IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v12y2024i9p136-d1465277.html
   My bibliography  Save this article

Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model

Author

Listed:
  • W. Brent Lindquist

    (Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-4012, USA)

  • Svetlozar T. Rachev

    (Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-4012, USA)

  • Jagdish Gnawali

    (Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-4012, USA)

  • Frank J. Fabozzi

    (Carey Business School, Johns Hopkins University, Baltimore, MD 21202, USA)

Abstract

We present a unified, market-complete model that integrates both Bachelier and Black–Scholes–Merton frameworks for asset pricing. The model allows for the study, within a unified framework, of asset pricing in a natural world that experiences the possibility of negative security prices or riskless rates. Unlike the classical Black–Scholes–Merton, we show that option pricing in the unified model differs depending on whether the replicating, self-financing portfolio uses riskless bonds or a single riskless bank account. We derive option price formulas and extend our analysis to the term structure of interest rates by deriving the pricing of zero-coupon bonds, forward contracts, and futures contracts. We identify a necessary condition for the unified model to support a perpetual derivative. Discrete binomial pricing under the unified model is also developed. In every scenario analyzed, we show that the unified model simplifies to the standard Black–Scholes–Merton pricing under specific limits and provides pricing in the Bachelier model limit. We note that the Bachelier limit within the unified model allows for positive riskless rates. The unified model prompts us to speculate on the possibility of a mixed multiplicative and additive deflator model for risk-neutral option pricing.

Suggested Citation

  • W. Brent Lindquist & Svetlozar T. Rachev & Jagdish Gnawali & Frank J. Fabozzi, 2024. "Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model," Risks, MDPI, vol. 12(9), pages 1-23, August.
    • Handle: RePEc:gam:jrisks:v:12:y:2024:i:9:p:136-:d:1465277
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/12/9/136/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-9091/12/9/136/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    2. Robert Brooks & Joshua A. Brooks, 2017. "An Option Valuation Framework Based On Arithmetic Brownian Motion: Justification And Implementation Issues," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 40(3), pages 401-427, September.
    3. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    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. Jaehyuk Choi & Minsuk Kwak & Chyng Wen Tee & Yumeng Wang, 2022. "A Black–Scholes user's guide to the Bachelier model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(5), pages 959-980, May.
    2. Jaehyuk Choi & Minsuk Kwak & Chyng Wen Tee & Yumeng Wang, 2021. "A Black-Scholes user's guide to the Bachelier model," Papers 2104.08686, arXiv.org, revised Feb 2022.
    3. W. Brent Lindquist & Svetlozar T. Rachev & Jagdish Gnawali & Frank J. Fabozzi, 2024. "Dynamic Asset Pricing in a Unified Bachelier-Black-Scholes-Merton Model," Papers 2405.12479, arXiv.org, revised Jun 2024.
    4. Kau, James B. & Keenan, Donald C., 1999. "Patterns of rational default," Regional Science and Urban Economics, Elsevier, vol. 29(6), pages 765-785, November.
    5. Carol Alexandra & Leonardo M. Nogueira, 2005. "Optimal Hedging and Scale Inavriance: A Taxonomy of Option Pricing Models," ICMA Centre Discussion Papers in Finance icma-dp2005-10, Henley Business School, University of Reading, revised Nov 2005.
    6. Weihan Li & Jin E. Zhang & Xinfeng Ruan & Pakorn Aschakulporn, 2024. "An empirical study on the early exercise premium of American options: Evidence from OEX and XEO options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(7), pages 1117-1153, July.
    7. Jun, Doobae & Ku, Hyejin, 2015. "Static hedging of chained-type barrier options," The North American Journal of Economics and Finance, Elsevier, vol. 33(C), pages 317-327.
    8. Thomas Kokholm & Martin Stisen, 2015. "Joint pricing of VIX and SPX options with stochastic volatility and jump models," Journal of Risk Finance, Emerald Group Publishing Limited, vol. 16(1), pages 27-48, January.
    9. Vorst, A. C. F., 1988. "Option Pricing And Stochastic Processes," Econometric Institute Archives 272366, Erasmus University Rotterdam.
    10. Antoine Jacquier & Patrick Roome, 2015. "Black-Scholes in a CEV random environment," Papers 1503.08082, arXiv.org, revised Nov 2017.
    11. Boyarchenko, Svetlana & Levendorskii[caron], Sergei, 2007. "Optimal stopping made easy," Journal of Mathematical Economics, Elsevier, vol. 43(2), pages 201-217, February.
    12. Robert C. Merton, 2006. "Paul Samuelson and Financial Economics," The American Economist, Sage Publications, vol. 50(2), pages 9-31, October.
    13. Ammann, Manuel & Kind, Axel & Wilde, Christian, 2003. "Are convertible bonds underpriced? An analysis of the French market," Journal of Banking & Finance, Elsevier, vol. 27(4), pages 635-653, April.
    14. Sergio Zúñiga, 1999. "Modelos de Tasas de Interés en Chile: Una Revisión," Latin American Journal of Economics-formerly Cuadernos de Economía, Instituto de Economía. Pontificia Universidad Católica de Chile., vol. 36(108), pages 875-893.
    15. Zhijian (James) Huang & Yuchen Luo, 2016. "Revisiting Structural Modeling of Credit Risk—Evidence from the Credit Default Swap (CDS) Market," JRFM, MDPI, vol. 9(2), pages 1-20, May.
    16. José Martins & Rui Cunha Marques & Carlos Oliveira Cruz & Álvaro Fonseca, 2017. "Flexibility in planning and development of a container terminal: an application of an American-style call option," Transportation Planning and Technology, Taylor & Francis Journals, vol. 40(7), pages 828-840, October.
    17. Marcelo F. Perillo, 2021. "Valuación de Títulos de Deuda Indexados al Comportamiento de un Índice Accionario: Un Modelo sin Riesgo de Crédito," CEMA Working Papers: Serie Documentos de Trabajo. 784, Universidad del CEMA.
    18. Kartono, Agus & Solekha, Siti & Sumaryada, Tony & Irmansyah,, 2021. "Foreign currency exchange rate prediction using non-linear Schrödinger equations with economic fundamental parameters," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    19. Jochen Bigus, 2002. "Investitionsanreize, Koalitionsverhalten und Gläubigerkonflikte," Schmalenbach Journal of Business Research, Springer, vol. 54(4), pages 317-342, June.
    20. Kim, Amy M. & Li, Huanan, 2020. "Incorporating the impacts of climate change in transportation infrastructure decision models," Transportation Research Part A: Policy and Practice, Elsevier, vol. 134(C), pages 271-287.

    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:gam:jrisks:v:12:y:2024:i:9:p:136-:d:1465277. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.