# Trading System Analysis: Learning from Perfection: We Can't Trade like the Perfect Trader, but by Analyzing the Statistical Characteristics of Maximum Profit Strategies, We Can Glean Lessons for Our Real-World Strategies

By Salov, Valerii | Modern Trader, November 2011 | Go to article overview

# Trading System Analysis: Learning from Perfection: We Can't Trade like the Perfect Trader, but by Analyzing the Statistical Characteristics of Maximum Profit Strategies, We Can Glean Lessons for Our Real-World Strategies

For 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:

1) A stop-and-reverse strategy alternating long and short positions of the same size;

2) A stop-and-reverse strategy reinvesting profits and increasing positions at swing points; and

3) 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 o (percentage of volatility) parameter in the geometric Brownian Motion equation: dP/P = [mu]dt + [sigma]dW. The o is non-observable. It is calibrated from prices of known derivative instruments and rates, as in the BlackScholes option-pricing model.

A large assortment of price and interest rate theories creates model risk and returns constants, deterministic functions of time or price [sigma](t), [sigma](t, P) or stochastic [sigma]. 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. …

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