What Geometric Brownian Motion Simulations Say About Future Prices
Summary
The document examines a long-horizon commodity-price simulation based on geometric Brownian motion, with drift and volatility estimated from historical log returns. It asks whether averaging many simulated paths offers a forecast and how the distribution could inform the amount to save for a future purchase. The answers explain that GBM produces a lognormal future price distribution, whose moments and event probabilities can be calculated analytically from the fitted parameters. Simulation is therefore mainly a numerical check when the model has a closed-form solution; estimates of parameter uncertainty can also inform uncertainty around those quantities.
For the stated GBM convention, the expected future price is available directly, so simulating paths and taking their mean does not add information. A decision about savings can instead be framed using a chosen probability threshold or distribution quantile, though the document does not prescribe a specific target. Its main caveat is model risk: commodity prices may not follow GBM, especially over long horizons, so analytical precision conditional on the model does not make the output a reliable real-world forecast.
Key ideas
- GBM implies a lognormal distribution for future prices, with moments available analytically.
- The expected price under the stated model can be calculated directly without Monte Carlo averaging.
- Simulation is useful as a numerical check when a closed-form distribution is already known.
- Saving decisions can use probabilities or quantiles of the modeled future price distribution.
- Long-range results are limited by the realism of GBM and the uncertainty in fitted parameters.
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# How to interpret and define statistics of GBM output
# How to interpret and define statistics of GBM output
I am trying to model the future prices of a number of commodities. For this, I am applying geometric Brownian motion, writing a Monte Carlo code in Python. Given that I want to estimate tommorows price $S_t$ of a commodity, I am using the equation:
$S_{t}=S_{t-1}\exp((\mu-\frac{\sigma^2}{2})+z\sigma)$
Where $z$ is determined from a stochastic number from a normal distribution. $\sigma$ and $\mu$ I have calculated using the log-"returns" based on annual price data from the respective commodity, i.e. from $t_0$ to $t$. The data I have is between 1950-2015.
Now, I want to model future prices until 2050, counting from 2015. So I have applied the aforementioned formula for each year, starting from 2015 until 2050. For instance, the price year 2041 would be:
$S_{2041}=S_{2040}\exp((\mu-\frac{\sigma^2}{2})+z\sigma)$
I have performed 1000 simulations, obtaining a wide range for the price in 2050. I understand this method is not appropiate for forecasting, but what can I do with my results to say something about the future price? If I take the mean of my results, and I run many simulations, it is equivalent to just apply the formula $S_t=S_{t-1}\exp(\mu*\Delta t)$, right? I cannot assume that this mean is a good forecast estimate, or can I?
Assume that I will buy this commodity in 2050, and there is a risk that the real price will increase, how can I apply this Monte Carlo simulation model to determine how much money I need to save in order to afford the commodity in 2050? I think that it is something like a risk calculation that I am looking for.
## Answer by userid is i (score 1)
https://quant.stackexchange.com/a/40507
The future price has a lognormal distribution, so its moments can be calculated exactly using the fitted parameter values (or looked up). Confidence intervals for parameter estimates will give c.i. for moments of future prices too.
## Answer by Kevin (score 0)
https://quant.stackexchange.com/a/47229
You are right, the mean is going to be $S_0 e^{\mu t}$. You may want to increase the number of simulations by the way, 1,000 ain’t that many. Since you know $(S_t)$ analytically in closed form, simulating and averaging does not really provide you with any further information. We know all moments of $(S_t)$ in closed-form anyway and can compute probabilities of all events analytically (e.g. how likely is it that $S_t$ is greater than $x$ for a certain $t$). Note that one typically uses approximations (simulations) if one does not have have analytical formulae. So, you can you interpret your results? All you can do is to 'numerically verify' that you arrive at the same values as the closed formula as you increase the number of simulations.
You can’t really use this method to forecast prices though. Asset prices almost certainly do not follow a geometric Brownian motion, so your results are going to be far off from the real world. You would need a more realistic model.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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