Martingale Optimal Transport and Robust Hedging

Transcription

1 Martingale Optimal Transport and Robust Hedging Ecole Polytechnique, Paris Angers, September 3, 2015

2 Outline Optimal Transport and Model-free hedging The Monge-Kantorovitch optimal transport problem Financial interpretation 1 Optimal Transport and Model-free hedging The Monge-Kantorovitch optimal transport problem Financial interpretation 2 Formulation and duality Optimal semi-static strategy and quasi-sure formulation 3 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem

3 The Monge-Kantorovitch optimal transport problem Financial interpretation

4 The Monge-Kantorovitch optimal transport problem Financial interpretation The problem of Déblais et Remblais (Monge 1781) Figure: Mass transport. Ref : C. Villani

5 Analytic formulation (Monge 1781) The Monge-Kantorovitch optimal transport problem Financial interpretation Initial distribution : probability measure µ on X Final distribution : probability measure ν on Y Problem : find an optimal transference plan T P M := sup T T (µ,ν) c ( x, T (x) ) µ(dx) where T (µ, ν) of all maps T : x y = T (x) such that ν = µ T 1

6 The Monge-Kantorovitch optimal transport problem Financial interpretation Randomization of mass transfert (Kantorovich 1942) Figure: Randomized mass transport

7 The Monge-Kantorovitch optimal transport problem Financial interpretation Probabilistic formulation (Kantorovich 1942) Randomization of transference plans : P K := sup E P[ c(x, Y ) ] P P(µ,ν) where Ω = X Y, X (x, y) = x, Y (x, y) = y, and P(µ, ν) := { P Prob Ω : X P µ and Y P ν }

8 Kantorovich duality The Monge-Kantorovitch optimal transport problem Financial interpretation Duality in linear programming, Legendre-Fenchel duality... D 0 { } := inf µ(ϕ) + ν(ψ) (ϕ,ψ) D 0 D 0 := { (ϕ, ψ) L 1 (µ) L 1 (ν) : ϕ ψ c } where µ(ϕ) := ϕdµ, ν(ψ) := ψdν, and ϕ ψ(x, y) := ϕ(x) + ψ(y), x X, y Y No reference probability measure on the product space

9 The duality Optimal Transport and Model-free hedging The Monge-Kantorovitch optimal transport problem Financial interpretation Theorem Let c 0 be measurable. Then duality : D 0 = P K existence holds for D 0 If in addition c USC, existence also holds for P K In this generality, the result is due to Kellerer 84

10 Financial interpretation The Monge-Kantorovitch optimal transport problem Financial interpretation X µ and Y ν prices of two assets at time 1 µ and ν identified from market prices of call options : C µ (K) = (x K) + µ(dx), C ν (K) = (y K) + ν(dy), K 0 Breeden-Litzenberger 1978 : µ = C µ and ν = C ν ϕ(x ) Vanilla position in X with market price µ(ϕ) ψ(y ) Y ν(ψ)

11 The Monge-Kantorovitch optimal transport problem Financial interpretation Financial interpretation : no reference probability c(x, Y ) payoff of derivative security Robust static hedging strategies for the derivative c(x, Y ) : D 0 := { (ϕ, ψ) L 1 (µ) L 1 (ν) : ϕ ψ c } Robust superhedging cost of c(x, Y ) is the Kantorovitch dual : D 0 { } = inf µ(ϕ) + ν(ψ) (ϕ,ψ) D 0 The primal Monge-Kantorovitch problem is : P K = sup E P [c(x, Y )] P P(µ,ν)

12 Outline Optimal Transport and Model-free hedging Formulation and duality Optimal semi-static strategy and quasi-sure formulation 1 Optimal Transport and Model-free hedging The Monge-Kantorovitch optimal transport problem Financial interpretation 2 Formulation and duality Optimal semi-static strategy and quasi-sure formulation 3 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem

13 Formulation and duality Optimal semi-static strategy and quasi-sure formulation MARTINGALE OPTIMAL TRANSPORT initiated by Pierre Henry-Labordère, Preprints Soc. Gén. Beiglböck, Henry-Labordère & Penkner Galichon, Henry-Labordère & NT

14 One asset observed at two future dates Formulation and duality Optimal semi-static strategy and quasi-sure formulation Our interest now is on the case where X = X 0 and Y = X 1 are the prices of the same asset at two future dates 0 and 1 Interest rate is reduced to zero This setting introduces a new feature : possibility of dynamic trading the asset between times 0 and 1

15 Superhedging problem Kantorovitch dual Formulation and duality Optimal semi-static strategy and quasi-sure formulation Robust super hedging problem naturally formulated as : D(µ, ν) := inf (ϕ,ψ,h) D { } µ(ϕ) + ν(ψ) where D := { (ϕ, ψ, h) L 1 (µ) L 1 (ν) L 0 : ϕ ψ + h c } where ϕ ψ(x, y) := ϕ(x) + ψ(y) and h (x, y) := h(x)(y x)

16 The Martingale Optimal Transport Problem Formulation and duality Optimal semi-static strategy and quasi-sure formulation Theorem (Beiglböck, Henry-Labordère, Penkner) Assume c USC and bounded from above. Then P = D, and existence holds for P(µ, ν). where the dual problem is : P(µ, ν) := sup E P[ c(x, Y ) ] P M(µ,ν) with M(µ, ν) := { P P(µ, ν) : E P [Y X ] = X } Strassen 65 : M(µ, ν) iff µ ν, i.e. µ(g) ν(g) g convex

17 Formulation and duality Optimal semi-static strategy and quasi-sure formulation Existence of optimal hedge does not hold in general There are easy examples where existence for the dual fails, even for bounded c, bounded support... (Beiglböck, Henry-Labordère & Penkner, Beiglböck, Nutz & NT) Let µ = ν, then M(µ, µ) = {P } where Y = X, P a.s. P(µ, µ) = E µ[ c(x, X ) ] The derivative is in fact c(x, X ), primal problem carries no information about c(x, y) outside the diagonal y = x One is only interested in hedging along the diagonal ϕ(x ) + ψ(x ) + h(x )(X X ) c(x, X ), µ a.s.

18 Duality under more general payoff functions Formulation and duality Optimal semi-static strategy and quasi-sure formulation The condition c USC is not innocent... Consider the following example of LSC payoff c(x, y) := 1I {x y}, x, y [0, 1] [0, 1] Let µ = ν = Lebesgue measure on [0, 1]. Then M(µ, µ) = {P } uniform distribution on the diagonal of the square [0, 1] 2 Then P(µ, µ) = 0 However, we may prove that D(µ, µ) = 1!

19 Quasi-sure robust superhedging Formulation and duality Optimal semi-static strategy and quasi-sure formulation Definition M(µ, ν) q.s. (quasi surely) means P a.s. for all P M(µ, ν) The quasi-sure robust superhedging cost D qs { } := inf µ(ϕ) + ν(ψ) (ϕ,ψ,h) D qs D qs := { (ϕ, ψ, h) : ϕ ψ + h c, M(µ, ν) q.s. } is more natural... Then, D(µ, ν) D qs (µ, ν) P(µ, ν) so if the duality P = D holds, it follows that D = D qs

20 Formulation and duality Optimal semi-static strategy and quasi-sure formulation Structure of polar sets in (standard) optimal transport Theorem (Kellerer) For N R R, TFAE : P[N] = 0 for all P P(µ, ν) N (N 1 R) (R N 2 ) for some N 1, N 2 R, µ(n 1 ) = ν(n 2 ) = 0 = no difference between the pointwise and the quasi-sure formulations in standard optimal transport

21 Pointwise versus Quasi-sure superhedging I Formulation and duality Optimal semi-static strategy and quasi-sure formulation Suppose Supp(µ) = [0, 2] =Supp(ν) = [0, 2], then M(µ, ν) q.s. only involves the values (x, y) [0, 2] 2 Pointwise superhedging involves all values (x, y) R 2

22 Pointwise versus Quasi-sure superhedging II Formulation and duality Optimal semi-static strategy and quasi-sure formulation Suppose Supp(µ) =Supp(ν) = [0, 2], and C µ (1) = C ν (1) E [ (X 1) +] = E [ (Y 1) +] E [ (X 1) +] by Jensen s inequality, and then {X 1} = {Y 1}

23 Formulation and duality Optimal semi-static strategy and quasi-sure formulation Structure of polar sets in martingale optimal transport Consider the partition : {C µ < C ν } = k 0 I k, I k = (a k, b k ), J k := I k {ν atoms} Theorem (Beiglböck, Nutz & NT 15) For N R R, TFAE : P[N] = 0 for all P M(µ, ν) N (N 1 R) (R N 2 ) { (I k J k ) } c for some N 1, N 2 R, µ(n 1 ) = ν(n 2 ) = 0

24 Formulation and duality Optimal semi-static strategy and quasi-sure formulation Duality and existence under quasi-sure robust superhedging Theorem (Beiglböck, Nutz & NT 15) Let µ ν and c 0 measurable. Then P(µ, ν) = D qs (µ, ν) and existence holds for D qs

25 Outline Optimal Transport and Model-free hedging Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem 1 Optimal Transport and Model-free hedging The Monge-Kantorovitch optimal transport problem Financial interpretation 2 Formulation and duality Optimal semi-static strategy and quasi-sure formulation 3 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem

26 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Geometry of optimal transport plans : cyclic monotonicity In standard optimal transport, optimality of a transport plan is a property of its support... Theorem For an optimal transport plan P, there exists Γ X Y : P [Γ] = 1, and for all finite subset (x i, y i ) i n Γ : n c(x i, y i ) i=1 n c(x i, y i+1 ) with y n+1 = y 1 i=1 (necessary and sufficient, under slight conditions)

27 Back to the original Monge formulation Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem P K P M : Kantorovitch formulation relaxation of Monge one Theorem (Rachev & Rüschendorf) µ without atoms, c C 1 with c xy > 0 (Spence-Mirrlees/Twist condition). Then there is a unique optimal transference plan : P (dx, dy) = µ(dx)δ {T (x)}(dy) with T = F 1 ν F µ Consequently P M = P K, and T solves both problems. T : monotone rearrangement, Fréchet-Hoeffding coupling Extension to R d (Brenier) : P concentrated on the graph of the gradient of some c convex function

28 On the Spence-Mirrlees condition Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem The solution of the Kantorovitch optimal transportation problem P K := sup P P(µ,ν) c(x, y)p(dx, dy) is not modified by the change of performance criterion : c(x, y) ĉ(x, y) := c(x, y)+a(x) + b(y) Spence-Mirrlees condition c xy > 0 stable by this transformation

29 Martingale monotonicity condition Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Optimality of martingale transport is also a property of its support... Theorem (Beiglböck & Juillet 12, Zaev 14) Let P be a solution of P(µ, ν). Then there exists Γ Ω, P [Γ] = 1 such that for all P 0 with Supp(P 0 ) = {ω 1,..., ω N } Γ, we have : [ ] E P 0 c(x, Y ) E P [ c(x, Y ) ] whenever P X 1 = P 0 X 1, P Y 1 = P 0 Y 1 and E P [Y X ] = E P 0 [Y X ]

30 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Necessary & sufficient Martingale monotonicity Theorem (Beiglböck, Nutz & NT 15) Let ( ˆϕ, ˆψ, ĥ) be solution of D qs (µ, ν), and set Γ := { ˆϕ ˆψ + ĥ = c} Then P solution of P(µ, ν) iff P [Γ] = 1

31 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Worst Case Financial Market Brenier Theorem Solution P M(µ, ν) exists for c USC. Is there an optimal transfert map, i.e. optimal transport of µ to ν through a map T? NO, unless µ = ν! Is there a transference plan along a minimal randomization? X Y = T u (X ) with probability q(x ) Y = Td (X ) with probability 1 q(x )

32 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Worst Case Financial Market Brenier Theorem Solution P M(µ, ν) exists for c USC. Is there an optimal transfert map, i.e. optimal transport of µ to ν through a map T? NO, unless µ = ν! Is there a transference plan along a minimal randomization? X Y = T u (X ) with probability q(x ) Y = Td (X ) with probability 1 q(x )

33 Left-monotone martingale transport Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Definition (Beiglbock & Juillet 2012) P M(µ, ν) is left-monotone if P[(X, Y ) Γ] = 1, for some Γ R R, and for all (x, y 1 ), (x, y 2 ), (x, y ) Γ : x < x = y (y 1, y 2 )

34 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Existence and uniqueness of left-monotone martingale transport Theorem (Beiglböck, Henry-Labordère & NT 15) Let P M(µ, ν) be solution of P(µ, ν). If c x strictly convex in y, µ a.e. x, then P is left-monotone Theorem (Beiglböck & Juillet 12, Beiglböck, Henry-Labordère & NT 15) Assume µ has no atoms. Then, there is a unique left-monotone P M(µ, ν) with distribution concentrated on two graphs Henry-Labordère & NT 13 provide an explicit description of the left-monotone transport plan and corresponding semi-static robust superhedging strategy

35 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem The martingale version of the Spence-Mirrlees condition... is c xyy > 0 Notice that the solution of the Martingale Transport problem is not altered by the change of payoff : c(x, y) ĉ(x, y) := c(x, y) + a(x) + b(y) + h(x)(y x) ĉ xyy = c xyy

36 Concluding remarks Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Extension to R d, i.e. µ and ν prob. meas. on R d : Duality for c USC and existence for P(µ, ν) Duality for c meas. and existence for D(µ, ν)?? Martingale version of the Brenier theorem, see Ghoussoub, Kim & Lim 2015 Extension to finite discrete-time, possibly finitely-many marginals constraints

37 Concluding remarks II Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Continuous-Time Martingale Transport : substitute for the return from a dynamic hedge h is the stochastic integral T 0 H sdx s But without reference probability?? = Two viewpoints Pointwise definition : restrict H to have finite variation, then : T 0 H sdx s := T 0 X sdh s + H T X T H 0 X 0 Dolinsky & Soner Quasi-sure definition : under any P M(µ, ν), T 0 H sdx s is defined by the standard stochastic analysis... Guo, Tan & NT 2015

38 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem THANK YOU FOR YOUR ATTENTION