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

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).