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Comparison of Option Prices in Semimartingale Models

Author

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  • Jan Bergenthum

    (University of Freiburg)

  • Ludger Rüschendorf

    (University of Freiburg)

Abstract

In this paper we generalize the recent comparison results of El Karoui et al. (Math Finance 8:93–126, 1998), Bellamy and Jeanblanc (Finance Stoch 4:209–222, 2000) and Gushchin and Mordecki (Proc Steklov Inst Math 237:73–113, 2002) to d-dimensional exponential semimartingales. Our main result gives sufficient conditions for the comparison of European options with respect to martingale pricing measures. The comparison is with respect to convex and also with respect to directionally convex functions. Sufficient conditions for these orderings are formulated in terms of the predictable characteristics of the stochastic logarithm of the stock price processes. As examples we discuss the comparison of exponential semimartingales to multivariate diffusion processes, to stochastic volatility models, to Lévy processes, and to diffusions with jumps. We obtain extensions of several recent results on nontrivial price intervals. A crucial property in this approach is the propagation of convexity property. We develop a new approach to establish this property for several further examples of univariate and multivariate processes.

Suggested Citation

  • Jan Bergenthum & Ludger Rüschendorf, 2006. "Comparison of Option Prices in Semimartingale Models," Finance and Stochastics, Springer, vol. 10(2), pages 222-249, April.
    • Handle: RePEc:spr:finsto:v:10:y:2006:i:2:d:10.1007_s00780-006-0001-9
    DOI: 10.1007/s00780-006-0001-9
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    References listed on IDEAS

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    Cited by:

    1. Fabio Bellini & Franco Pellerey & Carlo Sgarra & Salimeh Yasaei Sekeh, 2012. "Comparison results for Garch processes," Papers 1204.3786, arXiv.org.
    2. Köpfer, Benedikt & Rüschendorf, Ludger, 2023. "Markov projection of semimartingales — Application to comparison results," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 361-386.
    3. Jonas Al-Hadad & Zbigniew Palmowski, 2020. "Perpetual American options with asset-dependent discounting," Papers 2007.09419, arXiv.org, revised Jan 2021.
    4. Breton, Jean-Christophe & Privault, Nicolas, 2024. "Wasserstein distance estimates for jump-diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    5. Liu, Yating & Pagès, Gilles, 2022. "Monotone convex order for the McKean–Vlasov processes," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 312-338.

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    More about this item

    Keywords

    Contingent claim valuation; Semimartingale model; Price orderings; Propagation of convexity;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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