Monotone martingale transport plans and Skorokhod embedding

We show that the left-monotone martingale coupling is optimal for any given performance function satisfying the martingale version of the Spence-Mirrlees condition, without assuming additional structural conditions on the marginals. We also give a new interpretation of the left monotone coupling in terms of Skorokhod embedding which allows us to give a short proof of uniqueness.

[1]  Mathias Beiglböck,et al.  Model-independent bounds for option prices—a mass transport approach , 2011, Finance Stochastics.

[2]  L. Kantorovich.  On the Translocation of Masses , 2006 .

[3]  A. M. G. Cox.  Extending Chacon-Walsh: Minimality and Generalised Starting Distributions , 2005 .

[4]  R. M. Loynes,et al.  Stopping times on Brownian motion: Some properties of root's construction , 1970 .

[5]  Jan Ob lój.  The Skorokhod embedding problem and its offspring ∗ , 2004 .

[6]  Nizar Touzi,et al.  A Stochastic Control Approach to No-Arbitrage Bounds Given Marginals, with an Application to Lookback Options , 2013, 1401.3921.

[7]  Martingale Inequalities for the Maximum via Pathwise Arguments , 2014, 1409.6255.

[8]  N. Touzi,et al.  The maximum maximum of a martingale with given $\mathbf{n}$ marginals , 2012, 1203.6877.

[9]  Claus Griessler,et al.  An optimality principle with applications in optimal transport , 2014, 1404.7054.

[10]  R. Rockafellar.  Characterization of the subdifferentials of convex functions , 1966 .

[11]  L. Ambrosio,et al.  A User’s Guide to Optimal Transport , 2013 .

[12]  Mathias Beiglböck,et al.  Optimal transport and Skorokhod embedding , 2013, Inventiones mathematicae.

[13]  J. Obłój.  The Skorokhod embedding problem and its offspring , 2004, math/0401114.

[14]  A. Galichon,et al.  A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options , 2014, 1401.3921.

[15]  H. Soner,et al.  Martingale optimal transport and robust hedging in continuous time , 2012, 1208.4922.

[16]  M. Beiglbock,et al.  On a problem of optimal transport under marginal martingale constraints , 2012, 1208.1509.

[17]  D. A. Edwards.  On the existence of probability measures with given marginals , 1978 .

[18]  On the Monge–Kantorovich problem with additional linear constraints , 2014, 1404.4962.

[19]  Nizar Touzi,et al.  An explicit martingale version of the one-dimensional Brenier theorem , 2016, Finance Stochastics.

[20]  Dirk Helbing,et al.  Modelling and Optimisation of Flows on Networks , 2013 .

[21]  Itrel Monroe,et al.  On Embedding Right Continuous Martingales in Brownian Motion , 1972 .

[22]  D. Hobson.  The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices , 2011 .

[23]  L. Kantorovitch,et al.  On the Translocation of Masses , 1958 .

[24]  R. McCann.  A convexity theory for interacting gases and equilibrium crystals , 1994 .

[25]  Bruno Bouchard,et al.  Arbitrage and duality in nondominated discrete-time models , 2013, 1305.6008.

[26]  G. Burton.  TOPICS IN OPTIMAL TRANSPORTATION (Graduate Studies in Mathematics 58) By CÉDRIC VILLANI: 370 pp., US$59.00, ISBN 0-8218-3312-X (American Mathematical Society, Providence, RI, 2003) , 2004 .

[27]  Nizar Touzi,et al.  Maximum Maximum of Martingales Given Marginals , 2013 .

[28]  N. Touzi,et al.  An Explicit Martingale Version of Brenier's Theorem , 2013, 1302.4854.

[29]  Arthur Cayley,et al.  The Collected Mathematical Papers: On Monge's “Mémoire sur la théorie des déblais et des remblais” , 2009 .

[30]  N. Touzi,et al.  The maximum maximum of a martingale with given $n$ marginals , 2016 .

[31]  Nizar Touzi,et al.  Complete Duality for Martingale Optimal Transport on the Line , 2015 .

[32]  David Hobson,et al.  ROBUST BOUNDS FOR FORWARD START OPTIONS , 2012 .

[33]  Данила Андреевич Заев,et al.  О задаче Монжа - Канторовича с дополнительными линейными ограничениями@@@On the Monge - Kantorovich Problem with Additional Linear Constraints , 2015 .

[34]  Martin Klimmek,et al.  Model-independent hedging strategies for variance swaps , 2012, Finance Stochastics.

[35]  Claude Martini,et al.  Change of numeraire in the two-marginals martingale transport problem , 2017, Finance Stochastics.