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

Author

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  • 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 1-24, August.
    • Handle: RePEc:gam:jrisks:v:12:y:2024:i:9:p:136-:d:1465277
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    References listed on IDEAS

    as
    1. 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.
    2. 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..
    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.
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