Calculating the set of superhedging portfolios in markets with transactions costs by methods of vector optimization Andreas Löhne Martin-Luther-Universität Halle-Wittenberg Co-author: Birgit Rudlo, Princeton University Barcelona, October 24, 2011
Markets with transaction costs d assets, discrete time 0,..., T, (Ω, (F t ) T t=0, P ) portfolio vector in physical units: V t (# of units in d assets at time t) proportional transaction costs at time t: polyhedral convex cone R d + K t(ω) R d (solvency cone), positions transferrable into nonnegative positions (V t ) T t=0 self-nancing portfolio process if V t V t 1 K t P a.s. t {0,..., T } (V 1 0) attainable claims at zero cost A T := {V T : V is self-nancing portfolio process}
Superhedging A claim X L 0 d can be superhedged with initial endowment v Rd if V T A T such that X = v + V T. The set of all superhedging portfolios is SHP 0 (X) := { v R d : X v + A T }.
Theorem. If the probability space is nite and the robust no arbitrage condition (NA r ) holds true, the set of superhedging portfolios SHP 0 (X) R d of a claim X L 0 d (F T, R d ) satises = SHP 0 (X) R d and can be obtained recursively via ω Ω T : SHP T (X)(ω) = X(ω) + K T (ω) t {T 1,..., 1, 0}, ω Ω t : SHP t (X)(ω) = ω succ (ω) SHP t+1 (X)( ω) + K t (ω).
Recombining trees 2 Assets 3 Assets t = 2 t = 3 t = 2 t = 1 t = 1 t = 0 t = 0
Toy example
Toy example digital option with 2 assets: 1 currency (cash) and 1 stock bid and ask prices of the stock: 20, 26 18, 25 16, 23 t = 0 t = T payo: X(ω) = (X 1 (ω), X 2 (ω)) T = (0, I {S at K} (ω) ) T for strike price K = 24 X(ω 1 ) = (0, 1) T, X(ω 2 ) = (0, 0) T
Motivation of the set-valued approach stock P 1 SHP 0 (X) 6 4 100 P 1 K 0 P 3 60 20 P 4 20 2 P 2 cash
Linear vector optimization problem (P) C-min P x s.t. Bx b P... (q n)-matrix C... ordering cone in R q : polyhedral, convex, pointed B... (m n)-matrix b... vector in R m
SHP-algorithm
SHP-algorithm...... is a reformulation of the recursive description of SHP An LVOP is solved in each iteration step Input: for all t {0, 1,..., T }, for all ω Ω t : K t (ω), for all ω Ω T : X(ω), Output: for all t {0, 1,..., T }, for all ω Ω t : SHP t (X)(ω) = { x R d B ω x b ω}....
01: for all ω Ω T ; 02: B ω = GeneratingVectors(K t (ω) + ); 03: b ω = (B ω ) T X(ω); 04: end; 05: for t=t-1 downto 0 06: for all ω Ω t ; 07: B = { B ω ω succ (ω) } ; 08: b = { b ω ω succ (ω) } ; 09: P = LiquidationMap(K t (ω)); 10: C = P K t (ω); 11: compute solution { (u 1, w 1 ),..., (u k, w k ) } of (D ); 12: B ω = (P T w 1,..., P T w k ) T ; 13: b ω = (b T u 1,..., b T u k ) T ; 14: end; 15: end;
BENSOLVE... is a VLP solver based on Benson's algorithm with the following features: - primal and dual variant of Benson's algorithm - solves primal and dual problem - unbounded problems supported - polyhedral, pointed ordering cones with nonempty interior
Examples, numerical results
European call option Asset 0: riskless bond, interest rate 10%, no transaction costs Asset 1: stock, Cox-Ross-Rubinstein binomial model, constant transaction costs λ = 0.125% d = 2, n 1800 (> 1.600.000 Vektor-LPs)
Basket options (2 underlyings) 1 bond, 2 stocks Binomial tree approximates (d 1)-dimensional Black-Scholes model Solvency cone for d = 3 is induced by (S a t )1 (S b t )1 (S a t )2 (S b t )2 0 0 B b t B a t 0 0 (S a t )2 (S b t )2 0 0 B b t B a t (S b t )1 (S a t )1,
Base of solvency cone for d=4 solvency cone has 12 extreme directions
Solution concepts
Solution to (P) A nonempty set S R n together with a set S h R n \ {0} is called solution to (P) if (i) S is a nite subset of S, (ii) S h is a nite subset of S h, (iii) P [ S] Min P [S], (iv) P [ S h ] Min P [S h ], (v) P [S] conv P [ S] + cone P [ S h ] + R q +.
Example (P) Minimize P : R n R q w.r.t. over S := {x R n Bx b} P = ( 1 1 1 1 ), B = 2 1 1 2 1 0 0 1, b = 6 6 0 0
Example y 2 y 2 P = P [S] + R q + 6 4 S 2 y 1 2 4 6 6 y 1 Solution to (P): S = {( 0 6 ), ( 2 2 )}, S h = {( 0 1 )}
Example y 2 y 2 P = P [S] + R q + 6 4 S 2 y 1 2 4 6 6 y 1 Solution to (P): S = {( 0 6 ), ( 2 2 )}, S h = {( 0 1 )}
Example y 2 y 2 P = P [S] + R q + 6 4 S 2 y 1 2 4 6 6 y 1 Solution ( S, S h ) to (P): S = {( 0 6 ), ( 2 2 )}, S h = {( 0 1 )}
Solution to (D ) A nonempty set T R m+q is called solution to (D ) if (i) T is a nite subset of T, (ii) D [ T ] Max K D [T ], (iii) D [T ] conv D [ T ] K.
Example D = D [T ] K 6 P = P [S] + R q + y 1 y 2 y 2 y 1 6 6 Solution T to (D ): T = 1 31 3 0 0 0 1, 0 0 1 0 1 2 1 2, 1 2 0 0 0 1 4 3 4
Computing solutions by BENSOLVE B=[2 1 1 0;1 2 0 1]'; b=[6 6 0 0]'; P=[1-1; 1 1]; [S,Sh,T,PP,PPh,DD]=bensolve(P,B,b)
S = -0.0000 2.0000 6.0000 2.0000 Sh = 0 1 T = 0.3333 0 0.5000 0.3333 0 0 0 1.0000 0 0 0 0 0 0.5000 0.2500 1.0000 0.5000 0.7500 PP = -6.0000 0 6.0000 4.0000 PPh = -1 1 0 1 0 1 DD = 0 0.5000 0.2500 4.0000 0 3.0000
BENSOLVE: another Example B=[eye(3);ones(1,3);1 2 2;2 2 1;2 1 2]; b=[0 0 0 1 3/2 3/2 3/2]'; P=[1 1 0;0 1 1;1 0 1]'; Y=[1 0 0 ; 0 1 0 ; -1 0 2 ; 0-1 2]'; clear options; options.vert_enum='c'; [S,Sh,T,PP,PPh,DD]=bensolve(P,B,b,Y,[],[],options); plotresult(pp,pph,dd);
Download BENSOLVE at http://ito.mathematik.uni-halle.de/ loehne
Geometric duality Frank Heyde, A.L. (2008) Frank Heyde (2011)
Dual problem K := R + (0, 0,..., 0, 1) T, c int C, c q = 1 D : R m R q R q, D (u, w) := ( w 1,..., w q 1, b T u ) T (D ) maximize D : R m R q R q w.r.t. K over T T := { (u, w) R m R q u 0, w C +, B T u = P T w, w T c = 1 } upper image of (P): lower image of (D ): P := P [S] + C D := D [T ] K
Idea of geometric duality Duality of polytopes Two polytopes P and P in R q are said to be dual to each other if there is an inclusion reversing one-to-one map Ψ between the set of faces of P and the set of faces of P. P P
Idea of geometric duality (P) and (D ) in R q are said to be dual to each other if there is an inclusion reversing one-to-one map Ψ between the set of K-maximal proper faces of D and the set of the weakly C-minimal proper faces of P. y 3 y 3 y 2 y 2 P D y 1 y 1
Geometric duality ϕ : R q R q R, ϕ(y, y ) := q 1 i=1 y i y i + y q(1 q 1 i=1 c i y i ) y q H : R q R q, H(y ) := {y R q ϕ(y, y ) = 0} H : R q R q, H (y) := {y R q ϕ(y, y ) = 0} Ψ : 2 Rq 2 Rq, Ψ(F ) := y F H(y ) P Duality theorem. Ψ is an inclusion reversing one-to-one map between the proper K-maximal faces of D and the weakly C-minimal proper faces of P. The inverse map is Ψ 1 (F ) = H (y) D. y F
Contents Part I: General and convex problems 1. A complete lattice for vector optimization 2. Solution concepts 3. Duality Part II: Linear problems 4. Solution concepts and duality 5. Algorithms
Conclusions: The set of superhedging portfolios can be computed by solving a sequence of Vektor-LPs Existing algorithms [Roux & Zastawniak 2009] (for d = 2 only) can be interpreted via vectorial duality Vectorial duality leads to interesting insights and interpretations
Choice of literature Bensaid, Lesne, Pagès, Scheinkman: Derivative asset pricing with transaction costs. Math. Finance (1992) Hamel, Heyde: Duality for set-valued measures of risk. SIAM J. on Financial Mathematics (2010) Hamel, Heyde, Rudlo: Set-valued risk measures for conical market models. Mathematics and Financial Economics (2011) Heyde: Geometric duality for convex vector optimization problems, draft Heyde, Löhne: Geometric duality in multiple objective linear programming. SIAM J. Opt. (2008) Jouini, Kallal: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Th. (1995) Kabanov: Hedging and liquidation under transaction costs in currency markets. Fin.& Stoch. (1999) Löhne; Rudlo: An algorithm for calculating the set of superhedging portfolios and strategies in markets with transaction costs, submitted to Fin. & Stoch. Palmer: A note on the Boyle-Vorst discrete-time option pricing model with transaction costs. Math. Fin. (2001) Pennanen, Penner: Hedging of claims with physical delivery under convex transaction costs. SIAM J. on Financial Mathematics (2010) Perrakis, Lefoll: Derivative asset pricing with transaction costs: an extension. Computational Economics (1997) Roux, Tokarz, Zastawniak: Options under proportional transaction costs: An algorithmic approach to pricing and hedging. Acta Appl. Math. (2008) Roux, A., Zastawniak, T.: American Options under Proportional Transaction Costs: Pricing, Hedging and Stopping Algorithms for Long and Short Positions. Acta Appl. Math. (2009) Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in nite discrete time. Math. Fin. (2004)
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