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On changes of measure in stochastic volatility models

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  • Bernard Wong
  • C. C. Heyde

Abstract

Pricing in mathematical finance often involves taking expected values under different equivalent measures. Fundamentally, one needs to first ensure the existence of ELMM, which in turn requires that the stochastic exponential of the market price of risk process be a true martingale. In general, however, this condition can be hard to validate, especially in stochastic volatility models. This had led many researchers to “assume the condition away,” even though the condition is not innocuous, and nonsensical results can occur if it is in fact not satisfied. We provide an applicable theorem to check the conditions for a general class of Markovian stochastic volatility models. As an example we will also provide a detailed analysis of the Stein and Stein and Heston stochastic volatility models.

Suggested Citation

  • Bernard Wong & C. C. Heyde, 2006. "On changes of measure in stochastic volatility models," International Journal of Stochastic Analysis, Hindawi, vol. 2006, pages 1-13, December.
    • Handle: RePEc:hin:jnijsa:018130
    DOI: 10.1155/JAMSA/2006/18130
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    Cited by:

    1. Mbodji, O.S. & Nguyen-Huu, A. & Pirvu, T.A., 2019. "Optimal sharing rule for a household with a portfolio management problem," Mathematical Social Sciences, Elsevier, vol. 101(C), pages 88-98.
    2. Hyungbin Park, 2018. "Sensitivity analysis of long-term cash flows," Finance and Stochastics, Springer, vol. 22(4), pages 773-825, October.
    3. Andrey Itkin, 2023. "The ATM implied skew in the ADO-Heston model," Papers 2309.15044, arXiv.org.
    4. Peter Carr & Andrey Itkin, 2019. "ADOL - Markovian approximation of rough lognormal model," Papers 1904.09240, arXiv.org.
    5. Torricelli, Lorenzo, 2020. "Trade duration risk in subdiffusive financial models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    6. Hyungbin Park & Heejun Yeo, 2022. "Dynamic and static fund separations and their stability for long-term optimal investments," Papers 2212.00391, arXiv.org, revised Mar 2023.
    7. Simon Ellersgaard & Martin Tegnér, 2018. "Stochastic volatility for utility maximizers — A martingale approach," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(01), pages 1-39, March.
    8. Sascha Desmettre, 2018. "Change of Measure in the Heston Model given a violated Feller Condition," Papers 1809.10955, arXiv.org, revised Oct 2019.
    9. Crosby, John & Frau, Carme, 2022. "Jumps in commodity prices: New approaches for pricing plain vanilla options," Energy Economics, Elsevier, vol. 114(C).
    10. Damiano Brigo & Federico Graceffa & Alexander Kalinin, 2021. "Mild to classical solutions for XVA equations under stochastic volatility," Papers 2112.11808, arXiv.org.

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