Sampling of probability measures in the convex order and - PowerPoint PPT Presentation

Introduction Sampling in the convex order The dimension 1 case Sampling in the convex order in higher dimensions Numerical results Sampling of probability measures in the convex order and approximation of Martingale Optimal Transport problems Aurélien Alfonsi CERMICS, Ecole des Ponts, Paris-Est University Joint work with Jacopo Corbetta and Benjamin Jourdain Jim Gatheral’s conference, NY October 15, 2017 Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 1 / 37

Introduction Sampling in the convex order The dimension 1 case Sampling in the convex order in higher dimensions Numerical results Structure of the talk Introduction 1 The dimension 1 case 2 Sampling in the convex order in higher dimensions 3 Numerical results 4 Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 2 / 37

Introduction Sampling in the convex order The dimension 1 case Introduction Sampling in the convex order in higher dimensions Numerical results The convex order Let X , Y : Ω → R d two random variables with respective law µ and ν . X is smaller than Y in the convex order if ∀ φ : R d → R , E [ φ ( X )] ≤ E [ φ ( Y )] , for any convex function φ such that both expectations exist. In this case, we write X ≤ cx Y or µ ≤ cx ν . � R d | y | ν ( dy ) < ∞ . µ ≤ cx ν iff Strassen’s theorem : (1965) Assume there exists a martingale kernel Q ( x , dy ) such that µ Q = ν , i.e. � µ ( dx ) Q ( x , dy ) = ν ( dy ) . Notation : Π M ( µ, ν ) = { π ( dx , dy ) = µ ( dx ) Q ( x , dy ) : ∀ x ∈ R d , � R d | y | Q ( x , dy ) < ∞ and � R d yQ ( x , dy ) = x } Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 3 / 37

Introduction Sampling in the convex order The dimension 1 case Introduction Sampling in the convex order in higher dimensions Numerical results Martingale Optimal Transport in Finance We assume r = 0. ( S t ) t ≥ 0 : price process of d assets. Suppose that we know for 0 < T 1 < T 2 the law of S T 1 and S T 2 (denoted µ 1 and µ 2 ), and that we want to price an option that pays c ( S T 1 , S T 2 ) at time T 2 , with c : R d × R d → R . Price bounds for the option : � R d × R d c ( x , y ) π ( dx , dy ) , π ∈ Π M ( µ 1 , µ 2 ) → minimize / maximize . Multi-marginal case : payoff c ( S T 1 , . . . , S T n ) with c : ( R d ) n → R . Beiglböck, Henry-Labordère, Penkner (2013) : Duality and connection with super/subhedging strategies. Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 4 / 37

Introduction Sampling in the convex order The dimension 1 case Introduction Sampling in the convex order in higher dimensions Numerical results Sampling in the Convex order When approximating µ ≤ cx ν by discrete measures µ I = � I i = 1 p i δ x i and ν J = � J j = 1 q j δ y j (typically with i.i.d. samples) we may not have µ I ≤ cx ν J . Our goal : to construct ˜ µ I (resp. ˜ ν J ) “close” to µ I (resp. ν J ) such that µ I ≤ cx ˜ ˜ ν J . Motivation : Numerical methods for Martingale Optimal Transport (MOT) problems. We can use linear programming solvers to solve : I J � � r ij c ( x i , y j ) i = 1 j = 1 under the constraints I J J � � � r ij ≥ 0 , r ij = q j , r ij = p i and r ij y j = p i x i . i = 1 j = 1 j = 1 Monte-Carlo : calculate together option prices and their bounds. Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 5 / 37

Introduction Sampling in the convex order The dimension 1 case Introduction Sampling in the convex order in higher dimensions Numerical results Existing methods to approximate measures in the convex order Quantization : Dual quantization preserves the convex order in dimension one. In general, gives a measure ˆ ν such that ν ≤ cx ˆ ν . (Pagès Wilbertz 2012) Primal quantization gives a measure ˆ µ such that ˆ µ ≤ cx µ . Drawbacks : ν and thus µ must have a compact support. Only for 2 marginals. Computation time. Dimension 1 (Baker’s thesis, 2012) : Assume µ ≤ cx ν and let � I � I µ I = 1 ν I = 1 ˆ i = 1 δ µ ( u ) du and ˆ i = 1 δ ( u ) du . � i � i I I F − 1 F − 1 I I I I ν i − 1 i − 1 I I ν I for any I ∈ N ∗ . µ I ≤ cx ˆ Then, we have ˆ Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 6 / 37

Introduction Sampling in the convex order The dimension 1 case Introduction Sampling in the convex order in higher dimensions Numerical results A first idea : to equalize the means Suppose µ ≤ cx ν , X 1 , . . . , X I i.i.d. ∼ µ and Y 1 , . . . , Y J i.i.d. ∼ ν . We set ¯ � I i = 1 X i and ¯ � J X I = 1 Y J = 1 j = 1 Y j , and I J I J µ I = 1 ν J = 1 � � ˜ δ X i + m − ¯ X I , ˜ δ Y j + m − ¯ Y J , I J i = 1 j = 1 x µ ( dx ) if it is known explicitly or ¯ � with m = X I otherwise. Under suitable conditions, a.s., ∃ M , ∀ I , J ≥ M , ˜ µ I ≤ cx ˜ ν J . σ 2 = exp ( σ ν G − σ 2 law law For X 1 = exp ( σ µ G − 2 ) , Y 1 µ 2 ) with σ µ = 0 . 24, ν σ ν = 0 . 28, I = 100, P (˜ µ I ≤ cx ˜ ν I ) ≈ 0 . 45. = ⇒ need for a non asymptotic approach. Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 7 / 37

Introduction Sampling in the convex order The dimension 1 case The dimension 1 case Sampling in the convex order in higher dimensions Numerical results Introduction 1 The dimension 1 case 2 Sampling in the convex order in higher dimensions 3 Numerical results 4 Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 8 / 37

Introduction Sampling in the convex order The dimension 1 case The dimension 1 case Sampling in the convex order in higher dimensions Numerical results Characterization of the convex order in dimension 1 � We set P 1 ( R ) = { µ ∈ P ( R ) : R | x | µ ( dx ) < ∞} . For x ∈ R , F µ ( x ) = µ (( −∞ , x ]) , ¯ F µ ( x ) = µ ([ x , + ∞ )) . For p ∈ ( 0 , 1 ) , F − 1 µ ( p ) = inf { x ∈ R : F µ ( x ) ≥ p } . � t R ( t − x ) + µ ( dx ) and For t ∈ R , we define ϕ µ ( t ) = −∞ F µ ( x ) dx = � � + ∞ ¯ R ( x − t ) + µ ( dx ) . � ϕ µ ( t ) = ¯ F µ ( x ) dx = t Theorem 1 Let µ, ν ∈ P 1 ( R ) . The following conditions are equivalent : (i) µ ≤ cx ν , � � (ii) R x µ ( dx ) = R x ν ( dx ) and ∀ t ∈ R , ϕ µ ( t ) ≤ ϕ ν ( t ) , � � (iii) R x µ ( dx ) = R x ν ( dx ) and ∀ t ∈ R , ¯ ϕ µ ( t ) ≤ ¯ ϕ ν ( t ) , � 1 � 1 � 1 0 F − 1 0 F − 1 q F − 1 (iv) µ ( p ) dp = ν ( p ) dp and ∀ q ∈ ( 0 , 1 ) , µ ( p ) dp ≤ � 1 q F − 1 ν ( p ) dp . Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 9 / 37

Introduction Sampling in the convex order The dimension 1 case The dimension 1 case Sampling in the convex order in higher dimensions Numerical results Application to discrete random variables Corollary 2 Let µ = � I i = 1 p i δ x i and ν = � J j = 1 q j δ y j be two probability measures on R . Without loss of generality, we assume that x 1 < · · · < x I , y 1 < · · · < y J and p 1 p I q 1 q J > 0 . Then we have µ ≤ cx ν if, and only if (i) y 1 ≤ x 1 and y J ≥ x I , (ii) for all j such that x 1 ≤ y j ≤ x I , ϕ µ ( y j ) ≤ ϕ ν ( y j ) , (iii) � I i = 1 p i x i = � J j = 1 q j y j . We can replace ( ii ) by ( ii ′ ) for all j such that x 1 ≤ y j ≤ x I , ¯ ϕ µ ( y j ) ≤ ¯ ϕ ν ( y j ) . Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 10 / 37

Introduction Sampling in the convex order The dimension 1 case The dimension 1 case Sampling in the convex order in higher dimensions Numerical results The increasing/decreasing convex orders Definition/Proposition : The following statements are equivalent : (i) µ ≤ icx ν (resp. µ ≤ dcx ν ), (ii) ∀ t ∈ R , ¯ ϕ µ ( t ) ≤ ¯ ϕ ν ( t ) (resp. ϕ µ ( t ) ≤ ϕ ν ( t ) ), � 1 � 1 q F − 1 q F − 1 (iii) ∀ q ∈ [ 0 , 1 ] , µ ( p ) dp ≤ ν ( p ) dp (resp. � q � q 0 F − 1 0 F − 1 µ ( p ) dp ≥ ν ( p ) dp ). � � Rk 1 : µ, ν ∈ P 1 ( R ) , µ ≤ icx ν = ⇒ R x µ ( dx ) ≤ R x ν ( dx ) . Therefore, � � µ ≤ cx ν ⇐ ⇒ µ ≤ icx ν and x µ ( dx ) = x ν ( dx ) R R ⇐ ⇒ µ ≤ icx ν and µ ≤ dcx ν. ⇒ µ ∗ ≤ icx ν ∗ , where µ ∗ (( −∞ , x ]) := µ ([ − x , + ∞ )) Rk 2 : µ ≤ dcx ν ⇐ Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 11 / 37

Introduction Sampling in the convex order The dimension 1 case The dimension 1 case Sampling in the convex order in higher dimensions Numerical results The lattice structure (Kertz & Rösler 1992,2000) ( P 1 ( R ) , ≤ icx ) and ( P 1 ( R ) , ≤ dcx ) are complete lattices. From µ, ν ∈ P 1 ( R ) , we can define µ ∨ icx ν and µ ∧ icx ν such that µ ∧ icx ν ≤ icx η ≤ icx µ ∨ icx ν for η ∈ { µ, ν } , µ ∧ dcx ν ≤ dcx η ≤ icx µ ∨ dcx ν for η ∈ { µ, ν } . � t �� t � t � F µ ∨ dcx ν ( x ) dx = max F µ ( x ) dx , F ν ( x ) dx −∞ −∞ −∞ � t �� t � t � F µ ∧ dcx ν ( x ) dx = convex hull of min F µ ( x ) dx , F ν ( x ) dx −∞ −∞ −∞ � + ∞ �� + ∞ � + ∞ � ¯ ¯ ¯ F µ ∨ icx ν ( x ) dx = max F µ ( x ) dx , F ν ( x ) dx t t t � + ∞ �� + ∞ � + ∞ � ¯ ¯ ¯ F µ ∨ icx ν ( x ) dx = convex hull of min F µ ( x ) dx , F ν ( x ) dx . t t t � q � q � q 0 F − 1 0 F − 1 0 F − 1 Rk : We also have µ ∧ dcx ν ( p ) dp = min ( µ ( p ) dp , ν ( p ) dp ) . Aurélien Alfonsi (Ecole des Ponts) October 15, 2017 12 / 37