or any series of price ticks in any market, there is a chain of actions that will generate the maximum profit (MP). A trading strategy that results in this is the maximum profit strategy (MPS) for any series of ticks. Studying these hypothetical systems can offer clues to building trading strategies for the real world. Indeed, signals can be used to establish actual positions (see "Idealized models for real profits," May 2008). Here, we’ll examine parameters that can be used to better manage those positions.
The measure of MPS performance is simple. If price fluctuations exceed transaction costs, then the MPS makes money. If price fluctuations don’t exceed transaction costs, then the MPS does nothing and achieves zero profit/loss. Likewise, if costs equal zero, the MPS profit/loss equals the sum of the absolute value of every price change.
There are three types of MPS:
- A stop-and-reverse strategy alternating long and short positions of the same size;
- A stop-and-reverse strategy reinvesting profits and increasing positions at swing points; and
- A stop-and-reverse strategy reinvesting profits and increasing positions as soon as it is permitted by equity and margin parameters.
The analysis here focuses on the first type.
Why we speculate
The main law of the speculative market is that short periods of time provide huge opportunities in markets. The qualitative statement meets its quantitative counterpart in MPS’s measure of what is "short" and what is "huge." While many traditional laws of physics are expressed by equations, this law is expressed by algorithms. As long as the market creates profit opportunities, it will attract trading and, thus, continue to exist.
Still, while the reason for markets seems almost fundamental to the human condition, we don’t have a good explanation of why prices fluctuate. It could be George Soros’ reflexivity, emphasizing the cognitive function, or human knowledge about the world, inspiring the participating function to change it. As a result, the market consumes all information with the apparent goal to fool the majority of its participants.
A common refrain among traders of all sizes and backgrounds is their desire to "break the market code." Typically, this means finding a perfect measure of market trend and volatility. The trend describes predictability of price direction. Volatility defines the certainty of that direction. Both concepts are polar notions that defy both definition and evaluation. Here’s why:
First, there are many definitions. Technical analysis postulates trend existence and formulates partly subjective tools of its determination. It recognizes the role of volatility, suggesting gauges such as Welles Wilder’s true range and John Bollinger’s standard-deviation bands to measure it.
Pricing options with stochastic differential equations introduces a random process to volatility’s definition. An example of this is the σ (percentage of volatility) parameter in the geometric Brownian Motion equation: dP/P = μdt + σdW. The σ is non-observable. It is calibrated from prices of known derivative instruments and rates, as in the Black-Scholes option-pricing model.
A large assortment of price and interest rate theories creates model risk and returns constants, deterministic functions of time or price σ(t), σ(t, P) or stochastic σ. Changing the probability measure of the underlying price to an equivalent martingale measure, they avoid arbitrage and "ignore" price drift that is relevant for a trader.
Autoregressive models approximate conditional heteroscedasticity of time series. (Heteroscedasticity refers to varying levels of correlation between observed data.) They simulate effects such as volatility clusters, time intervals aggregating large and small price fluctuations and provide new definitions.
Second, there is a tendency to measure trend and volatility using price increments sometimes adjusted by time increments. It is strange that such polar properties are measured by the same quantities. Ironically, a price change cannot separate deterministic trend and random volatility contributions. We need further assumptions. For instance, whatever is responsible for the trend and volatility for one increment remains constant for others. Mapping notions expressed by the same words is especially confusing when economic data are treated by a mixture of methods.
The lesson here is that while mainstream market analysis offers many options for measuring trend and volatility, none of them is perfect and, perhaps, many are worse than random. Indeed, the best resolution may be to avoid terms such as "trend" and "volatility" altogether, admitting that we cannot define them well.
MPS does not try to decipher the market. It instead attempts to learn from whatever the market is teaching. Under equal conditions, MPS depends on costs. In futures, commissions and fees do not depend on price; they change based on the number of traded contracts. We can vary these costs to get a new MPS. In addition, futures prices are discrete. They rise and fall by whole numbers of minimal price fluctuations, specified by a contract’s specifications.
The first task of the MPS framework is to build the MPS for a chain of ticks and costs. This can be done manually or by developing an algorithm for a given set of price data.
The second task is to aggregate the MPS’s transactions into trades. A trade is a complete round trip in the market.
We describe MPS transactions using a unique system for trade identification. For instance, the strategy (11, 02, -23, 14) is buy one contract at t1, do nothing at t2, sell two contracts reversing long to short position at t3, exit buying one contract at t4. Three transactions (from four actions) are combined into two trades (11, -13) and (-13, 14). Their durations are t3 to t1 and t4 to t3.
The third task is statistical analysis of the optimal trades. Notice how we distinguish transactions and trades. The former are individual buy and sell actions and the latter are ordered sets of offsetting transactions. In all cases, the net trade action, algebraic sum, is equal to zero. MPS transactions can be combined into trades with zero net action each because the MPS is free to add the last mark-to-market transaction.
This leads us to the optimal trading element. This is a collective name for properties associated with an optimal trade returned by an MPS. This measure is an alternative, or complement, to traditional trend and volatility analysis.
Optimal trading elements
While the MPS is self-evident given the parameters placed on it by the analysis, discrete algorithms can be used to generate the MPS and define the optimal transactions and trades more efficiently. "Optimal trading elements" (below) describe the properties needed to study an optimal trade.
We begin the analysis by specifying a cost for building the MPS. This is not the trading cost (applied later) but a money goal filter. For example, if it is equal to $75 per transaction per contract, then each trade (two transactions) will seek a profitable price change exceeding 2 * $75 = $150. The final MPS is evaluated along with trading cost, for instance $7.64 per transaction, per contract or 2 * $7.64 = $15.28.
We can record the optimal trading elements from many sessions and subject them to statistical analysis. For instance, we can evaluate empirical distributions and sample moments of profits and durations. The value of this analysis should be clear. The best typical profit can be used to take an actual profit or set a trailing stop order tighter to the current price. The typical best duration can be used to identify when a current trade is not acting as expected (see "Optimal trade analysis," below).
The number of ticks and volume are liquidity characteristics of the optimal trading elements. They indicate how quickly transactions can be executed and how large they can be. Slippage or execution of limit orders critically depends on this.
The terms a- and b-increments denote the smallest time and price increments between neighboring ticks. The c-increments (differences between the first in the current and last price in the previous sessions) are not in the list of intraday trades. Building a-, b- and price distributions uncovers the time-price microstructure of the optimal trading elements ("Market Profile and the distribution of price," June 2011).
Directional efficiency (DE) is a new geometric and economic concept involving fractal properties of a price path (see "Optimize your MPS," last page).
Analyzing trading elements
In the table, there are 374 trading elements recorded from five business days before Memorial Day 2011. They returned from $142.13 to $933.22 per contract after commissions and fees. Lean hogs presented 18 elements, the least, while wheat provided the most with 110. This confirms an opinion of many professional intraday traders that success comes from many trades with relatively small profit from each trade. The expectation of thousands of dollars from an intraday position often leads to losses (usually after giving up a solid winner).
Many trades lasted from 200-300 seconds to one-two hours. Some trades took just a few seconds, perhaps because a quick liquidation was necessary to avoid a big loss. Trades stagnating for two to three hours are found for the E-mini S&P 500 (results of optimal trades in S&Ps and bonds are included online).
MP and MPS are objective market properties. They could be used as building blocks for creating profitable pattern recognition strategies. Quantitatively expressing the main law of the speculative market, they stand out as fundamental notions. Being well defined and oriented on specified trading goals, they introduce the optimal trading element — a powerful complement to modern trend and volatility analysis. They provide an analytical view of the best trades and reveal new horizons for studying repetitive market properties, the understanding of which is a clear path to trading success.
Optimize your MPS
Directional efficiency (DE) is a new concept that has a geometric and economic form involving fractal properties of a price path. They are complementary. The DE varies from 0 to 1. Geometrically, two arbitrary ticks on a price chart are the ends of a straight-line segment.
The interval of the ticks is the shortest path between two events. If the latter are not time neighbors, then intermediate events constitute an interval-wise path connecting the two events. In general, it is longer than the interval. Geometric directional efficiency (GDE) of an interval is the ratio of the lengths of the interval to the path. If ticks are on a straight line, then GDE is equal to one irrespective of scaling. Otherwise, it is less than one and greater than zero.
Economically, consider two examples of MPS, one adjusted for a trading cost and another for a larger filtering cost, and evaluate them with respect to the trading cost. The former MPS will have a greater MP than the latter because of a non-linear increase of the number of trades. Economic directional efficiency (EDE) is the ratio of the latter MP to the former one. It is between zero and one. GDE and EDE do not coincide, are complementary and depict the linearity of two events.
Valerii Salov wrote "Modeling Maximum Trading Profits with C++: New Trading and Money Management Concepts" (John Wiley & Sons, 2007). Email: firstname.lastname@example.org.