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Superreplication in stochastic volatility models and optimal stopping

Author

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  • RØdiger Frey

    (Swiss Banking Institute, University of Zurich, Zurich, Plattenstrasse 14, CH-8032 Zurich, Switzerland Manuscript)

Abstract

In this paper we discuss the superreplication of derivatives in a stochastic volatility model under the additional assumption that the volatility follows a bounded process. We characterize the value process of our superhedging strategy by an optimal-stopping problem in the context of the Black-Scholes model which is similar to the optimal stopping problem that arises in the pricing of American-type derivatives. Our proof is based on probabilistic arguments. We study the minimality of these superhedging strategies and discuss PDE-characterizations of the value function of our superhedging strategy. We illustrate our approach by examples and simulations.

Suggested Citation

  • RØdiger Frey, 2000. "Superreplication in stochastic volatility models and optimal stopping," Finance and Stochastics, Springer, vol. 4(2), pages 161-187.
    • Handle: RePEc:spr:finsto:v:4:y:2000:i:2:p:161-187
    Note: received: June 1998; final version received: April 1999
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    References listed on IDEAS

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    1. Rüdiger Frey & Carlos A. Sin, 1999. "Bounds on European Option Prices under Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 9(2), pages 97-116, April.
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    Cited by:

    1. Daniel Fernholz & Ioannis Karatzas, 2012. "Optimal arbitrage under model uncertainty," Papers 1202.2999, arXiv.org.
    2. Mykland, Per Aslak, 2019. "Combining statistical intervals and market prices: The worst case state price distribution," Journal of Econometrics, Elsevier, vol. 212(1), pages 272-285.
    3. Wei Chen, 2013. "G-Doob-Meyer Decomposition and its Application in Bid-Ask Pricing for American Contingent Claim Under Knightian Uncertainty," Papers 1401.0677, arXiv.org.
    4. Sebastian Herrmann & Johannes Muhle-Karbe, 2017. "Model uncertainty, recalibration, and the emergence of delta–vega hedging," Finance and Stochastics, Springer, vol. 21(4), pages 873-930, October.
    5. Joel Vanden, 2006. "Exact Superreplication Strategies for a Class of Derivative Assets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(1), pages 61-87.
    6. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Post-Print hal-03460952, HAL.
    7. Sebastian Herrmann & Johannes Muhle-Karbe & Frank Thomas Seifried, 2017. "Hedging with small uncertainty aversion," Finance and Stochastics, Springer, vol. 21(1), pages 1-64, January.
    8. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," SciencePo Working papers hal-03460952, HAL.
    9. Chenxu Li, 2016. "Bessel Processes, Stochastic Volatility, And Timer Options," Mathematical Finance, Wiley Blackwell, vol. 26(1), pages 122-148, January.
    10. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," SciencePo Working papers Main hal-03460952, HAL.
    11. A. Galichon & P. Henry-Labord`ere & N. Touzi, 2014. "A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options," Papers 1401.3921, arXiv.org.
    12. Huang, Haishi, 2010. "Convertible Bonds: Risks and Optimal Strategies," Bonn Econ Discussion Papers 07/2010, University of Bonn, Bonn Graduate School of Economics (BGSE).
    13. Sebastian Herrmann & Johannes Muhle-Karbe, 2017. "Model Uncertainty, Recalibration, and the Emergence of Delta-Vega Hedging," Papers 1704.04524, arXiv.org.
    14. Yuhong Xu, 2014. "Robust valuation and risk measurement under model uncertainty," Papers 1407.8024, arXiv.org.
    15. repec:spo:wpecon:info:hdl:2441/5rkqqmvrn4tl22s9mc0ck8ecp is not listed on IDEAS
    16. Sebastian Herrmann & Johannes Muhle-Karbe & Frank Thomas Seifried, 2016. "Hedging with Small Uncertainty Aversion," Papers 1605.06429, arXiv.org.
    17. repec:spo:wpmain:info:hdl:2441/5rkqqmvrn4tl22s9mc0ck8ecp is not listed on IDEAS
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    More about this item

    Keywords

    Stochastic volatility; optimal stopping; incomplete markets; superreplication;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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