Martingale Optimal Transport and Robust Finance

Transcription

1 Martingale Optimal Transport and Robust Finance Marcel Nutz Columbia University (with Mathias Beiglböck and Nizar Touzi) April 2015 Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 1 / 20

2 Outline 1 Monge Kantorovich Optimal Transport 2 Martingale Optimal Transport Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 1 / 20

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 = T 1 µ such as to maximize the total reward, max f (x, T (x)) µ(dx). T Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 2 / 20

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) Martingale Optimal Transport and Robust Finance 3 / 20

5 Example: Hoeffding Frechet Coupling Theorem: Let f (x, y) = g(y x) where g is strictly concave and sufficiently integrable. Then the optimal P is 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. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 4 / 20

6 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) Martingale Optimal Transport and Robust Finance 5 / 20

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) Martingale Optimal Transport and Robust Finance 5 / 20

8 Application: Fundamental Theorem of Optimal Transport 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) Martingale Optimal Transport and Robust Finance 6 / 20

9 Application: Fundamental Theorem of Optimal Transport 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) Martingale Optimal Transport and Robust Finance 6 / 20

10 Application: Fundamental Theorem of Optimal Transport 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) Martingale Optimal Transport and Robust Finance 6 / 20

11 Application: Fundamental Theorem of Optimal Transport 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) Martingale Optimal Transport and Robust Finance 6 / 20

12 Outline 1 Monge Kantorovich Optimal Transport 2 Martingale Optimal Transport Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 6 / 20

13 Dynamic Hedging Dynamically tradable underlying S = (S 0, S 1, S 2 ). 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 ). Formally, duality with P Π(µ, ν) satisfying the constraint that E P [h(x )(Y X )] = 0 h; i.e. E P [Y X ] = X. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 7 / 20

14 Dynamic Hedging Dynamically tradable underlying S = (S 0, S 1, S 2 ). 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 ). Formally, duality with P Π(µ, ν) satisfying the constraint that E P [h(x )(Y X )] = 0 h; i.e. E P [Y X ] = X. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 7 / 20

15 Martingale Transport Set of martingale transports: M(µ, ν) = {P Π(µ, ν) : E P [Y X ] = X }. Theorem (Strassen): M(µ, ν) is nonempty iff µ c ν; i.e., µ(φ) ν(φ) φ convex. Martingale Optimal Transport problem: Given µ c ν, sup E P [f (X, Y )]. P M(µ,ν) Beiglböck, Henry-Labordère, Penkner; Galichon, Henry-Labordère, Touzi; Hobson; Beiglböck, Juillet; Acciaio, Bouchard, Brown, Cheridito, Cox, Davis, Dolinsky, Fahim, Huang, Källblad, Kupper, Lassalle, Martini, Neuberger, Obłój, Rogers, Schachermayer, Soner, Stebegg, Tan, Tangpi, Zaev,... Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 8 / 20

16 Martingale Transport Set of martingale transports: M(µ, ν) = {P Π(µ, ν) : E P [Y X ] = X }. Theorem (Strassen): M(µ, ν) is nonempty iff µ c ν; i.e., µ(φ) ν(φ) φ convex. Martingale Optimal Transport problem: Given µ c ν, sup E P [f (X, Y )]. P M(µ,ν) Beiglböck, Henry-Labordère, Penkner; Galichon, Henry-Labordère, Touzi; Hobson; Beiglböck, Juillet; Acciaio, Bouchard, Brown, Cheridito, Cox, Davis, Dolinsky, Fahim, Huang, Källblad, Kupper, Lassalle, Martini, Neuberger, Obłój, Rogers, Schachermayer, Soner, Stebegg, Tan, Tangpi, Zaev,... Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 8 / 20

17 Example: Beiglböck Juillet Coupling Theorem (Beiglböck, Juillet): Let f (x, y) = g(y x) where g is differentiable, sufficiently integrable, and g is strictly concave. Then the optimal P is given by the Left-Courtain Coupling: x x y y y + Figure : Forbidden Configuration. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 9 / 20

18 Beiglböck Juillet Coupling for Uniform Marginals Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 10 / 20

19 Beiglböck Juillet Coupling for Uniform Marginals Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 10 / 20

20 Beiglböck Juillet Coupling for Uniform Marginals Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 10 / 20

21 Beiglböck Juillet Coupling for Uniform Marginals Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 10 / 20

22 Beiglböck Juillet Coupling for Uniform Marginals Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 10 / 20

23 Duality for Martingale Optimal Transport In analogy to Monge Kantorovich duality we want: (1) No duality gap: sup E P [f (X, Y )] = inf µ(ϕ) + ν(ψ). P M(µ,ν) ϕ,ψ,h (2) Dual existence: ˆϕ, ˆψ, ĥ. Theorem (Beiglböck, Henry-Labordère, Penkner): For upper semicontinuous f 0, there is no duality gap. Dual existence fails in general, even if f is bounded, continuous and µ, ν are compactly supported. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 11 / 20

24 Duality for Martingale Optimal Transport In analogy to Monge Kantorovich duality we want: (1) No duality gap: sup E P [f (X, Y )] = inf µ(ϕ) + ν(ψ). P M(µ,ν) ϕ,ψ,h (2) Dual existence: ˆϕ, ˆψ, ĥ. Theorem (Beiglböck, Henry-Labordère, Penkner): For upper semicontinuous f 0, there is no duality gap. Dual existence fails in general, even if f is bounded, continuous and µ, ν are compactly supported. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 11 / 20

25 An Example with Duality Gap Let f be the bounded, lower semicontinuous function { 0 on the diagonal, f (x, y) = 1 x y = 1 off the diagonal. Let µ = ν = Lebesgue measure on [0, 1]. There exists a unique martingale transport P, concentrated on the diagonal (T (x) = x). Primal value: sup P M(µ,ν) E P [f (X, Y )] = 0. Dual optimizers exist, ˆϕ = 1, ˆψ = 0, ĥ = 0 but there is a duality gap: dual value = 1 > 0. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 12 / 20

26 An Example with Duality Gap Let f be the bounded, lower semicontinuous function { 0 on the diagonal, f (x, y) = 1 x y = 1 off the diagonal. Let µ = ν = Lebesgue measure on [0, 1]. There exists a unique martingale transport P, concentrated on the diagonal (T (x) = x). Primal value: sup P M(µ,ν) E P [f (X, Y )] = 0. Dual optimizers exist, ˆϕ = 1, ˆψ = 0, ĥ = 0 but there is a duality gap: dual value = 1 > 0. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 12 / 20

27 Financial Intuition Earlier work considered model uncertainty over a set P of real-world models (with finitely many traded options). P-q.s. version of the FTAP: Theorem (Bouchard, N.): Let M = {calibrated martingale measures Q P}. Then No arbitrage NA(P) P and M have same polar sets and under this condition, quasi-sure duality holds with existence. Reverse-engineered: To avoid arbitrage, neglect events not seen by M. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 13 / 20

28 Financial Intuition Earlier work considered model uncertainty over a set P of real-world models (with finitely many traded options). P-q.s. version of the FTAP: Theorem (Bouchard, N.): Let M = {calibrated martingale measures Q P}. Then No arbitrage NA(P) P and M have same polar sets and under this condition, quasi-sure duality holds with existence. Reverse-engineered: To avoid arbitrage, neglect events not seen by M. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 13 / 20

29 Ordinary and Martingale 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) Martingale Optimal Transport and Robust Finance 14 / 20

30 Ordinary and Martingale 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) Martingale Optimal Transport and Robust Finance 14 / 20

31 Obstructions for Martingale Transport In martingale OT, some roads x y can be blocked. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 15 / 20

32 Potential Functions Potential u µ (x) := t x µ(dt) = E[ X x ] under any P M(µ, ν). µ c ν u µ u ν. If u µ (x) = u ν (x); i.e. E[ X x ] = E[ Y x ] ( ), then x is a barrier for any martingale transport: 1. Jensen: X x = E[Y X ] x = E[Y x X ] E[ Y x X ] 2. Under ( ), it follows that X x = E[ Y x X ] a.s. Hence, E[ Y x 1 X x ] = E[ X x 1 X x ] = E[(X x)1 X x ] = E[(Y x)1 X x ] so that Y x a.s. on {X x}. Partition R into intervals {u µ < u ν }. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 16 / 20

33 Potential Functions Potential u µ (x) := t x µ(dt) = E[ X x ] under any P M(µ, ν). µ c ν u µ u ν. If u µ (x) = u ν (x); i.e. E[ X x ] = E[ Y x ] ( ), then x is a barrier for any martingale transport: 1. Jensen: X x = E[Y X ] x = E[Y x X ] E[ Y x X ] 2. Under ( ), it follows that X x = E[ Y x X ] a.s. Hence, E[ Y x 1 X x ] = E[ X x 1 X x ] = E[(X x)1 X x ] = E[(Y x)1 X x ] so that Y x a.s. on {X x}. Partition R into intervals {u µ < u ν }. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 16 / 20

34 Structure of M(µ, ν)-polar Sets Theorem: These are precisely the M(µ, ν)-polar sets. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 17 / 20

35 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 ) M(µ, ν)-q.s. Then, (1) there is no duality gap, (2) dual optimizers ˆϕ, ˆψ, ĥ exist. The superhedge is pointwise on each component (e.g., µ = δ x0 ). Dual existence in the pointwise formulation typically fails as soon as there is more than one component. Application as in the FTOT. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 18 / 20

36 Key Idea for the Proof Core step: make almost-optimal ϕ n, ψ n converge. Control ϕ n, ψ n by a single, convex function χ n. Suppose there is only one component; thus ν µ > c 0. After a normalization, χ n (0) = χ n(0) = 0 and 0 χ n d(ν µ) const. This bounds the convexity of χ n. Relative compactness of (χ n ). Relative compactness of (ϕ n, ψ n ); Komlos. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 19 / 20

37 Conclusion The quasi-sure formulation ( model uncertainty ) seems to be a natural setup for the Martingale Optimal Transport problem. One may expect this to be true for a larger class of problems; in particular, if discontinuous reward functions are involved or attainment is desired. Thank you. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 20 / 20

38 Conclusion The quasi-sure formulation ( model uncertainty ) seems to be a natural setup for the Martingale Optimal Transport problem. One may expect this to be true for a larger class of problems; in particular, if discontinuous reward functions are involved or attainment is desired. Thank you. Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance 20 / 20