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

Paper: arXiv 2405.12479

Authors: W. Brent Lindquist, Svetlozar T. Rachev, Jagdish Gnawali, Frank J. Fabozzi

Abstract

We present a unified, market-complete model that integrates both the 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. In contrast to classical Black-Scholes-Merton, we show that option pricing in the unified model displays a difference 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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a mathematically rigorous continuous-time framework that unifies the Bachelier and Black-Scholes-Merton models using stochastic differential equations, PDEs, and equivalent martingale measures. However, the provided content is almost purely theoretical with minimal empirical backtesting or empirical validation, relying primarily on mathematical derivations.

Research Flowchart

  flowchart TD
    A[Research Goal: Unify Bachelier & Black-Scholes-Merton Models for Asset Pricing] --> B[Inputs: Asset Prices, Riskless Bank Account/Bonds & Negative Rate Possibilities]
    B --> C[Unified Framework Development: Continuous-Time & Discrete Binomial Models]
    C --> D[Derive Option Pricing Formulas & Dynamic Hedging Strategies]
    C --> E[Extend to Interest Rate Term Structure: Zero-Coupon Bonds, Forwards & Futures]
    D & E --> F[Findings: Replicating Portfolio Divergence Bank Account vs. Bonds & Perpetual Derivative Condition]
    F --> G[Outcomes: Model Simplifies to Classic BSM & Bachelier Limits + Mixed Deflator Proposal]

Cited References (24)

1. Steven Shreve: Stochastic Calculus and Finance (1997)
2. Futures trading with transaction costs (2010) — link
3. The Pricing of Options and Corporate Liabilities (1973) — link
4. The fundamental theorem of asset pricing for unbounded stochastic processes (1998) — link
5. Théorie de la spéculation (1900) — link
6. Option pricing: A simplified approach (1979) — link
7. HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK–MERTON–SCHOLES? (2007) — link
8. A New Set of Financial Instruments (2019) — link

0 of 24 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.