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Market completion using options

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  • Mark Davis
  • Jan Obloj

Abstract

Mathematical models for financial asset prices which include, for example, stochastic volatility or jumps are incomplete in that derivative securities are generally not replicable by trading in the underlying. In earlier work (2004) the first author provided a geometric condition under which trading in the underlying and a finite number of vanilla options completes the market. We complement this result in several ways. First, we show that the geometric condition is not necessary and a weaker, necessary and sufficient, condition is presented. While this condition is generally not directly verifiable, we show that it simplifies to matrix non-degeneracy in a single point when the pricing functions are real analytic functions. In particular, any stochastic volatility model is then completed with an arbitrary European type option. Further, we show that adding path-dependent options such as a variance swap to the set of primary assets, instead of plain vanilla options, also completes the market.

Suggested Citation

  • Mark Davis & Jan Obloj, 2007. "Market completion using options," Papers 0710.2792, arXiv.org, revised Oct 2008.
    • Handle: RePEc:arx:papers:0710.2792
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    References listed on IDEAS

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    1. Marc Romano & Nizar Touzi, 1997. "Contingent Claims and Market Completeness in a Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 399-412, October.
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    Cited by:

    1. Sebastian Herrmann & Johannes Muhle-Karbe, 2017. "Model Uncertainty, Recalibration, and the Emergence of Delta-Vega Hedging," Papers 1704.04524, arXiv.org.
    2. Dmitry Kramkov & Sergio Pulido, 2019. "Density of the set of probability measures with the martingale representation property," Post-Print hal-01598651, HAL.
    3. Falko Baustian & Katev{r}ina Filipov'a & Jan Posp'iv{s}il, 2019. "Solution of option pricing equations using orthogonal polynomial expansion," Papers 1912.06533, arXiv.org, revised Jun 2020.
    4. Patrick Beissner & Frank Riedel, 2018. "Non-implementability of Arrow–Debreu equilibria by continuous trading under volatility uncertainty," Finance and Stochastics, Springer, vol. 22(3), pages 603-620, July.
    5. Dmitry Kramkov & Sergio Pulido, 2017. "Density of the set of probability measures with the martingale representation property," Papers 1709.07329, arXiv.org, revised Jul 2019.
    6. Kramkov, Dmitry & Predoiu, Silviu, 2014. "Integral representation of martingales motivated by the problem of endogenous completeness in financial economics," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 81-100.
    7. Samuel N. Cohen & Christoph Reisinger & Sheng Wang, 2021. "Arbitrage-free neural-SDE market models," Papers 2105.11053, arXiv.org, revised Aug 2021.
    8. 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.
    9. Jan Obłój & Thaleia Zariphopoulou, 2021. "In memoriam: Mark H. A. Davis and his contributions to mathematical finance," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1099-1110, October.
    10. Dmitry Kramkov & Sergio Pulido, 2017. "Density of the set of probability measures with the martingale representation property," Working Papers hal-01598651, HAL.
    11. Alziary Chassat, Bénédicte & Takac, Peter, 2017. "On the Heston Model with Stochastic Volatility: Analytic Solutions and Complete Markets," TSE Working Papers 17-796, Toulouse School of Economics (TSE).
    12. Jean Jacod & Philip Protter, 2010. "Risk-neutral compatibility with option prices," Finance and Stochastics, Springer, vol. 14(2), pages 285-315, April.
    13. Mehdi El Amrani & Antoine Jacquier & Claude Martini, 2019. "Dynamics of symmetric SSVI smiles and implied volatility bubbles," Papers 1909.10272, arXiv.org, revised Feb 2021.

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