Abstract
By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge–Kantorovich mass transport problem was introduced in (Beiglböck et al. in Finance Stoch. 17:477–501, 2013; Galichon et al. in Ann. Appl. Probab. 24:312–336, 2014). Further, by suitable adaptation of the notion of cyclical monotonicity, Beiglböck and Juillet (Ann. Probab. 44:42–106, 2016) obtained an extension of the one-dimensional Brenier theorem to the present martingale version. In this paper, we complement the previous work by extending the so-called Spence–Mirrlees condition to the case of martingale optimal transport. Under some technical conditions on the starting and the target measures, we provide an explicit characterization of the corresponding optimal martingale transference plans both for the lower and upper bounds. These explicit extremal probability measures coincide with the unique left- and right-monotone martingale transference plans introduced in (Beiglböck and Juillet in Ann. Probab. 44:42–106, 2016). Our approach relies on the (weak) duality result stated in (Beiglböck et al. in Finance Stoch. 17:477–501, 2013), and provides as a by-product an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.
Similar content being viewed by others
References
Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W., Temme, J.: A trajectorial interpretation of Doob’s martingale inequalities. Ann. Appl. Probab. 23, 1494–1505 (2013)
Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance 26, 233–251 (2016)
Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices—a mass transport approach. Finance Stoch. 17, 477–501 (2013)
Beiglböck, M., Juillet, N.: On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44, 42–106 (2016)
Breeden, D.T., Litzenberger, R.H.: Prices of state-contingent claims implicit in options prices. J. Bus. 51, 621–651 (1978)
Brenier, Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sci. Paris, Ser. I Math. 305(19), 805–808 (1987)
Brown, H., Hobson, D., Rogers, L.C.G.: Robust hedging of barrier options. Math. Finance 11, 285–314 (2001)
Carlier, G.: On a class of multidimensional optimal transportation problems. J. Convex Anal. 10, 517–529 (2003)
Cousot, L.: Necessary and sufficient conditions for no static arbitrage among European calls. Courant Institute, New York University (2004). Available online at: http://www.en.affi.asso.fr/uploads/Externe/04/CTR_FICHIER_137_1226317002.pdf
Cousot, L.: Conditions on option prices for absence of arbitrage and exact calibration. J. Bank. Finance 31, 3377–3397 (2007)
Cox, A.M.G., Hobson, D., Obłój, J.: Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping. Ann. Appl. Probab. 18, 1870–1896 (2008)
Cox, A.M.G., Obłój, J.: Robust hedging of double touch barrier options. SIAM J. Financ. Math. 2, 141–182 (2011)
Cox, A.M.G., Obłój, J.: Robust pricing and hedging of double no-touch options. Finance Stoch. 15, 573–605 (2011)
Cox, A.M.G., Wang, J.: Root’s barrier: construction, optimality and applications to variance options. Ann. Appl. Probab. 23, 859–894 (2013)
d’Aspremont, A., El Ghaoui, L.: Static arbitrage bounds on basket option prices. Math. Program., Ser. A 106, 467–489 (2006)
Davis, M.H.A., Hobson, D.: The range of traded option prices. Math. Finance 17, 1–14 (2007)
Davis, M.H.A., Obłój, J., Raval, V.: Arbitrage bounds for prices of options on realized variance. Math. Finance 24, 821–854 (2014)
Dolinsky, Y., Soner, H.M.: Robust hedging and martingale optimal transport in continuous time. Probab. Theory Relat. Fields 160, 391–427 (2014)
Dolinsky, Y., Soner, H.M.: Robust hedging under proportional transaction costs. Finance Stoch. 18, 327–347 (2014)
Dubins, L.E., Schwarz, G.: On extremal martingale distributions. In: Le Cam, L.M., Neyman, J. (eds.) Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2: Contributions to Probability Theory, Part 1, pp. 295–299. University of California Press, Berkeley (1967)
Galichon, A., Henry-Labordère, P., Touzi, N.: A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Probab. 24, 312–336 (2014)
Gangbo, W., Świȩch, A.: Optimal maps for the multidimensional Monge–Kantorovich problem. Commun. Pure Appl. Math. 51, 23–45 (1998)
Henry-Labordère, P., Obłój, J., Spoida, P., Touzi, N.: The maximum maximum of martingales given marginals. Ann. Appl. Probab. 26, 1–44 (2016)
Hobson, D.: Robust hedging of the lookback option. Finance Stoch. 2, 329–347 (1998)
Hobson, D.: The Skorokhod embedding problem and model-independent bounds for option prices. In: Carmona, R.A., et al. (eds.) Paris–Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Mathematics, vol. 2003, pp. 267–318. Springer, Berlin (2011)
Hobson, D., Klimmek, M.: Model-independent hedging strategies for variance swaps. Finance Stoch. 16, 611–649 (2012)
Hobson, D., Neuberger, A.: Robust bounds for forward start options. Math. Finance 22, 31–56 (2012)
Hobson, D., Pedersen, J.L.: The minimum maximum of a continuous martingale with given initial and terminal laws. Ann. Probab. 30, 978–999 (2002)
Jacod, J., Shiryaev, A.N.: Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance Stoch. 2, 259–273 (1998)
Jacod, J., Yor, M.: Etude des solutions extrémales et représentation intégrale des solutions pour certains problèmes de martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb. 38, 83–125 (1977)
Kahalé, N.: Model-independent lower bound on variance swaps. Math. Finance. (2016, to appear). Available online at: http://onlinelibrary.wiley.com/doi/10.1111/mafi.12083/full
Laurence, P., Wang, T.H.: Sharp upper and lower bounds for basket options. Appl. Math. Finance 12, 253–282 (2005)
Oberhauser, H., dos Reis, G., Gassiat, P.: Root’s barrier, viscosity solutions of obstacle problems and reflected FBSDEs. Stoch. Process. Appl. 125, 4601–4631 (2015)
Obłój, J.: The Skorokhod embedding problem and its offspring. Probab. Surv. 1, 321–392 (2004)
Pass, B.: Uniqueness and Monge solutions in the multi-marginal optimal transportation problem. SIAM J. Math. Anal. 43, 2758–2775 (2011)
Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems, Vol. I: Theory, Vol. II: Applications. Probability and Its Applications. Springer, New York (1988)
Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965)
Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. AMS, Providence (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors are grateful to Mathias Beiglböck and Xiaolu Tan for fruitful comments, and for pointing out subtle gaps in a previous version. This work benefits from the financial support of the ERC Advanced Grant 321111, and the Chairs Financial Risk and Finance and Sustainable Development.
Rights and permissions
About this article
Cite this article
Henry-Labordère, P., Touzi, N. An explicit martingale version of the one-dimensional Brenier theorem. Finance Stoch 20, 635–668 (2016). https://doi.org/10.1007/s00780-016-0299-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00780-016-0299-x