Dynamically Consistent α-maxmin Expected Utility

Transcription

1 535 Cener for Mahemaical Economics Working Papers February 2016 Dynamically Consisen α-maxmin Expeced Uiliy Parick Beißner, Qian Lin and Frank Riedel Cener for Mahemaical Economics(IMW) Bielefeld Universiy Universiässraße 25 D Bielefeld Germany hp:// ISSN:

2 Dynamically Consisen α Maxmin Expeced Uiliy Parick Beissner Qian Lin Frank Riedel February 13, 2016 Absrac The α maxmin model is a prominen example of preferences under Knighian uncerainy as i allows o disinguish ambiguiy and ambiguiy aiude. These preferences are dynamically inconsisen for nonrivial versions of α. In his paper, we derive a recursive, dynamically consisen version of he α maxmin model. In he coninuous ime limi, he resuling dynamic uiliy funcion can be represened as a convex mixure beween wors and bes case, bu now a he local, infiniesimal level. We sudy he properies of he uiliy funcion and provide an Arrow Pra approximaion of he saic and dynamic cerainy equivalen. We derive a consumpion based capial asse pricing formula and sudy he implicaions for derivaive valuaion under indifference pricing. Key words: Dynamic consisency, α maxmin expeced uiliy, Knighian uncerainy, ambiguiy aiude JEL subjec classificaion: C60, D81, D90 We hank Shige Peng for fruiful discussions. Financial suppor hrough he Gran RI 1128/7-1 by he German Research Foundaion DFG. Cener for Mahemaical Economics, Bielefeld Universiy, Germany. parick.beissner@uni-bielefeld.de School of Economics and Managemen, Wuhan Universiy, Wuhan , China. linqian@whu.edu.cn. Cener for Mahemaical Economics, Bielefeld Universiy, Germany. frank.riedel@uni-bielefeld.de and Deparmen of Economic and Financial Sciences, Universiy of Johannesburg 1

3 1 Inroducion In an effor o differeniae concepually ambiguiy from ambiguiy aiude, Ghirardao, Maccheroni, and Marinacci (2004) inroduce he α maxmin model of preferences under Knighian uncerainy. These preferences can be represened by a uiliy funcion of he form U(X) = α min P P EP [u(x)] + (1 α) max P P EP [u(x)] (1) for a Bernoulli uiliy funcion u, a class of priors P, and an index of ambiguiy aiude α [0, 1]. Such preferences generalize he well known α maxmin rule of Hurwicz o seings of Knighian uncerainy where he subjecive percepion of ambiguiy can be described by a se of probabiliy measures and he aiude owards ambiguiy by a parameer α which describes he relaive weigh pu on pessimism versus opimism 1. In his paper, we discuss α maxmin uiliy in a dynamic framework. For he purely pessimisic case (α = 1), Epsein and Schneider (2003) have shown ha he muliple priors model of Gilboa and Schmeidler (1989) is dynamically consisen if and only if he se of priors is recangular, i.e. sable under pasing marginal and condiional disribuions. Our saring poin is he following fac: Even if he se of priors is recangular, α maxmin uiliy is no ime consisen for non rivial values of ambiguiy aiude α. We hus se ou o define a recursive version of α maxmin uiliy where we apply he logic of α maxmin uiliy condiionally upon he available informaion in every discree ime sep. Such a recursive consrucion leads o a ime consisen overall uiliy funcion. In discree ime, racable represenaions of he resuling uiliy funcion are usually no available. The coninuous ime limi of our recursive consrucion, however, admis a nice represenaion. The dynamic uiliy form saisfies he backward sochasic differenial equaion ( ) du (X) = α min θσ + (1 α) max θσ d + σ db θ Θ θ Θ where now he se Θ describes he perceived ambiguiy. We hus obain again an α maxmin represenaion, bu now locally, a he infiniesimal (one sep ahead) level 2. The represenaion of he uiliy funcional as backward sochasic differenial equaion allows a more deailed sudy of is properies and he economic 1 The neo addiive capaciies of Chaeauneuf, Eichberger, and Gran (2007) are anoher insance of such preferences. 2 This represenaion generalizes he represenaion for ime consisen pessimisic muliple prior preferences of Chen and Epsein (2002). 2

4 consequences for agens wih such preferences. We discuss he properies of he uiliy funcional and derive a represenaion for he cerainy equivalen. As an applicaion, we show he implicaions for consumpion based asse pricing models. Ambiguiy leads o an addiional premium for uncerain asses, similar o Chen and Epsein (2002). The ambiguiy premium is reduced by opimism. The shor ineres rae increases usually wih pessimism. The paper is se up as follows. The nex secion shows ha he naive version of α maxmin uiliy is no dynamically consisen. Secion 3 derives he recursive, dynamically consisen version and is coninuous ime limi. Secion 4 discusses he properies of he resuling preferences. Secion 5 discusses he implicaions for equilibrium asse prices in he framework of he consumpion based capial asse pricing model and he implicaions for derivaive valuaion if one uses he mehod of indifference pricing. 2 Dynamic Inconsisency of α maxmin Uiliy Gilboa Schmeidler uiliy funcions are dynamically consisen if (and only if) he se of priors is recangular (Epsein and Schneider (2003)). This resul carries over o opimisic, ambiguiy loving agens (α = 0). For inermediae values of α, recangulariy is no sufficien for dynamic consisency as we show in his secion wih he help of wo examples, in discree and coninuous ime. Consider he wo period binomial ree of Figure 1. The ransiion probabiliies of moving up in he ree are given by p, q, r [ 1, 3 ]. By consrucion, he 4 4 resuling se of priors { ( ) [ 1 P = pq, p(1 q), (1 p)r, (1 p)(1 r) (Ω) : r, p, q 4, 3 4] } is recangular. We wrie p = q = r = 1 4 and p = q = r = 3 4. Figure 1: Two period binomial model and recangular se of priors. For α = 1/2, he agen prefers X o Y ex ane and reverses he ranking in all nodes a ime 1. 3

5 Consider he wo payoffs X and Y depiced in he figure. Noe ha Y has a payoff which is known a ime 1 bu uncerain a ime 0. For simpliciy 3, we ake u(x) = x and α = 1/2. We firs show ha Y is uniformly preferred o X a ime 1. Indeed, in he upper node, he uiliy of X is U 1 [X] = 1 2 min q [ 1 4, 3 4 ] q 0 + (1 q) max q [ 1 4, 3 4 ] q 0 + (1 q) 8 = 4, which is sricly smaller han he uiliy of Y which is 4.1 (recall ha Y is known a ime 1). Similarly, in he lower node we obain U 1 (X) = 2 < U 1 (Y ) = 2.2. Now we show ha he ranking is reversed a ime 0. We compue U 0 [X] = 1 2 min P P EP [X] max P P EP [X] = 1 ( ) 8p(1 q) + 4(1 p)r + 1 ( ) 8p(1 q) + 4(1 p)r 2 2 = 1 ( ) ( ) 16 4 = ( and U 0 [Y ] = ) ( ) = A ime = 0, X is preferred o Y. Dynamic consisency is closely relaed o recursiviy, or he dynamic [ programming principle. The ieraed α-maxmin expeced uiliy is U 0 U1 [X] ]. We plug U 1 [X] = (4, 2) ino U 0 and ge ] U 0 [U 1 [X] = 1 [( ) ( )] 4 2 = 3 U 0 [X]. The inequaliy U 0 [U 1 [X]] U 0 [X] shows ha recursiviy fails. For our coninuous ime example, we choose he drif ambiguiy model of Chen and Epsein (2002). Fix a finie ime inerval [0, T ]. Le B be a Brownian moion on a probabiliy space (Ω, F, P) and (F ) [0,T ] he filraion generaed by B, and compleed by null ses. The se of priors consiss of all probabiliy measures P θ such ha B has drif θ under P θ. More specifically, we denoe by Θ he se of all adaped processes θ = (θ ) [0,T ] wih values in he inerval [ 1, 0]. The maringale dp θ dp ( = exp 1 F 2 0 θ 2 s ds + 0 θ s db s ) hen defines a measure P θ under which B has drif θ by Girsanov s heorem. 3 The resul carries over easily o arbirary isoone u and values of α (0, 1). (2) 4

6 Now le us consider he hree daes = 0, = 1 and T = 2. Again, we ake α = 1. Consider he payoffs X = 2 eb T and Y = 1(C + 2 ε)eb for C = e e 1 2 and ε > 0 sufficienly small. As we deal wih drif ambiguiy, and X is a monoone funcion of B T, he wors case prior P assigns drif 1 and he bes case prior drif 0, i.e. he reference measure P is he bes prior. A ime = 1, we hus ge U 1 [e B T ] = 1 ( ) E P [e B T ] + E P [e B T ] 2 = 1 ( ) e 1 2 T E P [e B T 1 2 T ] + e B+ 1 2 E P [e B T 1 2 T +B T ] 2 = 1 ( ) e 1 2 eb 2 (T ) + e 1 2 (T ) = 1. 2 CeB Noe ha his is sricly smaller han U 1 (Y ) = Y. A ime 0, we have U 0 [e B T ] = 1 2 whereas he uiliy of Y is ( E P [e B T ] + E P [e B T ] ) = 1 2 U 0 [Y ] = 1 2 (C + ε) ( 1 2 EP e B EP e B ) Again, he ranking is reversed a ime 0. ( e 1 + E P [e B T 1/2T +B T ] ) 1.54, C Recursive α maxmin uiliy and is Coninuous Time Limi This secion inroduces a dynamically consisen version of α-maxmin uiliy. We sar wih a recursive formulaion in discree ime. Then we inroduce he counerpar in coninuous ime. We show (Theorem 2) ha he discree ime version converges o he coninuous ime version. 3.1 General Seup in Coninuous Time Fix a finie ime inerval [0, T ]. Le B be a Brownian moion on a probabiliy space (Ω, F, P). Le (F ) [0,T ] be he filraion generaed by B, compleed by P-null ses. Se L = L 2 (Ω, F, P), for every [0, T ], he space of square inegrable and F measurable random variables. Le Θ : Ω [0, T ] R be an adaped se-valued process and assume for every (ω, ) he se Θ (ω) is a convex and closed subse of some compac se 5

7 K R. For a real valued process θ = (θ ) wih θ Θ we define he densiy process ( z θ exp θ 2 s ds + 0 θ s db s ). By he Girsanov heorem, z θ deermines a probabiliy measure P θ. Given Θ, we hus obain a corresponding se of priors: P = { P θ : θ Θ, P θ is defined by (2) }. (3) The induced se of priors P is weakly compac, convex and recangular (see Chen and Epsein (2002)). 3.2 Recursive α-maxmin uiliy in discree ime On he probabiliy space (Ω, F, P) wih se of priors P inroduced in (3), we consruc a recursive α-maxmin uiliy in discree ime. For an ineger N, we le = T N. The collecion of all adaped processes (a ) aking values in he uni inerval [0, 1] is denoed by [0, 1]. We define he (naive and ime inconsisen) nonlinear expecaion, for X L T and [0, T ]. I [X] = a min P P EP [X] + (1 a ) max P P EP [X]. Le u be a concave and increasing funcion. We consruc a recursive uiliy in discree ime as follows. For he erminal ime = T, we define U N T [X] = I T [u(x)] = u(x). For [i, (i + 1) ), where i = 0, 1,..., N 1, we define ] U N [X] = I [U(i+1) [X] N. (4) By consrucion he family (U i ) i=0,...,n is a family of recursive uiliies. Theorem 1 The family (Ui N ) i=0,...,n is dynamically consisen in he following sense: for all X, Y L T and all i < j, if Uj N (X) U j N (Y ), a.s., hen Ui N (X) U i N (Y ), a.s. 3.3 Coninuous-Time Limi of α-maxmin uiliy The coninuous ime seup allows o describe he recursive relaion of nonlinear condiional expecaions in differenial erms. This differenial formulaion in (5) is he coninuous-ime counerpar of (4). 6

8 For every a [0, 1] and every X L T, here exiss a unique soluion (E [X], σ ) of he backward sochasic differenial equaion (BSDE) wih 4 E T [X] = X. de [X] = a max θ σ + (1 a ) min θ σ d + σ db, (5) θ Θ θ Θ }{{} =e(,σ ) Example 1 1. If Θ = {θ} is a singleon, hen a is irrelevan and we obain he usual linear expecaion E [X] = E P θ [X], where he subjecive prior P θ is again given by (2). 2. If a 1, i.e., a form of maximal pessimism in beliefs, hen E [X] = min P P E P [X] reduces o he coninuous ime analog of Gilboa Schmeidler preferences of Chen and Epsein (2002). The case of an exremely opimisic expecaion E [X] is obained 5 wih a 0. Definiion 1 Le a [0, 1]. For every X L T, he α-maxmin condiional expecaion (E [X]) [0,T ] is he unique soluion of (5). The BSDE formulaion of E implies a dynamic sabiliy of he funcional form. In he noaion of Example 1, he comparison principle for BSDEs yields E [X] [ E [X], E [X] ] for any X and ime. Consequenly, here is a process (α X ) [0, 1], depending on (a ) and X, ha allows for a global represenaion E [X] = α X E [X] + (1 α X )E [X]. (6) The dynamically inconsisen expecaion of Secion 2 employed a consan weigh α α X. A sochasic and X-dependen α X provides he dynamic consisency of E. For perspecive, Proposiion 5 in Appendix B collecs a lis of furher properies. The following heorem esablishes he announced connecion beween he discree- and coninuous-ime formulaion. Theorem 2 Le u be concave and increasing. For every X L T and every, U N [X] from (4) converges o E [u(x)], in he norm of L T : lim [X] E [u(x)] = 0. U N N 4 Equaion (5) is a special BSDE. For more deails, see Appendix A and Peng (1997). 5 Noe ha he operaors max and min are inerchanged in he differenial formulaion; he maximum corresponds o he pessimisic par. 7

9 4 Properies of Recursive α maxmin Expeced Uiliy and he Cerainy Equivalen In his secion, we sudy he properies of he coninuous-ime α maxmin uiliy funcion as given by for he ime 0 uiliy and U(X) = E[u(X)] U (X) = E [u(x)] for he dynamic uiliy process. We sar wih he basic coninuiy properies and dynamic consisency. Proposiion 1 Le u be coninuous and sricly increasing. Then he nonlinear expeced-uiliy funcional U : L T R is (i) norm coninuous: if X n X in L T, hen lim n U(X n ) = U(X). (ii) order coninuous: if X n X, P-a.s., hen U(X n ) U(X). (iii) (sricly) monoone: if X Y hen U (X) U (Y ), for all [0, T ]. If also P(X > Y ) > 0 and u is sricly increasing, hen U(X) > U(Y ). (iv) dynamically consisen: le s, if U (X) U (Y ) hen U s (X) U s (Y ). Le us now come o risk aversion. As he uiliy funcional U is no concave, one migh wonder if U displays risk aversion. We will show ha for a naural exension of he concep of risk aversion o Knighian uncerainy, risk aversion is sill equivalen o he concaviy of he Bernoulli uiliy funcion u. Definiion 2 An agen is condiionally E-risk averse on L T if U (X) u (E [X]), for all X L T, [0, T ]. Proposiion 2 Le u C 2 (R) be increasing. The agen is condiionally E-risk averse if and only if u is concave. We coninue wih a discussion of he cerainy equivalen and exend he Arrow- Pra analysis. Definiion 3 C X R is called cerainy equivalen of X L T if holds rue. u(c X ) = E[u(X)] 8

10 For derivaion of he second-order approximaion of he cerainy equivalen consider a given wealh w R and denoe he absolue risk aversion by A(x) = u (x). For a Taylor expansion, we need some furher erminology: u (x) The expression var(x) = E [(X E[X]) 2 ] denoes he variance under E and co(x, Y ) = E[X Y ] E[X] + E[Y ] refers o he so-called coexpecaion and quanifies he compensaion for he nonlineariy of E. 6 The case a = 0 in Example 1, yields a sub linear expecaion, hence co(, ) 0. Theorem 3 Le u be concave and wice differeniable and E be an α-maxmin condiional expecaion. Then, C w+x w 1 ( 2 A(w)var(X) + co X, 1 ) 2 A(w)X2, (7) where X wih E[X] = 0 denoes a cenered disorion. For perspecive, we sae wo examples in a saic seup, ha invesigae he role of he coexpecaion and he resuling uncerainy premium. Example 2 Le he normally disribued disorion X be ambiguous in he volailiy parameer, i.e. Law Pσ (X) = N(0, σ). Wih P = {P σ : σ [σ, σ]}, each σ [σ, σ] induces a law P σ for X. Le u be of CARA ype such ha A(w) = 2. To calculae he co-par in (7) for arbirary α [0, 1], we begin wih E α [X X 2 ] = α min σ [σ,σ] EPσ [X X 2 ] + (1 α) max σ [σ,σ] EPσ [X X 2 ] = ασ + (1 α)σ and similarly E α [X 2 ] = ασ + (1 α)σ. Since E α [X] = 0, we have co (X, 12 ) ( ) A(w)X2 = (1 2α) σ σ. Every α 1 2 yields a posiive coexpecaion. The following example discusses he quaniaive differences of risk premia when comparing wih he sandard expeced uiliy model. Example 3 Le here be wo saes of he world Ω = {good, bad}. The nonlinear expecaion given by P = {P = (p, 1 p) (Ω) : p [ 2 5, 3 5 ]} and α = ( 1 3, 2 3 ). We compare he quaniaive effec wih a linear expecaion P = ( 1 2, 1 2 ) (Ω). 6 In comparison o he approximaion of he (second order) smooh ambiguiy cerainy equivalen in Maccheroni, Marinacci, and Ruffino (2013), he presen cerainy equivalen reveals he nonlinear srucure of he expecaion. In our approximaion his is exposed by he coexpecaion. 9

11 For expeced uiliies, ake u(x) = x as he uiliy index. Consider he gamble X = (1, 4). A direc calculaion yields E[X] = 1 3 min P P EP [X] max P P EP [X] 2.53 > 2.5 = E P [X] and similarly for he expeced uiliies we derive E( X) 1.53 > 1.5 = E P [ X]. Since he inverse of x is x 2, we hen have for he cerainy equivalens C X 2.34 > 2.25 = C X. In conras o he risk premium R(X) under P, he uncerainy premium R(X) = C X E[X] conains a nonlinear componen. The second erm in he decomposiion R(X) + (R(X) R(X)) resuls in an ambiguiy premium. For a small disorion, he example poins ou ha under a nonlinear expecaion he uncerainy premium may vary considerably in comparison o he linear case. This is consisen wih he derivaions in (7), where he coexpecaion co : L T L T R conrols his issue. The possibly negaive ambiguiy premium, caused by preferences for ambiguiy, is manifesed in he nonlinear behavior of he risk premium R(X). We exend he concep of cerainy equivalen now o he dynamic case and begin wih he complee descripion of he condiional cerainy equivalen C X = u 1 (U (X)). Proposiion 3 Le u C 2 (R) be sricly increasing and concave and X L T. The condiional cerainy equivalen C X saisfies he following: 1. C X E [X]. 2. Le (U (X), σ u ) be he unique soluion of du (X) = e(, σ u )d + σ u db, U T (X) = u(x), where e (, σ u ) = a max θ Θ θ σ u + (1 a ) min θ Θ θ σ u and denoe σ C = σu u (C X ). Then, he pair (CX, σ C ) solves ( 1 dc X = 2 A(CX )σ C σ C + e (, ) σu ) σ C σ u d + σ C db, CT X = X. (8) Since σ C σ C is he derivaive of he quadraic variaion C X of C X, he variance muliplier of (7) appears again in he condiional version (8). From his perspecive, (8) describes he local decomposiion of he condiional cerainy equivalen. The residual compensaion e(,σu ) σ C σ u sems from he nonlineariy of he expecaion and corresponds o he coexpecaion in he saic approximaion of Theorem 3. So far, we have fixed he ime T of he payoff. To discuss aspecs abou ime consisency of he cerainy equivalen, we need o vary he erminal ime. For fixed 0 <, define as in (5): de s, [u(x)] = e(s, σ s )ds + σ s db s, s [0, ], E, [u(x)] = u(x). (9) 10

12 For given X L, (9) has a unique soluion by he same argumens as before. The wo ime parameers in (9) corresponds o (5) via E,T = E. Definiion 4 Le u C 2 (R) be sricly increasing and concave and X L. The dynamic cerainy equivalen C s, : L L s a X, wih s [0, ], is defined by u(c s, (X)) = E s, [u(x)], where (E s, [u(x)], σ s ) s is he unique soluion of (9). The condiional cerainy equivalen of Proposiion 3 considers a fixed = T. The dynamic cerainy equivalen has he following properies. Proposiion 4 For 0 r s <, A F s and X, Y L, he following properies hold: (i) Consan-preserving: C, (X) = X. (ii) Recursiviy: C r, (X) = C r,s ( Cs, (X) ). (iii) Dynamic consisency: C r, (X) C r, (Y ), if C s, (X) C s, (Y ). (iv) Monooniciy: If X Y, hen C r, (X) C r, (Y ). (v) Zero-one law: C s, (X1 A ) = C s, (X)1 A and C s, (X1 A + Y 1 A c) = C s, (X)1 A + C s, (Y )1 A c. (vi) Dominance: C r, (X) E r,s [C s, (X)]. In paricular, C r, (X) E r, [X]. The ype of recursiviy for he dynamic cerainy equivalen is illusraed in Figure 2. The cerainy equivalen of X on [r, ] can be obained direcly. Figure 2: Time consisency of dynamic cerainy equivalen Anoher way deermines C s, for X in a firs sep and hen evaluae C s, (X) under he dynamic cerainy equivalen on [r, s]. 11

13 5 Applicaions o Dynamic Asse Pricing 5.1 A Consumpion-Based CAPM wih Mild Opimism Consider a single agen economy wih aggregae endowmen de = µ e d+σ e db and cumulaive dividend process dd = µ D d+σ D db of a long-lived asse wih iniial condiions e 0, D 0 R ++ and adaped inegrable processes µ e, µ D and σ e, σ D. Assume a varian of he maringale generaor condiion (see Secion 10 D in Duffie (1996) for a deailed accoun): σ D > 0 almos everywhere. A ime, he α-maxmin expeced uiliy of he represenaive agen is [ T ] U (e) = E u(e s )ds, where u is a concave and hree-imes differeniable. If a is sufficienly close o 1, as discussed in Example 4 below, σ e(, σ) is sub-linear, he generaor of E in (5). Then, E [X] = min P P Θ E P [X] resuls in a super-linear expecaion. Example 4 Consider only a consan weighing process a = a [0, 1] and he case of κ-ignorance, i.e., Θ = [ κ, κ] and κ R +. The generaor e of he α-maximin expecaion in (5) simplifies considerably: e(, σ ) = (1 a) min θ σ + a max θ σ = (2a 1)κ σ. (10) θ [ κ,κ] θ [ κ,κ] If a > 1, e is sub-linear and yields a super-linear expecaion given by 2 E [X] = min P P Θ E P X, where Θ = [ θ, θ ] = (2a 1) [ κ, κ]. Consequenly, opimism appears as a shrinkage of he size of ambiguiy in E. Deparing from Example 4, we resric he subsequen analysis o a weigh process a ha is sufficienly close o 1 in a way he nonlinear expecaion remains super-linear. In view of Example 4, he residual ambiguiy is denoed by Θ = [ θ, θ ]. The resuling concaviy of c U 0 (c) allows o follow Secion 2.4 of Beißner (2015) for he single-agen case and pars of Secion 5 in Chen and Epsein (2002). By he assumpions on Θ and he lineariy of P E P X, here is a minimizing densiy process θ Θ such ha 7 U (e) = min P P Θ E P [ T ] [ T ] u(e s )ds = E P θ u(e s )ds, 7 There is a prior in P ha is a minimizer of he super-linear expecaion. By consrucion, here is an associaed drif process θ, such ha (2) yields he densiy z θ. 12

14 for every [0, T ]. Under P θ P, he sae-price densiy a ime is given by ψ = u (e ). 8 Assuming a complee marke, sandard argumens yield a descripion of he risky asse by a sochasic Euler equaion [ 1 T ] S = u (e ) EP zs θ u (e s )dd s, [0, T ). (11) The process z θ dz θ = z θ z θ = dp θ F dp is given by (2) and, by virue of (10), solves θ db wih θ = (2a 1)θ sgn(σ U ). 9 The process σ U is he second componen in he BSDE formulaion of U (e) du (e) = (2a 1)θ σ U + u(e )d + σ U db, U T (e) = 0. (12) Wih he Euler equaion in (11) we follow he argumens in Secion 10 H of Duffie (1996) o derive a consumpion-based CAPM relaion. Furhermore, he asse price can be rewrien as he cumulaive reurn ds S = dr = µ R d+σ R db, for deails see Secion 6D in Duffie (1996). P (x) = u (x) denoes he degree of u (x) absolue prudence. The measure of abolue risk aversion is again denoed by A(x). Theorem 4 (CCAPM) Assume ha drif ambiguiy is symmeric, i.e., Θ = [ θ, θ ], wih θ > 0, and suppose opimism is mild, i.e., a [ 1 2, 1]. Then here exiss a securiy spo marke in which, a any ime, he excess reurn of he securiy saisfies µ R r = A(e ) σ R σ e + (2a 1)θ sgn(σ U ) σ R. (13) The equilibrium ineres rae saisfies [ r = A(e ) µ e + σ e (2a 1)θ sgn(σ U ) 1 ] 2 σe σ e P (e ). (14) The second erm of he righ hand side of (13) refers o he ambiguiy premium under mild opimism and yields a refined explanaion of he equiy premium. Specifically, he ambiguiy premium becomes a funcion of a. The comparaive saics are as follows: an increase in opimism, ha is a decrease of a, yields a smaller ambiguiy premium. This funcional dependency has an inuiive appeal, as preferences for ambiguiy, encoded in E and given by a, direcly quanifies he size of he ambiguiy premium via he opimism facor (2a 1). The boundary case a = 1 le he ambiguiy premium vanish. In he case of 2 no opimism, a = 1, we ge he CCAPM formula of Chen and Epsein (2002). In several cases, he process σ U in (13) can be wrien explicily. This is o some exend of imporance, as he sign of σ U deermines he form of he ambiguiy premium. 8 For deails on he necessary and sufficien firs order condiions of he resuling equilibrium we refer o Duffie (1996). 9 Here, sgn(x) = 1 if x > 0, = 1 for x and = 0 for x = 0. The form of θ follows from x sgn(x) = x. If σ U > 0, (12) is a linear BSDE. Example 5 relies on his aspec. 13

15 Example 5 Suppose ( he aggregae ) endowmen follows a geomeric Brownian moion de = e µ e d + σ e db saring in e0 = 1. The degree of ambiguiy is given by Θ [ κ, κ]. Consider u(x) = (x β 1)/β, wih β (, 1] \ {0} and le a [ 1, 1]. These addiional assumpions on he primiives allows for an 2 explici formulaion 10 of σ U = 1 exp(ρ(a)( T )) e β σ e, ρ(a) where ρ(a) = β ( µ e 1 β 2 (σe ) 2 (1 2a)κ σ e ) is linear and decreasing in a. Following Campbell (2003) abou he sylized facs on aggregae consumpion, se σ e = µ e = 2%. A moderae relaive risk aversion of 2 wih β = 1 yields a posiive ρ(a) κ 25 a and consequenly sgn(σu ) = 1 almos everywhere. The ambiguiy premium in (13) akes now he simple form (2a 1)κσ R. 5.2 Applicaion o Indifference Pricing The dynamic cerainy equivalen sudied in Secion 4 enables us o price coningen claims also via indifference pricing. This yields an alernaive ime consisen pricing principle. The novely of he presen modeling ress on he non-concave uiliy specificaion X E[u(X)]. Hodges and Neuberger (1989) firs use cerainy equivalens o price claims in a saic seing, i.e., from he seller poin of view he indifference price is he smalles amoun money π R ha he seller would willingly sell he claims X: u(π) = E P [u(x)]. Indifference pricing under E P can be exended o our dynamically consisen version of α-maxmin expeced uiliy. Le he uiliy index be wice differeniable, sricly increasing and concave. From Definiion 4, for fixed τ > s, he dynamic cerainy equivalen of a claim X L τ a ime s, π s (X) L s, saisfies u ( π s (X) ) = E s,τ [u(x)]. (E s,τ [u(x)], σ s ) s τ is he unique soluion of (9), wih erminal condiion E τ,τ [u(x)] = u(x). Thus we define he pricing rule as he cerainy equivalen: π s (X) = u 1( E s,τ [u(x)] ). π s (X) is he amoun of money ha he decision maker would pay a ime s for he claim X wih mauriy a ime τ. By virue of Proposiion 4, he indifference pricing rule π s : L τ L s is ime-consisen, monoone increasing and saisfies he zero-one law. Furhermore, by he Jensen inequaliy (see Appendix B) for E s,τ wih fixed τ he price of X saisfies π s (X) E s,τ [X]. We now consider he special case of exreme pessimism. In view of Example 1, we se a = 1. This paricular case capures a form of robus uiliy indifference pricing by incorporaing risk aversion and model uncerainy. 10 The volailiy of he uiliy σ U is he second par in he soluion of he BSDE (12). Since a is consan he argumen follows he same line as Secion 2.4 of Epsein and Miao (2003). 14

16 Example 6 For fixed > 0, le (U s ) s [0,τ] be a dynamic wors case expeced uiliy defined by U s (X) = min P P EP s [u(x)], s [0, τ]. For s < τ, we define he dynamic cerainy equivalen of a X L as above. The pricing rule becomes ) π s (X) = u (min 1 P P EP s [u(x)]. From his expression, i is apparen ha preferences for risk and ambiguiy are again disenangled. As argued above, π s : L τ L s is ime-consisen, monoone and saisfies he zero-one law. Moreover, he price of X defined by he cerainy equivalen is less han min P P E P s [X]. 6 Conclusion We have derived a dynamically consisen exension of he α maxmin model. In coninuous ime, he ime consisen version reains he α maxmin srucure and hus allows o disinguish ambiguiy and ambiguiy aiude, as he saic model does. We characerize risk aversion hrough he concaviy of Bernoulli uiliy funcions. The Arrow-Pra approximaion of he cerainy equivalen conains an addiional ambiguiy premium ha depends on he nonlineariy of he expecaion and herefore on local ambiguiy aiudes. We presen a consumpion based CAPM formula ha allows o explain how he inerplay of opimism and pessimism affecs he excess reurn in erms of an ambiguiy premium. Opimism can decrease he ambiguiy premium. We finally characerize he dynamic cerainy equivalen and use i o discuss he consequences for indifference pricing. A Backward sochasic differenial equaions For he convenience of he reader, we gaher some resuls on backward sochasic differenial equaions (BSDE) here. Pardoux and Peng (1990) inroduced he following equaion: dy = f(, y, σ )d + σ db, [0, T ], y T = X, (15) where he erminal condiion X L T and f : Ω [0, T ] R R R, so ha he generaor f(, y, z) of he BSDE is an adaped process for every y, z R. A pair of adaped real-valued processes (y, σ) is called a soluion of he above BSDE, if E P [sup y 2 ] <, E P [ T σ 0 2 d] < and (y, σ) saisfies (15). Pardoux and Peng (1990) obained he following exisence and uniqueness of he soluion of (15). 15

17 Lemma 1 If E P [ T 0 f(, 0, 0) 2 d] < and f(,, ) is Lipschiz coninuous on R R, hen he above BSDE has a unique adaped soluion (y, σ). B Properies of Equaion (5) In view of (15), he BSDE in (5) considers he following generaor f(, y, σ ) = a max θ Θ θ σ + (1 a ) min θ Θ θ σ, where Θ capures he muliple prior uncerainy P. Proposiion 5 Le a [0, 1] and P be an arbirary specificaion of an α- maxmin condiional expecaion E. For every X L T, here exiss a unique soluion (E [X], σ ) of equaion (5). Moreover, he following properies hold rue for every s, [0, T ], X, Y L T : (i) (sric) Monooniciy: If X Y, hen E [X] E [Y ]. If also P(X > Y ) > 0, hen E 0 [X] > E 0 [Y ]. (ii) Consan-preserving: E [η] = η, if η L and E [c] = c, for all c R. (iii) Tower propery: E s [X] = E s [E [X]], for all s. (iv) Condiional lineariy: E [X + η] = E [X] + η, for every η L. (v) Zero-one law: For any A F, we have E [X1 A ] = E [X]1 A. (vi) Posiive homogeneiy: E [ηx] = ηe [X], for all η 0. (vii) Jensen inequaliy: If u C 2 (R) is increasing and concave, hen E [u(x)] u (E [X]). By 1 A, for some A F, we denoe he usual indicaor funcion, being 1 on A and 0 on A c = Ω \ A. Proof: We sar wih he uniqueness and exisence of he soluion of he BSDE. For all x, y R, we have e(, x) e(, y) = a max θ Θ θ x + (1 a ) min θ Θ θ x a max θ Θ θ y (1 a ) min θ Θ θ y a max θ Θ θ x max θ Θ θ y + (1 a ) min θ Θ θ x min θ Θ θ y max θ Θ θ (x y) + max θ Θ θ (x y). Since Θ is compac, hen here exiss a posiive consan C such ha e(, x) e(, y) C x y. 16

18 Therefore, e(, ) is uniformly Lipschiz and e(, 0) = 0, hen from Lemma 1 in Appendix A, equaion (5) has a unique soluion, Properies (i) o (v) direcly follow from from Lemma 36.6 and Theorem 37.3 in Peng (1997). To show (vi), noe ha for all x R, β > 0, we have posiive homogeneiy of e(, x) e(, βx) = a max (βx) + (1 a ) min (βx) θ Θ θ Θ = βe(, x), Applicaion of Lemma 36.9 (see also Example 10 herein) in Peng (1997) gives us (vi). Since u C 2 (R) is increasing and concave, we can ge (vii) from Theorem 1 in Jia and Peng (2010). C Proofs Proof of Theorem 1 In order o prove ha he family (U i ) i=0,...,n is dynamically consisen, we only need o show ha for i = 0, 1,, N 1, and X, Y L T, if U(i+1) N [X] U (i+1) N [Y ] hen U i N [X] U i N [Y ]. From he definiion of I [X], [0, T ], we know ha I [X] is increasing in X. Therefore, ] [ ] Ui [X] N = I i [U (i+1) [X] N I i U(i+1) [Y N ] = Ui [Y N ], from which we complee he proof. Proof of Theorem 2 For [ N N 1, N N ) := [(N 1), N ), we have U N [X] = I [U N [X][X]] = I N [X]. N Le (E [u(x)], σ ) and (E [u(x)], σ ) be he soluions of he following BSDEs, respecively, and This implies de [u(x)] = min θ Θ θ σ d + σ db, E T [u(x)] = u(x), (16) de [u(x)] = max θ Θ θ σ d + σ db, E T [u(x)] = u(x). (17) E [u(x)] = min P P EP [u(x)], and E [u(x)] = max P P EP [u(x)]. 17

19 Le (E [u(x)], σ ) be he soluions of he following BSDEs. de [u(x)] = = e(, σ )d + σ db, E T [u(x)] = u(x). (18) Then, using he sandard esimaes of BSDEs (16) and (18), here exiss a consan C (C is independen of and can be differen from line o line) such ha E P [ sup E [u(x)] E [u(x)] 2 ] CE P [ s [,T ] CE P [ T T e(r, σ r ) max θ Θ θ rσ r dr] 2 max θ rσ r min θ rσ r dr] 2 θ Θ θ Θ T C(T )E P [ σ r 2 dr] T C E P [ σ r 2 dr] = C. 0 In a similar way, we have he following esimae of BSDEs (17) and (18) Therefore, E P [ sup E [u(x)] E [u(x)] 2 ] C. s [,T ] E P [ U N [X] E [u(x)] 2 ] 2E P [ E [u(x)] E [u(x)] 2 ] + 2E P [ E [u(x)] E [u(x)] 2 ] C. (19) For [ N N 2, N N 1 ), we have U N [X] = I [U N N N 1[X]]. Le (E [X], σ ) and (E [X], σ ) be he soluions of he following BSDEs, respecively, and Then de [X] = min θ σ d + σ db, E N θ Θ N 1 [X] = U N [X], (20) N N 1 de [X] = max θ σ d + σ db, E N θ Θ N 1 [X] = U N [X]. (21) N N 1 E [X] = min P P EP [U N [X]], N N 1 E [X] = max P P EP [U N [X]]. N N 1 18

20 Therefore, using he sandard esimaes of BSDEs (18) and (21), here exiss a consan C (C is independen of and can be differen from line o line) such ha E P [ sup E [X] E [u(x)] 2 ] s [,T ] N CE P N 1 [ e(r, σ r ) max θ rσ r dr] 2 + E P [ U N θ Θ [X] E N N 1 N N 1 [u(x)] 2 ] N CE P N 1 [ max θ rσ r min θ rσ r dr] 2 + E P [ U N θ Θ θ Θ N N 1 N N 1 C(T )E P [ σ r 2 dr] + E P [ U N E N 1[X] N N N 1 [u(x)] 2 ] T C E P [ σ r 2 dr] + E P [ U N E N 1[X] N N N 1 [u(x)] 2 ]. 0 From (19) i follows ha E P [ sup E [X] E [u(x)] 2 ] C. s [,T ] [X] E N N 1 [u(x)] 2 ] In a similar way, we have he following esimae of BSDEs (5) and (20) Therefore, E P [ sup E [X] E [u(x)] 2 ] C. s [,T ] E P [ U N [X] E [u(x)] 2 ] 2E P [ E [X] E [u(x)] 2 ] + 2E P [ E [X] E [u(x)] 2 ] C. Using he above approach, we can prove ha, for all [ N i, N i+1), i = 0, 1,..., N 2, E P [U N [X] E [u(x)] 2 ] C, and he resul follows by leing N. Proof of Proposiion 1 (i) The coninuiy follows from he presence of a dominaing subliner expecaion, which implies norm-coninuiy. (ii) We jus give he proof when {X n } n 1 is decreasing. From he monooniciy of he nonlinear expecaion E, we know ha {U(X n )} n 1 is decreasing. Since {X n } n 1 is decreasing and lim X n = X, P-a.s., we ge ha u(x n ) n u(x) u(x n ) + u(x) L T, and lim u(x n ) u(x) = 0, P-a.s. Then by n 19

21 virue of he dominaed convergence heorem we have, lim n E P u(x n ) u(x) 2 = 0. From (vii) in Proposiion 5 we know ha, here is a consan C > 0 such ha, U(X n ) U(X) 2 CE P [ u(x n ) u(x) 2], from which we can ge lim n U(X n ) = U(X). (iii) Since X Y, P-a.s., and u is increasing, we have u(x) u(y ), P-a.s. From (i) in Proposiion 5 i follows ha U (X) = E [u(x)] E [u(x)] = U (Y ). Moreover, if P(X > Y ) > 0 and u is sricly increasing, hen P(u(X) > u(y )) > 0. Using (i) in Proposiion 5 again U(X) = E[u(X)] > E[u(Y )] = U(Y ). (iv) By (i) and (iii) in Proposiion 5, i is easily o ge his. Proof of Proposiion 2 Since e(, σ) is a convex combinaion of an inf and sup operaion, e(, σ) is posiive homogeneous in σ. By an applicaion of Theorem 3.2 in Jia and Peng (2010) o e(, σ), which is independen of E [X], he condiional E-concaviy, i.e., u(e [X]) E [u(x)], can be characerized as follow 1 2 u (x) σ 2 + e(, u (x)σ) u (x)e(, σ) 0 By he posiive homogeneiy of e(, ) his is equivalen o u (x) 0, being equivalen o concaviy. Proof of Theorem 3 We consider he second-order Taylor expansion around w for u(w) E[u(X + w)] u(w) + E [u (w)x + 12 ] u (w)x 2 = u(w) + u (w)e [X 12 ] A(w)X2 ( 1 = u(w) + u (w) 2 A(w)E [ X 2] + co(x, 1 ) 2 A(w)X2 ), where we applied he concaviy of u via u 0, u 0 and Proposiion 5 (iv) and (iv). Using he firs-order Taylor expansion for u(c w+x ) around w: u(c w+x ) u(w) + u (w)(c w+x w). Combining boh approximaions esablishes he desired resul. 20

22 Proof of Proposiion 3 1. From Proposiion 5 (vii) we know ha E [u(x)] u (E [X]). Since u is sricly increasing, we have C (X) = u 1 (E [u(x)]) E [X]. 2. Le (E [u(x)], σ u ) [0,T ] be he unique soluion of he following equaion: de [u(x)] = e(, σ u )d + σ u db, [0, T ], E T [u(x)] = u(x). Then from Iô Lemma wih respec o u 1 (E [u(x)]) i follows ha ( e(, σ X dc (X) = ) u (C (X)) 1 ) u (C (X)) 2 u (C (X)) 3 (σu ) 2 σ u d + u (C (X)) db. We denoe σ C = dc (X) = σ u u (C (X)), hen ( σ C σ u e(, σ X ) 1 ) u (C (X)) 2 u (C (X)) (σc ) 2 d + σ C db = σc e ( ), σ U 1 ( ) σ u + σ C 2 2 A(C X )d + σ C db. Proof of Proposiion 4 (i) By (vi) in Proposiion 5 and he definiion of he dynamic cerainy equivalen, we have C (X) = u 1 (E, [u(x)]) = u 1 (u(x)) = X. (ii) By (iv) in Proposiion 5 and he definiion of he dynamic cerainy equivalen, we have (iii) From (ii) i follows ha C r, (X) = u 1 (E r, [u(x)]) = u 1 (E r,s [E s, [u(x)]]) = u 1 (E r,s [u(c s, (X))]) = u 1 (u(c r,s (C s, (X)))) = C r,s (C s, (X)). C r, (X) = C r,s (C s, (X)) C r,s (C s, (Y )) = C r, (Y ). (iv) We ake v = in (iii) and from (i) i follows ha, if X Y, hen C r, (X) C r, (Y ). 21

23 (v) Since u(x1 A + Y 1 A c) = u(x1 A + Y 1 A c)1 A + u(x1 A + Y 1 A c)1 A c = u(x)1 A + u(y )1 A c, we have C s, (X1 A + Y 1 A c) = u 1 (E s, [u(x1 A + Y 1 A c)]) = u 1 (E s, [u(x)1 A + u(y )1 A c]). (22) Le us consider he following wo BSDEs de s, [u(x)] = ê(σs)ds 1 + σsdb 1 s, E, [u(x)] = u(x), (23) and de s, [u(y )] = ê(σs)ds 2 + σsdb 2 s, E, [u(y )] = u(y ). (24) Then (23) 1 A + (24) 1 A c yields wih he erminal condiion Recall he following BSDE d(e s, [u(x)]1 A + E s, [u(y )]1 A c) = [ê(σ 1 s)1 A + ê(σ 2 s)1 A c]ds + (σ 1 s1 A + σ 2 s1 A c)db s = ê(σ 1 s1 A + σ 2 s1 A c)ds + (σ 1 s1 A + σ 2 s1 A c)db s, E, [u(x)]1 A + E, [u(y )]1 A c = u(x)1 A + u(y )1 A c. de s, [u(x)1 A + u(y )1 A c] = ê(σ s )ds + σ s db s, E, [u(x)1 A + u(y )1 A c] = u(x)1 A + u(y )1 A c. From he uniqueness of he soluion of he above equaions, we have E s, [u(x)]1 A + E s, [u(y )]1 A c) = E s, [u(x)]1 A + E s, [u(y )]1 A c. Therefore, from (22) we have C s, (X1 A + Y 1 A c) = u 1 (E s, [u(x)]1 A + E s, [u(y )]1 A c) = u 1 (E s, [u(x)])1 A + u 1 (E s, [u(y )])1 A c = C s, (X)1 A + C s, (Y )1 A c. Le Y = 0, hen by E s, [0] = 0 we have C s, (X1 A ) = C s, (X)1 A. (vi) From (ii) we have C r, (X) = C r,s (C s, (X)) = u 1 (E r,s [u(c s, (X))]). 22

24 Therefore, by Jensen inequaliy in Proposiion 5 we ge E r,s [u(c s, (X))] u(e r,s [C s, (X)]). From he above inequaliies i follows ha C r, (X) E r,s [C s, (X)]. In paricular, by using (i) we ge C r, (X) E r, [X]. Proof of Theorem 4 A usual applicaion of Iô s formula o u (e ) and ψ = u (e ) z θ (under P) allows o formulae he dynamics of he sae-price densiy as dψ ( = u (e )µ e z θ + σ e θ + 1 ) 2 σe σ e u (e ) d + u (e )σ e u (e )θ db. (25) The equilibrium shor rae mus saisfy r = µψ ψ, where µ ψ denoes he drif par in (25). By virue of (25), he excess reurn urns ou o be µ R r = σψ ψ σ R = A(e ) σ R σ e + (2a 1)θ sgn(σ U ) σ R, ) where σ ψ = z (u θ (e )σ e u (e )θ is he volailiy par in (25). As in Theorem 3, A(e ) denoes he Arrow Pra measure of absolue risk aversion a e. References Beißner, P. (2015): Brownian equilibria under Knighian uncerainy, Mahemaics and Financial Economics, 9(1), Campbell, J. Y. (2003): Consumpion-based asse pricing, Handbook of he Economics of Finance, 1, Chaeauneuf, A., J. Eichberger, and S. Gran (2007): Choice under uncerainy wih he bes and wors in mind: Neo-addiive capaciies, Journal of Economic Theory, 137(1), Chen, Z., and L. Epsein (2002): Ambiguiy, risk, and asse reurns in coninuous ime, Economerica, 70(4), Duffie, D. (1996): Press. Dynamic asse pricing heory, Princeon Universiy 23

25 Epsein, L. G., and J. Miao (2003): A wo-person dynamic equilibrium under ambiguiy, Journal of Economic Dynamics and Conrol, 27(7), Epsein, L. G., and M. Schneider (2003): Recursive muliple-priors, Journal of Economic Theory, 113(1), Ghirardao, P., F. Maccheroni, and M. Marinacci (2004): Differeniaing ambiguiy and ambiguiy aiude, Journal of Economic Theory, 118(2), Gilboa, I., and D. Schmeidler (1989): Maxmin expeced uiliy wih nonunique prior, Journal of Mahemaical Economics, 18(2), Hodges, S. D., and A. Neuberger (1989): Opimal replicaion of coningen claims under ransacion coss, Review of fuures markes, 8(2), Jia, G., and S. Peng (2010): Jensen s inequaliy for g-convex funcion under g-expecaion, Probabiliy Theory and relaed Fields, 147(1-2), Maccheroni, F., M. Marinacci, and D. Ruffino (2013): Alpha as ambiguiy: robus mean-variance porfolio analysis, Economerica, 81(3), Pardoux, E., and S. Peng (1990): Adaped soluion of a backward sochasic differenial equaion, Sysems and Conrol Leers, 14(1), Peng, S. (1997): Backward SDE and relaed g-expecaion, Piman research noes in mahemaics series, pp