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Convex order of discrete, continuous and predictable quadratic variation & applications to options on variance

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  • Martin Keller-Ressel
  • Claus Griessler

Abstract

We consider a square-integrable semimartingale and investigate the convex order relations between its discrete, continuous and predictable quadratic variation. As the main results, we show that if the semimartingale has conditionally independent increments and symmetric jump measure, then its discrete realized variance dominates its quadratic variation in increasing convex order. The results have immediate applications to the pricing of options on realized variance. For a class of models including time-changed Levy models and Sato processes with symmetric jumps our results show that options on variance are typically underpriced, if quadratic variation is substituted for the discretely sampled realized variance.

Suggested Citation

  • Martin Keller-Ressel & Claus Griessler, 2011. "Convex order of discrete, continuous and predictable quadratic variation & applications to options on variance," Papers 1103.2310, arXiv.org, revised Oct 2012.
    • Handle: RePEc:arx:papers:1103.2310
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    References listed on IDEAS

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    1. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    2. Mark Broadie & Ashish Jain, 2008. "The Effect Of Jumps And Discrete Sampling On Volatility And Variance Swaps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(08), pages 761-797.
    3. repec:dau:papers:123456789/1380 is not listed on IDEAS
    4. repec:dau:papers:123456789/1392 is not listed on IDEAS
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    Cited by:

    1. Carole Bernard & Zhenyu Cui, 2013. "Prices and Asymptotics for Discrete Variance Swaps," Papers 1305.7092, arXiv.org.
    2. Carole Bernard & Zhenyu Cui & Don McLeish, 2013. "Convergence of the discrete variance swap in time-homogeneous diffusion models," Papers 1310.0099, arXiv.org.
    3. Bernard, Carole & Jiang, Xiao & Wang, Ruodu, 2014. "Risk aggregation with dependence uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 93-108.

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