The first chart in "Tick-level distributions" (below) shows an empirical distribution of price b-increments expressed in δ = 0.03125 = $31.25. It is symmetrical and has one mode. Fitting it by a Gaussian bell curve is unsound because of the δ-discreteness and the following sample moments and extreme values: Mean, 0.000711; sample size, 18,296; variance, 0.175; standard deviation, 0.419; skewness, 0.107; excess kurtosis, 8.93; maximum value, 6; and minimum value, –5.
The Gaussian excess kurtosis is equal to zero. The value 8.93 implies larger deviations. Indeed, the observed extreme values expressed in standard deviations are equal to –5/0.419 ≈ –11.9 and 6/0.419 ≈ 14.3. The Gaussian probabilities are equal to P(x ≤ –11.9) = 6*10–33 and P(x ≥ 14.3) = 10–46.
To say these are tiny would be inexact. The age of the Earth is about 4.54 billion years. With the transactions occurring every second, only one of them could create the deviation 8.53 * 0.419 = 3.57δ with the Gaussian probabilities P(x ≤ –8.53) = P(x ≥ 8.53) = 7.3*10–18, which is the reciprocal of the age of the Earth in seconds. This is still enormously larger than our findings. A Gaussian model makes them "impossible," but an ordinary ZBH10 session on Friday, Jan. 8, 2010, had 5 - |3δ|, 4 - |4δ|, 1 - |5δ|, and 1 - |6δ| deviations.
Distributions of prices
Two building methods can be suggested. The first is to count how many times the same price occurs in the sequence P0, …, Pn–1 and plot a frequency histogram, such as those shown in "Tick-level distributions." The second is to summarize volumes corresponding to the same price (the third chart in the distribution graphic).
In general, the two histograms differ. They rarely resemble a bell curve, are δ-discrete because of the market conventions, often skewed, have a few peaks, and are limited. (Some futures have daily price limits.) Both distributions can be viewed as simplified versions of Market Profile.
In the original Market Profile histogram, a bar is built up of letters marking non-overlapping time intervals containing the price. These intervals (often 30 minutes) are referred to as time/price opportunities (TPOs) and are marked as A, B, … depending on the market. Original tables additionally present percent contributions of the four types of traders into the volume and the number of transactions. The latter are known after the market is closed. However, by contrast, the frequency histograms require only prices and, in the case of volume diagrams, volume. They can be built dynamically from a tick stream.
Experienced traders know that Market Profile can change within a single session. (Market Profile does not always follow the original conventions.) These same traders may remain puzzled whether the chart is a frequency or volume histogram. As long as statistics and trading rules are established for a particular type, it is important only to apply them consistently.
The TPO letters give an idea about the order of prices in a session and are more informative than a price distribution. Reordering observations in a sample does not change a distribution. The latter and other sample statistics are symmetric functions of arguments. During construction of a distribution, the information about the order of observations is lost. Is it partly recovered if both P and ΔP distributions are given? For concreteness and simplicity let us concentrate on frequency distributions and their means.
The sample mean price is equal to:
(2) μP = ∑i=0n–1Pi/n
Substituting Equation 1 into Equation 2 and evaluating the sum of sums gives the exact equation:
(3) μP = P0 + (n – 1)μΔP – ∑i=1n–1iΔPi/n
where μΔP = ∑i = 1n – 1ΔPi/(n – 1).
Due to the weights i in the sum the difference in the sample means is sensitive to the order of b-increments. Pair of distributions partly recovers the order of prices, while each of them individually loses it. Why?
Reordering prices does not influence μP, but changes the ΔP and μΔP. Reordering the ΔP does not affect μΔP, but keeping P0 intact creates different prices changing μP. Both distributions are built from one sequence. We could expect that neither has an informational advantage. In fact, both are complementary.