Martingale Transport, Skorokhod Embedding and Peacocks

Transcription

1 Martingale Transport, Skorokhod Embedding and CEREMADE, Université Paris Dauphine Collaboration with Pierre Henry-Labordère, Nizar Touzi 08 July, 2014 Second young researchers meeting on BSDEs, Numerics and IMB Martingale Transport, Skorokhod Embedding and

2 Outline Problem formulation Applications Martingale Transport, Skorokhod Embedding and

3 Outline Problem formulation Applications Martingale Transport, Skorokhod Embedding and

4 A peacock is a stochastic process (X t, t 0), if (i) it is integrable, i.e. E[ X t ] <, t 0 ; (ii) it increases in convex ordering, i.e. for every convex function φ : R R, the map t E[φ(X t )] is increasing. Kellerer s theorem : Every peacock has the same one-dimensional marginals as a martingale (M t, t 0), i.e E[M t M r, 0 r s] = M s and X t M t in law for every t 0. Remarks : PCOC : Processus Croissant pour l Ordre Convexe. A peacock is determined by the family of one-dimensional marginal distributions. Martingale Transport, Skorokhod Embedding and

5 Xiaolu Figure Tan: The Martingale book Transport, Skorokhod Embedding and

6 Martingale Transport, Skorokhod Embedding and

7 Optimal martingale peacocks Let µ = (µ t, t 0) be a peacock, ξ be a reward/cost function on the martingale M, we look for the optimal martingale associated to the peacocks µ : [ sup E ξ ( M ) ]. M martingale peacock Martingale Transport, Skorokhod Embedding and

8 Martingale Optimal Transport Monge-Kantorovich s Optimal Transportation Problem : sup E P[ c(x 0, X 1 ) ] = inf P P(µ 0,µ 1 ) { } µ 0 (λ 0 ) + µ 1 (λ 1 ) : λ 0 (x) + λ 1 (y) c(x, y). Martingale Transportation Problem : sup E P[ c(x 0, X 1 ) ] = inf P M(µ 0,µ 1 ) { } µ 0 (λ 0 ) + µ 1 (λ 1 ) : λ 0 (x) + λ 1 (y) + h(x)(y x) c(x, y). Martingale Transport, Skorokhod Embedding and

9 Martingale Optimal Transport and Finance Primal problem : finding extremal martingale given marginals : sup E [ ξ(m) ]. M M(µ 0,µ 1 ) Dual problem : finding the minimum super hedging cost : (Beiglbock, Henry-Labordère, Penkner) { } inf µ 0 (λ 0 ) + µ 1 (λ 1 ) : λ 0 (X 0 ) + λ 1 (X 1 ) + h(x 0 )(X 1 X 0 ) ξ(x 0, X 1 ), Martingale Transport, Skorokhod Embedding and

10 Main problems Given a peacock µ = (µ t ) t 0, [ sup E ξ ( M ) ]. M martingale peacock Main problems Duality Find the optimal martingale Find the dual optimazer The value Martingale Transport, Skorokhod Embedding and

11 Outline Problem formulation Applications Martingale Transport, Skorokhod Embedding and

12 Martingale Transportation Problem Monge-Kantorovich s Optimal Transportation Problem : sup E P[ c(x 0, X 1 ) ] = inf P P(µ 0,µ 1 ) { } µ 0 (λ 0 ) + µ 1 (λ 1 ) : λ 0 (x) + λ 1 (y) c(x, y). Martingale Transportation Problem : sup E P[ c(x 0, X 1 ) ] = inf P M(µ 0,µ 1 ) { } µ 0 (λ 0 ) + µ 1 (λ 1 ) : λ 0 (x) + λ 1 (y) + h(x)(y x) c(x, y). Martingale Transport, Skorokhod Embedding and

13 Martingale version of Brenier s theorem Brenier s theorem (Fréchet-Hoeffding coupling) in the one-dimensional case : when xy c > 0, the solution is given by the monotone transference plan T := F 1 1 F 0. Martingale version (Beiglbock-Juillet, Henry-Labordère -Touzi) : When xyy c > 0, the optimal solution is given by the left-monotone martingale transference plan (which is a binomial model). The transition kernel of the binomial model is, with T d (x) x T u (x), q(x) := x T d (x) T u(x) T d (x), T (x, dy) := q(x)δ Tu(x)(dy) + (1 q(x))δ Td (x)(dy). Martingale Transport, Skorokhod Embedding and

14 Martingale version of Brenier s theorem Determinate T u and T d : assume that δf := F 1 F 0 has only one local maximizer m. Coupled ODE, on [m, ), d(δf T d ) = (1 q)df 0, d(f 1 T u ) = qdf 0. Resolution of ODE : denote g(x, y) := F1 1 ( F0 (x) + δf (y) ), x [ F 1 1 (F 0 (ξ)) ξ ] df 0 (ξ) + Td (x) (g(x, ξ) ξ)dδf (ξ) = 0, T u (x) = g ( x, T d (x) ). Martingale Transport, Skorokhod Embedding and

15 Martingale version of Brenier s theorem Figure : An example of T u and T d. Martingale Transport, Skorokhod Embedding and

16 The optimal dual components The dynamic strategy h : h (x) = c x(x, T u (x)) c x (x, T d (x)), x [m, ), T u (x) T d (x) ( h (x) = h T 1 d (x)) ( + c y (x, x) c y T 1 d (x), x), x (, m). The static strategy (λ 0, λ 1 ) : λ 1 = c y (T 1, ) h T 1, T 1 = Tu 1 1 [m, ) + T 1 d λ 0 = q ( c(, T u ) λ 1 (T u ) ) + (1 q) ( c(, T d ) λ 1 (T d ) ). 1 (,m). Martingale Transport, Skorokhod Embedding and

17 The multi-marginals case An easy extension to the multi-marginals case sup P M(µ 0,,µ n) E P[ n k=1 ] c(x k 1, X k ). The extremal model is a Markov chain (martingale), and the optimal dual strategies are all explicit. What happens if n? Do they converge? the criteria function, the Markov chain, the super hedging strategy. Does the limit keep the optimality? Martingale Transport, Skorokhod Embedding and

18 Outline Problem formulation Applications Problem formulation Applications Martingale Transport, Skorokhod Embedding and

19 Limit of the criteria function Problem formulation Applications Assumption : c(x, x) = c y (x, x) = 0, c xyy (x, y) > 0. Quadratic variation (Föllmer) of a càdlàg path x : [0, 1] R, lim n 1 k n ( xtk x tk 1 ) 2δtk 1 (dt). It is proved in Hobson and Klimmek (2012) that n c(x tk 1, x tk ) C(x) := 1 2 k=1 1 0 c yy (x t, x t )d[x] c t + 0 t 1 c(x t, x t ). Martingale Transport, Skorokhod Embedding and

20 Problem formulation Applications Continuous-time martingale transport Let µ = (µ t ) 0 t 1 be increasing in convex ordering, right-continuous and unif. integrable. Let Ω := D([0, 1], R), M the set of martingale measures on Ω and M (µ) that subset of measures under which X fits all marginals. MT problem P (µ) := sup E P[ C(X ) ]. P M (µ) Martingale Transport, Skorokhod Embedding and

21 Dual formulation Problem formulation Applications Dynamic strategy : H 0 : [0, 1] Ω R denotes the set of all predictable, locally bounded processes, H := { H H 0 : H X is a P-supermartingale for every P M }. Λ := {λ(x, dt) = λ 0 (t, x)γ(dt), Λ(µ) := { λ Λ : µ( λ ) < }, µ(λ) := λ 0 (t, x)µ t (dx)γ(dt). Dual problem D (µ) := { (H, λ) : 1 0 λ(x t, dt) + (H X ) 1 C(X ), P-a.s., P M }. Martingale Transport, Skorokhod Embedding and

22 The limit of Markov chain Problem formulation Applications (i) Suppose that (µ t ) t [0,1] admits smooth density functions f (t, x). Denote by F (t, x) the distribution function. (ii) x t F (t, x) has only one local maximizer m(t). Define T d : [0, 1) [m(t), ) R by x T d (t,x) (x ξ) t f (t, ξ)dξ = 0 j d (t, x) := x T d (t, x) j u (t, x) := tf (t, T d (t, x)) t F (t, x). f (t, x) Martingale Transport, Skorokhod Embedding and

23 Technical Lemma Problem formulation Applications Lemma The functions j d and j u are both continuous in (t, x) and locally Lipschitz in x. Lemma We have the asymptotic estimates T ε u (t, x) = x + εj u (t, x) + O(ε 2 ), T ε d (t, x) = x j d(t, x) + O(ε). Martingale Transport, Skorokhod Embedding and

24 The limit of dual component Problem formulation Applications Dynamic strategy : h : [0, 1) R is defined by x h (t, x) := c x(x, x) c x (x, T d (t, x)), j d (t, x) x m(t), h (t, x) := h (t, T 1 d (t, x)) c y (T 1 d (t, x), x), x < m(t). Static strategy : let ψ and λ 0 be defined by x ψ (t, x) := h (t, x), ( λ 0 := t ψ + x ψ j u + (ψ ( ) ψ ( j d ( ) + c( j d ( )) j ) u 1 j x m(t). d the static strategy is given by ψ (1, x) ψ (0, x) λ 0(t, x)dt. Martingale Transport, Skorokhod Embedding and

25 Problem formulation Applications Theorem Let (π n ) n 1 be a sequence of partitions of [0, 1], and X n be the associated optimal Markov chain, then the law of X n converge to P M (µ), under which X is local Lévy process t X t = X 0 1 Xs >m(s)j d (s, X s ) ( dn s j u (s, X 0 j s )ds ), d where N is a pure jump process with predictable compensated process ju j d. Under further integrability conditions, we have = E P [ C(X ) ] = P (µ) = D (µ) = µ(λ ) 1 0 m(t) j u j d (t, x)c(x, x j d (t, x))f (t, x)dxdt. Martingale Transport, Skorokhod Embedding and

26 Robust hedging of variance swap Problem formulation Applications The payoff of variance swap : in discrete-time case n k=1 log2 X tk X tk 1 ; in continuous-time case Xt 2 d[x ] c t + 0<t 1 log 2 X t X t. Application of the main result with c(x, y) := log 2 (x/y), we find an optimal no-arbitrage bounds as well as the super-hedging strategies. Martingale Transport, Skorokhod Embedding and