1 Markets with convex transaction costs Irina Penner Humboldt University of Berlin Email: penner@math.hu-berlin.de Joint work with Teemu Pennanen Helsinki University of Technology Special Semester on Stochastics with Emphasis on Finance Concluding Workshop, Linz, December 2, 2008 Concluding Workshop, December 2, 2008
Markets with convex transaction costs 2 The Market Model • The market consists of d assets traded at t = 0 , . . . , T . • Filtered probability space (Ω , F , ( F t ) T t =0 , P ) . • The price of a portfolio is a non-linear function of the amount due to transaction costs, other illiquidity effects. . . → Modeling portfolio processes becomes an issue. • Kabanov (1999): Portfolios are vectors in R d , expressing the number of physical units of assets (or values of assets in terms of some num´ eraire). • The set of all portfolios that can be transformed to a vector in R d + is a random subset of R d : solvency region. The form of the solvency region is determined by the current price and transaction costs. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 3 The Market Model • A market model is a sequence ( C t ) T t =0 of F t -measurable set-valued mappings Ω ⇒ R d such that each C t ( ω ) is a closed subset of R d with R d − ⊆ C t ( ω ) . • For each t and ω C t ( ω ) denotes the set of all portfolios that are freely available in the market. • A market model is called convex, if each C t ( ω ) is convex. • A convex market model is called conical, if each C t ( ω ) is a cone. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 4 Example 1: Frictionless market If ( S t ) T t =0 is an adapted price process with values in R d + , then C t ( ω ) = { x ∈ R d | S t ( ω ) · x ≤ 0 } , t = 0 , . . . , T defines a conical market model. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 5 Example 2: Proportional transaction costs • [Kabanov (1999)]: If ( S t ) T t =0 is an adapted price process and (Λ t ) T t =0 an adapted matrix of transaction costs coefficients, the solvency regions are defined as d K t := { x ∈ R d | ∃ a ∈ R d × d ( a ji − (1 + λ ij ˆ � : x i S i t ) a ij ) ≥ 0 , 1 ≤ i ≤ d } . t + + j =1 • One can also define solvency regions directly in terms of bid-ask matrices (Π t ) T t =0 as in [Schachermayer (2004)]: K t = { x ∈ R d | ∃ a ∈ R d × d : x i + � d j =1 ( a ji − π ij ˆ t a ij ) ≥ 0 , 1 ≤ i ≤ d } . + • For each ω and t the set ˆ K t ( ω ) is a polyhedral cone and C t ( ω ) := − ˆ K t ( ω ) , t = 0 , . . . , T defines a conical market model. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 6 Example 3: Convex price processes [Astic and Touzi (2007)], [Pennanen (2006)] • A convex price process is a sequence ( S t ) T t =0 of R ∪ { + ∞} -valued functions on R d × Ω such that for each t the function S t is B ( R d ) ⊗ F t -measurable and for each ω the function S t ( · , ω ) is lower semicontinuous, convex and vanishes at 0 . • S t ( x, ω ) denotes the total price of buying a portfolio x at time t and scenario ω . • If ( S t ) T t =0 is a convex price process, then C t ( ω ) = { x ∈ R d | S t ( x, ω ) ≤ 0 } , t = 0 , . . . , T defines a convex market model. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 7 Example 3: Convex price processes Concluding Workshop, December 2, 2008
Markets with convex transaction costs 8 Example 3: Convex price processes • [C ¸etin and Rogers (2007)]: A market with one riskfree and one risky asset. The convex price process is given by S t (( y, x ) , ω ) = y + s t ( ω ) ϕ ( x ) for a strictly positive adapted price process of a risky asset ( s t ) T t =0 and a strictly convex and increasing function ϕ : R → ( −∞ , ∞ ] . (Example: ϕ ( x ) = e αx − 1 .) α • [C ¸etin, Jarrow and Protter (2004)]: A supply curve s t ( x, ω ) gives a price per unit of x units of a risky asset. Then the total price is given by S t (( y, x ) , ω ) = y + s t ( x, ω ) x . No assumptions about convexity, smoothness required. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 9 Example 4: Convex transaction costs • Replace a bid-ask matrix (Π t ) T t =0 by a matrix of convex price processes ( S ij t ) T t =0 (1 ≤ i, j ≤ d ) on R + . • S ij ( x, ω ) denotes the number of units of asset i for which one can buy x units of asset j . In a market with proportional transaction π ij ( ω ) x if x ≥ 0 , costs we have S ij ( x, ω ) = . 1 if x ≤ 0 π ji ( ω ) x • If ( S ij t ) (1 ≤ i, j ≤ d ) are sequences of convex price processes on R + , then d C t ( ω ) = { x ∈ R d | ∃ a ∈ R d × d : x i ≤ ( a ji − S ij � t ( a ij , ω )) , 1 ≤ i ≤ d } + j =1 defines a convex market model. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 10 Notation • A denotes the set of all adapted R d -valued processes. • A process x ∈ A is a self-financing portfolio processes if x t − x t − 1 ∈ C t P -a.s. for all t = 0 , . . . , T We always define x − 1 := 0 . • The set of all final values of self-financial portfolio processes (or, equivalently, of all claims that can be replicated at no cost) is denoted by A T ( C ) Concluding Workshop, December 2, 2008
Markets with convex transaction costs 11 Motivation: Hedging • We want to give a dual characterization of the set of all initial endowments that allow an investor to hedge a given claim → “Hedging theorem”. • A key to the hedging theorem is no-arbitrage condition and FTAP: In a “classical” frictionless model it – provides existence of “pricing” martingales (martingale measures) – provides closedness of the set A T ( C ) of all claims that can be replicated at no cost. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 12 Motivation: Hedging • In a market with proportional transaction costs several natural generalizations of the notions of arbitrage and martingale measures are possible [Kabanov and Stricker (2001)], [Schachermayer (2004)], [Grigoriev (2005)], [R´ asonyi (2008)]... • In a market with convex structure martingale measures are not sufficient for the dual characterization [F¨ ollmer and Kramkov (1997)]... • We are interested in a no-arbitrage notion that implies closedness of the set A T ( C ) . Concluding Workshop, December 2, 2008
Markets with convex transaction costs 13 No-arbitrage notions for conical models [Kabanov and Stricker (2001)], [Kabanov, R´ asonyi and Stricker (2001), (2003)], [Schachermayer (2004)] • A market model C has the no arbitrage property if A T ( C ) ∩ L 0 ( R d + ) = { 0 } , where A T ( C ) = { x T | x is self-financing } . • A market model ˜ C dominates a conical market model C if C t ⊆ ˜ t ⊂ ri ˜ C t \ C 0 C t and C t for all t = 0 , . . . , T, where C 0 t = C t ∩ − C t . • A conical market model C has the robust no-arbitrage property if C is dominated by another conical model ˜ C which has the no-arbitrage property. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 14 No-arbitrage notions for convex models • Given a convex market model C , we define a conical market model C ∞ by t ( ω ) = { x ∈ R d | C t ( ω ) + αx ⊂ C t ( ω ) ∀ α > 0 } , C ∞ t = 0 , . . . , T. • C ∞ t ( ω ) is the recession cone of C t ( ω ) : � C ∞ t ( ω ) = αC t ( ω ) α> 0 If C is conical then C ∞ = C . • The set C ∞ t ( ω ) describes the behavior of C t ( ω ) infinitely far from the origin. • We say that a convex market model C has the robust no scalable arbitrage property if the model C ∞ has the robust no-arbitrage property. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 15 No-arbitrage notions for convex models • Given a convex market model C , one can also consider the conical market model C ′ given by � C ′ t ( ω ) := cl αC t ( ω ) , t = 0 , . . . , T. α> 0 • C t ( ω ) is the tangent cone of C t ( ω ) . If C is conical then C ′ = C . • The set C ′ t ( ω ) describes the behavior of C t ( ω ) close to the origin. • We say that a convex market model C has the robust no marginal arbitrage property if the model C ′ has the robust no-arbitrage property. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 16 Main result Theorem 1 If the convex market model C has the robust no scalable arbitrage property then the set A T ( C ) of all claims that can be replicated with zero initial investment is closed in probability. Concluding Workshop, December 2, 2008
Markets with convex transaction costs 17 Applications: Hedging • A contingent claim processes with physical delivery c = ( c t ) T t =0 ∈ A is a security that gives its owner a random portfolio c t possibly at each time t = 0 , . . . , T . • The set of all claim processes that can be replicated with zero initial investment is A ( C ) = { c ∈ A | ∃ x ∈ A : x t − x t − 1 + c t ∈ C t , t = 0 , . . . , T, x T = 0 } . • We call a process p ∈ A a super-hedging premium process for a claim process c if c − p ∈ A ( C ) . • If c = (0 , . . . , 0 , c T ) and p = ( p 0 , 0 , . . . , 0) , then c − p ∈ A ( C ) iff there exists a self-financing portfolio process such that c T ≤ p 0 + x T . Concluding Workshop, December 2, 2008
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