Interpreting the Sample Space Behind Brownian Stock Models
Summary
The discussion clarifies what an outcome in the probability space represents when modeling stock returns with a Brownian motion. The sample space can be thought of as possible states of the world, including economic conditions, firm fundamentals, weather, and investor sentiment, though probability theory does not require assigning each outcome a concrete description.
The key modeling point is that a stochastic process is specified through its probability law, which lets analysts study patterns without modeling every underlying cause. Apparent randomness may reflect limited knowledge or complexity rather than fundamental randomness. The answer offers no derivation of the Black–Scholes equation or empirical evidence; it is a conceptual explanation of probability spaces and their role in financial models.
Key ideas
- A sample space contains the possible outcomes represented by a probability model.
- An outcome may stand for a complete state of the world affecting asset prices.
- A probability model can be useful without identifying the causes behind each outcome.
- Observed randomness can arise from missing knowledge or causes too complex to model.
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# Answer by byouness (score 1, accepted)
# What are the underlying events that the random variables map to the real line in the derivation of the Black-Scholes PDE?
When we first try and set up a model for the evolution of S, the value of the underlying stock, I have seen in a lot of textbooks that they model the evolution by the formula $$\frac{dS_t}{S_t}=\mu dt+\sigma dB_t$$ where $\mu$ is the mean average growth of S, $\sigma$ is the volatility of the stock and $dB_t$ is an increment of a Brownian motion.
My question is that if we view $dB\sim N(0,\sqrt{dt})$, what is the underlying event $\omega \in \Omega$ that $dB$ maps to the real line?
## Answer by byouness (score 1, accepted)
https://quant.stackexchange.com/a/43510
Think of it this way, $\Omega$ elements are the states of the world. This means the set of {macroeconomic data, geopolitic situation, weather, fundamentals of the firms, sentiment of market participants, etc.} and anything else that might have any influence on the stocks prices...
As Ezy explained, what makes probability theory useful is that it allows us to study seemingly random phenomena, and get to interesting conclusions with simply the probability law that seems to govern these phenomena, without having to concern ourselves with the underlying causes that drive them.
Please note that these phenomena are not necessarily really random, but might seem so because of our lack of knowledge of the causes that drive them, or because these causes are too complex for us to be able to get to any conclusion in a reasonable amount of time.
You can take as an example, stocks prices, but also weather forecasts, the waiting time in a bus line, etc.
## Answer by Ezy (score 1)
https://quant.stackexchange.com/a/43504
In probability theory $\Omega$ is called the « sample space » of possible outcomes. It does not have an actual representation and it does not matter much since the only thing that really matters is the probability measure $P$. More details here:
https://en.m.wikipedia.org/wiki/Probability_spaceShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.