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Financial markets with volatility uncertainty

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  • Joerg Vorbrink

Abstract

We investigate financial markets under model risk caused by uncertain volatilities. For this purpose we consider a financial market that features volatility uncertainty. To have a mathematical consistent framework we use the notion of G-expectation and its corresponding G-Brownian motion recently introduced by Peng (2007). Our financial market consists of a riskless asset and a risky stock with price process modeled by a geometric G-Brownian motion. We adapt the notion of arbitrage to this more complex situation and consider stock price dynamics which exclude arbitrage opportunities. Due to volatility uncertainty the market is not complete any more. We establish the interval of no-arbitrage prices for general European contingent claims and deduce explicit results in a Markovian setting.

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  • Joerg Vorbrink, 2010. "Financial markets with volatility uncertainty," Papers 1012.1535, arXiv.org, revised Dec 2010.
    • Handle: RePEc:arx:papers:1012.1535
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    Cited by:

    1. Larry G. Epstein & Shaolin Ji, 2013. "Ambiguous Volatility and Asset Pricing in Continuous Time," The Review of Financial Studies, Society for Financial Studies, vol. 26(7), pages 1740-1786.
    2. Frank Riedel, 2011. "Finance Without Probabilistic Prior Assumptions," Papers 1107.1078, arXiv.org.
    3. Vorbrink, Jörg, 2014. "Financial markets with volatility uncertainty," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 64-78.
    4. Patrick Bei{ss}ner, 2012. "Coherent Price Systems and Uncertainty-Neutral Valuation," Papers 1202.6632, arXiv.org.
    5. Daniel Fernholz & Ioannis Karatzas, 2012. "Optimal arbitrage under model uncertainty," Papers 1202.2999, arXiv.org.
    6. Yuhong Xu, 2014. "Robust valuation and risk measurement under model uncertainty," Papers 1407.8024, arXiv.org.
    7. Beißner, Patrick, 2013. "Coherent Price Systems and Uncertainty-Neutral Valuation," VfS Annual Conference 2013 (Duesseldorf): Competition Policy and Regulation in a Global Economic Order 80010, Verein für Socialpolitik / German Economic Association.
    8. Frank Riedel, 2015. "Financial economics without probabilistic prior assumptions," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 38(1), pages 75-91, April.
    9. Li, Hanwu & Peng, Shige & Soumana Hima, Abdoulaye, 2018. "Reflected Solutions of BSDEs Driven by $\textit{G}$-Brownian Motion," Center for Mathematical Economics Working Papers 590, Center for Mathematical Economics, Bielefeld University.
    10. Beißner, Patrick, 2016. "Radner Equilibria under Ambiguous Volatility," Center for Mathematical Economics Working Papers 493, Center for Mathematical Economics, Bielefeld University.

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