Greetings all --
The posting as originally entered is too long. This is part 1.
Some of my thoughts on using Monte Carlo techniques with trading systems.
First, some background.
Monte Carlo analysis is the application of repeated random sampling done in order to learn the characteristics of the process being studied.
Monte Carlo analysis is particularly useful when closed form solutions to the process are not available, or are too expensive to carry out. Even in cases when a formula or algorithm can supply the information desired, using Monte Carlo analysis can often be used.
Here is an example of Monte Carlo analysis. Assume that a student is unaware of the formula that relates the area of a circle to its diameter. A Monte Carlo solution is to conceptually draw a square with sides each one unit in length on a graph, with the origin at the lower left corner. The horizontal side goes from 0.0 to 1.0 along the x-axis and the vertical side goes from 0.0 to 1.0 along the y-axis. Draw a circle with a diameter of one unit inside the square. The center of the circle will be at coordinates 0.5, 0.5. The Monte Carlo process to compute the area of the circle is to generate many random points inside the square (each point a pair of number with the values of the x-coordinate and y-coordinate being drawn from a uniform distribution between 0.0 and 0.999999), then count the number of those points that are also inside the circle. The ratio between the number of points inside the circle to the number of points drawn gives an estimate of the constant pi. Running this experiment several times, each using many random points, allows application of statistical analysis techniques to estimate the value of pi to within some probable uncertainty. The process being studied in that example is stationary. The relationship between the area of the circle and the area of the square is always the same.
When we are developing trading systems, the ultimate question we are most often asking is "What is the future performance of this trading system?" Recall that the measure of goodness of a trading system is your own personal (or corporate) choice. Some people want highest compounded annual return with little regard for drawdown. Others value systems that have low drawdown, or infrequent trading, or whatever else may be important. But, in all cases, the goal is to have the trading system be profitable. Assume that many of us are trading a single issue over a period of several years, and that the price per share at the end of that period is the same as it was at the beginning of the period, with significant price variations in between. If we ignore frictional costs -- the bid - ask spread of the market maker and the commission of the broker -- we are playing a zero-sum game. Those of us who make money are taking it from those who lose money. If, instead of the final price being the same as the beginning price, the final price is higher, then the price has an upward bias and more money is made than lost. This is when we all get to claim it was our cleverness that made us money. If the final price is lower, the price has a downward bias and more money is lost than made.
The price data for the period we are trading has two components. One is the information contained in the data that represents the reason the price changes -- the signal component. The other is everything we cannot identify profitably -- the noise component. Note that there may be two (or more) signal components. Say one is a long term trend in profitability of the company, and the price follows profitability. Say the other is cyclic price behavior that goes through two complete cycles every month for some unknown but persistent reason. In every financial price series, there is always the random price variation that is noise. The historical price data that we see consists, in this case, of trend plus cycle plus noise. Each component has a strength that can be measured. If the signal is strong enough, relative to the noise, our trading system can identify the signal and issue buy and sell signals to us. If our trading system has coded into it logic that only recognizes changes in trend, the cycle component is noise as seen by that system. That is -- anything that a trading system does not identify itself, even though it may have strong signal characteristics when analyzed in other ways, is noise.
Over the recent decades, analysis of financial data has progressed from simple techniques applied by a few people in a few markets using proprietary tools to sophisticated techniques applied by many people in many markets using tools that are widely available at low cost. The techniques used successfully by Richard Donchian from the 1930s, and Richard Dennis and William Eckhart in the 1980s, were simple. To the extent that the markets they traded did not have strong trends, every profitable trade they made was at the expense of another trader. Today, every person hoping to have a profitable career in trading learns about techniques that did work at one time. They are well documented and are often included in the trading system examples when a trading system development platform is installed.
Assume that a data series is studied over a given date range. Using hindsight, we can determine the beginning price and the ending price. Continuing with hindsight, we can develop a trading system that recognizes the signal component -- some characteristic about the data series that anticipates and signals profitable trades. By trying many combinations of logic and parameter values, we will eventually find a system that is profitable for the date range analyzed. If we are lucky or clever, the system recognizes the signal portion of the data. Or, the system may have simply been fit to the noise. The data that was used to develop the system is called the in-sample data. If the system does recognize the signal and a few of us trade that system, while all the rest of the traders make random trades, those of us who trade the system will make a profit. On average, the rest lose. As more and more people join us trading the system, each of us earns a lower profit. In order to continue trading profitably, we must be earlier to recognize the signal, or develop better signal recognition logic and trade different signals or lower strength signals. By the time the date range we have studied has passed, most of the profit that could have been taken out of that price series using that system has been taken. Perhaps the future data will continue to carry the same signal in the same strength and some traders will make profitable trades using their techniques, or perhaps that signal changes, or perhaps so many traders are watching that system that the per-trade profit does not cover frictional costs.
Data that was not used during the development of the system is called out-of-sample data. But -- important point -- testing the profitability of a trading system that was developed using recent data on older data is guaranteed to over-estimate the profitability of the trading system.
Financial data is not only time-series data, but it is also non-stationary. There are many reasons related to profitability of companies and cyclic behavior of economies to explain why the data is non-stationary. But -- another important point -- every profitable trade made increases the degree to which the data is non-stationary. There is very little reason to expect that future behavior and profitability of well known trading systems will be the same as past behavior.
Thanks for listening,
Howard