The total number of MPS transactions varies from session to session. While the frequency of observations can be arbitrary, the transactions objectively fix the trading time, price and volume. Sorted by time, the ticks form an ordered sequence. Because of rounding off time to one second, a few transactions can appear to occur simultaneously. They are distinguished by the natural order of records.
Analysts using a constant frequency of observations have to assume that prices and volumes corresponding to artificially selected observation times ordinarily do not match transaction times. This process can be said to resemble an invisible and continuous process periodically highlighted by a strobe light. The real process expressed by the complete list of transactions is discrete.
We can classify price changes in three ways. The a-increment and b-increment are the time and price differences between neighboring transactions within the same trading session. Both typically vary from transaction to transaction and look random.
The time between the last transaction of one session and the first transaction of the next consists of a known change. However, two smaller random contributions comparable with a-increments can be counted from the end of one session to the beginning of a next one. The c-increment is the price difference between the last transaction of one session and the first transaction of a next session.
The a-, b- and c-increments are measures of the a-, b- and c-properties. Now, the price process can be described as an a-b-c-process: 1) The a-property determines the time of the next transaction within a session; 2) the b-property determines the price of the next transaction within a session; these increments are added to the time and price of the previous transaction until the last transaction in a session and, then, 3) the c-increment connects the prices of the current and next sessions. The a-increments are non-negative and obey Weibull and Kumaraswamy distributions. The b- and c-increments can be of any sign, or zero.
The b- and c-increments, as well as futures prices, are discrete because of market conventions. The minimal non-zero price increment for live cattle is 0.025, or $10. That is, there is no price between 123.000 and 123.025. (Continuous price models, which most traders and analysts employ, ignore this fact.) Live cattle contracts also respect daily price limits equal to three points, or $1,200.
A number of statistical distributions (Hurwitz zeta, multinomial, Zipf-Mandelbrot, Lattice) can be found in b-increments. The last-minus-first price equal to the algebraic sum of b-increments, or the average sum obtained after dividing by the number of b-increments, does not always obey a Gaussian distribution. This is counter to what we would assume relying on the central limit theorem for the average sum of a large number of random variables. Either the rate of convergence is low, or the time variation of distributions of the variables in the sum violates the conditions of the smooth theorem application.
Empirical a- and b-increment and price distributions are important properties of the OTE (see “Market Profile and distribution of price,” June 2011). BOTEs have positive and SOTEs have negative mathematical expectations of b-increments.