Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model
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 r...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-08-01
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| Series: | Risks |
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| Online Access: | https://www.mdpi.com/2227-9091/12/9/136 |
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| author | W. Brent Lindquist Svetlozar T. Rachev Jagdish Gnawali Frank J. Fabozzi |
| author_facet | W. Brent Lindquist Svetlozar T. Rachev Jagdish Gnawali Frank J. Fabozzi |
| author_sort | W. Brent Lindquist |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-7779ae2dec514e9e98b0dfcd6c90ce5d |
| institution | OA Journals |
| issn | 2227-9091 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Risks |
| spelling | doaj-art-7779ae2dec514e9e98b0dfcd6c90ce5d2025-08-20T01:55:49ZengMDPI AGRisks2227-90912024-08-0112913610.3390/risks12090136Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton ModelW. Brent Lindquist0Svetlozar T. Rachev1Jagdish Gnawali2Frank J. Fabozzi3Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-4012, USADepartment of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-4012, USADepartment of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-4012, USACarey Business School, Johns Hopkins University, Baltimore, MD 21202, USAWe 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.https://www.mdpi.com/2227-9091/12/9/136dynamic asset pricingBachelier modelBlack–Scholes–Merton modeloption pricingperpetual derivativebinomial model |
| spellingShingle | W. Brent Lindquist Svetlozar T. Rachev Jagdish Gnawali Frank J. Fabozzi Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model Risks dynamic asset pricing Bachelier model Black–Scholes–Merton model option pricing perpetual derivative binomial model |
| title | Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model |
| title_full | Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model |
| title_fullStr | Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model |
| title_full_unstemmed | Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model |
| title_short | Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model |
| title_sort | dynamic asset pricing in a unified bachelier black scholes merton model |
| topic | dynamic asset pricing Bachelier model Black–Scholes–Merton model option pricing perpetual derivative binomial model |
| url | https://www.mdpi.com/2227-9091/12/9/136 |
| work_keys_str_mv | AT wbrentlindquist dynamicassetpricinginaunifiedbachelierblackscholesmertonmodel AT svetlozartrachev dynamicassetpricinginaunifiedbachelierblackscholesmertonmodel AT jagdishgnawali dynamicassetpricinginaunifiedbachelierblackscholesmertonmodel AT frankjfabozzi dynamicassetpricinginaunifiedbachelierblackscholesmertonmodel |