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Volatility, two questions about GARCH

David Harper CFA FRM

David Harper CFA FRM
Staff member
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I received two related questions about GARCH(1,1)

1. Jay: "when discussing Garch (1,1) and it's applicability to stocks and talking about yesterday's volatility, I wonder if that is the standard deviation of yesterdays' price change intraday?...i struggle with how you can have a volatility of just one day?"

2. Matthew: "In your tutorial that shows the Monte Carlo Simulation for Geometric Brownian Motion as:
ΔS = St-1 ( μ Δt + σε√ Δt) So is the σ in this case the daily volatilty, annual volatility, or standard deviation? Also, how does one treat negative values with GBM for forecasting commodity prices, as you can't have negative commodity values"

These are good questions. Jay is correct, you need a series of periodic returns to calculate historical volatility. If we say, "the volatility right now is 15%," we aren't quite specific and we may mean "the annualized volatility of daily (periodic) returns is 15%." For the periodic returns, we can estimate daily returns, weekly returns, intra-daily returns; the periodic return is just the return from price-point S0 to price-point S1. Periodic return (continuous compounding) = LN(S1/S0).

Jay is correct because we do need a SERIES of these periodic returns to calculate historical volatility (the length of which, the window, offers a trade-offs: do we want more data or more recency?). Each return in the series is squared (sidebar: we can do this simple thing because we assume the average return is zero, otherwise it's a full on variance), and those squares are averaged to get what Jorion calls the moving average (MA). See how the MA is just the average squared return, over a selected window?

The exponentially weighted moving average (EWMA) simply weights each squared return, but the weights decline in constant ratio lambda. The elegant math of this produces the recursive EWMA, where the entire series is reduced to two simple terms: the recent variance and the recent square return.

So, as I wrote Jay, "for both GARCH(1,1) and EWMA, the lagged variance itself can be 'unpacked' into a historical series of squared returns. In GARCH(1,1), yesterday's WEIGHTED daily variance, for example, is just the recursive substitute for a series of (exponentially declining) squared returns, starting with the day before yesterday."

In other words, the series is impounded into yesterday's variance. This is the elegance of a recursive function: a lot of information is contained in yesterday's variance.

So, don't stop there, once you get this about the EWMA, you are pretty much understanding the GARCH(1,1). It adds one more term to the EWMA: a weight multiplied by the long-term variance. And so, with this additional term the EWMA morphs into the GARCH(1,1) by incorporating both prior information and mean reversion (they have much in common: both are parametric, both estimate conditional volatility, both apply exponentially declining weights to historical squared returns).

For Matthew's quote, note that he happens to refer to the DISCRETE stochastic process rather than continuous-time. But the answer is that the volatility can be any periodicity; e.g., daily, annual. That's because time gets "converted" by the square root of delta-time. You will see in the Hull reading that examples tend to assume an annual volatility, say 30% per annum. You then get to decide what is the TIME INTERVAL for each step in the GBM. For example, if the time interval is one week, then the 30% annual volatility is multiplied by the square root of (1/52). As such, the volatility is rescaled to the selected time step. (I am asked about the square root rule--why multiply by the square root of time?. The short answer, not really helpful, is that the variance is a direct function of time, so we are taking the square root of both sides. Finally, don't forget this square root rule only applies under i.i.d. If returns are autocorrelated, it's wrong)

The last part of this question: the GBM won't go negative because, at each interval, it (effectively) gives a percentage change in the stock price. So, you can go negative (-) in steps forever but you will never reach zero because your base keeps getting smaller. Notice there is no limit on the upside. And this helps explain why, that, while the percentage returns are normally distributed, the price levels are lognormally (nonzero) distributed.
 

JPslx

New Member
David,
I followed your tutorial on the various volatilty calculation methods...and to take it one step further, after having read the wp linked to http://www.voxeu.org/index.php?q=node/876 I was, wondering if you could provide an example of a tGarch calculation on Excel. Apparently this modified version of the Garch is efficiently more precise in modelling volatilty...

Thanks,
JPSlx
 
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