December 28th, 2015

Stock Trading Randomness

A stock price series is the same for everyone. Everyone trading it wants to profit from it. Anyone wishing to trade it, implying short term, understandably, will have some kind of method to do so. Trading one stock or instrument at a time might not be enough. One has to have some perspective, a long term plan, not only to build up a portfolio, but also on how to manage it over time.

A no plan scenario could be viewed as just gambling, even if gambling could be interpreted as some kind of plan.

Millions of traders, thousands upon thousands of trading methods. Some are discretionary methods, others automated, and some in between, we could say computer assisted. A lot of them have to be similar or have many traits in common as if variations on the same theme.

There is no way to do an inventory of such things. Almost no one with a good trading strategy would dare provide their code free for public scrutiny and analysis. It is understandable, why should anyone give away their work? In any other field of activity people get paid for the work they do, and it should be the same for those developing software code, even if their programs could be relatively short.

Which method is the most profitable?

I really don't know. In the first place, it is not a good question. All one could answer, maybe and at most, would be: which one was, and not which one will. Some don't want to acknowledge this slight difference in perception.

We know the past, it is part of historical records, but the future is a new frontier with a lot of uncertainty. We can use the past as basis to extrapolate what might happen. And, in many fields, we can do this, but when trying to forecast the markets the whole picture changes. You enter a probability space with quite blurry boundaries.

Price Series

The following chart shows a stock price series:

   #1  Price Series
Random Price Series

(click chart to enlarge)

Anyone can see short term trends, mean reversions, and all kinds of patterns in that chart. Some would even venture they can see what is coming their way from any of the 5,000 data points in the above series, and could trade such a chart, not only that, but will claim they will win: saying they have the math on their side, or a trading strategy that will make them win.

Well, sorry, they won't, except by luck alone.

That price series is unique and was randomly generated using the flip of a fair coin. Acquiescing to the concept of a 50/50 game is the same as admitting that your trading strategy, based on this Gaussian distribution, has an overall profit expectancy of zero. It is like shooting oneself in the foot, contradicting any profitability claims.

You can find profitable strategies over past data, but when it comes to trading forward, it is another game altogether. And this whatever math one might want to apply! No surprise why so many trading programs eventually fail, they try to extract meaning from what is a random process or set what are effectively 50/50 trading rules.

The above price series is the result of an Excel random function: G8 = G7 + (rand() - 0.5)*2.5. It reads: p(t) = p(t-1) + (rand() - 0.50)*2.5, a recursive formula. From whatever point on that chart, looking forward you would be at its right edge with probability 0.50 to go either up or down by an amount within +/- $1.25. Even if the chart looks smooth, its regression polynomial almost cyclical, with a trend that is easily definable, it was still randomly generated.

Pressing F9 will produce, each time, a totally different chart. Therefore, whatever trading method you might have applied to a previous chart might simply not be profitable on a new one. The price could easily fall to zero on any chart. You could not predict what a future chart would look like! Except maybe in general statistical terms.

You could make a guess, for sure, or multiple ones, but that is not a dependable forecast, it would be just that: a guess. And, even with such guesses, you would not escape the pull of the expected long term outcome where expected profits from trading a bunch of these charts would still tend to zero.

I chose that one chart among the gazillions of possible choices because it did look predictable with its almost cyclic pattern. One could have taken a moving average crossover system and traded it profitably.

That is where the deception is: it looked feasible!

A trend following method might probably work out on such a chart just as a mean reversion strategy would. Anyone could design a profitable trading strategy based on chart #1. It is past data, and with this hindsight, a huge number of profitable trading strategies could emerge, all based on that one chart. You could view any displayable stock chart as in chart #1 where after the fact, historical facts, you could design a profitable trading strategy.

If your short term trading strategy can not survive a bunch of such price series as presented in chart #1, or if you view the game as a 50/50 game, I would not give much for your strategy going forward. I would tend to say: it is doomed to fail. Not because I am not ready to wish you all the luck possible, you would need it, but because you don't respect the game you want to play for what it is.

I can not stop you from trading "your" way. How could I? But you should nonetheless also realize that the expected value playing the above chart is zero. Practically with no regards to whatever trading strategy you might want to use. So when I see people promoting their ability to see the next short term move of what could be considered a random price series, they better bring really strong arguments because I will not be easily persuaded.

Nonetheless, one could find trading procedures, at the portfolio level, where the long term outcome would be positive, almost surely.

Price Series With Linear Drift

Can you design a trading strategy where you can win anyway?

I would have to answer that with a resounding "yes". But first, you will need a slight change in perspective by adding a long term drift to the above chart, and this alone will change the game. 

   #2  With Linear Drift
Random Price Series with drift

(click chart to enlarge)

Now, it becomes evident what should be done: you should play predominantly to the upside.

Even sitting on your hands, a Buy & Hold strategy, would make you win the game. The expected gain: the drift, for the time you are in a position.

There is an upward bias, a trend added to the price series in chart #1. The Excel formula for the added line in chart #2 is: G8 = G7 + (rand() - 0.50)*2.5 + 0.02. Two cents per day (5/250 = 0.02) was added as underlying long term drift, representing about a 10% increase over the first year (5/50). One could approximate the output, it is about the same as having a slightly biased coin to the upside: p(t) = p(t-1) + (rand() - 0.49)*2.5.

The stock's price series as a stochastic differential equation would be: dp = µdt + σdW where µdt is the drift ($0.02 per day). It is a more realistic representation of price movement even if it is just an approximation. At least it recognizes that there is a drift, a long term trend, added to the random component of the equation used in chart #1.

After having started at the same price, the upper price series diverges more and more on the positive side as time increases. 2 cents per day might not seem like much, but over time it adds up.

If your trading strategy is ready to respect this "drift", I would say: you won the game. The how might be unimportant since we can not know which methodology will perform best going forward. But, at least, you would know before hand that you would win the game, and not blow away your trading account like so many do trading as if playing chart #1.

All you would have to do is sit tight for the duration. Or, if you wanted to sell at some time, make arrangements to buy back those shares later on. On average, stocks are going up!

You could view this upward trend bias as the long term trend for stocks in general. You might not know what one stock will do going forward, but taking a sufficiently large number of stocks you could forecast an average for the group, and this estimate should come relatively close to the market's average secular trend. It would be up to you to design your trading strategy so that it respects what is out there and takes advantage of it.

Price Series With Compounding Drift

But even the above chart is not that good a representation of what is out there. The stock market is a CAGR game. As such, you would need to change the stochastic differential equation to: dp = ((1+µ)^d - 1)dt + kσdW to make the first term compounding, and then raise or scale the random volatility component σdW by an increasing factor k. Even that would not be enough since all 3 terms: µ, k and σ are themselves processes having stochastic differential equations with chaotic drifts and random components of their own, meaning everything varies quasi-randomly with time.

But still, even as an approximation, putting the trend on exponential growth and applying it to the same price series as in the first chart would result in something like this: 

   #3  With Compounding Drift
Random Price Series with compounding drift

(click chart to enlarge)

It becomes even more evident that holding on to shares, on average, might not be that bad a course of action. Underneath, it is the same random component for all three charts. The differences only comes from how you view the drift component: from none, to linear, to exponential. I say the trend is there; and it is exponential. In chart #3, the upper line is closer to reality than what you see in chart #2 or chart #1.

The Excel formula used was: p(t) = p(t-1) * (1 + 0.0003815) + (rand() - 0.5) * 2.5 . It is a recursive formula, and at t = 5,000, it will give about the same results as a 10% CAGR drift.

Using the exponential drift is a game changer.

The yellow line might have stayed relatively close to the green or blue line in the beginning, but as can be seen, the divergence expands with time. And now, it becomes even more beneficial to consider holding on to the shares for the duration.

Playing the blue line would require exceptional timing abilities while you could easily achieve more just by doing less: simply by holding on. Also, trading the blue line would need to produce enough to at least exceed what the yellow line is practically giving away for free.

To Resume

Chart #1 was purely random. You would press F9 and get a totally new chart as if picked from an infinite stock universe. Some iterations would go to zero, go bankrupt. Expected average profit from trading: zero. Expected profit from a portfolio of such stocks: zero. That is the most probable outcome.

Most effective method to win that game: none. The best course of action: don't trade.

There is no need to trade chart #1 at all, except for the fun of it. There would be frictional costs to pay: commissions, fees and the like which would technically reduce expectations to below zero: a losing game, the more one played, and the longer one played.

Chart #2 benefits from market observations. On average, given time, stocks do go up, on average. And simply diversifying a portfolio could be sufficient to make, on average, someone win over the long term. All one might have to do is put in the money in a sufficient number of stocks and then put in the time. One can trade over the process but should have his/her trading biased to the upside. Basically, there is an edge, a small one, and it is built in.

Chart #3 has knowledge. The stock market is not a linear world as in chart #2, nor is it a random one as in chart #1. It is compounding. And if you want to accelerate or increase the performance level, what you have to do is find ways to increase the inventory over the investment period.

It should be evident that if you trade and end up with twice the inventory than in a Buy & Hold, it will be more than just going for a Buy & Hold.

There are many ways to improve performance by designing trading strategies that can take advantage of the underlying drift. It is given away almost free, why not grab more than just a part of it.

One of the most basic ingredients required by a stock trading strategy might just be time.

Created... December 28th, 2015 © Guy R. Fleury. All rights reserved.

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