Some new results on homothetic forward performance processes

Transcription

1 Some new resuls on homoheic forward performance processes WCMF, Sana Barbara Sepember 2014 Thaleia Zariphopoulou The Universiy of Texas a Ausin

2 Represenaion of homoheic forward performance processes via ergodic and infinie horizon quadraic BSDE Join work wih G. Liang Connecion of homoheic forward performance processes wih he fracional Kelly crierion under model uncerainy Join work wih S. Källblad and J. Ob lój

3 Homoheic forward performance processes and ergodic and infinie horizon quadraic BSDE

4 The sochasic facor model W = (W 1,, W d ) T is a d-dimensional BM on (Ω, F, F = (F ) 0, P) The marke consiss of a riskless bond wih price 1 and n socks S = (S 1,, S n ) T ds/s i i = b i (V )d + σ i (V )dw d = b i (V )d + σ ij (V )dw j j=1 The d-dim process V = (V 1,, V d ) T models sochasic facors dv = η(v )d + ΣdW Predicabiliy models (Review aricle, Z. (2009))

5 Assumpions The vecor b(v) = (b 1 (v),, b n (v)) T, and he volailiy marix σ(v) = (σ ij (v)) 1 i n,1 j d are uniformly bounded for v R d, and moreover, he volailiy marix σ(v) has full row rank n. Then, he marke price of risk equaion σ(v)θ(v) = b(v) admis a soluion θ(v) = σ(v) T [σ(v)σ(v) T ] 1 b(v) The marke price of risk θ(v) is uniformly bounded and Lipschiz coninuous for v R d The drif of he sochasic facor η(v) saisfies he dissipaive condiion (η(v) η( v)) T (v v) C η v v 2 for a posiive large enough consan C η. Hence, V v V v 2 e 2Cη v v 2 The volailiy marix of he sochasic facor is consan wih Σ = 1 and ΣΣ T is uniformly ellipic

6 Wealh and admissible sraegies The wealh equaion is where dx π = X π π T (b(v )d + σ(v )dw ) = X π π T (θ(v )d + dw ) π T = π T σ(v ) Admissible sraegies in he rading inerval [, s] A [,s] = { (π u) u [,s] π L 2 BMO[, s], π u Π, and X π u D } where Π R d is closed and convex, D he wealh admissibiliy domain and L 2 BMO[, s] = { (π u) u [,s] : π is F-progressively measurable and { ( s ) } ess sup E π u 2 du Fτ for any F-sopping ime τ [, s] < } τ τ For A [0, ), i means π L 2 BMO[0, ] for any 0

7 The classical expeced uiliy The key ingrediens are he choices of he rading horizon [0, T ] and he invesor s uiliy u T (x) a T The objecive is o maximize he expeced uiliy of erminal wealh over he admissible sraegies: u(x, ; T ) = ess sup π A [,T ] E P (u T (X π T ) F, X = x) If u T (x) is power, log or exponenial, hen he value process u(x, ; T ) and he opimal sraegy can be characerized in erms of he soluion o a quadraic BSDE defined on [0, T ]; see, El Karoui and Rouge (2000), Hu e al (2005), Mania and Schweizer (2005),... For Markovian cases, see Z. (1999), Pham (2002),...

8 Quadraic BSDE Le u T (x) = xδ δ (2005)), for 0 < δ < 1, and D = R+. Then, (Hu e al. u(x, ; T ) = xδ (T ) δ ey and he opimal sraegy is given by π = P roj Π Z (T ) + θ(v ) (1 δ) where (Y (T ), Z (T ) ) is he unique bounded soluion of he quadraic BSDE dy (T ) = F (V, Z (T ) )d + (Z (T ) ) T dw wih Y (T ) T = 0 and he driver F (, ) is given by } + 1 δ 2 1 δ z+θ(v ) z 2 F (V, z) = 1 2 δ(1 δ)dis2 { Π, z + θ(v ) 1 δ

9 Quadraic BSDE (con d) Condiions on he coefficiens: and F (v, z) F (v, z) C z (1 + z + z ) z z F (v, 0) K Kobylanski (2001) shows ha here exiss a unique Markov soluion (Y (T ), Z (T ) ), Y (T ) = y(v, ; T ), wih y(, ; T ) being uniformly bounded I can be shown ha y(v, ; T ) solves y (v, ; T ) Trace(ΣΣT D 2 y(v, ; T )) + η(v) T Dy(v, ; T ) wih y(v, T ; T ) = 0 + F (v, Σ T Dy(v, ; T )) = 0,

10 The forward performance process

11 The forward performance process Musiela and Z. (2003),... Definiion A forward performance process is an F-progressively measurable process U(x, ) such ha for any 0, he map x U(x, ) is sricly increasing and sricly concave for x D for any admissible sraegy π A [0, ) and for any 0 s < U(X π, ) E P (U(X π s, s) F ) and here exiss an opimal sraegy π A [0, ) such ha ( ) U(X π, ) = E P U(Xs π, s) F

12 The forward performance SPDE Suppose ha he SPDE du(x, ) = b(x, )d + a(x, ) T dw admis an F-progressively measurable soluion U(x, ), which is, for 0, sricly increasing, sricly concave and smooh in x, and saisfies cerain inegrabiliy condiions, where he drif F (x, ) is given by b(x, ) = 1 { 2 x 2 xx U(x, )dis 2 Π, } xa(x, ) + θ(v ) x U(x, ) x xx U(x, ) + xa(x, ) + θ(v ) x U(x, ) 2 2 xx U(x, ) Then U(x, ) is a forward performance process. In general, solving he above fully nonlinear ill-posed SPDE is a formidable ask; see El Karoui and M rad (2013a, 2013b), Nadochiy and Z. (2013), Nadochiy and Tehranchi (2014)

13 Time monoone case and no rading consrains If a(x, ) 0, he SPDE reduces o du(x, ) = 1 2 θ(v ) 2 U x(x, 2 ) U xx (x, ) d ( Then U(x, ) = u x, ) 0 θ(v ) s 2 ds where u (x, ) = 1 u 2 x(x, ) 2 u xx (x, ) The solvabiliy of he above PDE is closely relaed o he ill-posed hea equaion h (x, ) h xx(x, ) = 0 The key ingredien is Widder s heorem; see, Musiela and Z. (2010), Berrier e al. (2009)

14 Homoheic forward performance processes

15 The power forward performance case U(x, ) = xδ δ ek where 0 < δ < 1, K is F-measurable and independen of x For he Markovian case, U(x, ) = xδ δ ek(v,), one needs o find K(v, ) ha solves K (v, ) Trace(ΣΣT D 2 K(v, )) +η(v) T DK(v, ) + F (v, Σ T DK(v, )) = 0 In a special (one dimensional) case, a similar equaion was sudied in Nadochiy and Z. (2013), and in higher dimensions in Nadochiy and Tehranchi (2014)

16 Represenaion via ergodic quadraic BSDE

17 Ergodic quadraic BSDE represenaion (Liang-Z.) Consider he ergodic quadraic BSDE wih dy = ( F (V, Z ) + λ)d + Z T dw F (V, Z ) = 1 { 2 δ(1 δ)dis2 Π, Z } + θ(v ) 1 δ + 1 δ 2 1 δ Z + θ(v ) Z 2 The equaion admis a unique Markovian soluion in he sense ha Y = y(v ) for some funcion y( ) wih linear growh, and Z The uniqueness refers o he uniqueness of λ. Cv C η C v

18 Ergodic quadraic BSDE represenaion (con d) The process U(x, ) = xδ δ ey λ is a power forward performance process and saisfies he forward SPDE du(x, ) = U(x, )( F (V, Z ) Z 2 )d + U(x, )Z T dw The opimal sraegy is given by π = Proj Π Z + θ(v ) (1 δ)

19 Examples

20 Time-monoone case wihou rading consrains In he ergodic BSDE, le Z = 0. Then, for (Y, λ) such ha, ( dy = 1 ) δ 2 1 δ θ(v ) 2 + λ d yields he soluion Y λ = Y δ 2 1 δ θ(v s) 2 ds Then U(x, ) = xδ δ ey λ = xδ 1 δ δ ey δ θ(vs) 2 ds is he familiar ime monoone power forward performance process. The opimal sraegy is π = θ(v ) 1 δ

21 Markovian case wihou rading consrains Le Y = y(v ) be he Markovian soluion. If y( ) has enough regulariy, Iô s formula yields ( ) 1 dy(v ) = 2 Trace(ΣΣT D 2 y(v )) + η(v) T Dy(V ) d + (Σ T Dy(V )) T dw Then, Z = Σ T Dy(V ) and (y( ), λ) solves, a leas formally, he ergodic PDE 1 2 Trace(ΣΣT D 2 y(v)) + η(v) T Dy(v) + F (v, Σ T Dy(v)) = λ The funcion K(v, ) = y(v) λ solves he ime-reversed PDE K (v, ) Trace(ΣΣT D 2 K(v, )) + η(v) T DK(v, ) + F (v, Σ T D(v, )) = 0

22 Verificaion

23 Idea of he proof Need o show ha for any π A [0, ), and 0 s <, and for some π, ( (X π s E δ ( (X π E s ) δ ) e Ys λs F δ ) δ e Ys λs F ) (Xπ ) δ e Y λ δ = (Xπ ) δ e Y λ δ Iô s formula o (Xs π ) δ on [, s] yields ( s (Xs π ) δ = (X π ) δ exp δ (π Tu θ(v u ) 12 ) π u 2 du + The ergodic quadraic BSDE gives ( s e Ys λs = e Y λ exp F (V u, Z u )du + s s Z T u dw u ) δπ T u dw u )

24 Idea of he proof (con d) Hence, ( s ( (Xs π ) δ e Ys λs = (X π ) δ e Y λ exp δ (π Tu θ(v u ) 12 ) π u 2 ) F (V u, Z u ) du + s (δπ T u + Z T u )dw u ) Take condiional expecaion wih respec o F on boh sides, ( ( s ( E((Xs π ) δ e Ys λs F ) = (X π ) δ e Y λ E exp δ (π Tu θ(v u ) 12 ) π u 2 ) F (V u, Z u ) du + s (δπ T u + Z T u )dw u ) F ) For any admissible sraegy π A [0, ), we define a new probabiliy measure Q Z s = dq ( ) = E (δπu T + Zu T )dw u dp Fs 0 s

25 Idea of he proof (con d) The condiional expecaion on he righ hand side of he above equaion is ( ( s ( ) ) ) E exp F π Zs (V u, Z u ) F (V u, Z u ) du Z F ( s ( ) ) ) = E (exp Q F π (V u, Z u ) F (V u, Z u ) du F, where F π (V, Z ) = 1 2 δ(1 δ) π 2 + δπ T (Z + θ(v )) Z 2 = 1 2 δ(1 δ) π Z + θ(v ) 1 δ For any π A [0, ), F π (V, Z ) F (V, Z ), so δ 2 1 δ Z + θ(v ) Z 2 E((X π s ) δ e Ys λs F ) (X π ) δ e Y λ Take π = Z+θ(V) 1 δ, hen F π (V, Z ) = F (V, Z ) and, in urn, E Q ((X π s ) δ e Ys λs F ) = (X π ) δ e Y λ

26 Solving he ergodic quadraic BSDE

27 Solving he ergodic quadraic BSDE Recall he ergodic quadraic BSDE dy = ( F (V, Z ) + λ)d + Z T dw wih he sochasic facor V dv = η(v ) + ΣdW Condiions on he coefficens F (v, z) F ( v, z) C v (1 + z ) v v and F (v, z) F (v, z) C z (1 + z + z ) z z F (v, 0) K

28 Truncaed ergodic BSDE Define a runcaion funcion: q(z) = min{ z, C v/(c η C v )} z1 {z 0} z Consider he runcaed ergodic BSDE Condiions on he coefficiens: dy = ( F (V, q(z )) + λ)d + Z T dw F (v, q(z)) F ( v, q(z) C ηc v C η C v v v F (v, q(z)) F (v, q( z) C z C η + C v C η C v z z

29 Truncaed ergodic BSDE (con d) If we can prove he runcaed ergodic BSDE admis a soluion (Y, Z, λ) wih Z C v C η Cv hen q(z ) = Z and herefore, (Y, Z, λ) is also a soluion o he ergodic quadraic BSDE The runcaed ergodic BSDE has been solved by Fuhrman, Hu and Tessiore (2009) Perurbaion argumen + Girsanov s ransformaion

30 Represenaion of he power forward performance process via infinie horizon quadraic BSDE

31 Infinie horizon quadraic BSDE Le ρ > 0. The infinie horizon quadraic BSDE on [0, ) dy ρ = ( F (V, Z ρ ) + ρy ρ )d + (Z ρ ) T dw admis a unique Markovian soluion, Y ρ = y ρ (V ), for some funcion y ρ ( ) bounded by K ρ, and Zρ Cv C η C v See Briand and Conforola (2008)

32 Infinie horizon quadraic BSDE represenaion The process U ρ (x, ) = xδ ρ δ ey ρy ρ 0 s ds is a power forward performance process and saisfies du ρ (x, ) = U ρ (x, )( F (V, Z ρ Zρ 2 )d The opimal sraegy is given by +U ρ (x, )(Z ρ ) T dw π = P roj Π Z ρ + θ(v ) (1 δ)

33 Connecion beween he ergodic and he infinie horizon represenaions

34 Connecion beween he ergodic and he infinie horizon represenaions There exiss a subsequence ρ n 0 such ha for a.e. (, ω) R + Ω which is U ρn (x, ) U(x, ) exp Y v,ρn Y v 0,ρn 0 Y v }{{} 0 U ρn (x, ) lim = consan ρ n 0 U(x, ) = Idea of he proof 0 ( exp Y v,ρn exp(y v 0 ρ nyu v,ρn λ) ) du ρ n(y v,ρn u Y v 0,ρn 0 ) du (ρ ny v 0,ρn 0 λ) + Y v 0,ρn 0 }{{} 0 }{{} 0

35 Connecion beween he power classical expeced uiliy and he forward performance process

36 An auxiliary expeced uiliy problem Consider he problem of maximizing erminal expeced uiliy in he presence of a muliplicaive random endowmen e ξ T where u ρ (X, ; T ) = ess sup π A [,T ] E ξ T = 1 δ T 0 ( (X π T e ξ T ) δ ) δ F ρyu ρ,t du for ρ > 0, and Y ρ,(t ) being he soluion of he quadraic BSDE on [0, T ] Y ρ,(t ) = T (F (V u, Zu ρ,(t ) ) ρyu ρ,(t ) )du T (Zu ρ,(t ) ) T dw u

37 Connecion beween he classical and forward seings For each ρ > 0, u ρ (x, ; T ) lim T U ρ = 1 (x, ) for a.e. (, ω) R + Ω Indeed, he resuls of Hu e al (2005) yield u ρ (x, ; T ) = xδ (T ) δ ey, where Y (T ) is he unique bounded soluion of he quadraic BSDE on [0, T ] Y (T ) = δξ + T F (V u, Z u (T ) )du T ( On he oher hand, Y ρ,(t ) ) ρ,(t ) ρy 0 u du, Z ρ,(t ) he (erminal horizon) quadraic BSDE. Therefore, and Y (T ) = Y ρ,(t ) ( u ρ (x, ; T ) = xδ δ exp Y ρ,(t ) 0 ρyu ρ,(t ) du 0 (Z u (T ) ) T dw u ) ρyu ρ,(t ) du also saisfies

38 Connecion beween he backward and forward uiliy maximizaion Therefore, u ρ (x, ; T ) U ρ (x, ) = exp Y ρ,t (con d) Y ρ } {{ } 0 (ρy ρ,t 0 } u ρyu ρ ) du {{} 0 Finally, for a.e. (, ω) R + Ω lim lim u ρn (x, ; T ) ρ n 0 T U(x, ) = lim ρn 0 lim T u ρn (x, ; T ) U ρn (x, ) = consan U ρn (x, ) U(x, )

39 Remarks Ergodic quadraic BSDE and infinie horizon quadraic BSDE are naural candidaes for he represenaion of power/log/exponenial forward performance processes Forward performance process moivaes he sudy of non-unique soluions of he BSDE The unique Markovian soluion of he BSDE gives Markovian forward performance process In he presence of a random endowmen, he classical value processes converge o he forward ones as he rading horizon becomes infinie

40 Homoheic forward uiliies under Knighian uncerainy

41 Knighian uncerainy (model ambiguiy) Frank Knigh (1921) The hisorical measure P migh no be a priori known Gilboa and Schmeidler (1989) buil an axiomaic approach for preferences owards boh risk and model ambiguiy. They proposed he robus uiliy form X π T inf Q Q E Q (U (X π T )), where U is a classical uiliy funcion and Q a family of subjecive probabiliy measures Sandard criicism: he above crierion allows for very limied, if a all, differeniaion of models wih respec o heir plausibiliy

42 Knighian uncerainy Maccheroni, Marinacci and Rusichini (2006) exended he above approach o X π T inf Q Q (E Q (U (X π T )) + γ (Q)) where he funcional γ (Q) serves as a penalizaion weigh o each Q-marke model Enropic penaly and enropic robus uiliy dq γ (Q) = H (Q P) wih H (Q P) = dp ln ( dq dp ( ) inf (E Q (U (XT π )) + γ (Q)) = ln E P e U(X T ) Q Q Hansen, Talay, Schied, Föllmer, Frielli, Weber... ) dp

43 Maxmin sochasic opimizaion problem Sock dynamics: S = ( ) S 1,..., S d, [0, T ], semimaringales Wealh dynamics: X α = x + 0 α s ds s, X α 0, 0 Objecive: v (x) = sup X X (x) where Q = {Q P γ (Q) < } inf (E Q (U (X T )) + γ (Q)), Q Q where u (y) = Ũ (y) = sup (U (x) xy) x>0 Dualiy approach inf inf ( (Ũ EQ (YT ) ) + γ (Q) ) Y Y Q (y) Q Q

44 Forward robus porfolio crierion (Källbald, Ob lój, Z.) allow flexibiliy wih respec o he invesmen horizon incorporae learning produce opimal invesmen sraegies closer o he ones used in pracice Forward robus crierion A pair (U (x, ), γ,t (Q)) of a uiliy process and a penaly crierion which saisfies, for all 0 T, ( ) ) T U(x, ) = ess sup ess inf (E Q U(x + α s ds s, T ) α Q Q,T F + γ,t (Q) wih Q T = {Q Q : Q FT P FT } This crierion gives rise o an ill-posed SPDE corresponding o a zero-sum sochasic differenial game

45 Connecion wih he (fracional) Kelly crierion

46 Is here a pair (U (x, ), γ,t ) ha yields he Kelly porfolio? ( ) True model ds = S λ d + σ dw 1, (W 1, W 2 ) under P Proxy model: ds = S (ˆλ d + σ dŵ 1 ), (W 1, W 2 ) under ˆP For each Q ˆP and each T > 0, le η = (η 1, η 2 ), 0 T, ( dq ) = E ηsdŵ 1 s 1 + ηsdŵ 2 s 2 dˆp FT 0 0 T Candidae penaly funcionals ( ) T γ,t (Q) = E Q g(η s, s)ds F

47 Logarihmic risk preferences and quadraic penaly U(x, ) = ln x δ s 1 + δ s ˆλ2 s ds, 0, x > 0 γ,t (Q η ) = E Q η ( T ) δ s 2 η s 2 ds F The process δ is adaped, non negaive and conrols he srengh of he penalizaion I models he confidence of he invesor re he rue model

48 (Fracional) Kelly sraegies and forward opimal conrols Invesor chooses proxy model (ˆλ ) and confidence level (δ ) Opimal measure Q η ( η = ˆλ ) ( (1 + δ ), 0 dq η and dˆp = E Opimal forward Kelly porfolio π = δ 1 + δ ˆλ σ 0 ˆλ 1 + δ dŵ 1 If δ (infinie rus in he esimaion), hen π ˆλ σ, which is he Kelly sraegy associaed wih he mos likely model ˆP If δ 0 (no rus in he esimaion), hen π 0 and he opimal behavior is o inves nohing in he sock ) T

49 Bon anniversaire Jean-Pierre!