The Annals of Applied Probability

Model-free superhedging duality

Matteo Burzoni, Marco Frittelli, and Marco Maggis

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In a model-free discrete time financial market, we prove the superhedging duality theorem, where trading is allowed with dynamic and semistatic strategies. We also show that the initial cost of the cheapest portfolio that dominates a contingent claim on every possible path $\omega \in \Omega$, might be strictly greater than the upper bound of the no-arbitrage prices. We therefore characterize the subset of trajectories on which this duality gap disappears and prove that it is an analytic set.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 3 (2017), 1452-1477.

Dates
Received: June 2015
Revised: May 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1500451228

Digital Object Identifier
doi:10.1214/16-AAP1235

Mathematical Reviews number (MathSciNet)
MR3678476

Zentralblatt MATH identifier
1370.60004

Subjects
Primary: 60B05: Probability measures on topological spaces 60G42: Martingales with discrete parameter 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 28B20: Set-valued set functions and measures; integration of set-valued functions; measurable selections [See also 26E25, 54C60, 54C65, 91B14] 46A20: Duality theory 91B70: Stochastic models 91B24: Price theory and market structure

Keywords
Superhedging theorem model independent market model uncertainty robust duality finite support martingale measure analytic sets

Citation

Burzoni, Matteo; Frittelli, Marco; Maggis, Marco. Model-free superhedging duality. Ann. Appl. Probab. 27 (2017), no. 3, 1452--1477. doi:10.1214/16-AAP1235. https://projecteuclid.org/euclid.aoap/1500451228


Export citation

References

  • [1] Acciaio, B., Beiglböck, M., Penkner, F. and Schachermayer, W. (2016). A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance 26 233–251.
  • [2] Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed. Springer, Berlin.
  • [3] Beiglböck, M., Henry-Labordère, P. and Penkner, F. (2013). Model-independent bounds for option prices—A mass transport approach. Finance Stoch. 17 477–501.
  • [4] Bertsekas, D. P. and Shreve, S. E. (1978). Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering 139. Academic Press, New York.
  • [5] Bouchard, B. and Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25 823–859.
  • [6] Brown, H., Hobson, D. and Rogers, L. C. G. (2001). Robust hedging of barrier options. Math. Finance 11 285–314.
  • [7] Burzoni, M., Frittelli, M. and Maggis, M. (2016). Universal arbitrage aggregator in discrete-time markets under uncertainty. Finance Stoch. 20 1–50.
  • [8] Cox, A. M. G. and Obłój, J. (2011). Robust pricing and hedging of double no-touch options. Finance Stoch. 15 573–605.
  • [9] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520.
  • [10] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential B. Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam.
  • [11] Dolinsky, Y. and Soner, H. M. (2014). Martingale optimal transport and robust hedging in continuous time. Probab. Theory Related Fields 160 391–427.
  • [12] Dolinsky, Y. and Soner, H. M. (2015). Martingale optimal transport in the Skorokhod space. Stochastic Process. Appl. 125 3893–3931.
  • [13] El Karoui, N. and Quenez, M.-C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29–66.
  • [14] Galichon, A., Henry-Labordère, P. and Touzi, N. (2014). A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Probab. 24 312–336.
  • [15] Henry-Labordère, P., Obłój, J., Spoida, P. and Touzi, N. (2016). The maximum maximum of a martingale with given $n$ marginals. Ann. Appl. Probab. 26 1–44.
  • [16] Hobson, D. (2011). The Skorokhod embedding problem and model-independent bounds for option prices. In Paris-Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Math. 2003 267–318. Springer, Berlin.
  • [17] Hobson, D. G. (1998). Robust hedging of the lookback option. Finance Stoch. 2 329–347.
  • [18] Kabanov, Y. and Stricker, C. (2001). A teachers’ note on no-arbitrage criteria. In Séminaire de Probabilités, XXXV. Lecture Notes in Math. 1755 149–152. Springer, Berlin.
  • [19] Karatzas, I. (1997). Lectures on the Mathematics of Finance. CRM Monograph Series 8. Amer. Math. Soc., Providence, RI.
  • [20] Obłoj, J. and Hou, Z. (2015). On robust pricing-hedging duality in continuous time. Preprint.
  • [21] Riedel, F. (2015). Financial economics without probabilistic prior assumptions. Decis. Econ. Finance 38 75–91.
  • [22] Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer, Berlin.
  • [23] Tan, X. and Touzi, N. (2013). Optimal transportation under controlled stochastic dynamics. Ann. Probab. 41 3201–3240.