Markets with proportional transaction costs and shortsale - PowerPoint PPT Presentation

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Markets with proportional transaction costs and shortsale restrictions Przemysław Rola Jagiellonian University in Kraków 6th General AMaMeF and Banach Center Conference June 10-15, 2013

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Overview Model and definitions 1 Necessary and sufficient conditions 2 Super-replication 3 Sketch of the proof 4

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Model (Ω , F , P ) equipped with the filtration F = ( F t ) T t = 0 such that F T = F risky asset S = ( S t ) T t = 0 = ( S 1 t , . . . , S d t ) T t = 0 - d -dimensional process adapted to F risk free asset B = ( B t ) T t = 0 , B t ≡ 1 for all t = 0 , . . . , T trading strategy H = ( H t ) T t = 1 = ( H 1 t , . . . , H d t ) T t = 1 -predictable with respect to F Let us denote the set of all strategies as P .

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Short selling Define P + = { H ∈ P | H ≥ 0 } . λ = ( λ 1 , . . . , λ d ) , µ = ( µ 1 , . . . , µ d ) where 0 < λ i , µ i < 1 λ < µ if and only if λ i < µ i for i = 1 , . . . , d Let ϕ := ( ϕ 1 , . . . , ϕ d ) where ϕ i ( x ) := x + λ i x + + µ i x − Denote t � ( H · S ) t := H j · ∆ S j j = 1

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof gain or loss process GLP is a process x = ( x λ,µ ) T t = 1 of the form t t � x λ,µ := x λ,µ ( H ) = − ϕ (∆ H j ) · S j − 1 − ϕ ( − H t ) · S t = t t j = 1 t d d � � � ϕ i (∆ H i j ) S i ϕ i ( − H i t ) S i = − j − 1 − t j = 1 i = 1 i = 1 where ∆ H i 1 = H i 1 .

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof gain or loss process GLP is a process x = ( x λ,µ ) T t = 1 of the form t t � x λ,µ := x λ,µ ( H ) = − ϕ (∆ H j ) · S j − 1 − ϕ ( − H t ) · S t = t t j = 1 t d d � � � ϕ i (∆ H i j ) S i ϕ i ( − H i t ) S i = − j − 1 − t j = 1 i = 1 i = 1 where ∆ H i 1 = H i 1 . Substituting ϕ we get t t � � x λ,µ λ (∆ H j ) + S j − 1 − µ (∆ H j ) − S j − 1 − µ H t S t . = ( H · S ) t − t j = 1 j = 1 (U. Çetin, L.C.G. Rogers, "Modelling liquidity effects in discrete time" )

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof The set of hedgeable claims T ( λ, µ ) := { x λ,µ Let us define R + ( H ) | H ∈ P + } and the set of T hedgeable claims as follows A + T ( λ, µ ) := R + T ( λ, µ ) − L 0 + . + T ( λ, µ ) the closure of A + Denote A T ( λ, µ ) in probability. Remark A + T ( λ, µ ) is a convex cone.

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof absence of arbitrage Definition (NA + ) We say that there is no arbitrage in the market if and only if R + T ∩ L 0 + = { 0 } . (NA + ) is equivalent to the condition A + T ∩ L 0 + = { 0 } .

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof absence of arbitrage Definition (NA + ) We say that there is no arbitrage in the market if and only if R + T ∩ L 0 + = { 0 } . (NA + ) is equivalent to the condition A + T ∩ L 0 + = { 0 } . Now we give the definition of robust no arbitrage Definition (rNA + ) We say that there is robust no arbitrage in the market if and only if ∃ ε > 0 : ( ε < λ, A + T ( ε, µ ) ∩ L 0 + = { 0 } ) or ( ε < µ, A + T ( λ, ε ) ∩ L 0 + = { 0 } ) . (W. Schachermayer "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time" )

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof ( λ, µ ) -consistent price system Definition ( λ, µ ) -CPS We say that a pair (˜ S , Q ) is ( λ, µ ) -consistent price system when Q is a probability measure equivalent to P and ˜ S = (˜ S t ) T t = 0 is an d -dimensional process, adapted to the filtration F which is Q -martingale and the following inequalities are satisfied ˜ S i t 1 − µ i ≤ ≤ 1 + λ i , P -a.s. S i t for all i = 1 , . . . , d and t = 0 , . . . , T . (P . Guasoni, M. Rásonyi, W. Schachermayer "The fundamental theorem of asset pricing for continuous processes under small transaction costs" )

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof ( λ, µ ) -supermartingale consistent price system Definition ( λ, µ ) -supCPS We say that a pair (˜ S , Q ) is ( λ, µ ) -supermartingale consistent price system when Q is a probability measure equivalent to P and S = (˜ ˜ S t ) T t = 0 is an d -dimensional process, adapted to the filtration F which is Q -supermartingale and the following inequalities are satisfied ˜ S i t 1 − µ i ≤ ≤ 1 + λ i , P -a.s. S i t for all i = 1 , . . . , d and t = 0 , . . . , T .

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof right-sided λ -consistent price system Definition λ -CPS + We say that a pair (˜ S , Q ) is right-sided λ -consistent price system when Q is a probability measure equivalent to P and ˜ S = (˜ S t ) T t = 0 is an d -dimensional strictly positive process, adapted to the filtration F which is Q -martingale and the following inequalities are satisfied ˜ S i t ≤ 1 + λ i , P -a.s. S i t for all i = 1 , . . . , d and t = 0 , . . . , T .

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Necessary conditions for the absence of arbitrage Main theorem The implications (a) ⇒ (b) ⇒ (c) ⇒ (d) are satisfied where: (a) A + T ( λ, µ ) ∩ L 0 + = { 0 } (NA + ); + (b) A + + = { 0 } and for any ε > λ : A + T ( λ, µ ) ∩ L 0 T ( ε, µ ) = A T ( ε, µ ) ; + T ( ε, µ ) ∩ L 0 (c) for any ε > λ : A + = { 0 } ; (d) for any ε > λ there exists ε -CPS + (˜ S , Q ) with d Q d P ∈ L ∞ .

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Necessary conditions for the absence of arbitrage Corollary The implications (a) ⇒ (b) ⇒ (c) ⇒ (d) are satisfied where: (a) A + T ( λ, µ ) ∩ L 0 + = { 0 } ; (NA + ) + (b) A + + = { 0 } and for any ε > µ : A + T ( λ, µ ) ∩ L 0 T ( λ, ε ) = A T ( λ, ε ) ; + T ( λ, ε ) ∩ L 0 (c) for any ε > µ : A + = { 0 } ; (d) for any ε > µ there exists λ -CPS + (˜ S , Q ) with d Q d P ∈ L ∞ .

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Necessary conditions for the absence of arbitrage Corollary The implications (a) ⇒ (b) ⇒ (c) ⇒ (d) are satisfied where: (a) A + T ( λ, µ ) ∩ L 0 + = { 0 } ; (NA + ) + (b) A + + = { 0 } and for any ε > µ : A + T ( λ, µ ) ∩ L 0 T ( λ, ε ) = A T ( λ, ε ) ; + T ( λ, ε ) ∩ L 0 (c) for any ε > µ : A + = { 0 } ; (d) for any ε > µ there exists λ -CPS + (˜ S , Q ) with d Q d P ∈ L ∞ . Main corollary (rNA + ) ⇒ ∃ λ -CPS + .

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Example The existence of λ -CPS + is not a sufficient condition for (NA + ). Let T = 2, d = 1, λ = µ < 1 1 A , S 2 = 1 + λ 3 and S 0 = 1, S 1 = 1 + 1 1 − λ where A ∈ F 1 and 0 < P ( A ) < 1. Furthermore, assume that F 0 = {∅ , Ω } , F 1 = {∅ , A , Ω \ A , Ω } . Notice that there exists λ -CPS + in the model. Define ˜ S t := ( 1 − µ ) E Q ( S 2 |F t ) where Q ∼ P and t ∈ { 0 , 1 , 2 } . The measure Q can be any probability measure equivalent to P due to the fact that ( 1 − λ ) E Q ( S 2 |F 1 ) = ( 1 − λ ) E Q ( S 2 |F 0 ) = 1 + λ. On the other hand notice that there exists an arbitrage in the model. Define a strategy as follows ∆ H 1 = H 1 = 1 and ∆ H 2 = − 1 1 A . Then 1 A +( 1 + λ 1 − λ − λ 1 + λ x λ,µ = − 1 − λ +( 2 − 2 λ ) 1 1 − λ ) 1 1 Ω \ A = ( 1 − 3 λ ) 1 1 A . 2 Finally A + 2 ( λ ) ∩ L 0 + ( F 2 ) � = { 0 } despite of existing λ -CPS + .

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Sufficient condition for the absence of arbitrage Theorem Let the pair (˜ S , Q ) will be ( λ, µ ) -supCPS. Then we have the absence of arbitrage in our model, i.e. A + T ( λ, µ ) ∩ L 0 + = { 0 } .

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Sufficient condition for the absence of arbitrage Theorem Let the pair (˜ S , Q ) will be ( λ, µ ) -supCPS. Then we have the absence of arbitrage in our model, i.e. A + T ( λ, µ ) ∩ L 0 + = { 0 } . Proof. Let ξ ∈ A + T ( λ, µ ) ∩ L 0 + , i.e. 0 ≤ ξ ≤ T T T � � � λ (∆ H t ) + S t − 1 − µ (∆ H t ) − S t − 1 . ≤ − ∆ H t S t − 1 +( 1 − µ ) H T S T − t = 1 t = 1 t = 1 t ≤ ˜ We use the inequalities − µ i S i S i t − S i t ≤ λ i S i t , P -a.s. and show that E Q ( H · ˜ S ) T ≤ 0.

Model and definitions Necessary and sufficient conditions Super-replication Sketch of the proof Implications Actually due to the above theorem and the previous example the existence of λ -CPS + do not imply the existence of ( λ, µ ) -supCPS.