The value of the high, low and close in the estimation of Brownian motion
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The value of the high, low and close in the estimation of Brownian motion Kurt Riedel1 Received: 14 August 2019 / Accepted: 11 November 2020 © The Author(s) 2020
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) where the close is the final value: B(t = 1) = c and t ∈ [0, 1]. The conditional expectation and conditional variance of Brownian motion are evaluated subject to one or more of the statistics: 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). Keywords Brownian motion · Bronwnian maximum · Brownian paths Mathematic Subject Classification 60J65 · 60J70 · 91G60
1 Introduction In today’s financial markets, every tick is archived. In analyzing events in the ancient past (1970s) or less automated markets like credit default swaps or emerging market bonds (roughly pre-2013), the only data that typically is available is the open, high, low, close data. An entire field, chartist analysis, uses these descriptors as the “sufficient statistics” for prediction. This paper defines the probability distribution of B(t|high, low, close) and calculates its expectation. Our formulas allow us to interpolate the price signal as E[B(t|open, high, low, close)] over all time in [0,1] given any data source that only has open, high, low, close data. We think of the open, high, low and close as “statistics” which we will use to estimate the mean and variance of the process.
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Kurt Riedel [email protected] Millennium Partners LLC, 399 Park Ave., New York, NY 10022, USA
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Statistical Inference for Stochastic Processes
Classically, most of the financial forecasting based on charts uses at most four pieces of information for each day, the opening price (open), the closing price (close), the maximum price (high) and the minimum price, (low) Morris (2006). We address the issue of how much additional information the high and low carry beyond that of the open and close. We measure the “value” by the reduction of the variance of the Brownian motion given one or both of the high, h, and low, . The variance of the path of a Brownian motion is V (t) = t which 1 integrates to 0 V (s) = 1/2. For the Brownian bridge pinned to B(t) = c, the variance is independent of the terminal value, c, and satisfies V (t) = t(1 − t). Integrating variance 1 of the Brownian bridge from zero to one yields an average variance, 0 V (s) = 1/6. Thus knowledge of both the open and the close significantly reduces
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