Deriving the Binomial Probability for a Brownian Motion Limit
Summary
This note explains why a binomial price model uses an up-move probability of one half adjusted by the ratio of drift to volatility and the square root of the time step. The probability is chosen so the expected binomial return over a small interval equals the drift times that interval: the up and down moves have equal size and opposite signs, so their probability-weighted mean is determined by the difference between the two probabilities.
The explanation connects this mean-matching condition to convergence toward geometric Brownian motion and says the variance should also match. It offers intuition for the formula, rather than a worked convergence proof or empirical evidence. The construction assumes small time steps and a probability within valid bounds; it is a simplified discrete approximation, not a complete account of continuous-time price dynamics.
Key ideas
- The up and down move sizes scale with the square root of the time interval.
- Choosing the up-move probability adjusts the expected return to match the specified drift.
- The binomial model is intended to converge to geometric Brownian motion as the time step shrinks.
- Matching mean and variance motivates the approximation, but the note does not prove convergence.
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# Brownian Motion as a Limit of Simpler Models
# Brownian Motion as a Limit of Simpler Models
Let $Δ$ be a small increment of time, and consider a process such that every $Δ$ time units the value of the process either increases by the amount $σ \cdot sqrt(Δ)$ with probability $p$ or decreases by the amount $σ \cdot sqrt(Δ)$ with probability $1-p$, where:
$p = \frac{1}{2}[1+(μ/σ)*sqrt(Δ)]$
This is taken from Chapter 3 in "Elementary Introduction to Mathematical Finance" by Sheldon Ross. Although I can follow the exposition throughout the rest of the chapter, I fail to understand why the above formula is defined the way it is.
I would greatly appreciate a resource or explanation to help me intuitively understand this.
## Answer by Valter (score 2)
https://quant.stackexchange.com/a/77638
The general idea is that we want to connect the binomial model to Geometric Brownian Motion. A binomial model for stock price movements assume that, in a small time increment $\Delta$, price and either increase with a probability of $p$ or decrease with a probability of $1-p$.
Geometric Brownian Motion (GBM) is a continuous-time stochastic processes used to model stock price. The general parameters are $\mu$ for the drift, and $\sigma$ for the volatility.
In order to connect the binomial model to GBM, we need to ensure that over a large number of steps, the binomial model converges to GBM. This is simply done by matching the mean and variance.
If we match the mean, in the binomial model, the return after a small time $\Delta$, the expected return for an up-move is $p\sigma \sqrt{\Delta}$, and for a down-move $(1-p)\sigma \sqrt{\Delta}$. The expected return for GBM is $\Delta\mu$. Thus, we have $$p\sigma \sqrt{\Delta} - (1-p)\sigma\sqrt{\Delta} = \Delta \mu$$
If we solve the above equation for $p$, we get the desired formula $$ p=\frac{1}{2}(1+\frac{\mu}{\sigma}\sqrt{\Delta}) $$ For the intuition behind it, is that the resulting value of $p$ will ensure that both the mean and variance of the binomial model match that of the GBM over small time increments.
If you want to read more about it, I would say reading about stochastic processes and understanding how to model stock price using GBM is important.Shown 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.