Estimating a Future Closing Price with Geometric Brownian Motion

Summary

The answer frames a forecast of a later stock price as a conditional expectation based on the latest observed price and an assumed price model. It uses geometric Brownian motion to illustrate the approach: under that model, the expected future price is the current price multiplied by an exponential function of the assumed drift and forecast horizon. The volatility parameter shapes the distribution of possible prices, while the expected value depends on drift.

The drift can be estimated with standard statistical methods such as maximum likelihood, but the response emphasizes that drift estimates are highly uncertain. It does not recommend a particular time-series alternative, define a data or execution schedule, or show a forecast evaluation. Thus, the model provides a basic mathematical framework rather than a practical answer to when an interday trader should sample prices or place orders. Forecast usefulness depends on model fit and reliable parameter estimates, especially for drift.

Key ideas

• A price forecast can be expressed as a conditional expectation under an assumed model.
• Under geometric Brownian motion, expected price grows exponentially with the drift and time horizon.
• Maximum likelihood is one way to estimate drift, but drift estimates are notoriously noisy.
• The example gives no evidence that this model improves trading forecasts or execution timing.

Tags

# How to estimate today's closing price?


# How to estimate today's closing price?












I'm working on interday trading algorithm and I have a basic question:

How can I estimate today's closing price? I need it to predict tomorrow closing price. Should I use the price few moments before market close? However this could be too late to place orders and use my prediction of tomorrow price. Is my logic incorrect somehow? What is the common practice?

## Answer by bcf (score 4, accepted)

https://quant.stackexchange.com/a/22461

Since you're asking on a quant finance forum, the mathematical approach would be

- Decide on a model that the stock price follows, and

- Compute the expected value of the price, conditional on the most recent price.

A famous model, made ubiquitous by Black, Scholes and Merton, is a geometric Brownian motion. Under this model, the stock price $S_T$ at time $T$ given the price $S_0$ at time $0$ is $$ S_T = S_0e^{\left(\mu - \frac{\sigma^2}{2}\right)T + \sigma \sqrt{T}Z}, $$ where $Z$ is a standard normal random variable. The expected value of the stock price under this model is $$ E(S_T) = S_0e^{\mu T}. $$ You may use standard parameter estimation techniques to estimate the parameter $\mu$, such as maximum likelihood estimation (MLE). But, I should warn you that estimating the drift rate $\mu$ is notoriously inaccurate. Perhaps a user with more time series experience could suggest more robust models based on time series models, if that interests you.

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