Simulating Tomorrow’s Prices with Geometric Brownian Motion
Summary
The document explains how to generate possible future prices when an asset is modeled as following geometric Brownian motion. It gives the stochastic differential equation for price dynamics and its analytical solution: the future price is the current price multiplied by an exponential involving drift, volatility, time, and a Brownian increment. Since that increment is normally distributed, simulations can draw normal variates and transform them into price scenarios.
It distinguishes this forecasting exercise from risk-neutral simulation for option pricing, where the drift is tied to the risk-free rate. For a real-world forecast, the assumed drift controls directional tendency; repeated draws produce a distribution of possible outcomes rather than a single prediction. The document gives one numerical illustration, but does not evaluate parameter estimation or compare the model against data. Its scenarios rely on the lognormal price assumption and chosen drift and volatility, so they inherit those modeling limits and should not be read as proof that actual returns follow the assumed process.
Key ideas
- Geometric Brownian motion models price changes using drift, volatility, and a Brownian shock.
- Its analytical solution expresses future prices as an exponential transformation of a normal random variable.
- Monte Carlo scenarios can be generated by drawing normal variates for the Brownian increment.
- Risk neutral option pricing uses a risk-free drift, while real-world scenarios require an assumed expected return.
- The resulting distribution depends on the lognormal model and the selected parameters.
Tags
Full text
# Should future price scenarios be symmetric around the current market price?
# Should future price scenarios be symmetric around the current market price?
Assume a financial instrument which has a (roughly) log-normal price distribution and behaves like a random walk. I would like to generate some possible scenarios for where the price might be tomorrow.
Knowing that the prices are log-normally distributed (and hence skewed, i.e. an increase in price is usually greater in magnitude than a decrease in price), would it be more correct to generate asymmetric scenarios (with equal probability) around today's market price, for ex. by estimating a log-normal distribution based on the historical data? Or would it be more correct to generate symmetric scenarios around today's market price (in a way assuming that the log-returns are normally distributed and hence symmetric)?
Does generating asymmetric scenarios in any way bias my future view of the market, by making me believe that one direction is "more likely" than the other?
## Answer by Julie Taylor (score 1, accepted)
https://quant.stackexchange.com/a/69750
Numerical simulation is used in Monte Carlo simulation with applications to option pricing but in option pricing, we simulate prices (which is assumed to follow a log normal distribution) have what is known as a drift equal to the risk free rate.
In your case, since it doesn't seem like you are doing this for the sake of pricing options, if prices are assumed to follow a log normal distribution, then its dynamic follows the following stochastic differential equation:
$$dS_t = \mu S_t dt + \sigma S_t dW_t$$
where $W_t$ is Brownian motion. This fortunately has an analytical solution which can be solved using Ito's lemma:
$$S_t = S_o \exp\bigg(\big(\mu - \frac{\sigma^2}{2}\big)t+\sigma W_t\bigg)$$
You can simulate $W_t$ using random normal variates. If you have positive drift $\mu$ then future price will drift upwards.
Example, today's spot price, $S_0 = 100$, drift $\mu = 0.2$, volatility $\sigma = 0.01$, time (in years) $t=1/252$ and $W_t=0.28499$ (generated using normal random number generator), plug all that into the equation to generate one possible scenario. Generating thousands of scenarios will give you a distribution of where tomorrow's price may lie.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.