Geometric Brownian motion is the stochastic process most token models use to generate price paths: a constant drift, a constant volatility term, and a fresh random shock at every step, producing prices that are lognormally distributed and never negative. It is the default because it is cheap to calibrate and easy to explain to a board. Applied to a token, both of its central assumptions are the ones the market breaks. Volatility in crypto is regime dependent rather than constant, and prices move in gaps that a continuous process cannot generate. Use it to produce the spread of conditions your treasury has to survive, not to say where a price goes.
The single sigma you feed GBM is the whole model. Calibrate it on a calm window and the simulation says the design holds; calibrate it on a stressed one and the same design fails, and nothing in the output tells a reader which window you used.
The two parameters, and what they quietly assume
The process is usually written dS = mu S dt + sigma S dW. In plain terms: the price grows at a constant expected rate, gets shaken at every instant by a random draw scaled by a constant volatility, and both effects are proportional to the price you already have. That proportionality is why the output is lognormal, why the price can approach zero without crossing it, and why the process has been the workhorse of option pricing for decades.
Three assumptions come bundled with it and none of them is optional. Volatility is constant across the horizon you simulate. Steps are independent, so what happened last month tells you nothing about this month. And the path is continuous, meaning the price passes through every value between two points rather than gapping across them. Each one is a modelling convenience. Each is also a claim about your token's market that you are making whether or not you noticed.
Constant volatility is the assumption crypto breaks first
Volatility clustering is the observed pattern where violent days arrive next to other violent days and quiet stretches sit next to quiet ones. Caporale and Zekokh fit Markov-switching GARCH models to cryptocurrency returns for exactly this reason, finding volatility that is regime dependent rather than described by one number.1 The working paper was later published in the International Review of Economics and Finance.2
The practical consequence is that your GBM result is a function of the calibration window, and that dependence is invisible in the chart. Fit sigma on the twelve months after a launch and you get one number. Fit it on a drawdown and you get a different one, often by a multiple. The distribution looks equally confident either way. Run it on both windows and treat the gap between the two answers as the real output.
Continuous paths, and a market made of gaps
GBM cannot produce a discontinuity. Token markets are built out of them: a cliff clearing, a listing or a delisting, a depeg, an exploit disclosed overnight, a regulator's filing landing on a Tuesday. Those are not extreme draws from a lognormal distribution, they are a different kind of event, and a model that can only reach them by sampling an improbable tail will underprice them every time.
The documented response in the literature is to change the process rather than stretch the parameters. Singh, Jha and Kumar build their cryptocurrency price simulation on Merton's jump diffusion model, adding a compound Poisson jump process on top of the diffusion term, because the plain continuous specification does not fit the behaviour they are modelling.3 If the failure you are worried about is a gap rather than a slide, the answer is a jump term or an explicit scenario, not a larger sigma.
What GBM is still worth inside a token model
None of this makes it useless, and it is still where we start. Its job is to generate a defensible spread of price conditions for everything downstream: treasury runway denominated in a volatile asset, the value of an unlock landing in a given month, whether a fee-funded buyback still funds anything at the tenth percentile, how much reserve survives a year of bad draws. For those questions the shape of the distribution carries the decision, not its precision.
It also does not have to work alone. One published crypto risk framework treats lognormal Monte Carlo price paths as a single component sitting beside volatility stress testing and correlation modelling rather than as the whole risk answer.4 That is the right posture. Use GBM for the middle of the distribution and add named stress scenarios for the parts of the world it cannot generate.
Scenario modelling is not a price forecast
This distinction matters more here than anywhere else in a token model, because GBM produces something that looks exactly like a price prediction and is not one. A simulated path is a conditional statement: if drift is this, and volatility is that, and both hold for four years, here is one way the price could travel. Change sigma and the entire picture changes. The output is downstream of inputs you selected, which means it can support a design decision and cannot support a price target.
So the defensible questions are all about your own design. How often does the treasury run out. How deep does the drawdown go before the buyback stops working. Which month is the structure most fragile in. We do not publish price forecasts and we do not build them for clients. A percentile that leaves the model and turns up in a pitch deck as an expected price has stopped being analysis.
What to settle before you press run
Four decisions, worth writing on the same page as the results. Which window sigma is calibrated on, and what the answer looks like on a second window. Whether drift is set to zero, because a positive drift assumption silently does the work of your entire thesis. Whether the token price is drawn independently of assets it is correlated with, which is rarely true for a treasury held in ETH. And whether the failure you fear is a slide or a gap, because GBM can only model the first.
Then keep the assumption sheet attached to every number the model produces. In our experience the model is rarely where a founder gets hurt. The damage comes six months later, when a tenth-percentile figure is quoted in a board meeting by someone who never saw the parameters that generated it, and the token's actual business case gets argued from a number nobody can reconstruct.
Common questions
Why is geometric Brownian motion used to model crypto prices?
It is the default because it takes two parameters, calibrates in minutes on any price history, and keeps prices positive by construction. That makes it a practical engine for generating thousands of price paths inside a token model. The reason to be careful is that it buys that convenience with assumptions of constant volatility and continuous movement, and cryptocurrency returns show regime-dependent volatility instead.1
What are the limitations of GBM for token modelling?
Three. Constant volatility, when crypto volatility is regime dependent and clusters.1 No jumps, so unlock cliffs, listings, depegs and exploit disclosures cannot be generated by the process at all. And independent steps, so a stressed month does not raise the odds of another stressed month. Each limitation pushes the model toward understating the tail, which is the part of the distribution a treasury decision actually depends on.
Is geometric Brownian motion the same thing as a Monte Carlo simulation?
No. GBM is the process that generates one price path. Monte Carlo is the practice of running many paths and reading the distribution of outcomes. A Monte Carlo run over a token model usually uses GBM for the price component, and randomises other inputs too, such as demand growth, staking participation and how much of an unlock gets sold. Swapping GBM for a jump diffusion process leaves the Monte Carlo method unchanged.
Can a GBM simulation predict a token's price?
No, and treating it as a prediction is the most common misuse. The output is conditional on a drift and a volatility you chose, and changing either changes every number. Simulation is for testing whether a design survives a range of conditions, which is a design question. Forecasting what a specific token will be worth is not something we publish, model for clients, or consider answerable by this method.
See Tokenomics Design for how this applies in practice.
Sources
- Modelling Volatility of Cryptocurrencies Using Markov-Switching GARCH Models
Guglielmo Maria Caporale and Timur Zekokh, Brunel University London, Economics and Finance Working Paper No. 18-10, 2018
Fits Markov-switching GARCH specifications to cryptocurrency returns, the empirical basis for treating crypto volatility as regime dependent rather than constant. - Modelling volatility of cryptocurrencies using Markov-Switching GARCH models (published version record)
IDEAS/RePEc record for International Review of Economics and Finance, vol. 48, pp. 143-155, 2019
Journal publication record for the working paper cited above. - Prediction of Cryptocurrency Prices Through a Path Dependent Monte Carlo Simulation
Ayush Singh, Anshu K. Jha and Amit N. Kumar, arXiv:2405.12988, 2024
Builds on Merton's jump diffusion model with a compound Poisson jump process, the documented alternative when continuous lognormal paths do not fit crypto price behaviour. - Quantifying Crypto Portfolio Risk: A Simulation-Based Framework
arXiv:2507.08915, 2025
Treats lognormal Monte Carlo price simulation as one module beside volatility stress testing and correlation modelling rather than as a complete risk answer.
Last reviewed 2026-08
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