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Robust pricing and hedging of double no-touch options

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Abstract

Double no-touch options are contracts which pay out a fixed amount provided an underlying asset remains within a given interval. In this work, we establish model-independent bounds on the price of these options based on the prices of more liquidly traded options (call and digital call options). Key steps are the construction of super- and sub-hedging strategies to establish the bounds, and the use of Skorokhod embedding techniques to show the bounds are the best possible.

In addition to establishing rigorous bounds, we consider carefully what is meant by arbitrage in settings where there is no a priori known probability measure. We discuss two natural extensions of the notion of arbitrage, weak arbitrage and weak free lunch with vanishing risk, which are needed to establish equivalence between the lack of arbitrage and the existence of a market model.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Alexander M. G. Cox

  2. Mathematical Institute and Oxford-Man Institute of Quantitative Finance, University of Oxford, 24-29 St. Giles, Oxford, OX1 3LB, UK

    Jan Obłój

Authors
  1. Alexander M. G. Cox
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  2. Jan Obłój
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Correspondence to Jan Obłój.

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Cox, A.M.G., Obłój, J. Robust pricing and hedging of double no-touch options. Finance Stoch 15, 573–605 (2011). https://doi.org/10.1007/s00780-011-0154-z

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  • DOI: https://doi.org/10.1007/s00780-011-0154-z

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