The Bachelier Model
Arithmetic Brownian Motion, Normality, and an Alternative View of Price Dynamics
Long before the development of the Black–Scholes framework, Louis Bachelier proposed a model for asset price behaviour that would later become one of the earliest mathematical descriptions of financial markets.
In his 1900 thesis, Bachelier introduced the idea that asset prices evolve as a continuous stochastic process. His formulation, based on what is now known as arithmetic Brownian motion, laid the foundation for modern quantitative finance.
Although later overshadowed by geometric models, the Bachelier framework remains conceptually important. It offers an alternative perspective on price dynamics, particularly in environments where traditional assumptions may not hold. Understanding this model provides insight into both the evolution of financial theory and the nature of uncertainty in markets.
Arithmetic Brownian Motion
The Bachelier model assumes that asset prices follow an additive stochastic process.
In practical terms, this means price movements are treated as fixed absolute changes, regardless of the current price of the asset. A £5 movement, for example, is considered equally likely whether the asset is trading at £20 or £200.
This differs fundamentally from the approach used in the Black–Scholes model, where price changes are assumed to scale proportionally with the asset’s current value.
One of the most important implications of the Bachelier framework is that prices are theoretically capable of falling below zero, since the model does not constrain asset values to remain strictly positive. As such, this formulation differs fundamentally from the geometric Brownian motion used in Black–Scholes. As, in the Bachelier model, changes in price are absolute, not proportional. The magnitude of fluctuations does not depend on the current price level.
This leads to a key implication:
prices can, in theory, become negative.
Normal Distribution of Prices
Under arithmetic Brownian motion, asset prices are normally distributed. This contrasts with the log-normal distribution implied by geometric models. The assumption of normality simplifies analysis. It treats price changes symmetrically, with equal likelihood of upward and downward movements of the same magnitude.
However, this symmetry also introduces limitations. In many financial markets, returns exhibit skewness and fat tails. Extreme events occur more frequently than predicted by a normal distribution. Despite this, the normal framework remains useful in certain contexts.
Comparison with Black–Scholes
The differences between the Bachelier and Black–Scholes models reflect two distinct ways of conceptualising price dynamics.
The Black–Scholes model assumes multiplicative changes, ensuring that prices remain positive and that volatility scales with the level of the asset. The Bachelier model assumes additive changes, allowing for constant absolute volatility and the possibility of negative prices.
These differences lead to distinct implications for pricing and risk.
Black–Scholes captures proportional uncertainty
Bachelier captures absolute uncertainty
Each framework is suited to different types of assets and market conditions.
Relevance in Modern Markets
While the Bachelier model is less commonly used for equity pricing, it has found renewed relevance in certain areas of modern finance.
In particular, it is applied in markets where:
prices can approach or fall below zero
absolute changes are more meaningful than proportional ones
Examples include interest rate markets, where negative rates have been observed in recent years. In such environments, the assumptions of geometric Brownian motion become less appropriate. The Bachelier framework provides a more natural representation of price behaviour.
Volatility in the Bachelier Framework
In the Bachelier model, volatility is expressed in absolute terms. This differs from the percentage-based volatility used in geometric models. Absolute volatility reflects the expected magnitude of price changes, independent of the current price level.
This has practical implications.
It simplifies interpretation in certain contexts, particularly when dealing with instruments whose value is not strictly positive or where relative changes are less meaningful.
Option Pricing Under Bachelier
The Bachelier framework can be extended to option pricing. In this setting, the distribution of the underlying asset’s price at maturity is normal rather than log-normal.
This leads to pricing formulas that differ from Black–Scholes, particularly in how tail risk is represented. While less commonly used for equities, Bachelier-based pricing is employed in certain derivatives markets, especially those involving rates.
Conceptual Insights
Beyond its specific applications, the Bachelier model provides important conceptual insights.
It highlights that the choice of model reflects assumptions about how prices behave.
are changes proportional or absolute?
are distributions bounded or unbounded?
how should uncertainty be represented?
These questions are central to quantitative finance. The model also illustrates that no single framework is universally applicable. Different markets and conditions require different representations.
Limitations of the Model
Despite its simplicity, the Bachelier model has limitations. The assumption of normality does not capture the fat tails observed in real markets. The possibility of negative prices may be unrealistic for certain assets.
Additionally, constant volatility remains an idealisation. These limitations mirror those found in other models. They reflect the broader challenge of representing complex, evolving systems with simplified mathematical structures.
The MorMag Perspective
At MorMag, the Bachelier model is viewed as part of a broader toolkit.
It provides an alternative lens through which to understand price dynamics, particularly in environments where traditional assumptions are less applicable. Its relevance lies not only in its direct application, but in the insight it offers into the nature of modelling.
Quantitative frameworks are chosen based on context.
no model is universally correct
each captures certain aspects of reality
interpretation is as important as formulation
This perspective supports a flexible and adaptive approach to analysis.
From Historical Insight to Modern Application
The Bachelier model represents an early attempt to formalise financial markets. Although later developments introduced more complex frameworks, its core ideas remain relevant.
It reminds us that:
models are abstractions
assumptions shape outcomes
simplicity can provide clarity
At the same time, it highlights the need to adapt models to changing conditions.
Conclusion
The Bachelier model offers a foundational perspective on financial markets, based on arithmetic Brownian motion and normally distributed price changes. While it differs from the geometric frameworks that dominate modern finance, it provides valuable insight into the nature of uncertainty and the role of assumptions in modelling.
At MorMag, this perspective reinforces a disciplined approach to quantitative analysis, in which models are selected and interpreted within the context of real-world conditions.
In financial markets, understanding does not come from relying on a single framework. It comes from recognising the strengths and limitations of each, and applying them with clarity and precision within an evolving system.
