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LAW OF LARGE NUMBERS
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LAW OF LARGE NUMBERS

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LAW OF LARGE NUMBERS

Besides loving to trade and possessing an extremely compulsive personality, I am ardent subscriber to the theorem, the law of large numbers, and on an occasion when I have too much too drink, a literal believer in the infinite monkey theorem.

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

A great deal of traders interpret this law in a negative way, reasoning that the more they trade, the greater the chance they will have losing trades, which would result in exacerbating current losses, or giving back current profits. What they don’t take into consideration is that the law is postulated on the premise of random events, i.e., a coin flip or the spin of a roulette wheel. Obviously, if you flip a coin 6 times, you may see 6 consecutive heads or 6 consecutive tails, but if you flip that same coin 100 times, you will probably realize app. 50 heads and 50 tails.

However, while trades are statistically independent events, and one trade has no bearing on another trade, they are not simple random events. Trading decisions may appear to be binary; either buy or sell or up or down, but they are not. There are a critical variables which must be accounted for, such as how much higher or lower is the market is going to move, or in other words, what is the risk/reward of the trade and how do I manage the trade. So, there is something other than chance that comes into play when trading, and that is skill and technique.

It stands to reason then, that the better your skills and technique, the more you should trade. While LLN is important because it "guarantees" stable long-term results for random events, it follows that it is also important that your sample of trades is large enough to maximize the number of successful outcomes from your skillful trades and therefore maximize your earning potential.

You can't find yourself subscribing to the theory that " you're only as good as your last trade. " If you are going to trade for a living, there is no last trade, only the next trade. Whether, your last trade was a winner or a loser, it has absolutely no bearing on the outcome of your next trade. Look forward to taking your next trade, because it's going to be the best trade you have ever made. At least, that's how you should be approaching it.

Unlike gambling, a winning streak by a trader will NOT eventually be overcome by the parameters of the game, unless he somehow convinces himself that this is his inevitable outcome. Trading is not gambling where the house has the edge; trading is a performance based activity that requires skill, technique, experience, and practice. Most important though, the trader must have the right attitude, focus, patience, and self-confidence, and then the trader will be the one who possesses the edge.





Last edited by tigertrader; October 30th, 2010 at 01:07 PM.
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I agree with theory. What has taken trading to gambling is the lack of technique, the # of lots that is changing that changes the odds and breaks the LLN because it assumes constant variables.
Awesome article! Thanks for taking the time to write.

M

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Conversely, most traders who believe they have an edge are fooled by randomness, to coin the title of an excellent book on the subject.

You see, the edge for a casino--a real one where the punter is always up against a negative expectancy--never evaporates. A trader's edge can be here today and gone tomorrow, never to return.

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Zoethecus View Post
Conversely, most traders who believe they have an edge are fooled by randomness, to coin the title of an excellent book on the subject.

You see, the edge for a casino--a real one where the punter is always up against a negative expectancy--never evaporates. A trader's edge can be here today and gone tomorrow, never to return.


only the traders that don't adjust and dominate

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There are many traders who think that if they make 4 or 5 loser in a row, that now the odds are standing on their side. The habbits of most trader are statisticly ilogical.

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ArnieM View Post
There are many traders who think that if they make 4 or 5 loser in a row, that now the odds are standing on their side. The habbits of most trader are statisticly ilogical.

IMO, if you take 4 or 5 losers in a row I would question if they are following their rules and trading with discipline or not.

Ryan

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Complexity and Evolution

The law of large numbers assumes that there is an underlying probability or expectancy that does not change during the sample period for your statistical experiment . This is true for Roulette, as the house had an edge 50 years ago and will have the same edge in 50 years.

Trading is only similar to gambling, but there are some major distinctions:

Complexity

There is no fixed set of rules. If you look at roulette, it is a simple 2-person game. The rules and the expectancies can be easily determined. I know that my best option is not to play, and my second best option is to play only once. For trading the rules are complex. A larger sample is required to determine whether my system has an edge. In case that I have used a lot of parameters to optimize my system, there is a risk of curve fitting. The law of large numbers now becomes the law of very large numbers. The number of results required to optimize my system further depends on the degrees of freedom, as given by my optimization parameters. The forward optimization tells me, whether my approach was simple enough to be valid.

Evolution

But this is not the worst problem. To get that very large sample of numbers required to verify my edge, I need to wait until the number of trades could be observed. And here comes the main point. The universe of trading strategies that existed during the beginning of the sample period has already changed. So my sample is no longer representative for the future. One of the laws of trading tells us that probably any strategy that has worked too well in the past will not work in the future! All strategies can be exploited. Successful strategies attract predators, and if they further spread, the predators will fare well and finally attract new predators which feed on predators. The breakout strategies that worked for the turtles do not work today, as new predators were born that fed on them. Trend following strategies lead to bubbles and increased volatility, attracting malicious predators not allowing them to rush to the exits and leaving behind them destruction and fear.

The fact that the rules of the game and the species feeding on each other quickly evolve, means that there may be no stable edge over the period during which the statistical sample is collected. This is particularly true for strategies that rely on daily data. For micro timeframes automated trading algorithms permanently redesigned by highly paid mathematicians and executed by robots using neuronal nets also imply a fast evolution of species. Difficult to compete here as well. Where is the niche for the individual trader?

The law of large numbers can easily lead to false conclusions. It can only be used, if the universe which is observed remains stable during the sample period. Sophistication does not help, but simplicity. Which are the behavioural patterns that have not changed during the sample period?


Last edited by Fat Tails; October 31st, 2010 at 08:20 AM.
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Fat Tails View Post
The law of large numbers assumes that there is an underlying probability or expectancy that does not change during the sample period for your statistical experiment . This is true for Roulette, as the house had an edge 50 years ago and will have the same edge in 50 years.

Trading is only similar to gambling, but there are some major distinctions:

Complexity

There is no fixed set of rules. If you look at roulette, it is a simple 2-person game. The rules and the expectancies can be easily determined. I know that my best option is not to play, and my second best option is to play only once. For trading the rules are complex. A larger sample is required to determine, whether my system has an edge. In case that I have used a lot of parameters to optimize my system, there is a risk of curve fitting. The law of large numbers now becomes the law of very large numbers. The number of results required to optimize my systemdepends on the degrees of freedom, as given by my optimization parameters. The forward optimization tells me, whether my approach was simple enough to be valid.

Evolution

But this is not the worst problem. To get that very large sample of numbers required to verify my edge, I need to wait until the number of trades could be observed. And here comes the main point. The universe of trading strategies that existed during the beginning of the sample period has already changed. So my sample is no longer representative for the future. One of the laws of trading tells us that probably any strategy that has worked too well in the past will not work in the future! All strategies can be exploited. Successful strategies attract predators, and if they further spread, the predators will fare well and finally attract new predators which feed on predators. The breakout strategies that worked for the turtles do not work today, as new predators were born that fed on them. Trend following strategies lead to bubbles and increased volatility, attracting malicious predators not allowing them to rush to the exits and leaving behind them destruction and fear.

The fact that the rules of the game and the species feeding on each other quickly evolve, means that there may be no stable edge over the period during which the statistical sample is collected. This is particularly true for strategies that rely on daily data. For micro timeframes automated trading algorithms permanently redesigned by highly paid mathematicians and executed by robots using neuronal nets also imply a fast evolution of species. Difficult to compete here as well. Where is the niche for the individual trader?

The law of large number can easily lead to false conclusions. It can only be used, if the universe which is observed remains stable during the sample period. Sophistication does not help, but simplicity. Which are the behavioural patterns that have not changed during the sample period?

i knew you would say that...lol

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tigertrader View Post

You can't find yourself subscribing to the theory that " you're only as good as your last trade. " If you are going to trade for a living, there is no last trade, only the next trade. Whether, your last trade was a winner or a loser, it has absolutely no bearing on the outcome of your next trade. Look forward to taking your next trade, because it's going to be the best trade you have ever made. At least, that's how you should be approaching it.

Great article, this section reminds me of one of my fave trading quotes by Tom Basso in his New Market Wizards interview, "think of each trade as just one of the next thousand trades you will take."

IMHO, on gambling, the similarities between it and trading are closer than most would think if we consider poker to be "gambling" as there are those that can win consistently both because they have great control of their emotions in addition to knowing the probabilities of having or drawing the best possible hand based on what the dealer(market) is giving them from card to card (tick to tick). The turn card gives you three of a kind, ok, you bet accordingly. The river throws the 4th heart and you have none with 2 others still in at your table, if anyone bets, you are stopped out without one iota of attachment to what was certain profit 1 minute before.

Absolutely true though, roulette, craps, keno, power ball, frog jumping, pit bull fightin', or betting on what time the spider eggs will hatch and/or what percentage by midnight tomorrow are all complete chance without any real edge...OK, maybe an entomologist would have the edge in the spider egg bet and there is a right way to play the casino games to increase your odds but still, to be consistent at any of them for any length of time is extremely unlikely.

Trading is playing when the probabilities are favorable...adapting and keeping one step ahead of the all devouring, ever moving spyder. Poker, once you know it, you don't have to learn anything new besides the characteristics of the players at each new table.
In support of your "large number" idea, I play as many high expectancy setups as I can. Then again, my win rate is like 30% with tight stops so that necessitates lots of trades for me to be profitable. I reckon it depends on the type of trading one does but if the risk to reward numbers look good, why not throw in a few chips


Last edited by Myshkin; October 31st, 2010 at 12:38 AM.
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