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Change of drift in one-dimensional diffusions

Author

Listed:
  • Sascha Desmettre

    (Johannes Kepler University Linz)

  • Gunther Leobacher

    (University of Graz)

  • L. C. G. Rogers

    (University of Cambridge)

Abstract

It is generally understood that a given one-dimensional diffusion may be transformed by a Cameron–Martin–Girsanov measure change into another one-dimensional diffusion with the same volatility but a different drift. But to achieve this, we have to know that the change-of-measure local martingale that we write down is a true martingale. We provide a complete characterisation of when this happens. This enables us to discuss the absence of arbitrage in a generalised Heston model including the case where the Feller condition for the volatility process is violated.

Suggested Citation

  • Sascha Desmettre & Gunther Leobacher & L. C. G. Rogers, 2021. "Change of drift in one-dimensional diffusions," Finance and Stochastics, Springer, vol. 25(2), pages 359-381, April.
    • Handle: RePEc:spr:finsto:v:25:y:2021:i:2:d:10.1007_s00780-021-00451-w
    DOI: 10.1007/s00780-021-00451-w
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    References listed on IDEAS

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    1. Hardy Hulley & Eckhard Platen, 2008. "A Visual Classification of Local Martingales," Research Paper Series 238, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Carole Bernard & Zhenyu Cui & Don McLeish, 2017. "On The Martingale Property In Stochastic Volatility Models Based On Time-Homogeneous Diffusions," Mathematical Finance, Wiley Blackwell, vol. 27(1), pages 194-223, January.
    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    4. Aleksandar Mijatović & Mikhail Urusov, 2012. "Deterministic criteria for the absence of arbitrage in one-dimensional diffusion models," Finance and Stochastics, Springer, vol. 16(2), pages 225-247, April.
    5. Guo, Zhi Jun, 2008. "A note on the CIR process and the existence of equivalent martingale measures," Statistics & Probability Letters, Elsevier, vol. 78(5), pages 481-487, April.
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    Cited by:

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    2. David Criens & Mikhail Urusov, 2022. "Separating Times for One-Dimensional Diffusions," Papers 2211.06042, arXiv.org, revised May 2023.
    3. David Criens & Mikhail Urusov, 2023. "Criteria for the absence of arbitrage in general diffusion markets," Papers 2306.11470, arXiv.org, revised Sep 2024.
    4. Yukihiro Tsuzuki, 2023. "Pitman's Theorem, Black-Scholes Equation, and Derivative Pricing for Fundraisers," Papers 2303.13956, arXiv.org.

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    More about this item

    Keywords

    One-dimensional diffusions; Change of measure; Heston model; Feller condition; Free lunch with vanishing risk;
    All these keywords.

    JEL classification:

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

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