The Value of the High, Low and Close in the Estimation of Brownian Motion: Extended Version

TL;DR

This paper analyzes how additional extremal information like high, low, and close values improves the estimation of Brownian motion's conditional expectation and variance, with computational results highlighting the value of high and extremal data.

Contribution

It extends the analysis of Brownian motion by evaluating conditional densities and variances given various extremal and final data, providing new insights into their relative usefulness.

Findings

Knowing the high is more informative than knowing the final value for estimation.

Including extremal data reduces the variance significantly.

Knowledge of open, high, low, and close reduces variance to 42% of that with only open and close.

Abstract

The conditional density of Brownian motion is considered given the max, B(t|\max), as well as those with additional information: B(t|close, max), B(t|close, max, min) and B(t|max, min) where the close is the final value: B(t=1)=c and t in [0,1]. The conditional expectation and conditional variance of Brownian motion are evaluated subject to one or more of the the close (final value), the high (maximum), the low (minimum). Computational results displaying both the expectation and variance in time are presented and compared with the theoretical values. We tabulate the time averaged variance of Brownian motion conditional on knowing various extremal properties of the motion. The final table shows that knowing the high is more useful than knowing the final value among other results. Knowing the open, high, low and close reduces the time averaged variance to 42% of the value of knowing only the open and close (Brownian bridge).