<b>Pareto-Levy distribution :</b> The Pareto-Levy distribution follows a rather complex formula that describes an entire family of distributions that run the gamut from the usual normal distribution to the very fat-tailed Cauchy distribution. (Technically, these are what are called stable Pareto-Levy distributions because when you add two variables that follow a stable Pareto-Levy distribution you get another variable that follows a stable Pareto-Levy distribution -- they are "stable" under addition) One major issue with the Pareto-Levy family of distributions is that the non-normal members of the family have infinite variance. This implies that the larger the data sample (or the longer one trades) the larger the observed standard deviation (the square-root of variance). Having infinite variance also explains the lack of convergence to a normal distribution (the central limit theorem only works for distributions of finite variance). Pareto-Levy fell out of favor in academic circles for three reasons. First, Pareto-Levy is hard to work with analytically (and because academics need published papers more than profits, easy-to-work-with equations are important). Second, infinite variance throws a monkey-wrench in a bunch of other quantitative finance stuff (and since our risk management equations cannot handle infinite risk, we'll just pretend that the risk cannot be infinite -- should comfort owners of ENE, ETYS, WCOM, etc.). Third, empirical evidence suggests that the standard deviation of returns doesn't seem to grow too much with increasing sample sizes (OK, maybe the variances are really infinite after all?) <b>t-Distribution:</b> The t-Distribution is used in statistical comparison tests for means values with small sample sizes of n < 25. For large samples sizes, the t-Distribution is identical to the z-distribution (the normal distribution with mean of zero and standard deviation of 1). As n gets smaller, the t-Distribution gets heavier tails. At n = 1 the t-Distribution is identical to the Cauchy distribution, which is one of the stable Pareto-Levy distributions with an infinite variance. By choosing the appropriate value of n, we can approximate a distribution that has slightly heavy tails. (Almost all statistics books have stuff on the t-distribution) For now, I won't get into distributions where the variance changes over time (heteroskedasticity). <b>Using Distributions for Risk Management:</b> Understanding the distribution helps understand the risk of a trading technique. Given some limited data on returns from a trading technique, we can find the distribution parameters that best fit those data (e.g., mean, standard deviation, skewness, etc.). Then, assuming that future performance will follow that distribution (yeah, right!), we can look at the probability of extreme drawdowns, expected gain, range of likely outcomes, etc. Using a distribution, a trader can make informed decisions about margin and leverage based on the probability of large drawdowns. <b>Using Distributions for Setting Exits</b> Intuitively, everyone knows that too tight a stop leads to too many premature exits. Understanding the distribution of price moves lets a trader choose the probability of exit. Or, having set a dollar-value exit level, the trader can estimate the chance of the stop being triggered before even entering the trade. In fact, if the stock is too volatile, then a trader might avoid the trade entirely or scale back the position size/margin allocated to that trade (scale back if the volatility either forces the trader to set too risky a stop or incur too high a probability of hitting the stop.) Likewise, the trader can use the distribution to estimate the probability of hitting the profit target. <b>Using Distributions for Predicting % Equity</b> If we know the probability of entering a trade on any given bar (based on the probability of getting a setup) and we know the probability of exit on any given bar, we can estimate the average equity %. A simple 2x2 Markov model lets us estimate the average %-cash:%-equity levels over the long-term. The higher the probability of entry and the lower the probability of exit, the more time the system will spend in the market and the greater the exposure of the system to market risk. <b>Testing Validity</b> Perhaps the most powerful aspect of all this is that making predictions about the behavior of a trading system lets us test our models. For example, say we predict that a given trade system will hit its stop 30% of the time (using some measured overall distribution of price changes) but in actual trading, it hits the stop 50% of the time (for a sufficient sample size). If we reject the possibility that random chance created the anomalously high frequency of stop-outs, we can conclude that either: 1) a different distribution of price changes is occurring when we are "in trade." or, 2) the distribution of price changes has changed since we measured it Either way, we learn something about the trading system. The point is that understanding the distribution of outcomes helps us develop better "expectations" for the outcomes of trades. If actual trading fails to meet those expectations, then we have good reason to question our models and our understanding of the trading system. If you don't know what to expect out of your trading system, you have no basis for being either disappointed or excited by its results. Hope this helps, -Traden4Alpha P.S. for more about basic statistics as it applies to finance, look at "Quantitative Methods in Finance" by Watsham and Parramore.
Stu, if you are an experienced trader I'm willing to bet you know exactly what he's talking about, you probably just aren't familiar with the language and definition specifics used. Concepts are concepts, whether shaped with simple analogies or advanced scientific terms. The intrinsic accuracy and usefulness of the trader's mental model is much more important than whether or not he can accurately express it with words. This is not meant to denigrate the value of rigorous research or the value of precise expression, only to point out that what really matters is where the rubber meets the road. "You don't have to know about the physics of tides, resonance, and fluid dynamics to catch a good wave. You just have to be able to sense when it's happening and then have the drive to act at the right time." -Ed Seykota
Hey, now that makes two of us All of my postings presume that the reader is interested in thinking about trading from a more explicit, rule-based, quantitative perspective. I personally mistrust intuition because I've spent too much time studying decision theory, biology, and cognitive science (there's some amazingly weird sh!t that goes between peoples ears). Thus, I take a very scientific/engineering approach to trading -- and use a 100% mechanical system using software. But if your wetware works for you, then more power (and profits) to you. I also want to make sure that nobody takes all this statistical stuff too seriously. No matter how complex and precise the equations, they are ALL approximations. Mathematics is extremely powerful for codifying and exploring the precise theoretical implications of mathematical models. But this precision is actually delusional since the real-world seldom obeys the assumptions that underpin the mathematical model. (I think few people understand this extreme power of mathematics, and those that do do not understand the extreme weakness of mathematics.) I look at all this stuff as tools in a tool box. Tools are powerful, but we need to be careful about using them. Each tool has some purpose and some limitations. Thinking about the statistical distributions of price movements helps traders think about how a trading system might behave. Predicting the behavior of a trading system helps a trader set expectations and gauge actual performance against those expectations. Statistics also helps traders judge when a trading system is misbehaving. But darkhorse is exactly right, ultimately, most traders measure success in terms of financial gains, not the number of lines of code in their software, not the elegance of their theories, not the complexity of their statistical models. All the tools, statistics, concepts, mental models, etc. are simply a means to that end. That different traders can use very different means to achieve the same end is one of the most intriguing aspects of trading. Hope everyone's having a great trading day, -Traden4Alpha
I'll comment on Traden4Alpha question ,thru my own personality. Concerning AAPL which I' ve traded before. Get two gamebirds with one shot like they do in Argentina. Lots of opportunity in AAPL. However for most of the week,too much of a sideways trend around $15. Too much of the month sideways trend of $15,for me that is.[at time of question] May or may not close around $15 today;for me its a non random price to close to a sideways trend.[Pretty orderly for NASDAQ] -------------------------------------------- ''I wasn't particularly adept at higher math. I [think] I excelled at which statistics had meaning'' John Henry-ESPN quote 8/14/02 P.S. Had to over RULE the systems speller ''AAPL not in dictionary''
Thank you very much for your excellent reply Traden4Alpha I certainly understood that. darkhorse, you dude ! Now I'm welcome at parties and people are pleased to meet me, after attending another of your deductive reasoning classes Thanks again for the explanations. Trading exists in a set of conditions which the average person thinks they believe but wishes they were certain of. If they would mechanise them, they may become certain. I bet Mark Twain could have said that if he 'd wanted to