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Robust pricing and hedging under trading restrictions and the emergence of local martingale models

  • Alexander M. G. Cox
  • Zhaoxu Hou
  • Jan Obloj
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    We consider the pricing of derivatives in a setting with trading restrictions, but without any probabilistic assumptions on the underlying model, in discrete and continuous time. In particular, we assume that European put or call options are traded at certain maturities, and the forward price implied by these option prices may be strictly decreasing in time. In discrete time, when call options are traded, the short-selling restrictions ensure no arbitrage, and we show that classical duality holds between the smallest super-replication price and the supremum over expectations of the payoff over all supermartingale measures. More surprisingly in the case where the only vanilla options are put options, we show that there is a duality gap. Embedding the discrete time model into a continuous time setup, we make a connection with (strict) local-martingale models, and derive framework and results often seen in the literature on financial bubbles. This connection suggests a certain natural interpretation of many existing results in the literature on financial bubbles.

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    Paper provided by in its series Papers with number 1406.0551.

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    Date of creation: Jun 2014
    Date of revision: Jun 2015
    Handle: RePEc:arx:papers:1406.0551
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    1. Arash Fahim & Yu-Jui Huang, 2014. "Model-independent Superhedging under Portfolio Constraints," Papers 1402.2599,, revised Jun 2015.
    2. Jarrow, Robert & Protter, Philip, 2012. "Discrete versus continuous time models: Local martingales and singular processes in asset pricing theory," Finance Research Letters, Elsevier, vol. 9(2), pages 58-62.
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    5. Mathias Beiglb\"ock & Pierre Henry-Labord\`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929,, revised Feb 2013.
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    8. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
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    14. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    15. Alexander Cox & Jan Obłój, 2011. "Robust pricing and hedging of double no-touch options," Finance and Stochastics, Springer, vol. 15(3), pages 573-605, September.
    16. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    17. Mark H. A. Davis & David G. Hobson, 2007. "The Range Of Traded Option Prices," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 1-14.
    18. Erhan Bayraktar & Zhou Zhou, 2014. "On Arbitrage and Duality under Model Uncertainty and Portfolio Constraints," Papers 1402.2596,, revised Mar 2015.
    19. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October.
    20. Hugonnier, Julien, 2012. "Rational asset pricing bubbles and portfolio constraints," Journal of Economic Theory, Elsevier, vol. 147(6), pages 2260-2302.
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