<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.
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.
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]