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Analytics on conditional moment generating functions of stochastic volatility models

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  • Lilian Hu
  • Yue Kuen Kwok

Abstract

Moment generating functions (MGFs) of the terminal log-asset price and integrated variance of stochastic volatility models conditional on the terminal variance value are required in exact simulation algorithms and Fourier based algorithms for pricing European path dependent options. Broadie and Kaya (Exact simulation of stochastic volatility and other affine jump diffusion processes. Oper. Res., 2006, 54(2), 217–231) initiate the analytic derivation of conditional MGF of integrated variance of the Heston model based on related analytic results for the Bessel bridge. Kang et al. (Exact simulation of the Wishart multidimensional stochastic volatility model. Oper. Res., 2017, 65(5), 1190–1206) and Zeng et al. (Analytical solvability and exact simulation in models with affine stochastic volatility and Lévy jumps. Math. Finance, 2023, 33, 842–890) employ different techniques of measure changes to obtain the conditional MGFs of the Heston model, multidimensional Wishart stochastic volatility model, 4/2-model and Ornstein–Uhlenbeck-driven stochastic volatility model. In this paper, we develop systematic and comprehensive measure change techniques that provide effective derivation procedures for the associated conditional MGFs. We establish an interesting linkage between joint conditional MGFs and their unconditional counterparts. Interestingly, the conditional MGFs under the 4/2-model can be deduced from those under the Heston model via an appropriate measure change. Besides, we employ the partial transform method to derive conditional MGFs that go beyond the Heston-type stochastic volatility models.

Suggested Citation

  • Lilian Hu & Yue Kuen Kwok, 2025. "Analytics on conditional moment generating functions of stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 25(12), pages 1991-2007, December.
    • Handle: RePEc:taf:quantf:v:25:y:2025:i:12:p:1991-2007
    DOI: 10.1080/14697688.2025.2587079
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    References listed on IDEAS

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    1. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    2. 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.
    3. Li, Chenxu & Wu, Linjia, 2019. "Exact simulation of the Ornstein–Uhlenbeck driven stochastic volatility model," European Journal of Operational Research, Elsevier, vol. 275(2), pages 768-779.
    4. del Baño Rollin, Sebastian & Ferreiro-Castilla, Albert & Utzet, Frederic, 2010. "On the density of log-spot in the Heston volatility model," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 2037-2063, September.
    5. Cheridito, Patrick & Filipovic, Damir & Kimmel, Robert L., 2007. "Market price of risk specifications for affine models: Theory and evidence," Journal of Financial Economics, Elsevier, vol. 83(1), pages 123-170, January.
    6. Monique Jeanblanc & Marc Yor & Marc Chesney, 2009. "Mathematical Methods for Financial Markets," Springer Finance, Springer, number 978-1-84628-737-4, March.
    7. Rainer Schöbel & Jianwei Zhu, 1999. "Stochastic Volatility With an Ornstein–Uhlenbeck Process: An Extension," Review of Finance, European Finance Association, vol. 3(1), pages 23-46.
    8. Pingping Zeng & Ziqing Xu & Pingping Jiang & Yue Kuen Kwok, 2023. "Analytical solvability and exact simulation in models with affine stochastic volatility and Lévy jumps," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 842-890, July.
    9. Jan Baldeaux, 2012. "Exact Simulation Of The 3/2 Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(05), pages 1-13.
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