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The purpose of this exercise is to look at potential techniques for analysing the non idea properties of volatility. This is important because estimation of volatility and differences in models can suggest trade opportunities which may or may not be realized.
 | Note The actual results obtained should be viewed as less significant than the techniques used and the approach taken to test those techniques |
I will look at apparently odd behaviour in the volatility of FTSE 100 components and attempt to determine if the odd behaviour is an artefact of either the process used, the samples size or some other factor.
 | Warning Past history cannot be taken as a guarantee of further trading profits. In all these strategies the potent for real loss is often higher than the real gain, over a long enough time period. Even successful strategies need to be carefully tuned not to commit too much money in any one trade/period or this quickly leads to loosing most or all of your capital. |
An example of the long term risk of model based trading. http://en.wikipedia.org/wiki/Long-Term_Capital_Management
A look at the sample size.
I have decided to choose a sample size going back 5 years. This is done because I believe going back further has less relevance today and secondly because this data is freely available for easily for 96 out 100 components of the FTSE from finance.yahoo.com for your comparison if your wish.
The sample size suggest an inherent error of around 2.8% or (5*252)^-0.5 . I assume this to mean that the error is going to be over 3%, hopefully not more than 5%.
This factor is important in judging any likely trading opportunities as an apparent imbalance of less than 5% may actually be in the opposite direction even in the long term.
Testing the techniques.
Each technique needs to be tested to ensure the result is not an artefact of the approach taken rather than a real feature. These techniques should also be tested because they are likely to be sub-optimal to demonstrating the features the technique is right to bring out. However, sufficiently tuned analysis can achieve any result. The tuning should be simple and not overly arbitrary. Where possible, I will try to test whether the apparent behaviour can be seen using another approach to determine its validity.
The main approach taken will be to compare the results against 10000 randomly generated company histories with a range of values which is similar to those of the FTSE 100 but assume constant volatility and Brownian motion. The intent is to then take a 99% range of these values (adding a 1% error) and say the error should under the target 5%.
This approach will itself be compared with other random walk samples of the same size to see if this approach is self consistent.
Tests for examination
How does the volatility vary with the period sampled?
A common approach to determining the volatility of an instrument is to take the standard deviation of daily changes and annualise it by multiplying a factor of the square root of the number of business days.
This assumes a very low correlation between the movement from one day to the next. However a more general equation could be:
volatility_function(time_period) = annualised_volatility * (days_per_year/time_period) ^ power.
| correlation | power |
|---|---|
| -1 | 0.0 |
| 0 | 0.5 |
| +1 | 1.0 |
This test will examine what this power is for the instruments considered and whether it is a function of the time_period.
The power should be between (0, 1) Additionally if we look at the power over a range of intervals it should never decrease. (otherwise the power for the comparison of the longer periods vs a shorter period should be positive)
| Note A negative power between successive time-periods could indicate cyclic behaviour in the price. |
How does the mean return and volatility vary for different days of the week?
Equity markets contain an element of speculation. People tend to feel differently at the start of the week, the end and on the week end. This may or may not effect the typical behaviour over a given week day. If this factor is taken into account it could reduce the apparent volatility (removing a systematic periodic element reduces the standard deviation)
e.g. The standard deviation of a sin wave is non-zero however its path is not random. Determining the sin waves characteristics and removing it from the signal or path gets us closer to the volatility of random motion.
How is the mean return and volatility effected by whether an instrument is perceived to be falling or rising, high or low?
It is often observed that as an instrument falls its realised volatility increases. Can we quantify this and what behaviour might we see consider rising instruments or instruments at a recent high or low.