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On martingale measures when asset returns have unpredictable jumps

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  • Bardhan, Indrajit
  • Chao, Xiuli

Abstract

We study financial market incompleteness induced by discontinuities in asset returns. When there are multiple outcomes for a discontinuity, it is shown that this incompleteness cannot be removed by the introduction of extra securities. Claims cannot be hedged and are thereby not uniquely priced by arbitrage. We characterize the family of martingale measures associated with this form of incompleteness and discuss issues of existence and uniqueness for important special cases. Finally, using methods of stochastic control, we apply these results to derive replicating policies for arbitrary contingent claims and thereby relate the prices of contingent claims to the family of measures.

Suggested Citation

  • Bardhan, Indrajit & Chao, Xiuli, 1996. "On martingale measures when asset returns have unpredictable jumps," Stochastic Processes and their Applications, Elsevier, vol. 63(1), pages 35-54, October.
    • Handle: RePEc:eee:spapps:v:63:y:1996:i:1:p:35-54
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    References listed on IDEAS

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    Cited by:

    1. Jin, Xing & Zhang, Kun, 2013. "Dynamic optimal portfolio choice in a jump-diffusion model with investment constraints," Journal of Banking & Finance, Elsevier, vol. 37(5), pages 1733-1746.
    2. Morten Christensen & Eckhard Platen, 2004. "A General Benchmark Model for Stochastic Jump Sizes," Research Paper Series 139, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Hong, Yi & Jin, Xing, 2018. "Semi-analytical solutions for dynamic portfolio choice in jump-diffusion models and the optimal bond-stock mix," European Journal of Operational Research, Elsevier, vol. 265(1), pages 389-398.
    4. Jérôme Detemple, 2014. "Portfolio Selection: A Review," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 1-21, April.
    5. Ralf Korn & Frank Oertel & Manfred Schäl, 2003. "Notes and Comments: The numeraire portfolio in financial markets modeled by a multi-dimensional jump diffusion process," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 26(2), pages 153-166, November.
    6. Carla Mereu & Robert Stelzer, 2015. "A BSDE arising in an exponential utility maximization problem in a pure jump market model," Papers 1508.07561, arXiv.org, revised Jan 2016.
    7. Anne Eyraud-Loisel, 2005. "Backward stochastic differential equations with enlarged filtration: Option hedging of an insider trader in a financial market with jumps," Post-Print hal-01298905, HAL.
    8. Carverhill, Andrew & Luo, Dan, 2023. "A Bayesian analysis of time-varying jump risk in S&P 500 returns and options," Journal of Financial Markets, Elsevier, vol. 64(C).
    9. Xing Jin & Xudong Zeng, 2018. "Dynamic Asset Allocation with Uncertain Jump Risks: A Pathwise Optimization Approach," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 347-376, May.

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