Efficient portfolios in financial markets with proportional transaction costs

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

Contents 1 2 3 4

: 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.

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)

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.

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.

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 }.

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)

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

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.

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.

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.

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)

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.

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.

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.