Canonical Supermartingale Couplings

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1 Canonical Supermartingale Couplings Marcel Nutz Columbia University (with Mathias Beiglböck, Florian Stebegg and Nizar Touzi) June 2016 Marcel Nutz (Columbia) Canonical Supermartingale Couplings 1 / 34

2 Outline 1 Classical Optimal Transport 2 Constrained Transport and Finance 3 Supermartingale Couplings 4 Duality for Supermartingale Optimal Transport Marcel Nutz (Columbia) Canonical Supermartingale Couplings 1 / 34

3 Monge Optimal Transport Given: Probabilities µ, ν on R. Reward (cost) function f : R R R. Objective: Find a map T : R R satisfying ν = µ T 1 such as to maximize the total reward, max f (x, T (x)) µ(dx). T Marcel Nutz (Columbia) Canonical Supermartingale Couplings 2 / 34

4 Monge Kantorovich Optimal Transport Relaxation: Find a probability P on R R with marginals µ, ν such as to maximize the reward: max E P [f (X, Y )], where Π(µ, ν) := {P : P 1 = µ, P 2 = ν} P Π(µ,ν) and (X, Y ) = Id R R. P Π(µ, ν) is a Monge transport if of the form P = µ δ T (x). Marcel Nutz (Columbia) Canonical Supermartingale Couplings 3 / 34

5 Example: Hoeffding Frechet Coupling Theorem: Let f satisfy the Spence Mirrlees condition f xy > 0. Then the optimal P is unique and given by the Hoeffding Frechet Coupling: P is the law of ((F µ ) 1, (F ν ) 1 ) under the uniform measure on [0, 1]. If µ is diffuse, P is of Monge type with T = (F ν ) 1 F µ. P is characterized by monotonicity: if (x, y), (x, y ) supp(p) and if x < x, then y y. Symmetrically, the Antitone Coupling is optimal for f xy < 0. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 4 / 34

6 Example: Hoeffding Frechet Coupling Theorem: Let f satisfy the Spence Mirrlees condition f xy > 0. Then the optimal P is unique and given by the Hoeffding Frechet Coupling: P is the law of ((F µ ) 1, (F ν ) 1 ) under the uniform measure on [0, 1]. If µ is diffuse, P is of Monge type with T = (F ν ) 1 F µ. P is characterized by monotonicity: if (x, y), (x, y ) supp(p) and if x < x, then y y. Symmetrically, the Antitone Coupling is optimal for f xy < 0. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 4 / 34

7 Kantorovich Duality Buy ϕ(x ) at price µ(ϕ) := E µ [ϕ] and ψ(y ) at ν(ψ) to superhedge, f (X, Y ) ϕ(x ) + ψ(y ). Then for all P Π(µ, ν), E P [f (X, Y )] E P [ϕ(x ) + ψ(y )] = µ(ϕ) + ν(ψ). Theorem (Kantorovich, Kellerer): For any measurable f 0, sup E P [f (X, Y )] = inf µ(ϕ) + ν(ψ) P Π(µ,ν) ϕ,ψ and dual optimizers ˆϕ, ˆψ exist. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 5 / 34

8 Kantorovich Duality Buy ϕ(x ) at price µ(ϕ) := E µ [ϕ] and ψ(y ) at ν(ψ) to superhedge, f (X, Y ) ϕ(x ) + ψ(y ). Then for all P Π(µ, ν), E P [f (X, Y )] E P [ϕ(x ) + ψ(y )] = µ(ϕ) + ν(ψ). Theorem (Kantorovich, Kellerer): For any measurable f 0, sup E P [f (X, Y )] = inf µ(ϕ) + ν(ψ) P Π(µ,ν) ϕ,ψ and dual optimizers ˆϕ, ˆψ exist. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 5 / 34

9 Application: Monotonicity Principle Let Γ = {(x, y) : ˆϕ(x) + ˆψ(y) = f (x, y)} and P Π(µ, ν). TFAE: (1) P is optimal. (2) P(Γ) = 1. (3) supp(p) is f -cyclically monotone P-a.s.; i.e., n f (x i, y i ) i=1 n f (x i, y σ(i) ) (x i, y i ) supp(p), σ Perm(n). i=1 1)(2) If P(Γ) < 1, then P charges {(x, y) : ˆϕ(x) + ˆψ(y) > f (x, y)} and thus µ( ˆϕ) + ν( ˆψ) > E P [f (X, Y )]. 2)(1) If P(Γ) = 1, then µ( ˆϕ) + ν( ˆψ) = E P [f (X, Y )], hence P, ˆϕ, ˆψ are optimal. 2)(3) This argument even shows: if P(Γ) = 1, then P is an optimal transport between its own marginals. Apply this with discrete P Γ is cyclically monotone. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 6 / 34

10 Application: Monotonicity Principle Let Γ = {(x, y) : ˆϕ(x) + ˆψ(y) = f (x, y)} and P Π(µ, ν). TFAE: (1) P is optimal. (2) P(Γ) = 1. (3) supp(p) is f -cyclically monotone P-a.s.; i.e., n f (x i, y i ) i=1 n f (x i, y σ(i) ) (x i, y i ) supp(p), σ Perm(n). i=1 1)(2) If P(Γ) < 1, then P charges {(x, y) : ˆϕ(x) + ˆψ(y) > f (x, y)} and thus µ( ˆϕ) + ν( ˆψ) > E P [f (X, Y )]. 2)(1) If P(Γ) = 1, then µ( ˆϕ) + ν( ˆψ) = E P [f (X, Y )], hence P, ˆϕ, ˆψ are optimal. 2)(3) This argument even shows: if P(Γ) = 1, then P is an optimal transport between its own marginals. Apply this with discrete P Γ is cyclically monotone. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 6 / 34

11 Outline 1 Classical Optimal Transport 2 Constrained Transport and Finance 3 Supermartingale Couplings 4 Duality for Supermartingale Optimal Transport Marcel Nutz (Columbia) Canonical Supermartingale Couplings 6 / 34

12 Semi-Static Hedging Hedge a derivative f by trading dynamically in underlying S = (S 0, S 1, S 2 ). trading statically in call/put options (S 1 K) ± and (S 2 K) ±. Semi-static superhedge: f ((S t ) t ) ϕ(s 1 ) + ψ(s 2 ) + H 0 (S 1 S 0 ) + H 1 (S 2 S 1 ). With S 0 = 0, S 1 = X µ, S 2 = Y ν and normalization H 0 = 0: f (X, Y ) ϕ(x ) + ψ(y ) + h(x )(Y X ). Marcel Nutz (Columbia) Canonical Supermartingale Couplings 7 / 34

13 Constrained Transport If h : R R is arbitrary, formal duality with P Π(µ, ν) satisfying E P [h(x )(Y X )] = 0 h; i.e. E P [Y X ] = X. Martingale Optimal Transport, cf. Beiglböck, Henry-Labordère, Penkner; Galichon, Henry-Labordère, Touzi; Hobson; Beiglböck, Juillet; Acciaio, Bouchard, Brown, Campi, Cheridito, Cox, Davis, Dolinsky, Fahim, Ghoussoub, Huang, Källblad, Kim, Kupper, Lassalle, Lim, Martini, Neuberger, Obłój, Rogers, Schachermayer, Soner, Stebegg, Tan, Tangpi, Zaev,... If range of h is a subset of R: constrained transport problem. E.g., supermartingale if h : R [0, ). Marcel Nutz (Columbia) Canonical Supermartingale Couplings 8 / 34

14 Outline 1 Classical Optimal Transport 2 Constrained Transport and Finance 3 Supermartingale Couplings 4 Duality for Supermartingale Optimal Transport Marcel Nutz (Columbia) Canonical Supermartingale Couplings 8 / 34

15 Supermartingale Couplings Given µ, ν on R, the set of supermartingale couplings is S(µ, ν) = {P Π(µ, ν) : E P [Y X ] X }. Theorem (Strassen): S(µ, ν) is nonempty iff µ cd ν; i.e., µ(φ) ν(φ) φ convex, decreasing. Given µ cd ν, is there a canonical choice for a coupling? Marcel Nutz (Columbia) Canonical Supermartingale Couplings 9 / 34

16 Known Special Cases Special Case 1: supp ν is to the left of supp µ. Every coupling is a supermartingale: S(µ, ν) = Π(µ, ν). Hoeffding Frechet and Antitone Couplings are canonical. Optimal Monge Kantorovich transports for f xy > 0 (resp. f xy < 0). Special Case 2: bary(µ) = bary(ν). Every supermartingale coupling is a martingale: S(µ, ν) = M(µ, ν). Left-Curtain and Right-Curtain Couplings are canonical. Optimal martingale transports for f xyy > 0 (resp. f xyy < 0). candidates for special f are sign combinations of f xy / f xyy : +/+, +/, /+, / Marcel Nutz (Columbia) Canonical Supermartingale Couplings 10 / 34

17 Known Special Cases Special Case 1: supp ν is to the left of supp µ. Every coupling is a supermartingale: S(µ, ν) = Π(µ, ν). Hoeffding Frechet and Antitone Couplings are canonical. Optimal Monge Kantorovich transports for f xy > 0 (resp. f xy < 0). Special Case 2: bary(µ) = bary(ν). Every supermartingale coupling is a martingale: S(µ, ν) = M(µ, ν). Left-Curtain and Right-Curtain Couplings are canonical. Optimal martingale transports for f xyy > 0 (resp. f xyy < 0). candidates for special f are sign combinations of f xy / f xyy : +/+, +/, /+, / Marcel Nutz (Columbia) Canonical Supermartingale Couplings 10 / 34

18 Known Special Cases Special Case 1: supp ν is to the left of supp µ. Every coupling is a supermartingale: S(µ, ν) = Π(µ, ν). Hoeffding Frechet and Antitone Couplings are canonical. Optimal Monge Kantorovich transports for f xy > 0 (resp. f xy < 0). Special Case 2: bary(µ) = bary(ν). Every supermartingale coupling is a martingale: S(µ, ν) = M(µ, ν). Left-Curtain and Right-Curtain Couplings are canonical. Optimal martingale transports for f xyy > 0 (resp. f xyy < 0). candidates for special f are sign combinations of f xy / f xyy : +/+, +/, /+, / Marcel Nutz (Columbia) Canonical Supermartingale Couplings 10 / 34

Left-Curtain Coupling for Martingale Transport Theorem (Beiglböck, Juillet): Suppose that µ c ν. Let f satisfy the martingale Spence Mirrlees condition f xyy > 0.

19 Left-Curtain Coupling for Martingale Transport Theorem (Beiglböck, Juillet): Suppose that µ c ν. Let f satisfy the martingale Spence Mirrlees condition f xyy > 0. Then the optimal P is given by the Left-Curtain Coupling: Symmetrically, the Right-Curtain Coupling is optimal for f xyy < 0. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 11 / 34

The Increasing Supermartingale Coupling P Theorem: Let f satisfy 1) f xy < 0 and f xyy > 0 e.g., f (x, y) = exp(x y); Then the optimal P is exists, is unique and independent of f.

20 The Increasing Supermartingale Coupling P Theorem: Let f satisfy 1) f xy < 0 and f xyy > 0 e.g., f (x, y) = exp(x y); Then the optimal P is exists, is unique and independent of f. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 12 / 34

The Decreasing Supermartingale Coupling P Theorem: Let f satisfy 2) f xy > 0 and f xyy < 0 e.g., f (x, y) = exp(x y); Then the optimal P is exists, is unique and independent of f.

21 The Decreasing Supermartingale Coupling P Theorem: Let f satisfy 2) f xy > 0 and f xyy < 0 e.g., f (x, y) = exp(x y); Then the optimal P is exists, is unique and independent of f. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 13 / 34

22 A Family of Supermartingale Couplings Send µ 1 µ to some destination ν 1 ν by a supermartingale. A special choice: minimize ν 1 with respect to cd. If µ 1 = kδ x is a Dirac, that amounts to: 1. If possible, achieve bary(ν 1 ) = x. If not, bring bary(ν 1 ) as close to x as possible. 2. Keep ν 1 as concentrated as possible. If µ is discrete, we can process one atom after the other. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 14 / 34

23 A Family of Supermartingale Couplings Send µ 1 µ to some destination ν 1 ν by a supermartingale. A special choice: minimize ν 1 with respect to cd. If µ 1 = kδ x is a Dirac, that amounts to: 1. If possible, achieve bary(ν 1 ) = x. If not, bring bary(ν 1 ) as close to x as possible. 2. Keep ν 1 as concentrated as possible. If µ is discrete, we can process one atom after the other. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 14 / 34

24 Example 1 Marcel Nutz (Columbia) Canonical Supermartingale Couplings 15 / 34

25 Example 1 (cont d) Marcel Nutz (Columbia) Canonical Supermartingale Couplings 16 / 34

26 Example 2 Marcel Nutz (Columbia) Canonical Supermartingale Couplings 17 / 34

27 Example 2 (cont d) Marcel Nutz (Columbia) Canonical Supermartingale Couplings 18 / 34

28 Example 3 Marcel Nutz (Columbia) Canonical Supermartingale Couplings 19 / 34

29 Example 3 (cont d) Marcel Nutz (Columbia) Canonical Supermartingale Couplings 20 / 34

30 Example 3 (cont d) Marcel Nutz (Columbia) Canonical Supermartingale Couplings 21 / 34

31 Example 4 Marcel Nutz (Columbia) Canonical Supermartingale Couplings 22 / 34

32 Example 4 (cont d) Marcel Nutz (Columbia) Canonical Supermartingale Couplings 23 / 34

33 Example 4 (cont d) Marcel Nutz (Columbia) Canonical Supermartingale Couplings 24 / 34

34 The Canonical Couplings Roughly: P is obtained by processing in increasing order (left-to-right). P is obtained by processing in decreasing order (right-to-left). Theorem: There exists a unique measure P on R R which transports ] µ (,x] arg min [µ (,x] cd ν for all x R. Similarly for P. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 25 / 34

35 Outline 1 Classical Optimal Transport 2 Constrained Transport and Finance 3 Supermartingale Couplings 4 Duality for Supermartingale Optimal Transport Marcel Nutz (Columbia) Canonical Supermartingale Couplings 25 / 34

36 Dual Problem for Supermartingale Optimal Transport Proceed through duality to analyze geometry of optimal transports. Formally, the dual problem is inf µ(ϕ) + ν(ψ), ϕ,ψ,h taken over ϕ, ψ : R R and h : R [0, ) such that ϕ(x) + ψ(y) + h(x)(y x) f (x, y). Relaxations are necessary to make duality work. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 26 / 34

37 Ordinary and Constrained OT: What is the Difference? In ordinary OT, all roads x y can be used. E.g., in the discrete case, µ ν already has full support. Theorem (Kellerer): A R R is Π(µ, ν)-polar if and only if A (N 1 R) (R N 2 ), where µ(n 1 ) = ν(n 2 ) = 0. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 27 / 34

38 Ordinary and Constrained OT: What is the Difference? In ordinary OT, all roads x y can be used. E.g., in the discrete case, µ ν already has full support. Theorem (Kellerer): A R R is Π(µ, ν)-polar if and only if A (N 1 R) (R N 2 ), where µ(n 1 ) = ν(n 2 ) = 0. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 27 / 34

39 Obstructions for Martingale Transport In martingale OT, some roads x y can be blocked by barriers. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 28 / 34

40 Structure of M(µ, ν)-polar Sets Theorem (Beiglböck, N., Touzi): These are the M(µ, ν)-polar sets. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 29 / 34

41 Back to Supermartingales: Structure of S(µ, ν)-polar Sets Theorem: There exist a maximal barrier x such that: martingale transport on (, x ], single component of strict supermartingale transport on [x, ). Marcel Nutz (Columbia) Canonical Supermartingale Couplings 30 / 34

42 Duality Result Theorem: Let f 0 be measurable and consider the quasi-sure relaxation of the dual problem: f (X, Y ) ϕ(x ) + ψ(y ) + h(x )(Y X ) S(µ, ν)-q.s. (1) There is no duality gap: (2) Dual existence holds: ˆϕ, ˆψ, ĥ. sup E P [f (X, Y )] = inf µ(ϕ) + ν(ψ). P S(µ,ν) ϕ,ψ,h The range of h also needs to be relaxed, as well as the integrability of ϕ, ψ. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 31 / 34

43 Monotonicity Principle Theorem: There exist Borel sets Γ R 2 and M R such that for any P S(µ, ν), the following are equivalent: (i) P is optimal, (ii) P is concentrated on Γ and P M R is a martingale. Roughly: Take dual optimizer (ϕ, ψ, h) and let M = {h > 0}, Γ = set where hedge is perfect. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 32 / 34

44 Extremal Decomposition of Optimal Supermartingales There is a decomposition R = M N such that for optimal P S(µ, ν): P M R is an optimal martingale transport from µ M and P(µ M ). P N R is an optimal Monge Kantorovich transport from µ N to P(µ N ), Marcel Nutz (Columbia) Canonical Supermartingale Couplings 33 / 34

45 Conclusion Interesting new couplings arise from problems in mathematical finance. Duality in a quasi-sure sense is useful for their analysis. We expect other constraints to be tractable as well: ongoing work. Thank you. Marcel Nutz (Columbia) Canonical Supermartingale Couplings 34 / 34

Monotone Martingale Transport Plans and Skorohod Embedding

Monotone Martingale Transport Plans and Skorohod Embedding Mathias Beiglböck Pierre Henry-Labordère Nizar Touzi arxiv:1701.06779v1 [math.pr] 24 Jan 2017 March 28, 2018 Abstract We show that the left-monotone