Geometric vs Arithmetic Brownian Motion
Modelling Price Dynamics, Scaling of Risk, and the Structure of Uncertainty
Modern quantitative finance is built on stochastic models of price behaviour.
At the core of these models lies a fundamental choice: how should asset prices evolve over time? The answer to this question determines how uncertainty is represented, how risk is measured, and how derivatives are priced.
Two canonical frameworks dominate this discussion: arithmetic Brownian motion and geometric Brownian motion.
While mathematically related, they reflect fundamentally different assumptions about how markets behave. Understanding the distinction between them is essential for interpreting models such as Bachelier and Black–Scholes, and for applying quantitative methods in real financial systems.
Two Views of Price Evolution
Arithmetic Brownian motion assumes that price changes are additive rather than proportional.
Under this framework, asset prices evolve through fixed absolute movements over time. The size of a price change is independent of the current value of the asset. A one-unit movement is treated the same whether the asset is trading at 10 or 100.
Geometric Brownian motion, by contrast, assumes that price changes are proportional to the asset’s current price level. In this framework, both expected returns and volatility scale with the value of the asset itself. Price movements are therefore interpreted in percentage terms rather than absolute terms.
Although the distinction appears subtle, it has major implications for how financial markets are modelled, how risk is measured, and how derivative instruments are priced.
Distributional Consequences
The choice between arithmetic and geometric dynamics determines the distribution of future prices.
Under arithmetic Brownian motion, prices are normally distributed. This implies symmetry: upward and downward movements of equal magnitude are equally likely.
Under geometric Brownian motion, prices are log-normally distributed. This ensures that prices remain strictly positive and introduces asymmetry, with a longer right tail reflecting the potential for large positive moves.
These differences shape how risk is perceived. Normal distributions allow for negative values, while log-normal distributions do not. The latter aligns more naturally with assets such as equities, where prices cannot fall below zero.
Scaling of Volatility
One of the most important distinctions lies in how volatility is interpreted.
In the arithmetic framework, volatility is absolute. It measures the expected magnitude of price changes in fixed units. In the geometric framework, volatility is relative. It measures the expected percentage change in price.
This difference affects both intuition and application.
Absolute volatility may be appropriate when price levels are constrained within a narrow range or when negative values are possible. Relative volatility is more suitable for assets where proportional changes better reflect underlying dynamics.
Implications for Modelling
Geometric Brownian motion underpins the Black–Scholes model and much of modern derivative pricing. Its assumptions ensure tractability and align with key features of equity markets, such as positive prices and proportional risk. Arithmetic Brownian motion, associated with the Bachelier model, offers an alternative perspective.
It is particularly relevant in environments where:
prices may approach or cross zero
absolute changes are more meaningful than proportional ones
traditional assumptions of log-normality are less appropriate
Each framework captures different aspects of market behaviour.
Risk Representation
The choice of model influences how risk is represented. In geometric Brownian motion, risk scales with price. As an asset increases in value, the magnitude of potential fluctuations increases proportionally. In arithmetic Brownian motion, risk is constant in absolute terms. Fluctuations do not depend on the price level.
This distinction has practical implications.
It affects how portfolios are constructed, how volatility is measured, and how derivatives are priced. It also shapes intuition about how markets behave under different conditions.
Time Evolution and Compounding
Geometric Brownian motion naturally incorporates compounding. Returns accumulate multiplicatively, reflecting the way capital grows over time. Arithmetic Brownian motion does not capture compounding in the same way. It treats changes as linear increments.
This difference becomes significant over longer time horizons, where compounding effects dominate.
Model Limitations
Neither framework fully captures the complexity of real markets.
Both assume:
continuous price evolution
constant volatility
absence of structural breaks
In practice, markets exhibit:
volatility clustering
jumps and discontinuities
changing distributions
These features require extensions beyond both arithmetic and geometric Brownian motion. The choice between the two models is therefore not about correctness, but about contextual suitability.
Conceptual Interpretation
At a deeper level, the distinction reflects two ways of thinking about uncertainty. Arithmetic Brownian motion views uncertainty as absolute variation. It focuses on the magnitude of change. Geometric Brownian motion views uncertainty as proportional variation. It focuses on relative change.
These perspectives correspond to different economic intuitions.
absolute changes may dominate in certain fixed-income or rate environments
proportional changes are central to equity and growth-based assets
Understanding which perspective is appropriate is a key element of model selection.
The MorMag Perspective
At MorMag, the distinction between arithmetic and geometric Brownian motion is treated as a fundamental modelling choice.
It reflects the broader principle that:
models are abstractions
assumptions define outcomes
context determines applicability
Quantitative frameworks are selected based on the nature of the asset, the structure of the market, and the behaviour being analysed. Rather than relying on a single paradigm, the approach emphasises flexibility and interpretation.
From Model Choice to Market Insight
The comparison between geometric and arithmetic Brownian motion illustrates a broader theme in quantitative finance. Different models provide different lenses through which to view markets.
No single framework captures all aspects of reality. Understanding the assumptions and implications of each model allows for more informed analysis and better decision-making.
Conclusion
Geometric and arithmetic Brownian motion represent two foundational approaches to modelling price dynamics. They differ in how they describe change, how they represent risk, and how they structure uncertainty.
Geometric Brownian motion emphasises proportional change and underpins much of modern finance. Arithmetic Brownian motion emphasises absolute change and provides an alternative perspective, particularly in specific market contexts.
At MorMag, these frameworks are understood not as competing truths, but as complementary tools.
In financial markets, insight is derived not from selecting a single model, but from understanding the assumptions behind each and applying them with clarity within an evolving system.
