Choosing Random Walk and Brownian Motion Models for Stock Prices
Summary
The document compares random walks with and without drift, standard Brownian motion, and geometric Brownian motion as simplified models for stock prices. One answer frames the distinction between discrete random walks and continuous Brownian motion largely as a choice of time scale, and discusses drift as a key assumption because nominal stock prices tend to trend and are nonstationary. It also contrasts normal and lognormal price assumptions, noting that drift and variance effects can make the distinction less decisive over a chosen horizon.
A second answer recommends choosing a model based on the market behavior a simulation needs to capture, such as liquidity shocks or long memory, then calibrating to that feature. It advises comparing plausible candidate models to expose sensitivity to assumptions rather than declaring one universally best. These models remain approximations: the document provides conceptual guidance, not empirical validation or a detailed calibration procedure.
Key ideas
- Random walks and Brownian motion differ mainly in whether time is represented discretely or continuously.
- The assumed drift is central when simulating nominal stock price levels.
- Standard and geometric Brownian motion imply different distributions for modeled prices.
- Choose a stochastic model based on the market feature the simulation is meant to represent.
- Comparing suitable models helps reveal how results depend on modeling assumptions.
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Full text
# Simulating artificial asset prices: Random walk vs Brownian motion? # Simulating artificial asset prices: Random walk vs Brownian motion? How well can each simulate the real-life behavior of stock prices, and what considerations or (dis-)advantages must we be aware of when deciding to use each: - Random walk with drift - Random walk without drift - standard Brownian motion - geometric Brownian motion ## Answer by demully (score 3) https://quant.stackexchange.com/a/58894 Echoing some of the comments to the OP above, the only real difference between random walks and Brownian motions is a question of time frequency. IE a Brownian motion is just an aggregation of a (binary) random walk with higher frequency. Given both will always be at best an approximation of reality, asking for which is "better" becomes a bit of a superfluous question. How pixelated do you want your thumbnail of the Mona Lisa? ;-) The real question is the degree of drift you want to assume. A simple perusal of stock price charts will tell you that there is clearly drift, at least in headline nominal terms. As such, stock prices, as quoted, are non-stationary. Maybe you could argue that stock prices are "real-stationary" (with respect to say money supply) or "output-stationary" (with respect to earnings growth being cointegrated with respect to GDP, investment, consumption, etc.). But then you'll probably end up arguing more about the correct economic deflator to correct for this drift than about useful conclusions from the model ;-( [Been there; done that; no T-shirts]. So the drift exists; but almost becomes a bigger problem handling it than the problem of stock returns... crazy, but sadly all too common. The "standard" versus "geometric" Brownian motion distinction boils down to whether you believe that prices are normal versus lognormal in nature. Which ceases to matter if you allow drift, because a "variance drag" (of half sigma squared) will make the two equivalent. At least over the timeframe you've chosen to measure this over, reference comments above about binary random walks versus normal Brownians. The short - and I'm sorry - answer is that there really isn't that much of a distinction between the choices above. Another way of saying this is that the errors of ALL of these models compared to reality are so correlated, it maybe doesn't matter which one you choose. I know probably not what you were hoping for here... ## Answer by wgajate (score 1) https://quant.stackexchange.com/a/58890 Echoing the comments earlier in the thread, I suggest we work backwards by first picking the key aspect of the market (e.g. flash crashes, liquidity shocks, long-memory) we seek to replicate. Then we select the stochastic model that best approximates our target behavior. We'll need to decide how to calibrate our models to the market and picking a salient feature can inform our choice of calibration method. Inevitably, we'll discover several suitable candidates that offer good enough approximations of the target market behavior. I would advise against picking a winner. Instead, we can surface model risk and better understand exposure to modeling assumptions, when we evaluate the results of competing models and keep a skeptical eye towards results.
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