Efficient portfolios in financial markets with proportional transaction costs

Transcription

1 Joint work E. Jouini and V. Portes Conference in honour of Walter Schachermayer, July 2010

2 Contents

3 : An efficient portfolio is an admissible portfolio which is optimal for at least one agent. Our goal is to characterize such as in Dybvig 88 and Jouini-Kallal 01 In Dybvig : multiperiod, finite Ω, no frictions. Main result: a pf X is efficient iff X and the state-price density are anticomonotonic. It gives a method to improve strategies (stop-loss, lock-in etc). Jouini-Kallal generalised Dybvig to a market different kind of frictions, e.g.. In this talk we allow for continuous time, that may vary time, be random and have jumps.

4 Market model (Kabanov, Schachermayer) Main features of the model : All is measured in physical units, d risky assets (e.g. foreign currencies), the terms of trading are given by a bid-ask process {Π t (ω), t [0, T]} : an adapted, càdlàg, d d matrix-valued process s.t. Π ij > 0, 1 i, j d Π ii = 1, 1 i d Π ij Π ik Π kj, 1 i, j, k d Meaning : To buy 1 unit of currency j one has to pay Π ij t (ω) units of i (at time t when the state of world is ω)

5 Solvency cones & consistent price systems solvency cone: K t = cone{e i, Π ij t e i e j : 1 i, j d} K t is the cone of available at price 0 polar of K t : K t = {w R d : v, w 0, v K t } Financial interpretation : w Kt iff w R d + and Π ij t w i w j Π ij t wj Π ij w i t = (1 + λ ij t ) wj for some w i λ ij t 0 Every w Kt (resp. in its interior) is called consistent (resp. strictly consistent) price system.

6 Solvency cones & consistent price systems solvency cone: K t = cone{e i, Π ij t e i e j : 1 i, j d} K t is the cone of available at price 0 polar of K t : K t = {w R d : v, w 0, v K t } Financial interpretation : w Kt iff w R d + and Π ij t w i w j Π ij t wj Π ij w i t = (1 + λ ij t ) wj for some w i λ ij t 0 Every w Kt (resp. in its interior) is called consistent (resp. strictly consistent) price system.

7 Solvency cones & consistent price systems solvency cone: K t = cone{e i, Π ij t e i e j : 1 i, j d} K t is the cone of available at price 0 polar of K t : K t = {w R d : v, w 0, v K t } Financial interpretation : w Kt iff w R d + and Π ij t w i w j Π ij t wj Π ij w i t = (1 + λ ij t ) wj for some w i λ ij t 0 Every w Kt (resp. in its interior) is called consistent (resp. strictly consistent) price system.

8 Solvency cones & consistent price systems solvency cone: K t = cone{e i, Π ij t e i e j : 1 i, j d} K t is the cone of available at price 0 polar of K t : K t = {w R d : v, w 0, v K t } Financial interpretation : w Kt iff w R d + and Π ij t w i w j Π ij t wj Π ij w i t = (1 + λ ij t ) wj for some w i λ ij t 0 Every w Kt (resp. in its interior) is called consistent (resp. strictly consistent) price system.

9 Solvency cones & consistent price systems solvency cone: K t = cone{e i, Π ij t e i e j : 1 i, j d} K t is the cone of available at price 0 polar of K t : K t = {w R d : v, w 0, v K t } Financial interpretation : w Kt iff w R d + and Π ij t w i w j Π ij t wj Π ij w i t = (1 + λ ij t ) wj for some w i λ ij t 0 Every w Kt (resp. in its interior) is called consistent (resp. strictly consistent) price system.

10 Strictly consistent price processes Z is a random dynamic analogue of SCPS s. Definition An R d + \ {0}-valued, adapted process Z is a strictly consistent price process if is a càdlàg martingale If Z τ intk τ τ stopping time, and Z σ intk σ σ predictable stopping time. Relations the usual concept of EMM: choose a numéraire Z 1, define S t = (1, Zt 2 /Zt 1... Zt d /Zt 1 ) and set dq/dp = ZT 1/Z1 0, then S is a Q-martingale.

11 Admissible Let Π t be a given Bid-Ask process. A d-dim process V is an admissible self-financing portfolio process if is predictable and finite variation (may have left as well as right jumps!) dv t K t, more precisely: V τ V σ K σ,τ = conv( σ u<τ K u, 0) is bounded from below by some threshold Interpretation: V t = (V 1 t,..., V d t ), V i t = quantity of asset i held by the agent at time t. We denote A x the set of all admissible portfolio processes V s.t. V 0 = x, and A x T := {V T : V A x }.

12 Let us come to efficient Assumption No-Arbitrage condition: There exists a SCPP Z. Our probability space is atomless. Let U be the family of all usc, concave, strictly R d +-increasing functions U : R d R supported on R d + A portfolio X A x T is called efficient wrt an initial pf x if it solves u(x) := sup{e [U(X)] : X A x T} for some U U. (Notice that we don t need Inada or RAE s condition)

13 Let us come to efficient Assumption No-Arbitrage condition: There exists a SCPP Z. Our probability space is atomless. Let U be the family of all usc, concave, strictly R d +-increasing functions U : R d R supported on R d + A portfolio X A x T is called efficient wrt an initial pf x if it solves u(x) := sup{e [U(X)] : X A x T} for some U U. (Notice that we don t need Inada or RAE s condition)

14 Duality (C.-Owen, 2010) Many authors: Karatzas, Cvitanic, Kamizono, Touzi, Pham, Deelstra, Bouchard... Set C = A 0 T L (R d ) and consider the dual cone of C D := {m ba(r d ; P) : m(x) 0 X C}. In C.-Owen (2010): Let U U, the following hold 1 int(dom u) = {x R d : x A 0 T } 2 If x int(dom u) then { u(x) = min E m D [U ( dm c dp )] } + m(x) R where U is the conjugate of U. Denote I := int(dom u), which doesn t depend on U

15 Duality (C.-Owen, 2010) Many authors: Karatzas, Cvitanic, Kamizono, Touzi, Pham, Deelstra, Bouchard... Set C = A 0 T L (R d ) and consider the dual cone of C D := {m ba(r d ; P) : m(x) 0 X C}. In C.-Owen (2010): Let U U, the following hold 1 int(dom u) = {x R d : x A 0 T } 2 If x int(dom u) then { u(x) = min E m D [U ( dm c dp )] } + m(x) R where U is the conjugate of U. Denote I := int(dom u), which doesn t depend on U

16 Multivariate comonotonicity To generalize Dybvig 88 and Jouini-Kallal 01, we need a good notion of multivariate anticomonotonicity. d = 1: X, Y are anticomonotonic if A, P[A] = 1, s.t. (X(ω) X(ω ))(Y(ω) Y(ω )) 0, (ω, ω ) A When d 1, many definitions are available (see Puccetti-Scarsini 10 for a review) From Rockafellar s book : X, Y are called cyclically anticomonotonic if A, P[A] = 1 s.t. for every p 2 and (ω 1, ω 2,..., ω p ) A p, we have p X(ω i ), Y(ω i ) Y(ω i+1 ) 0. (3.1) i=1 where we set ω p+1 = ω 1.

17 Multivariate comonotonicity To generalize Dybvig 88 and Jouini-Kallal 01, we need a good notion of multivariate anticomonotonicity. d = 1: X, Y are anticomonotonic if A, P[A] = 1, s.t. (X(ω) X(ω ))(Y(ω) Y(ω )) 0, (ω, ω ) A When d 1, many definitions are available (see Puccetti-Scarsini 10 for a review) From Rockafellar s book : X, Y are called cyclically anticomonotonic if A, P[A] = 1 s.t. for every p 2 and (ω 1, ω 2,..., ω p ) A p, we have p X(ω i ), Y(ω i ) Y(ω i+1 ) 0. (3.1) i=1 where we set ω p+1 = ω 1.

18 Multivariate comonotonicity To generalize Dybvig 88 and Jouini-Kallal 01, we need a good notion of multivariate anticomonotonicity. d = 1: X, Y are anticomonotonic if A, P[A] = 1, s.t. (X(ω) X(ω ))(Y(ω) Y(ω )) 0, (ω, ω ) A When d 1, many definitions are available (see Puccetti-Scarsini 10 for a review) From Rockafellar s book : X, Y are called cyclically anticomonotonic if A, P[A] = 1 s.t. for every p 2 and (ω 1, ω 2,..., ω p ) A p, we have p X(ω i ), Y(ω i ) Y(ω i+1 ) 0. (3.1) i=1 where we set ω p+1 = ω 1.

19 Main result: characterization of efficient Theorem Let x I. A positive pf X is efficient for the initial portfolio x if and only if m = m c + m p D, such that: 1 Ŷ := d mc dp int Rd + a.s. 2 m c ( X) = E[ XŶ] = m(x). 3 X and Ŷ = d mc dp are cyclically anticomonotonic. 4 The following properties hold i, ess supŷi = + j, ess inf X j = 0 j, ess sup X j < + i, ess infŷi > 0.

20 Sketch of the proof X efficient 3) X and Ŷ = d mc dp By def of U, U(X) XŶ are cyclic anticomonotonic U (Ŷ) 0 for all X 0. By optimality E [U( X) XŶ ] U (Ŷ) = 0 so that... U( X) XŶ U (Ŷ) = 0, which is equivalent to Ŷ U( X) equivalent to the fact that X, Ŷ are cyclic anticomonotonic (see Rockafellar) Let X and m satisfy properties 1-4. Inspired by Thm 24.8 in Rockafellar, consider the function U(x) = inf( x X(ω m ), Ŷ(ω m) + + X(ω 1 ) X(ω 0 ), Ŷ(ω 0) ) where inf is taken wrt all finite sets (ω 0,..., ω m ), m 1. We verify that U is a utility function in U.

21 Sketch of the proof X efficient 3) X and Ŷ = d mc dp By def of U, U(X) XŶ are cyclic anticomonotonic U (Ŷ) 0 for all X 0. By optimality E [U( X) XŶ ] U (Ŷ) = 0 so that... U( X) XŶ U (Ŷ) = 0, which is equivalent to Ŷ U( X) equivalent to the fact that X, Ŷ are cyclic anticomonotonic (see Rockafellar) Let X and m satisfy properties 1-4. Inspired by Thm 24.8 in Rockafellar, consider the function U(x) = inf( x X(ω m ), Ŷ(ω m) + + X(ω 1 ) X(ω 0 ), Ŷ(ω 0) ) where inf is taken wrt all finite sets (ω 0,..., ω m ), m 1. We verify that U is a utility function in U.

22 I : Preliminaries We want to measure the inefficiency of a given pf X A x T Super-replication (C.-Schachermayer 06): tfae X A x T E [ X, Z T ] x, Z 0 for all SCPP Z m c (X) m c (x) for all m D. Let x R. Consider D (x) := { m D : m c (Ω) = x } x 2 + y, y, x = 0 The (minimal) amount of pf x needed to attain X 0 is given by π(x, x) = sup m c (X) [0, 1] m D (x)

23 I : Preliminaries We want to measure the inefficiency of a given pf X A x T Super-replication (C.-Schachermayer 06): tfae X A x T E [ X, Z T ] x, Z 0 for all SCPP Z m c (X) m c (x) for all m D. Let x R. Consider D (x) := { m D : m c (Ω) = x } x 2 + y, y, x = 0 The (minimal) amount of pf x needed to attain X 0 is given by π(x, x) = sup m c (X) [0, 1] m D (x)

24 I : Preliminaries We want to measure the inefficiency of a given pf X A x T Super-replication (C.-Schachermayer 06): tfae X A x T E [ X, Z T ] x, Z 0 for all SCPP Z m c (X) m c (x) for all m D. Let x R. Consider D (x) := { m D : m c (Ω) = x } x 2 + y, y, x = 0 The (minimal) amount of pf x needed to attain X 0 is given by π(x, x) = sup m c (X) [0, 1] m D (x)

25 II : Utility price Let X A x T be an admissible pf x I. The utility price of X respect to the initial portfolio x is the minimum percentage of x needed for any agent to fund a strategy giving at least the same expected utility as X, i.e. where B U ( X) = P U ( X, x) := sup U U [ ]} {X A x T E [U(X)] E U( X) π(x, x) = sup m D (x) mc (X) inf π(x, x) (4.1) X B U ( X) The inefficiency size is then I U ( X, x) := 1 P U ( X, x). Notice that if X is efficient, then I U ( X, x) = 0.

26 II : Utility price Let X A x T be an admissible pf x I. The utility price of X respect to the initial portfolio x is the minimum percentage of x needed for any agent to fund a strategy giving at least the same expected utility as X, i.e. where B U ( X) = P U ( X, x) := sup U U [ ]} {X A x T E [U(X)] E U( X) π(x, x) = sup m D (x) mc (X) inf π(x, x) (4.1) X B U ( X) The inefficiency size is then I U ( X, x) := 1 P U ( X, x). Notice that if X is efficient, then I U ( X, x) = 0.

27 II : Utility price Let X A x T be an admissible pf x I. The utility price of X respect to the initial portfolio x is the minimum percentage of x needed for any agent to fund a strategy giving at least the same expected utility as X, i.e. where B U ( X) = P U ( X, x) := sup U U [ ]} {X A x T E [U(X)] E U( X) π(x, x) = sup m D (x) mc (X) inf π(x, x) (4.1) X B U ( X) The inefficiency size is then I U ( X, x) := 1 P U ( X, x). Notice that if X is efficient, then I U ( X, x) = 0.

28 III : Characterization of I U Theorem Take x I and X A x T positive. The following hold 1 2 P U ( X, x) = min X B( X) π(x, x) = π( X, x) for some X B( X) := U U BU ( X). Furthermore, X is in the closed convex hull of random vectors X X. P U ( X, x) = sup m D (x) min m c (X) X B( X) where min is attained by some X m X and cyclically anticomonotonic w.r.t. dmc dp.

29 III : Characterization of I U Theorem Take x I and X A x T positive. The following hold 1 2 P U ( X, x) = min X B( X) π(x, x) = π( X, x) for some X B( X) := U U BU ( X). Furthermore, X is in the closed convex hull of random vectors X X. P U ( X, x) = sup m D (x) min m c (X) X B( X) where min is attained by some X m X and cyclically anticomonotonic w.r.t. dmc dp.

30 IV : Some remarks A consequence is existence of an efficient trading strategy whose final payoff X is as good as X X has not necessarily the same law (and so the same expected utility) as X, maybe E [ U( X) ] > E [ U( X) Under some technical condition, we can prove that sup is attained in P U ( X, x) = max P( X, m) m D (x) ] Next: Look at some concrete trading strategies in more specific models as B&S model.