Trading is attractive because markets provide huge profit opportunities in short periods of time. The maximum profit strategy (MPS) provides a quantitative measure of this opportunity relative to frequency and size. A market’s MPS is the sequence of optimal buy, sell and do-nothing actions that generates the maximum profit, with respect to trading times, prices, execution costs, capital, margins and government regulations.
MPS is an objective market property associated with any sequence of ticks. Positive and negative numbers are used to express long and short trades. A do-nothing, or non-action, is recorded as zero. This strategy is a hypothetical analytical tool that never loses money. (For a fundamental description of MPS and more on how it can be used to build real trading strategies, see “Trading system analysis: Learning from perfection,” November 2011.)
The optimal trading element (OTE) includes all the market properties associated with an optimal trade within an MPS. In actual application, a trader who is applying an MPS-based strategy “buys” and “sells” OTEs rather than futures contracts or shares.
MPS, OTE: A review
There are three types of MPS described in the November 2011 article. Here, we’re going to examine a stop-and-reverse approach, which alternates between long and short positions of the same size.
This system will be constructed by the l- or r-algorithm (see “Modeling Maximum Trading Profits with C++,” John Wiley & Sons, 2007). For example, let’s say that four consecutive time periods of live cattle futures prices are 123.025, 123.275, 122.900 and 122.875, while the cost of a transaction association with each price would be 7.40, 7.40, 7.40 and 7.40. Given this information, then the optimal trading strategy would be: 1, -2, 0, 1.
In English, the transactions are grouped into two trades: Buy, sell one to go flat, sell one to go short, do nothing, buy one to go flat. The buy OTE (BOTE) and the sell OTE (SOTE) are determined by the sign of the transactions. Their profits are $85.20 and $145.20, and their durations are t2 – t1 = 1 and t4 – t2 = 2. (Purely mathematically speaking, this would be (–123.025 * 1t1 – 123.275 * -1t2) * 400 – 2 * 7.40 + (–123.275 * -1t2 – 122.875 * 1t4) * 400 – 2 * 7.40 = $85.20 + $145.20 = $230.40).
MPS is an analytical tool used in hindsight. An MPS and the prices determining it are unknown in advance. However, signals from a pair of MPSs can be used for developing real trading rules (see “Idealized models for real profits,” May 2008). OTEs offer even more possibilities. Here, we’ll create an MPS employing an arbitrary filtering cost, or f-cost, approach and evaluate it with a smaller transaction cost, or t-cost.
Time and sales data are presented daily on the CME Group web site. The intraday ticks of real transactions of electronic trading were studied for the most liquid live cattle contracts. These included February, April, June, August, October and December 2011 and the February 2012 contracts.
The total numbers of sessions are 225 and there are 1,134,304 ticks. The electronic trading hours are 9:05 a.m. Monday to 1:55 p.m. Friday (Central time) with halts from 4 p.m.-5 p.m. each day. Each electronic transaction tick includes time, price and size.
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.
With our stop-and-reverse MPS, a BOTE always follows a SOTE and vice versa. Monitoring the arrival of ticks, it is easy to detect the advent of a new OTE. However, new ticks can affect other properties of the OTE, such as profit and duration. As for the profit, the following three scenarios exist after the initial tick of an OTE is detected: 1) The profit of the current OTE will grow, 2) the current OTE will be replaced by a new one of the opposite type or 3) the session will terminate.
The financial difference between the ticks is double the filtering cost (f-cost) minus the transaction cost (t-cost) per contract. Because the ending tick is unknown in advance, the result will depend on how frequently and how far OTE profits grow after the initial tick. The answer is given by empirical distributions of OTE profits, and under other equal conditions, it depends on the f-cost.
The number of OTEs depends on what we set as our filtering cost. The greater the f-cost, the fewer OTEs for a trading session. If the f-cost becomes too large, this number becomes zero. With a zero f-cost, the profit is maximized. We do not consider this case, however, because all computations are done with t-cost = $7.40, which is less than the f-cost. By applying MPS algorithms to individual trading sessions, we ignore the potential of the c-increments. This corresponds to intraday trading, where all positions are opened and closed in a single session.
If we set the f-cost to $49.99, we consider price moves exceeding 2 * $49.99 = $99.98 as significant. This produces a sequence of alternating BOTE and SOTE with a price differential corresponding to this dollar amount. On a chart, they appear as adjacent time intervals, where the starting and ending ticks mark local minimum and maximum prices.
Then, the OTEs are evaluated using the t-cost. Because live cattle’s minimum tick is 0.025, or $10, each filtered-out OTE will have a price differential of $100 or larger. Because of the t-cost, the smallest OTE profit will be equal to $100 – 2 * $7.40 = $85.20. Other possible profit values are obtained by adding $10 to this amount: $95.20, $105.20, …
Because of discrete prices, it is sufficient to examine the t-cost $7.40 and other values with the increment of $5: $9.99, $14.99, ..., $124.99. The last will target price moves of $250 or larger. The 801 moves of this magnitude are found in 225 electronic sessions over our testing data. The mean (801/225 = 3.56) indicates that a trader should be prepared to get at least three moves exceeding $250 in a single electronic session for the most liquid contract (see “OTE means,” below).
OTE profits are proportional to the algebraic sums of b-increments. For an f-cost of $49.99 and a t-cost of $7.40, the dollar profits can be in increments of $10: 85.20, 95.20, 105.20, … There are 4,630 OTEs found in 225 sessions corresponding to this f-cost. The greatest OTE profit of $875.20 occurred once on Jan. 18, 2011. The least OTE profit of $85.20 occurred 466 times. (Interestingly, 466/4630 ~ 10%.) The mean OTE profit is $176.
While mean, minimum and maximum values together with variance, skewness and kurtosis can give us ideas, a better understanding comes from an empirical cumulative distribution function. “Profit frequency” (below) illustrates such functions for live cattle OTE profits.
In addition to the parameters of MPS, we also have to deal with the reality of trading. For instance, to buy BOTE at the initial tick, we must place an order in advance. If we enter on a stop, then price can be worse than desirable because of slippage. This will decrease the profit potential or result in a loss. Using a stop-limit ensures we get our price; however, the order may not be filled.
The new MPS and OTE approach is designed to organize trading, provide a framework for creating new trading rules and manage risk. However, it cannot eliminate losses or replace the art of trading. Attention to a- and b-increments, their sequences, distributions and patterns have new applications when these quantities are considered within OTE intervals and on the boundaries between BOTE and SOTE.
The time difference between the ending and starting ticks is the OTE duration. The durations for f-cost = $49.99 in our live cattle data are between zero and 59,400 seconds. Most trades (80%) are between zero and 3,120 seconds. Half are between zero and 840 seconds. The varying durations are shown in “Trade lengths” (below). This dependence in logarithmic coordinates is almost linear ln(Duration, s) = 1.753 * ln(f-Cost, $) + 1.123, r = 0.9990. It also is found that ln(mean OTE profit, $) = 0.647 * ln(mean OTE duration, s) – 0.00625, r = 0.997, where a slope larger than 1/2 may indicate non-Gaussian dependence between mean price and time increments.
MPS is the quantitative measure of the frequency and size of potential OTEs. MPS, OTE and their trading applications are a new area of study. The high frequency and large size of OTE profits are inherent qualities in the markets. Their appearance is a phenomenological and main law of the speculative market.
Valerii Salov is the author of “Modeling Maximum Trading Profits with C++: New Trading and Money Management Concepts” (John Wiley and Sons Inc., 2007). E-mail him at firstname.lastname@example.org.