IDEAS home Printed from https://ideas.repec.org/a/wly/apsmbi/v24y2008i6p507-524.html
   My bibliography  Save this article

Bayesian parameter inference for models of the Black and Scholes type

Author

Listed:
  • Henryk Gzyl
  • Enrique ter Horst
  • Samuel W. Malone

Abstract

In this paper, we describe a general method for constructing the posterior distribution of the mean and volatility of the return of an asset satisfying dS=SdX for some simple models of X. Our framework takes as inputs the prior distributions of the parameters of the stochastic process followed by the underlying, as well as the likelihood function implied by the observed price history for the underlying. As an application of our framework, we compute the value at risk (VaR) and conditional VaR (CVaR) measures for the changes in the price of an option implied by the posterior distribution of the volatility of the underlying. The implied VaR and CVaR are more conservative than their classical counterpart, since it takes into account the estimation risk that arises due to parameter uncertainty. Copyright © 2008 John Wiley & Sons, Ltd.

Suggested Citation

  • Henryk Gzyl & Enrique ter Horst & Samuel W. Malone, 2008. "Bayesian parameter inference for models of the Black and Scholes type," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 24(6), pages 507-524, November.
    • Handle: RePEc:wly:apsmbi:v:24:y:2008:i:6:p:507-524
    DOI: 10.1002/asmb.709
    as

    Download full text from publisher

    File URL: https://doi.org/10.1002/asmb.709
    Download Restriction: no
    File URL: https://libkey.io/10.1002/asmb.709?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Karolyi, G. Andrew, 1993. "A Bayesian Approach to Modeling Stock Return Volatility for Option Valuation," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(4), pages 579-594, December.
    2. Darsinos, T. & Satchell, S.E., 2001. "Bayesian Analysis of the Black-Scholes Option Price," Cambridge Working Papers in Economics 0102, Faculty of Economics, University of Cambridge.
    3. Wang, Shaun S. & Young, Virginia R. & Panjer, Harry H., 1997. "Axiomatic characterization of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 173-183, November.
    4. repec:bla:jfinan:v:59:y:2004:i:1:p:407-446 is not listed on IDEAS
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Darsinos, T. & Satchell, S.E., 2001. "Bayesian Forecasting of Options Prices: A Natural Framework for Pooling Historical and Implied Volatiltiy Information," Cambridge Working Papers in Economics 0116, Faculty of Economics, University of Cambridge.
    2. Cangrejo Esquivel, Álvaro Javier & Tovar Cuevas, José Rafael & García, Isabel Cristina & Manotas Duque, Diego Fernando, 2022. "Estimación clásica y bayesiana de la volatilidad en el modelo de Black-Scholes [Classical and Bayesian estimation of volatility in the Black-Scholes model]," Revista de Métodos Cuantitativos para la Economía y la Empresa = Journal of Quantitative Methods for Economics and Business Administration, Universidad Pablo de Olavide, Department of Quantitative Methods for Economics and Business Administration, vol. 34(1), pages 237-262, December .
    3. Shu Wing Ho & Alan Lee & Alastair Marsden, 2011. "Use of Bayesian Estimates to determine the Volatility Parameter Input in the Black-Scholes and Binomial Option Pricing Models," JRFM, MDPI, vol. 4(1), pages 1-23, December.
    4. Khaledi, Baha-Eldin & Shaked, Moshe, 2010. "Stochastic comparisons of multivariate mixtures," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2486-2498, November.
    5. Furman, Edward & Landsman, Zinoviy, 2010. "Multivariate Tweedie distributions and some related capital-at-risk analyses," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 351-361, April.
    6. Schumacher Johannes M., 2018. "Distortion risk measures, ROC curves, and distortion divergence," Statistics & Risk Modeling, De Gruyter, vol. 35(1-2), pages 35-50, January.
    7. Nan Zhang & Heng Xu, 2024. "Fairness of Ratemaking for Catastrophe Insurance: Lessons from Machine Learning," Information Systems Research, INFORMS, vol. 35(2), pages 469-488, June.
    8. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    9. Pesenti, Silvana M. & Tsanakas, Andreas & Millossovich, Pietro, 2018. "Euler allocations in the presence of non-linear reinsurance: Comment on Major (2018)," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 29-31.
    10. Chen, Zengjing & Kulperger, Reg, 2006. "Minimax pricing and Choquet pricing," Insurance: Mathematics and Economics, Elsevier, vol. 38(3), pages 518-528, June.
    11. Ghossoub, Mario & Zhu, Michael B., 2024. "Stackelberg equilibria with multiple policyholders," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 189-201.
    12. Negi, Shekhar Singh & Torra, Vicenç, 2022. "Δ-Choquet integral on time scales with applications," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    13. Wentao Hu & Cuixia Chen & Yufeng Shi & Ze Chen, 2022. "A Tail Measure With Variable Risk Tolerance: Application in Dynamic Portfolio Insurance Strategy," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 831-874, June.
    14. Massimiliano Barbi & Silvia Romagnoli, 2016. "Optimal hedge ratio under a subjective re-weighting of the original measure," Applied Economics, Taylor & Francis Journals, vol. 48(14), pages 1271-1280, March.
    15. Nendel, Max & Streicher, Jan, 2023. "An axiomatic approach to default risk and model uncertainty in rating systems," Journal of Mathematical Economics, Elsevier, vol. 109(C).
    16. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    17. Freddy Delbaen, 2021. "Commonotonicity and time-consistency for Lebesgue-continuous monetary utility functions," Finance and Stochastics, Springer, vol. 25(3), pages 597-614, July.
    18. Leitner, Johannes, 2005. "Dilatation monotonous Choquet integrals," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 994-1006, December.
    19. Asimit, Alexandru V. & Bignozzi, Valeria & Cheung, Ka Chun & Hu, Junlei & Kim, Eun-Seok, 2017. "Robust and Pareto optimality of insurance contracts," European Journal of Operational Research, Elsevier, vol. 262(2), pages 720-732.
    20. Birghila, Corina & Pflug, Georg Ch., 2019. "Optimal XL-insurance under Wasserstein-type ambiguity," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 30-43.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:apsmbi:v:24:y:2008:i:6:p:507-524. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://doi.org/10.1002/(ISSN)1526-4025 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.