More than 25 years ago, J. Peter Steidlmayer and the Chicago Board of Trade (CBOT) developed and popularized the market analysis method known as Market Profile. Market Profile, and its companion tool, the Liquidity Data Bank, emphasize price distributions. This is in contrast to most techniques that accentuate price changes and time series of prices themselves. With an exception being those methods that integrate mean reversion, these models lack what may be described as a nondiscriminatory broad view of market price relationships. Instead, a significant premium is placed on the last price, while price history is de-emphasized.
Detailed information about transactions on stocks and futures arrives as ticks. A tick is a record of time, price and volume (t, P, V). These are the result of cooperation between human consciousness and technology governed by partly unknown laws of nature. Study of price and volume vs. time stretches from science to astrology because of a wide spectrum of underlying phenomena. Profits blind justifying means. Losses sober, making traders selective. Above all is a potential profit attracting traders and driving speculation.
Here, we’ll examine relationships between distributions of prices and increments, and show how to build price distributions, simple versions of Market Profile, from ticks. In doing so, we formulate an alternative point of view on prices.
Tick sequences
Ticks (t0, P0, V0), …, (tn–1, Pn–1, Vn–1) are ordered t0 ≤ … ≤ tn–1. Inequalities are not strictly because of rounding of the time scale. Each trading session brings a different number of ticks n. The n times prices and volumes, and n – 1 waiting times Δt1 = t1 – t0, …, Δtn–1 = tn–1 – tn– 2 and price changes ΔP1 = P1 – P0, …, ΔPn–1 = Pn–1 – Pn–2 are not constant. One way to reference time and price changes is as a- and b-increments. Prices and b-increments are discrete. Contract specifications set minimal price fluctuations δ reaching percents of daily price ranges Pmax – Pmin.
"Ticks through time" (below) plots a typical sequence of tick prices and volumes. The 30-year U.S. Treasury bond futures are electronically traded on CME Group’s Globex system under ticker ZB from 5:30 p.m. (Central) of one day until 4 p.m. of the next regular business day. The total number of ticks reported for the session between Thursday and Friday, Jan. 7-8, 2010, for the contract expired in March 2010, ZBH10, is equal to 32,779. Some of the ticks contain indicative, bid, ask and other prices. Real transactions of 18,297 are extracted from Jan. 8, 2010, from 7:20 a.m. to 2 p.m. This is the time of the contract’s open outcry session.
The relationships between prices and their increments are trivial: ΔPi = Pi – Pi–1 or ΔPi+1 = Pi+1 – Pi. They raise a question: What is more fundamental, the price level or the price increment?
Some theories assume that a price increment is independent of the past. Being added to the last price, the increment creates a new price instance. Louis Bachelier (1900) considered independent ΔP normally distributed with the zero mean, variance and standard deviation proportional to the elapsed time and square root of it. These are Weiner processes or Brownian motion fluctuations. Because the Gaussian shocks can be negative, their sum theoretically leads to negative prices.
In 1953, Paul Samuelson suggested a geometric or economic Brownian motion with Gaussian asset-returns, log(Pi+1/Pi) ~ ΔPi+1/Pi, and the variance proportional to time, where higher prices give birth to higher gains or losses. This also prevents negative prices: While logarithms can be negative, their arguments are positive. The Black-Scholes option pricing formulas are based on this proposal. Advanced models, with a local volatility dependent on time and stock price, or stochastics, do a better job of fitting option prices.
Irrespective of the models, the following is true for any price in a sequence:
(1) Pi = P0 + ∑j = 1iΔPj
The matters are simpler if the b-increments ΔPj are independent and identically distributed random variables. This simplicity likely is inadequate. Short-term dependencies occur and distributions change in time (see Benoit Mandelbrot and Richard Hudson’s "The (mis) Behavior of Markets," New York: Basic Books, 2004).
In addition, prices and b-increments are discrete. Discrete alternatives have to replace Gaussian, lognormal and other continuous distributions. Zipf-Mandelbrot Law, Riemann, Hurwitz Zeta and multinomial distributions are applied. Weibull and Kumaraswamy distributions model random a-increments. Both a- and b-increments are elementary and there is no clear dependence between them.
Proportionality of asset returns, or price increments, to the square root of elapsed time is used for estimations of value at risk and expected value at risk. These ideas were popular until the recent financial crisis.
Non-Gaussian properties
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.
Click the chart above to enlarge.
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.
Price distributions: Illogical?
Building a distribution from a sequence implies that price is a random variable. A trading session becomes a device producing random quantities similar to a dice drawing among numbers one to six. The δ-discreteness strengthens the analogy. Such a picture does not correspond to theories claiming that price increments are standalone random quantities and current price is the sum of increments drawn until now and added to an initial price. They may consider construction of price distributions and Market Profile illogical.
Indeed, each term in Equation 2 is equal to a constant P0 plus a sum of a different number of random price increments. This implies that the terms are drawn from different distributions and combining them into one does not make sense. Stable distributions provide for some justification.
If a distribution of individual, independent, random variables is stable, then their sum obeys the same distribution type. For instance, the sum of independent Gaussian variables is again a Gaussian one. We should see a bell curve for the sum, if there is a bell curve for the increments. Thus, if ΔPj are Gaussian, then Pi are Gaussian, too. However, this justification is weak because price distributions are non-Gaussian.
The picture changes if a meaning is assigned to some prices. They become primary and increments-derived quantities. A reverse stochastic process oscillating around a single price or rate demonstrates that an answer to the chicken or egg question does not have to be simple.
A meaningful price is the point of control, POC. Traders denote the mode, where the main number of transactions on a frequency histogram or the main volume on a volume histogram takes place. If the main number of positions has been established in the previous day’s trading session at POC, then further moves from this level will make actions of the majority dependent on this price. Losing traders will experience pain dependent on distributions of capital and relative losses. The latter will determine a distribution of critical price of closing losing positions. Personal risk preferences add to this distribution.
While price increments are random, the mechanism seems to involve certain prices dependent on previous prices, where the massive volume or number of transactions occurred. Because of the non-uniform distributions of account sizes and risk preferences, the discussed dependencies likely are non-linear. Evaluations of small or zero correlation coefficients can result in misleading conclusions about independence because they do not guarantee it.
A textbook example is random variables x uniformly distributed on the interval [–1, 1] and x2. The coefficient of linear correlation between them is equal to 0. However, if x is known, then x2 is known too. We see that the behavior can be quite sophisticated, where random price increments coexist with temporary price attractors, which may obey their own distributions and be partly dependent. It is better to study these effects using price distributions and Market Profile in combination with increment distributions.
Price distributions and Market Profile used for the computation of the value area and price increment distributions used for the risk estimations are complementary tools that together provide more information about the sequence. Despite the more than 25-year-history, Market Profile remains a modern tool, which deserves extensive research.
Valerii Salov is the author of "Modeling Maximum Trading Profits with C++: New Trading and Money Management Concepts" (John Willey and Sons Inc., 2007). Email him at v7f5a7@comcast.net.
