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Combinatorial implications of nonlinear uncertain volatility models: the case of barrier options

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  • Marco Avellaneda
  • Robert Buff

Abstract

Extensions to the Black-Scholes model have been suggested recently that permit one to calculate worst-case prices for a portfolio of vanilla options or for exotic options when no a priori distribution for the forward volatility is known. The Uncertain Volatility Model (UVM) by Avellaneda and Paras finds a one-sided worstcase volatility scenario for the buy resp. sell side within a specified volatility range. A key feature of this approach is the possibility of hedging with options: risk cancellation leads to super resp. sub-additive portfolio values. This nonlinear behaviour causes the combinatorial complexity of the pricing problem to increase significantly in the case of barrier options. In the paper, it is shown that for a portfolio P of n barrier options and any number of vanilla options, the number of PDEs that have to be solved in a hierarchical manner in order to solve the UVM problem for P is bounded by O (n2). A numerically stable implementation is described and numerical results are given.

Suggested Citation

  • Marco Avellaneda & Robert Buff, 1999. "Combinatorial implications of nonlinear uncertain volatility models: the case of barrier options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(1), pages 1-18.
    • Handle: RePEc:taf:apmtfi:v:6:y:1999:i:1:p:1-18
    DOI: 10.1080/135048699334582
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    References listed on IDEAS

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    1. G. O. Roberts & C. F. Shortland, 1997. "Pricing Barrier Options with Time–Dependent Coefficients," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 83-93, January.
    2. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    3. Peter Carr & Katrina Ellis & Vishal Gupta, 1998. "Static Hedging of Exotic Options," Journal of Finance, American Finance Association, vol. 53(3), pages 1165-1190, June.
    4. Hélyette Geman & Marc Yor, 1996. "Pricing And Hedging Double‐Barrier Options: A Probabilistic Approach," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 365-378, October.
    5. Marco Avellaneda & Antonio ParAS, 1996. "Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(1), pages 21-52.
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    Cited by:

    1. Lu, Xiaoping & Putri, Endah R.M., 2020. "A semi-analytic valuation of American options under a two-state regime-switching economy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).
    2. Max Nendel, 2021. "Markov chains under nonlinear expectation," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 474-507, January.
    3. 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.
    4. 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.
    5. Sebastian Herrmann & Johannes Muhle-Karbe, 2017. "Model Uncertainty, Recalibration, and the Emergence of Delta-Vega Hedging," Papers 1704.04524, arXiv.org.
    6. Sebastian Herrmann & Johannes Muhle-Karbe & Frank Thomas Seifried, 2016. "Hedging with Small Uncertainty Aversion," Papers 1605.06429, arXiv.org.

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