Orhan Karsligil's Ideas, Thoughts and Collection of Resources

Musings with Volatility

Volatility is a vague term. There are strict definitions but within limited context. There is always something "implied".

I will try to do some research and find some answers for my questions.

Financial Volatility:
Volatility most frequently refers to the standard deviation of the continuously compounded returns of a financial instrument with a specific time horizon. It is often used to quantify the risk of the instrument over that time period. Volatility is typically expressed in annualized terms, and it may either be an absolute number ($5) or a fraction of the mean (5%). from Wikipedia
Stochastic Volatility
The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the underlying, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others. from Wikipedia

My Take

Now there is so much about volatility that depends on the context that it is hard to talk about the same thing in different areas of interest (finance, process control, signal processing etc).

Since I will be introducing my bias, too, at least I should define some of the concepts that I base my view.
<<Fig: S&P 500 daily close-to-close returns>>
Assume there is a timeseries data T (like stock data as above). Then the consensus is to convert it to a daily percentage change format:
$$ P_{i}=\frac{T_{i}-T_{i-1}}{T_{i}}$$
Then the day to day percentage changes can be treated as components of the volatility. People then describe the above diagram with phrases like volatility clustering. Day to day changes are interesting (or one can also find day to day ratios which would fluctuate around 1) but it assumes that the expected percentage change is zero, which is not the case, actually the expectation is that it is not zero and that is why people invest (both short and long).

In my mind the definition of volatility is variability around a modelled behavior. It is basically an indication of the accuracy of the underlying model. If the model is good then the volatility should be low and uniform. If the model is bad then a fluctuating volatility should be expected. So for me:
$$ \Delta_{i}=M_{i}-T_{i}$$
$$ \mu=\frac{1}{n}\sum_{k=1}^n{\Delta_{k}}$$
$$ \sigma^2=\frac{1}{n-1}\sum_{k=1}^n{(\Delta_{k}-\mu)^2}$$


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