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 (t_{0}, P_{0}, V_{0}), …, (t_{n–1}, P_{n–1}, V_{n–1}) are ordered t_{0} ≤ … ≤ t_{n–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 Δt_{1} = t_{1} – t_{0}, …, Δt_{n–1} = t_{n–1} – t_{n– 2} and price changes ΔP_{1} = P_{1} – P_{0}, …, ΔP_{n–1} = P_{n–1} – P_{n–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 P_{max} – P_{min}.

"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: ΔP_{i} = P_{i} – P_{i–1} or ΔP_{i+1} = P_{i+1} – P_{i}. 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(P_{i+1}/P_{i}) ~ ΔP_{i+1}/P_{i}, 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) P _{i} = P_{0} + ∑_{j = 1}^{i}ΔP_{j}**

The matters are simpler if the b-increments ΔP_{j} 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.