OHLC volatility (Part 2): Rogers-Satchell and Yang-Zhang (Excel) - YouTube

Hello everyone, and welcome again to NEDL - the best platform around for distance learning in Business, Finance, Economics and much much more!

My name is Savva,

and today we are continuing our discussion of volatility estimators that can be calculated using open, high, low, close data, or to be short, OHLC.

In the last video we have already discussed the Parkinson and the Garman and Klass volatility estimators that have been derived independently of each other in 1980,

that incorporate high and low, and high and low, close and open prices in different ways to factor in different sources of stock market volatility.

Some criticism can be directed to those two volatility measures, as both of those do not account for opening price jumps.

Opening price jumps are very common in stock markets, when the previous day's close is not necessarily equal to next day's open.

For example, in our data set even, we can see that some of the stock price movement,

for example from the 2nd of January 2015 to 5th of January 2015,

was not incorporated in just from opening price to the closing price, some of this movement has been already there,

in the over-the-counter trading during the night hours, or it had occurred immediately because there are a lot of orders have been satisfied immediately when the market opened,

so the price actually decreased substantially, the index fell almost four full points while the market was not opened yet,

so those opening jumps are something that is not incorporated in the Garman and Klass volatility, and surely it's not incorporated in the Parkinson volatility.

It is implicitly accounted for in the close and close volatility, because we do not actually use any opening prices in here, so we implicitly assume

that the movement is from close to close not from open to close,

and that is why some people still argue for the easiest volatility measure being used,

but recently, and by recently I mean in the past 30 years, a couple of more advanced volatility estimators using OHLC have been devised that seek to incorporate all possible sources of stock market volatility,

that manifest themselves in the OHLC data, that is high divided by low chaotic intraday volatility that's useful and important for intraday high frequency traders that care about margining requirements being satisfied,

that is drift that some of the intraday volatility is just attributable to long term or medium term, at least, stock market trend,

and opening price jumps when some of the stock price movement occurs immediately from past close until the next days open,

when the markets just open for trading because of over-the-counter trading during night hours, or because of a high volume of orders being satisfied immediately when the market opens.

The first measure that sought to incorporate all three, was the Rogers and Satchell 1991 measure, or for short, it is also called the RS volatility estimate

it uses normalized measures of high, U is for up here, normalized up, normalized close and normalized down, so normalized low.

So how would we normalize open, high, low, close?

Well, for the normalized open, the only thing we need is to take the natural logarithm of this day's open divided by past day's close,

so that would on its own account for any opening price jumps.

Then for normalized up, so normalized high, we need to take the natural logarithm of this day's high divided by this day's open.

For the normalized down, we need to take the natural logarithm of this day's low divided by this day's open, and normalized close is just the logarithm of this day's close divided by this day's open.

And having computed those, we can calculate this squared sum component for the Rogers and Satchell volatility estimator,

which is equal to normalized up times normalized up minus normalized close,

plus normalized down times the normalized down minus the normalized close.

Easy to remember, easy to digest.

Now, we can enforce all those formulas and bottom right click them all the way down, and very easily calculate the Rogers and Satchell volatility estimate, or for short RS volatility.

so to figure out the Rogers and Satchell volatility estimator, we first need to take the square root,

as this formula gives us variance, not the volatility as in standard deviation,

and then accounting for this 1/n, we can just take the average of this column that says RS, and get 0.65%.

And we can see here that this estimation is quite a bit lower than all three estimations that we investigated so far,

so again it might underestimate some of the volatility sources that are present on the market,

and surely one of the sources it doesn't take into account again,

is the opening price jump, as it doesn't use the normalized open. So what has been done later on by Yang and Zhang in the year 2000,

is that they basically strive to take best of both, worlds or even the best of four worlds, that is the close on close, Parkinson,

Garman and Klass, and Rogers and Satchell volatility estimators,

and actually they explicitly use the Rogers and Satchell, RS, volatility as one of the components of their volatility.

So if we look at the formula that has been proposed by those,

it consists of the open volatility, then weighted close volatility, weighted by some constant k,

and 1- k times the Rogers and Satchell volatility estimator. So to calculate the Yang-Zhang volatility that, as argued by those authors, perfectly accounts for all three sources of stock market volatility,

we have to first figure out the volatility of the open,

the volatility of the close, and also estimate this coefficient k,

the weighting coefficient for closing volatility, and the Rogers and Satchell volatility.

So the opening volatility, we have already got it implicitly, we've got the normalized opens. So what we need to do, is we need just to figure out the sample standard deviation of those normalized opens,

of those logarithms, of opening price jumps, and we get 0.29%. For the volatility of the closing prices,

we just need to take the sample standard deviation of the normalized closes. And then, we just need to estimate the constant k, the weighting factor.

And Yang and Zhang come up with a neat algorithm that allows them to estimate k,

so that the efficiency of the overall variance estimator is maximized,

so that this estimator is as precise as you can get,

so they argue that the weighting factor k should be dependent on the number of observations,

so it's a usual, basically degrees of freedom adjustment, sample size adjustment, and it should be equal to 0.34 in the numerator,

divided by 1.34, plus the number of observations which is 1258, plus 1,

divided by the total number of observations minus 1,

so 1258 minus 1, and close the brackets, close the denominator brackets again and get 0.15,

so that's the weighting factor that we should use for the closing price volatility,

and 1 minus that, so 1 minus 0.15, roughly is the weightening factor that we should use for the Rogers and Satchell volatility.

And now, we can estimate the Yang and Zhang volatility, finally,

and that would be equal to the square root of the opening price variance, so that squared, plus k times the closing price volatility so that squared,

plus 1, minus k times the Rogers and Satchell variance, so that squared again, and we close the brackets,

and we figure out the square root of this weighted sum, and we get 0.73%, which is precisely the Yang-Zhang volatility estimator for our S&P500 data.

So how do all of the five volatility measures we have derived and calculated during these two videos correspond with each other?

Well, the smallest are the Rogers and Satchell and Parkinson volatility estimators, that don't account for either drift, or opening price jumps, or both. The highest is the Garman and Klass volatility estimator,

that does not account for opening price jumps, and to some extent overestimates the drift and the chaotic price behavior that is occurring during the day in terms of high and low,

somewhere in the middle are the close and close volatility estimator and Yang and Zhang volatility estimators

that, I mean, first of all, first of them just doesn't actually care about any intraday movements and simply estimates the volatility of close and close daily returns, and we can see that it does a decent job,

but if we account for all of the information that we've got,

and we account for it properly, so we actually calculate all of the appropriate weighting factors, and we synthesize all three potential sources of variance,

which are high divided by low chaotic intraday stock price movement,

that is relevant for intraday high frequency traders that care about marginal requirements.

If we incorporate the close divided by open drift, that is the share price movements

that are attributable to long-term trend, not some chaotic intraday shocks,

and finally if we account for opening price jumps that is basically accounted for by this variance of the opening price, that is unaccounted for any of the previous measures,

we get a value that is relatively close to the close and close volatility estimator, no pun intended,

but a little bit lower. So what it shows, is that in this case, in case of S&P500, the close and close volatility actually overestimates the daily stock market variance, while the real variance is a little bit low at 0.73%.

And that's all there is for volatility estimation using open, high, low and closing prices.

Please leave a like under this video if you found it helpful, in the comments below I will be eager to see

any suggestions for future videos on Business, Economics,

and Finance that you want me to record, and finally,

please don't forget to subscribe to our channel! thank you very much,

and stay tuned!