Research output: Contribution to journal › Article

Original language | English |
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Journal | Stochastic Processes and Their Applications |

DOIs | |

Accepted/In press | 26 Mar 2018 |

Published | 11 Apr 2018 |

**Pathwise superhedging_FORDE_Accepted 26 Mar 18_GREEN_AAM**Pathwise_superhedging_FORDE_Accepted_26_Mar_18_GREEN_AAM.pdf, 185 KB, application/pdf

Uploaded date:04 Apr 2018

Version:Accepted author manuscript

**Pathwise superhedging for time-dependent_FORDE_Accepted26March2018_GREEN AAM (CC BY-NC-ND)**Pathwise_superhedging_for_time_dependent_FORDE_Accepted26March2018_GREEN_AAM_CC_BY_NC_ND_.pdf, 748 KB, application/pdf

Uploaded date:13 Apr 2018

Version:Accepted author manuscript

Licence:CC BY-NC-ND

© <2017> This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode

We establish pathwise duality using simple predictable trading strategies for the robust hedging problem associated with a barrier option whose payoff depends on the terminal level and the infimum of a càdlàg strictly positive stock price process, given tradeable European options at all strikes at a single maturity. The result allows for a significant dimension reduction in the computation of the superhedging cost, via an alternate lower-dimensional formulation of the primal problem as a convex optimization problem, which is qualitatively similar to the duality which was formally sketched using linear programming arguments in Duembgen and Rogers [10] for the case where we only consider continuous sample paths. The proof exploits a simplification of a classical result by Rogers (1993) which characterizes the attainable joint laws for the supremum and the drawdown of a uniformly integrable martingale (not necessarily continuous), combined with classical convex duality results from Rockefellar (1974) using paired spaces with compatible locally convex topologies and the Hahn–Banach theorem. We later adapt this result to include additional tradeable One-Touch options using the Kertz and Rösler (1990) condition. We also compute the superhedging cost when in the more realistic situation where there is only finite tradeable European options; for this case we obtain the full duality in the sense of quantile hedging as in Soner (2015), where the superhedge works with probability where can be arbitrarily small), and we obtain an upper bound for the true pathwise superhedging cost. In Section 5, we extend our analysis to include time-dependent barrier options using martingale coupling arguments, where we now have tradeable European options at both maturities at all strikes and tradeable forward starting options at all strikes. This set up is designed to approximate the more realistic situation where we have a finite number of tradeable Europeans at both maturities plus a finite number of tradeable forward starting options.1

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