaical poin of view, he requiremen of he exisence of an equivalen sigma maringale measure under NFLVR) is weakened in a NUPBR) marke by he exisence of a process W, such ha any possible porfolio in he marke ha is discouned by W is a supermaringale under he original marke measure. This process is called he numéraire and is exisence, as is shown in [11, is linked o he characerisic riple of he sochasic basis of he marke. The firs aim of his paper is o find and explici link beween he characerisic riple of he underlying semimaringales and he differen noions of arbirage in he marke. Assuming 1

2 a weak form of he srucure condiion, firs inroduced in [13, and under addiional ye necessary) condiions, we obain a represenaion of he underlying semimaringales ha depends on he marke price of risk or informaion drif). Under his represenaion we show ha he exisence of arbirage depends, firsly on he jump srucure of he semimaringales and secondly on he inegrabiliy of he process describing he marke price of risk. Having sudied his link, we exend our resuls o markes wih asymmeric informaion. Furhermore, we sudy he uiliy advanage ha a beer informed invesor may have in erms of he underlying marke srucure, and inerpre i as in [2 by enropy noions such as he Shannon enropy in he case of logarihmic uiliy. In conras o previous work his is o be achieved in a seing as general as possible, in he sense of [11. The paper is organised as follows. The marke se-up is explained in Secion 2. The special semimaringales playing he role of underlying price dynamics for he marke are discussed in Secion 3. The link beween characerisics of he underlying semimaringales and he differen noions of arbirage, i.e. he exisence of he numéraire, is sudied in Secion 4. The advanages of our analysis is illusraed in he examples of Secion 4.1, in which he explici form of he numéraire porfolio can be given. Wih he ensuing descripions especially of he relaed numéraire porfolios, we discuss in Secion 5 he srucure of he informaion drif of an iniially enlarged) filraion G, and herefore he expeced logarihmic uiliy advanage of he beer informed invesor. We are able o idenify he exra expeced logarihmic uiliy in a purely disconinuous seing, in which he squares of he jumps are hedgeable, wih he Shannon enropy of he addiional informaion, hereby exending his sriking equaliy beyond he coninuous case, see [3, [1. Inroducory remarks and noaion In he analysis hereafer he noaion and resuls on semimaringales are based on [1. Le Ω, F, F, P) be a complee probabiliy space, where F = F R+ saisfies he usual condiions. Wih PR d ) we denoe he se of R d valued predicable processes in he given probabiliy space. For any adaped càdlàg process X we define he jump process X = X X, where X denoes he lef-hand limi of X. Le π PR d ) and Y be a d dimensional semimaringale, hen π Y = πdy denoes he sochasic inegral whenever his is well defined. Furhermore, we define he quadraic covariaion process of wo semimaringales X, Y as [X, Y = XY X Y Y X. If X, Y are locally square inegrable maringales, hen [X, Y is locally inegrable and has a predicable compensaor X, Y. Lasly, for a semimaringale X saring a zero, wih EX) we denoe he Dolean-Dade exponenial. The exponenial has he form EX) = exp X 1 ) 2 [Xc, X c Π s 1 + X s i) exp X s ), where X c denoes he coninuous par of he process, and saisfies he inegral equaion Z = 1 + Z Y. 2 Marke se-up We work in a marke characerized by a complee probabiliy space Ω, F, F, P), where F = F [,T saisfies he usual condiions and he ime horizon T is finie. The marke consiss of a risk free asse S and d risky asses S 1,..., S d. Wih no loss of generaliy we assume 2

3 ha S 1,..., S d are sricly posiive semimaringales and S = 1. Therefore we may sae ha for every i here exiss a semimaringale X i, wih X i = and Xi > 1 such ha S i = S i EX i ), wih S i >. Wih he d+1 asses of our marke we creae a porfolio W π, where π denoes he invesmen sraegy. As in [11, we impose a credi limi in order o avoid doubling sraegies. This limi is a uniform lower bound on he wealh process W π, which we se equal o zero, i.e. we impose W π >. Furhermore we normalize he iniial value W = 1. Le X = X 1,..., X d ), π = π 1,..., π d ), wih π i denoing he proporion of he porfolio value invesed in asse i, i = 1,..., d, and π = 1 d i=1 πi denoing he proporion of he porfolio invesed in he risk free asse. Then he dynamics of he porfolio saisfy he equaion dw π W π = d i=1 π i ds i S i = π dx, hence W π = Eπ X). For he laer o make sense he inegral π dx has o be well defined. Furhermore from he credi limi W π has o be posiive. For hese he se of admissible porfolios, denoed by W, is defined as W = W π = Eπ X) π LX) and π X > 1, wih LX) denoing he se of R d valued predicable processes ha are inegrable wih respec o X. Of specific ineres are he admissible porfolios ha ouperform any oher porfolio in W. More precisely we focus on he porfolios inroduced by he following definiion. Definiion 2.1 i). An admissible porfolio W π is called he numéraire porfolio, if he process W ρ W π is a supermaringale for every W ρ W. ii). An admissible porfolio W π is called relaive) growh opimal GOP), if [ W ρ ) T E log WT π for all W ρ W. iii). An admissible porfolio W π wih E[ln W π < is called log-uiliy-opimal porfolio if E[ln W ρ E[ln W π for every W ρ W. The exisence and properies of hese opimal porfolios is closely relaed o he exisence of differen forms of arbirage in he marke, ha are presened in he following definiion. Definiion 2.2 [11 We consider he following ypes of arbirage. i). A porfolio W π W is said o generae an arbirage opporuniy, if i saisfies P [WT π 1 = 1 and P [WT π > 1 >. If such a porfolio does no exis, we have no arbiragena). 3

4 ii). A sequence W πn ) n N of admissible porfolios is said o generae an unbounded profi wih bounded risk UPBR), if he collecion of posiive random variables W πn T ) n N is unbounded in probabiliy, i.e. if lim sup P [W πn m T > m >. n N If such a sequence does no exis, we say ha here is no unbounded profi wih bounded risk NUPBR). iii). A sequence W πn ) n N of admissible porfolios is said o be a free lunch wih vanishing risk FLVR), if here exis an ɛ > and an increasing sequence δ n ) n N wih δ n 1, such ha P [W πn T > δ n = 1 as well as P [W πn T > 1 + ɛ ɛ. If such a sequence does no exis, we say ha here is no free lunch wih vanishing risk NFLVR). iv). An admissible porfolio W π is said o generae an unbounded increasing profi if he wealh process is increasing, i.e., if P [Ws π W π, s < T = 1, and if P [WT π > 1 >. If such a porfolio does no exis, no unbounded increasing profi NUIP) is said o hold. The connecion beween he differen forms of arbirage is an ineresing subjec by iself, however for our purposes i suffices o consider only heir hierarchical ordering. According o [11 and [7, we can sae ha NUIP) is a weaker noion han NUPBR) and NA), which in urn are weaker noions han NFLVR). Furhermore, NFLVR) holds if and only if NUPBR) and NA) hold. However here is no apparen connecion beween NA) and NUPBR). The link beween he opimal porfolios and differen forms of arbirage, which has been sudied in he aforemenioned papers, has been summarised in [9, from where we have he following heorem, modulo some changes ha fi our noaion. Theorem 2.1 For an R d -values semimaringale S, he following are equivalen: i). S saisfies NUPBR) ii). The numéraire porfolio exiss. iii). The growh-opimal porfolio exiss. Furhermore, he numéraire and he growh-opimal porfolio are unique and idenical. In he case ha supe[log WT π W π W wih E[log WT π < <, he above saemens are equivalen o iv). The log-uiliy opimal porfolio exiss. Furhermore i is unique and idenical o he numéraire growh opimal porfolio. Remark 2.1 In his secion we sared by describing he asses in he marke as semimaringales. This assumpion can be omied in markes generaed by coninuous price dynamics under he condiion of finie logarihmic uiliy. In his seing i is proven by [3 ha for simple buy and hold sraegies, finieness of he logarihmic uiliy implies ha he coninuous processes in he marke are semimaringales, wih no assumpion on he exisence of arbirage. [12 elaboraes on his by showing ha finie uiliy no only implies ha S is a semimaringale for any admissible rading sraegy, bu ha i also has a canonical decomposiion of he form S = M + α M, where M is a local) maringale and α a square inegrable 4

5 predicable process. Furhermore i is proven ha here exiss a GOP ha is given by W α, i.e. by invesing on S according o he sraegy α. Hence from Theorem 2.1 we can conclude ha finieness of he logarihmic uiliy in his marke implies NUPBR), or even NFLVR) if S saisfies some furher echnical condiions. However hese nice properies do no ranslae o he non-coninuous seing, as is illusraed in [12 by a counerexample. The auhors show ha finieness of logarihmic uiliy no only does no imply a decomposiion for he process as saed before, bu no even ha he semimaringale propery of he underlying process S holds. 3 Semimaringale decomposiion Having inroduced he seing of he marke, in his secion we urn our aenion o he dynamics of he underlying semimaringale. More specifically we inroduce new assumpion ha allow us o reach an explici form of he numéraire. 3.1 Marke price of risk We assume ha X is a d-dimensional special semimaringale, wih he unique represenaion X = M + L, 1) where M = M 1,..., M d ) is a d-dimensional local square inegrable maringale and L = L 1,..., L d ) is a d-dimensional predicable process wih finie variaion. Hence [X, X is locally inegrable, and he predicable process X = M is well defined. In he case of a marke generaed by a coninuous semimaringale X, he exisence of arbirage is closely linked o he properies of he process L. A number of papers deals wih his subjec. In [7 he auhors prove ha if NFLVR) holds hen X is a semimaringale and dl i d X i for 1 i d. If heres exiss a d dimensional process α in L 2 X) such ha dl = α dx, he marke saisfies he srucure condiionsc), see [14. In he case of coninuous semimaringale i has been proven in [9, ha NUPBR) is equivalen o he SC). In he more general seing of disconinuous semimaringales, a weaker form of SC) is necessary for NUPBR) bu no sufficien. This is denoed by SC ) and differs from SC), in ha α is assumed o only be a predicable d dimensional process, see [9. In order o illusrae his argumen we need o inroduce he noion of immediae arbirage. The definiion we provide is a sligh modificaion of he one in [11, ha fis our seing. Definiion 3.1 A sraegy ξ is called an immediae arbirage opporuniy, if for all [, T i saisfies ξ d X c =, ξ X and ξ dl P a.s. Immediae arbirage is he weakes noion of arbirage and is exisence in he marke leads o he violaion of NA) and NUPBR), and consequenly of NFLVR). The auhors in [7 have proven ha he marke has no immediae arbirage iff dl i d X i. In he case of disconinuous semimaringales, as is poined ou in [11, Remark 3.13, he condiion dl i d X i for i = 1,..., d, is necessary for he absence of immediae arbirage, and hence he absence of UPBR) and FLVR), bu no sufficien. Therefore we inroduce he following assumpion. Assumpion 1 There exiss a predicable process α wih values in R d such ha dl i = α i d X i for i = 1,..., d, i.e. SC ) is saisfied. 5

6 This assumpion provides us wih a process ha capures he marke price of risk. Moreover, i is no resricive, since if i fails, here is already immediae arbirage in he marke and here is no much ha we can say abou i. Moving on from he assumpion of he exisence and predicabiliy of he marke price of risk α, we come o he quesion of is inegrabiliy and is impac on arbirage in he marke. In he coninuous case i is proven, see [3, ha NFLVR) is violaed in case α is no inegrable. As he nex heorem illusraes, he inegrabiliy of α is only relevan, if he sraegy α produces a posiive porfolio W α >. Theorem 3.1 Le α be he marke price of risk such ha W α > P -a.s. for all [, T. Then, if P T α sd X s α s = ) >, NUPBR) is violaed. Proof Since here is a posiive probabiliy ha T α sd X s α s =, we have α / LX). From Proposiion 4.16 in [11 he non-inegrabiliy of α implies P WT α = ) >. This in urn implies ha NUPBR) is violaed. Remark 3.1 In he case he. Hence, from he definiion of W and he previous heorem, we conclude ha: α W iff W α > P a.s. for all [, T, and α L 2 X). Which implies ha he SC) holds rue. 3.2 Characerisics of he marke Our aim is o sudy he exisence of UPBR) in he marke and explicily calculae he numéraire porfolio. For his reason we inroduce furher assumpions, ha provide us wih a more explici form of he d-dimensional semimaringale X. Assumpion 2 The filraion F = F [,T is quasi-lef coninuous. Assumpion 3 The d-dimensional locally square inegrable maringale M has he following represenaion: M = M c + H µ η), where M c is he coninuous par of he maringale, µ is a d-dimensional random measure on R + R d, η is he d-dimensional compensaor of µ, H is a d-dimensional predicable process ha is in G loc µ) 1 and H µ = R Hs, z)µds, dz). d Hence X is a quasi-lef coninuous semimaringale, wih a characerisic riple B, C, η), where db = αd X, C = M c, M c. From [1 Proposiion II.2.9, he characerisics of X can ake he form C = c A 2) ηd, dz) = ν dz)da, 3) where c is a predicable processes in R d d and posiive definie. A is a d-dimensional coninuous predicable process in R d, wih A i = for i = 1,..., d and non decreasing pahs. 1 For a definiion see Definiion II.1.27 p.72 in [1. 6

7 and X = Hence dm c s + d X = ) c + H 2, z)ν dz) da 4) Hs, z)µdz, ds) ν s dz)da s )+ ) α s c s + H 2 s, z)ν s dz) da s Abusing he noaion in wha follows, we denoe by α c + H 2 ν ), c, H ν) he characerisics of X. Remark 3.2 The characerisic riple B, C, η) of any special semimaringale can be represened as in he sysem of equaions 2) and 3). The condiion of Y being quasi-lef coninuous in Assumpion 2 is necessary for A o be a coninuous process. This condiion is inroduced in order o ease he presenaion and he analysis in he forhcoming secions, since he choice of a coninuous A provides a racable version of X as given in 4). 4 Numéraire porfolio. So far we have inroduced a marke, he asses of which are driven by special semimaringales. We have also presened differen noions of arbirage and defined he numéraire porfolio. In his secion we sudy he relaionship beween he characerisics of X and he exisence of arbirage in is various forms. More specifically we are ineresed in including porfolios ha can poenially violae NUPBR). For his reason we need o consider a larger class of porfolios han he admissible ones denoed by W. This class is defined and sudied in his secion afer some inroducory resuls. From his poin onwards, o simplify noaion and compuaions, we consider a marke consising of only one risky asse. The only requiremen ha is imposed on he considered inves sraegies π is he credi limi, i.e. W π > P a.s. The following definiion presens he inerval in which hese sraegies live. Definiion 4.1 Le [, T. Define he max fracion π and he min fracion π as π = infπ 1 + π H, z) >, ν almos everywhere π = supπ 1 + π H, z) >, ν almos everywhere. The limi se of invesmen sraegies Π saisfying he credi limi is defined as Π := π π PR) and π [π, π, [, T and he se of invesmen sraegies π is defined as Π := π π Π and < 1 + π H, z) < P a.s., for all [, T. In he case ha π and π are bounded processes he limi se Π coincides wih he se of invesmen sraegies Π. However, since π, π can ake values in ± or values ha lead o a zero value porfolio, he se Π is included in Π. Having inroduced he space of invesmen sraegies, in order o impose an opimaliy condiion in his class we proceed wih he sudy of he raio of wo porfolios in Π. 7

8 Lemma 4.1 Le X be a semimaringale wih characerisics α c + H 2 ν ), c, H ν), such ha X > 1. Then for π Π we have and for any ρ Π d W ρ W π W π = expπx c + [ln1 + πx) µ) exp 1 ) 2 ππ 2α)c + [απh2 πh ν A, = W ρ W π ρ π )dx c + ρ π ) + π ρ ) Proof For π Π we have π α )c + π H, z) µdz, d) 1 + π H, z) π H 2 ), z) 1 + π H, z) α H 2, z) ) ν dz) da, [, T. W π = EπX) = exp πx 1 ) 2 π2 X c, X c Π s 1 + π X s ) exp π X s ) = exp πx c + [πh µ ν) + απc + [H 2 ν) A 1 ) 2 π2 c A exp[ln1 + πh) µ [πh µ) = expπx c + [ln1 + πh) µ) exp 1 ) 2 ππ 2α)c + [απh2 πh ν A For ρ Π we herefore have W ρ W π = exp ρ π)x c + exp [ ln 1 + ρh ) µ 1 + πh 1 2 ρ π)ρ + π 2α)c + [ρ π)αh2 H) ν ) A. 8

9 Applying Iô s formula he dynamics of he porfolio are given by d W ρ W π = W ρ ) 1 + W π ρ π )dx c ρ H, z) π H, z) 1 µdz, d) ρ π ) 2 c da 1 ρ π ) 2 ρ + π 2α )c c α H 2, z) H, z) ) ) ν dz) da = W ρ W π ρ π )dx c ρ π )H, z) + µdz, d) 1 + π H, z) + ρ π ) α π )c + α H 2, z) H, z) ) ) ν dz) da = W ρ W π ρ π )dx c ρ π )H, z) + ) ν dz)da [, T. + ρ π ) α H 2, z) H, z) + = W ρ W π ρ π )dx c + ρ π ) + π ρ ) π α )c π H, z) µdz, d) + ρ π )α π )c da H, z) 1 + π H, z) H, z) µdz, d) 1 + π H, z) π H 2 ), z) 1 + π H, z) α H 2, z) ) ν dz) da, Having he explici form of he raio W ρ W, we wan o find, if i exiss, a porfolio ha has π he greaes reurns in he marke relaive o any possible invesmen sraegy in Π. For his reason he main objec of ineres from he las lemma is he drif of he raio process W ρ W, π namely π H 2 ) ), z) D ρ, π ) = π ρ ) π α )c π H, z) α H 2, z) ν dz), [, T. More specifically, we are ineresed in he exisence of a porfolio W π such ha he drif erm is negaive for any ρ Π a any fixed poin, ω) [, T Ω. Definiion 4.2 A porfolio W π is called Π-opimal if π Π and D ρ, π ) P a.s. for all [, T and ρ Π. As a example, we sudy he case when he jump measure is rivial, i.e. ν = for all [, T. Then he drif has he form P a.s. D ρ, π ) = π ρ )π α )c, [, T. Assuming ha α Π, he Π-opimal porfolio clearly is he one ha follows he sraegy α. If α LX), hen W α is no only he Π-opimal porfolio bu also he numéraire. Furhermore, from Remark 3.1 i follows ha 1 W is a maringale and he densiy of an equivalen α maringale measure, implying NFLVR) in he marke. Oherwise, from Theorem 3.1 porfolio W α akes advanage of arbirage opporuniies in he marke, leading o he violaion of NUPBR). 9

10 and In general he jump measure is no rivial, hence we need o sudy he funcions π H 2 ), z) E π ) = 1 + π H, z) α H 2, z) ν dz), E F π ) = π α )c + E π ), [, T. Boh x E x) and x F x) are increasing funcions, a propery ha is criical for he analysis in he sequel. Le Π = Π, i.e. π, π Π, and fix ω, ) Ω [, T. Then: 1. If < E π ), E π ) > holds for any π [π, π. Hence he sign of F ) depends on he marke price of risk α : a) If α < π, hen F π ) > for any π [π, π. For he Π opimal porfolio o exis we need o have D ρ, π ) < for every ρ [π, π, which makes π = π he Π opimal sraegy. The analysis hereafer follows he same logic. b) If π α, since he funcion F ) is increasing, he following cases are possible: i. If F π ) >, he Π opimal sraegy is π = π. ii. If F π ) <, he Π opimal sraegy is π = π. iii. Oherwise, F akes boh posiive and negaive values in π [π, π, hence he Π opimal sraegy is he unique soluion of he equaion F π ) =. 2. If E π ) E π ), he drif behaves as follows: a) If α < π, hen F π ). The sign of F π ) is crucial for he possible scenarios. Since F ) is an increasing funcion, here exis wo cases i. If F π ) F π ) he equaion F π ) = has a soluion in [π, π, which is also he Π opimal sraegy. ii. If F π ) > he Π opimal sraegy is π. b) If π α π he conclusion is he same as in a).i). c) If π < α, hen F π ). Again he sign of F π ) is crucial. Since F ) is an increasing funcion, here exis wo cases i. If F π ) F π ) he equaion F π ) = has a soluion in [π, π, which is also he Π opimal sraegy. ii. If F π ) < he Π opimal sraegy is π. 3. E π ) <. In his case E π ) < for all π [π, π. Then we have he following cases: a) Le α > π, hen he Π opimal sraegy is given by π = π. b) π α. Since he funcion F ) is increasing, we face he following cases. i. Le F π ) >. Then he Π opimal sraegy is given by π = π. ii. Le F π ) <. Then he Π opimal sraegy is π = π. iii. Oherwise, here exis a soluion of he equaion F π ) =. 1

11 Remark 4.1 As is obvious from he previous analysis in he case Π = Π, here exiss a Π opimal porfolio for any ω, ) Ω [, T. However, his does no imply he exisence of a numéraire in he marke. The laer depends on he inegrabiliy of he Π opimal sraegy. Remark 4.2 In he special case in which π, π LX), he opimal sraegy belongs o he se of admissible ones, making he opimal porfolio also he numéraire. The resuls of his analysis are summarized in he following heorems, afer addiional noaion is inroduced. We define he following predicable subses of Ω [, T : I =, ω) F π ) F π ) I =, ω) F π ) = I =, ω) F π ) = The following heorem is he firs main resul of his paper. Theorem 4.1 Le X be a special semimaringale wih characerisic riple α c + H 2 ν ), c, H ν), π, π Π LX). Then here exis a numéraire porfolio WT π <, hence NUPBR) is saisfied. Moreover, i). If I c has measure T, hen he fracion π invesed in he numéraire a ime akes 1 values in π, π for all [, T. Furhermore, W is a sric supermaringale. π ii). If I has measure T, hen he fracion π invesed in he numéraire a ime is he 1 soluion of F π) = for all [, T. Furhermore, W is he densiy of an equivalen π local maringale measure implying ha NFLVR) is also saisfied. iii). Le α [π, π for all [, T. Then W α is he numéraire porfolio and a) if X is a coninuous semimaringale, NFLVR) is saisfied and 1 of he equivalen maringale measure; W α is he densiy b) if Eα ) =, P d-a.s., NFLVR) is saisfied and here exiss an equivalen minimal maringale measure Q, such ha dq dp = 1 W. α Proof The fac ha he numéraire exiss and NUPBR) is saisfied follows from Remarks 4.1 and 4.2. Par i) follows from cases 1a), 1b),i), 1b),ii), 2a),ii), 2c),ii), 3a), 3b),i), 3b),ii). Par ii) follows from he combinaion of cases 1b),iii), 2a),i), 2b), 2c),i), 3b),iii). Par iii), a) follows from he pre-exising analysis. Pariii), b) is a combinaion of par 2,b), he Remark 3.1 and he definiion of he Fölmer-Schweizer minimal maringale measure. The following heorem covers he case in which π and/or π are no inegrable. Theorem 4.2 Le X be a special semimaringale wih characerisic riple α c + H 2 ν ), c, H ν) and π, π Π. 11

12 i). If π, π are no in LX) and I c or I I has measure T, hen NUPBR) is violaed. ii). If π resp. π) is no in LX) and I resp.i) has measure T, hen NUPBR) is violaed. Proof This follows from Theorem 3.1 and he cases of he analysis of he drif, where π or π is seleced as an opimal sraegy. The las case ha we would like o explore is wha happens o he drif of W ρ W, when Π π is a sric subse of he limi sraegy se Π. In his case here exis [, T such ha lim π π 1 + π H, z)) = and/or lim π π 1 + π H, z)) = P a.s.. The analysis of when he drif D ρ, π ) is negaive for ρ Π and π Π, follows he same seps as in he case π Π, wih he only excepion being ha we now need o sudy he behaviour of he limis lim π π F π) and lim π π F π). The conclusions are also he same modulo ha here is no Π opimal sraegy in he case when only in he limi sraegies π, π he drif is non-posiive. Le us define J =, ω) lim F π ) lim F π ) π π π π J = J = Then we have he following corollary., ω) lim F π ) = π π, ω) lim F π ) =. π π Corollary 4.1 Le X be a special semimaringale wih characerisic riple α c + H 2 ν ), c, H ν) and π, π Π. i). If π, π are no in Π and I c or I I has a posiive measure, hen here exiss no Π opimal sraegy and NUPBR) is violaed. ii). If π resp. π) is no in Π and J resp.j ) has a posiive measure, hen here exiss no Π opimal sraegy and NUPBR) is violaed. Remark 4.3 In he previous analysis we sared wih he characerisics of he underlying semimaringales and found an explici link beween hem and various forms of arbirage. The quesion could also be reversed, as is in he case of [13, where he auhors assume NFLVR) and find necessary condiions on he characerisics of he semimaringales. 4.1 Examples In he following examples we examine he properies of he characerisics of X and heir relaionship o arbirage properies. Example 4.1 From Karazas and Kardaras[11 Le us assume ha S = EN ), where N is a Poisson process wih inensiy λ = 1. The marke is characerized by he riple 1,, 1) and he range of he P i invesemen sraegies is [ 1, + for all [, T. Furhermore, he marke price of risk is α = 1 for all [, T, and we have E π ) = 1 1+π, where π 1, + ). Clearly E ) is sricly negaive wih lim π + E π ) =. Thus we are in case ii) of Corollary 4.1, and we conclude ha NUPBR) is violaed. 12

13 Example 4.2 From Becherer [4 This example is a coninuous ime version of ex. 6 in [4. Le S = Π s Y s, where [, T and Y is lognormally disribued, log Y N µ, σ 2 ). The semimaringale ha generaes he marke is given by ) X = e z σ2 µ+ 1) µdz, ds) + e 2 1, σ2 µ+ wih he characerisic riple e 2 1,, ) e z 1)νdz), where ν is he densiy of he sandard normal disribuion. α = e µ+ σ e σ2 1)e µ+σ2 µ+ σ2 + e 2 1 I follows ha he marke price of risk is given by ) 2. There is no shor sale in he marke, hence he range of he Π-opimal sraegies is [, 1. Under hese assumpions he condiions of Theorem 4.1 are saisfied. This implies ha NUPBR) is saisfied and a numéraire porfolio exiss. Since his σ2 µ+ is a pure jump marke, we sudy he properies of E ). We have E ) = 1 e 2 and σ2 µ+ E 1) = e 2 1, [, T. If µ σ2 2, E ) > and α < = π for all [, T. Hence we are in case 1,i), or case i) of Theorem 4.1, which implies ha he opimal sraegy, which also describes he numéraire, is given by π =, and he numéraire is a sric supermaringale. For σ2 2 µ σ2 2, since E ) < < E 1) for all [, T we are in case ii) of Theorem 4.1, he numéraire porfolio exiss and 1 W is a maringale. π For µ σ2 2, we are in case 3,b),ii), since E 1) < and π > α for all [, T. This implies ha he Π opimal sraegy is π = 1 and he numéraire is a sric supermaringale. Example 4.3 Chrisensen-Plaen[6 Here we consider a one dimensional version of he seing in [6. The marke asse saisfies he sde ds S = + θ 2 + E E ψ 2 ), z) 1 ψ, z) νdz) d + θ dw ψ, z) µdz, d), 1 ψ, z) where θ is a predicable and square inegrable process, ψ, ) is predicable and ψ, z) < 1 a.e.. Furhermore, he Lévy measure ν is finie. θ2 + ψ 2,z) E 1 ψ,z) νdz) In his case α = he characerisics are given by θ 2+ ψ 2,z) E 1 ψ,z)) 2 νdz), θ 2 + ) ψ 2,z) E 1 ψ,z) νdz), θ 2, ψ,z) E 1 ψ,z) ), νdz) and he range of Π opimal sraegies is [, 1. Then ) 2 F π ) = π 1)θ 2 π ψ,z) 1 ψ,z) + ψ2, z) ψ,z) νdz) E 1 + π 1 ψ, z) 1 ψ,z) = π 1) θ 2 + E ψ 2, z) π )ψ, z) νdz). Hence i is easy o see ha we are in case ii) of Theorem 4.1, and he numéraire porfolio is given by π = 1. In his case 1 S and π Sρ S are local maringales for all ρ W. π 13

14 5 Enlarged filraion In his secion we are ineresed in idenifying he difference in reurn due o asymmeric informaion. The classical approach o his problem compares he logarihmic uiliies under differen informaion srucures. To his end, under he assumpion of finie logarihmic uiliies, we calculae he addiional logarihmic uiliy of a rader wih larger informaion flow G han he res of he marke, possessing informaion described by a smaller filraion F G. Opimal logarihmic uiliy is linked o he exisence of a GOP and in essence o he exisence of a numéraire, see Theorem 2.1. For his reason subsecion 5.1 summarizes resuls on he link beween he he opimal logarihmic uiliy of he porfolio and he numéraire. In subsecion 5.2 he characerisics of he underlying semimaringale X under G are derived, he available resuls on he relaionship beween he characerisics of X and he exisence of he numéraire porfolio are exended o he seing in he large filraion G. In a final sep we aim a comparing he addiional logarihmic uiliy wih he relaive enropy of he filraions. From [2 we know ha in a coninuous semimaringale framework he exra logarihmic uiliy of an insider is equal o he Shannon enropy of his addiional informaion. This propery also holds rue in markes wih purely disconinuous semimaringale basis under furher assumpions. 5.1 Log-uiliy The descripion of he logarihmic uiliy under NFLVR) involves he se of local) equivalen maringale measure, and in he exended framework of NUPBR) he se of supermarinale densiies. The definiion of hese ses is aken from [4. Definiion Wih M we denoe he se of all probabiliy measures Q, such ha Q P and W ρ is a Q-local maringale for any W ρ W. 2. The se of all P -supermaringale densiies is denoed by SM := Z Z, Z = 1, ZW ρ is a P -supermaringale for all W ρ W. Then he following basic resuls hold. Proposiion 5.1 Le NUPBR) be saisfied and u <. Then here exiss a numéraire porfolio W π Wi.e. a GOP)), ha saisfies Furhermore, if NFLVR) holds, we have E [log W π = sup E [log W ρ W ρ W = inf Z SM E [ log 1 Z T E [log W π = inf Q M HP Q). Lemma 5.1 Le NUPBR) hold and u <. Then he reurn of he GOP) for a marke wih characerisics α c + H 2 ν ), c, H ν) is given by [ T E[log WT π = E 1 ) π 2 2 2α c da [ + E ln1 + π H, z)) + π H, z)α H, z) 1)) ν dz)da. E 14

15 5.2 Asymmeric filraion To describe he addiional logarihmic uiliy, in his subsecion sar in he following enlargemen of filraions seing. Le G be a filraion such ha F G. We work under he following assumpion concerning he decomposiion of he underlying X in he larger filraion. Assumpion 4 X is a quasi-lef-coninuous semimaringale under G and has he represenaion, X = N + β X, X, where N is a local square inegrable maringale wih respec o he filraion G and β is a predicable process wih respec o G. In he previous secions, under Assumpions 1 and 2 we have deduced he characerisics of X wih respec o F, sudied heir relaionship wih arbirage properies, and evaluaed he opimal logarihmic uiliy in Lemma 5.1. To exend his o he enlarged filraion framework we deermine he characerisics of X under G in he following heorem. Theorem 5.1 Le X be a semimaringale wih characerisics αc + H 2 ν), c, H ν) wih respec o he filraion F. Le G be a filraion such ha F G. Then he characerisic riple of X under G is given by βc + H 2 ν), c, H[1 α β)h ν). Proof From he represenaions of X under he differen filraions we have N = M + α β) X, X = M c + [H µ ν A) + α β) c A + [H 2 ν A ) = M c + α β)c A + [H µ [H 1 α β)h) ν A. Using orhogonaliy argumens he resul follows. From he previous heorem, we conclude ha he srucure of he jump size wih respec o he original filraion is preserved in he enlarged filraion. Hence he following lemma is immediae. Lemma 5.2 The limi se of invesmen sraegies wih respec o he filraion G coincides wih he se of limi sraegies Π under F. The lemma implies ha also he se of invesmen sraegies under G coincides wih he se of invesmen sraegies Π under F. Proposiion 5.2 Le X be as in Theorem 5.1, such ha X > 1. Then for π Π we have W π = exp πn c + [ln1 + πh) µ) exp 12 ) ππ 2β)c + [πhαh 1) ν A, 15

16 and for any ρ Π Proof For π Π we have d W ρ [ W π + α β)h 2 π = π ρ) π β)c + W ρ 1 + πh W π + ρ π)dn c + [ ρ π)h 1 + πh µ G, [, T. αh 2 ν da W π = EπX) = exp πn c + [πh µ [πh1 α β)h) ν + βπc + [H 2 ν) A 1 ) 2 π2 c A exp[ln1 + πh) µ [πh µ) = expπn c + [ln1 + πx) µ) exp 12 ) ππ 2β)c + [πhαh 1) ν A. Le ρ Π. Then W ρ [ W π = exp ρ π)n c + ln 1 + ρh 1 + πh exp ) µ 12 ρ π)ρ + π 2β)c + [ρ π)hαh 1) ν ) A and by applying Iô s rule we have d W ρ W π = W ρ ) 1 + W π ρ π )dx c ρ H, z) π H, z) 1 µdz, d) ρ π ) 2 c da ) 1 ρ π ) 2 ρ + π 2β )c H, z) α H, z) 1) ν dz) da = W ρ W π ρ π )dx c ρ π )H, z) + µdz, d) 1 + π H, z) ) + ρ π ) β π )c + H, z) α H, z) 1) ν dz) da = W ρ W π ρ π )dx c ρ π )H, z) π H, z) µg dz, d) + ρ π ) π + β )c da + ρ π ) α H 2, z) H, z) + H, z)1 α ) β )H, z)) ν dz)da 1 + π H, z) = W ρ W π ρ π )dx c π H, z) + ρ π ) 1 + π H, z) µg dz, d) π + α β )H 2, z) + π ρ ) π β )c + α H 2, z) 1 + π H, z) ), ) ν dz) da. From Theorem 5.1 we know he characerisics of X under G. Following he same seps as in Proposiion 4 we obain he desired resuls. 16

17 Hence he drif under G is given by [ D π + α β )H 2, z) ρ ) = π ρ ) π β )c π H, z) As in secion 4 we inroduce he funcions E π + α β )H 2, z) π ) = 1 + π H, z) and E α H 2, z) ν, [, T. ) α H 2, z) ν dz), F π ) = π β )c + E π ), [, T. To proceed wih he analysis, we inroduce he following assumpion. Assumpion 5 The informaion drifs α, β saisfy 1 + β α )H, z) > P -a.s. for all [, T. 2 Under his assumpion, he funcions x E x) and x F x) are increasing. Using he characerisic riple under G and he properies of he funcions E ), F ), he analysis of he drif is idenical wih he one under under F, and he resuls ransfer accordingly. In case he jump measure is rivial, i.e. ν = P a.s. for all [, T, he opimal porfolio is he one ha follows sraegy β. If β LX), hen W β is he numéraire, 1 is a maringale and he densiy of an equivalen maringale measure, implying NFLVR) W β in he marke. Oherwise, he porfolio W β akes advanage of arbirage opporuniies in he marke, leading o he violaion of NUPBR). Le π, π Π and fix ω, ) Ω [, T. Then we have 1. if E π ) and a) β < π, he opimal sraegy is given by π = π. b) π β, hen i. for F π ) > he opimal sraegy is described by π = π, ii. for F π ) <, he opimal sraegy is π = π, iii. oherwise, he opimal sraegy is he unique soluion of he equaion F π ) =. 2. If E π ) E π ) and a) β < π, hen i. for F π ) F π ) he opimal sraegy is he unique soluion of he equaion F π ) =, ii. if F π ) > he opimal sraegy is π. b) π β π, he conclusion is he same as in a),i). c) π < β, hen i. if F π ) F π ) he opimal sraegy is he unique soluion of he equaion F π ) =, 2 As will become eviden in he nex secion, Assumpion 5 is also necessary for he definiion of he enropy and hence no resricive. 17

18 ii. if F π) < he opimal sraegy is π. 3. If E π ) < and a) β > π, he opimal sraegy is π = π, b) π β, i. for F π ) >, he opimal sraegy is π = π, ii. for F π ) <, iii. oherwise, he opimal sraegy is he unique soluion of he equaion F π ) =. In analogy o secion 4 we define he following predicable subses of Ω [, T : I =, ω) F π ) F π ), J = I =, ω) F π ) =, I =, ω) F π ) =., ω) lim J = J = F π π π ) lim, ω) lim F π π π ) =, ω) lim F π ) = π π F π ), π π, We have he following resul abou he exisence of numéraire porfolios. Theorem 5.2 Le X be a special semimaringale as in Theorem If he marke price of risk β saisfies W β >, P a.s. for all [, T, and P T β2 s d X s ) > NUPBR) is violaed. 2. If π, π Π LX), here exis a numéraire porfolio WT π saisfied. Moreover,. <, hence NUPBR) is i) If I ) c has measure T, hen he fracion π invesed in he numéraire a ime 1 akes values in π, π for all [, T. Furhermore, W is a sric supermaringale. π ii) If I has measure T, hen he fracion π invesed in he numéraire a ime is he soluion of F 1 π) = for all [, T. Furhermore, W is a maringale implying π ha NFLVR) is also saisfied. iii) Le β [π, π for all [, T. Then W β is he numéraire porfolio and a). if X is a coninuous semimaringale, NFLVR) is saisfied and 1 is he W β densiy of he equivalen maringale measure. b). If Eβ ) =, P d-a.s., NFLVR) is saisfied and here exiss an equivalen minimal maringale measure Q, such ha dq dp = 1. W β 18

19 3. Le π, π Π a). If π, π are no in LX) and I ) c or I I has posiive measure, hen NUPBR) is violaed. b). If π resp. π) is no in LX) and I resp.i ) has a posiive measure, hen NUPBR) is violaed. 4. Le π, π Π. i). If π, π are no in Π and J ) c or J J has a posiive measure, hen here exiss no Π opimal sraegy and NUPBR) is violaed. ii). If π resp. π) is no in Π and J resp.j ) has a posiive measure, hen here exiss no Π opimal sraegy and NUPBR) is violaed. Proof The argumens are equivalen he proofs of Theorem 4.1, Theorem 4.2, Corollary 4.1 and Theorem 3.1. Proposiion 5.3 Le X be a semimaringale wih characerisic riple α X, C, Hη) wih respec o a filraion F where NUPBR) holds, and G a filraion such ha NUPBR) holds and F G for all [, T. Furhermore, if W π and W ρ are he numéraire porfolios under F and G respecively, he difference in reurn is given by [ T u G u F = E 1 ) 2 π π 2β ) ρ ρ 2α )) c da [ T + E π ρ )H, z)α H, z) 1) + ln 1 + π H, z) 1 + ρ H, z) + β α )H, z) ln1 + π H, z))) ν dz)da. Proof We have [ T T E[log WT π = E π N c d + log1 + π H, z))µdz, d) [ T 1 T + E 2 π π 2β )c da + π H, z)α H, z) 1)ν dz)da [ T T = E π N c d + log1 + π H, z)) µdz, d) [ T 1 T + E 2 π π 2β )c da + [π H, z)α H, z) 1)ν dz)da [ T + E 1 α β )H, z)) log1 + π H, z))ν dz)da [ T [ 1 T = E 2 ππ 2β )c da + E β H, z)) log1 + π H, z))ν dz)da [ T + E [α H, z) 1)π H, z) log1 + π H, z))ν dz)da. Combining he previous formula wih Lemma 5.1, he resul follows. 19

20 5.3 Enropy In his secion we describe he enropy of he addiional informaion ha a larger filraion provides wih respec o a smaller one. To simplify our presenaion we assume ha G is obained by an iniial enlargemen of F. Under he Assumpion 2 he local semimaringale M generaes he filraion F ) [,T. Le F ) [,T be a filraion he σ-algebras of which are counably generaed, and under which M is a local maringale. Assume ha F ) [,T is he smalles filraion conaining F ) [,T and saisfying he usual condiions. Also, le G ) [,T be a filraion wih counably generaed σ-algebras, and G ) [,T he smalles filraion saisfying he usual condiions and conaining F, i.e. G F for all. The inroducion of he smaller filraions F, G is a necessary condiion for he regulariy of he condiional probabiliy P, A) given F, where A GT, [, T. From [2 and [1 i follows ha P ω, A) is a F )-maringale. And by he maringale represenaion propery, for [, T, P, A) has he form P, A) = P A) + γ s, A)dM c s + δ u z,, A) µdz, d), 5) where γ, δ are predicable processes belonging o L 2 P ) and L 2 P η) respecively. coninue our analysis, we inroduce he following assumpion. To Assumpion 6 For T le γ ω, ) G and δ z, ω, ) G be signed measures on G, such ha γ ω, ) G << P ω, ) G, P a.s., and δ z, ω, ) G << P ω, ) G P η a.s. Theorem 5.3 Under Assumpion 6 here exis F G predicable processes and c ω, ω ) = γ ω, dω ) P ω, dω ) d z, ω, ω ) = δ z, ω, dω ) P ω, dω ) G G P a.s., P π a.e. Furhermore c ω, ω) = β ω) α ω) and d z, ω, ω) = β ω) α ω))h, z, ω) P -a.s. Proof From he beginning of his subsecion we know ha he informaion drif for he coninuous par of he semimaringale is given by β α and ha for [, T ηg dz,ω) η = 1 + β dz,ω) ω) α ω))h, z, ω). Then using he orhogonaliy of M c and µ he resul follows easily from Lemma 2.3 and Theorem 2.6 in [2, as well as Lemma 2.5 and Theorem 2.6 from [1. The preceding heorem is insrumenal in he compuaion of addiional informaion, ha is inroduced in he following. 2

21 Definiion 5.2 Le A be a sub-σ-algebra of F and P, Q wo probabiliy measures on F. Then we define he relaive enropy of P wih respec o Q on he sigma field A by log dp H A P Q) = dqdp, if P Q, else. Moreover, he addiional informaion of A relaive o he filraion F u ) on [s,, where s < T, is defined by H A s, ) = H A P ω, ) P s ω, ))dp ω). The explici form of H G is provided by he following Lemma. Lemma 5.3 The addiional informaion of G relaive o he filraion F u ) on [s, is given by [ β α ) 2 H G s, ) = E d M c s 2 + β α )H, z)νdz)da s β α )H, z)) ln 1 + β α )H, z)) νdz)da. Proof Using Iô s rule for semimaringales we ge d ln P, A) = + = + + γ P, A) dm c γ 2 2P, A) 2 d M c [lnp, x) + δ z)) ln P, A) µdz, d) + [ γ P, A) dn c + ln 1 + δ z) ˆµdz, d) P, A) ) γ β α d M c P, A) γ 2P, A) [ δ z) P, A) β α )H, z)) ln δ z) P, A) νdz)da 1 + δ ) z) νdz)da. P, A) Since N c and ˆµ are local maringales under G ) we have [ E P, A) log P [, A) γ = E 1 A β α P s, A) s P, A) + 1 A From Theorem 5.3 we infer [ E P, A) log P [, A) = E P s, A) + s s s ) γ d M c 2P, A) [ δ z) P, A) β α )H, z)) ln β α ) 2 1 A d M c 2 1 A β α )H, z)νdz)da β α )H, z)) ln 1 + β α )H, z)) νdz)da δ ) z) νdz)da. P, A)

22 Using he same seps as in he proof of Lemma 5.3 of [2 we reach our resul. In he case of a coninuous marke, as explained in [2, he addiional informaion H G, T ) equals he expeced logarihmic uiliy incremen beween he wo filraions, since u G u F = E[ T β α ) 2 2 d. However, in he case of a sochasic basis wih boh a coninuous and a jump componen his is no obvious. 5.4 Purely disconinuous semimaringales The expeced logarihmic uiliy incremen, as was noed before, is no always equal o he enropy of he addiional informaion. However, as he nex heorem shows, in purely disconinuous markes in which he jumps are hedgeable, he equaliy holds. Theorem 5.4 Le X be a quasi-lef coninuous semimaringale under he filraion F, wih characerisic riple αh 2 ν,, H ν), such ha 1 α H, z) > P a.s. for all [, T and α Π. Then he Π-opimal porfolio sraegy is π =. Furhermore, for a α 1 α H,z) filraion G F, where X has he characerisic riple βh 2 ν,, H[1 + β α)h ν), he Π-opimal porfolio sraegy is given by ρ =. If α, β LX), hen β 1 α H,z) u G u F = H G, T ). Proof Given he characerisic riple under F, from secion 4 we have π H 2 ), z) F π ) = E π ) = 1 + π H, z) α H 2, z) ν dz). Clearly E α 1 α H,z) E ) = P a.s. for all [, T, so we are in case 2. of he analysis. α 1 αh, and W π saisfies The opimal porfolio sraegy is hen given by π = d 1 W π = α H, z)d µ W π. If α LX), hen W π is a local maringale and is boh he numéraire porfolio and he densiy of he minimal maringale measure. The expeced logarihmic uiliy of W π is given by u F = E[ln WT π = E [ [ln1 + πh) µ + E [ [παh 2 πh + ln1 + πh) ν [ T = E [α H, z) + ln 1 α H, z)) νdz)da. For a larger filraion G we have F π ) = E π + α β )H 2, z) π ) = 1 + π H, z) E ) α H 2, z) ν dz). From Assumpion 5 ρ = β 1 αh is in Π. Furhermore E ρ ) = and d 1 W ρ = 1 β H, z) W ρ 1 + β α )H, z) d µ. 22

23 1 If β LX) he soluion of he previous equaion is a local maringale, hence densiy of maringale measure, and logarihmic uiliy given by W ρ is he T u G = E[ln W ρ T = E [[1 + β α)h)ln1 + β α)h) ln1 αh)) βh ν. Noe ha Hence ln1 α H, z)) µ T ln1 α H, z)) µ = T [ T E β α ) ln1 α )H, z)νdz)da =. β α ) ln1 α )H, z)νdz)da. We have [ T u G u F = E 1 + β α)h, z)) [ln1 + β α)h, z)) ln1 αh, z)) νdz)da [ T + E [ β α )H, z) + ln1 α H, z) νdz)da [ T ) = E 1 + β α)h, z)) ln 1 + β α)h, z) β α )H, z) νdz)da. Hence under hese assumpions we recover he resul of he coninuous marke, namely ha he expeced logarihmic uiliy incremen is equal o he Shannon enropy of he addiional informaion. Remark 5.1 From 5.3 onwards we have assumed ha he filraion G is an iniial enlargemen of F. This assumpion can be relaxed o include progressive enlargemens, as is shown in [2. However, his exceeds he scope of his paper. References [1 Ankirchner, S. On filraion enlargemens and purely disconinuous maringales. Sochasic Processes and heir Applicaions ), [2 Ankirchner, S., Dereich, S., and Imkeller, P. The Shannon informaion of filraions and he addiional logarihmic uiliy of insiders. Ann. Probab ), [3 Ankirchner, S. and Imkeller, P. Enlargemen of filraions and coninuous Girsanovype embeddings. Séminaire de Probabilis XL. Lecure Noes in Mahemaics, Volume 1899/27. Springer: Berlin 27. [4 Becherer, D. The numeraire porfolio for unbounded semimaringales. Finance and Sochasics 5 21), [5 Chrisensen, M. and Larsen, K. No arbirage and he growh opimal porfolio. Sochasic Analysis and Applicaions 25 27), [6 Chrisensen, M. and Plaen, E. A general benchmark model for sochasic jump sizes. Sochasic Analysis and Applicaions, 235):

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