Scaling Volatility with Autocorrelation and Forecast Models

Summary

The document asks whether daily volatility can be annualized with the square-root-of-time rule when daily volatility observations are autocorrelated. Its answer says autocorrelation makes that scaling inappropriate, and suggests modeling the dependence with an ARMA/GARCH or GARCH framework before estimating volatility. The discussion then relates the issue to Black–Scholes: the model uses a volatility assumption over the option’s maturity, so a forecast for that horizon can be used as an input.

This is a brief Q&A rather than a derivation or empirical study. It gives no data, comparison of alternative scaling methods, or details on how to fit or validate a volatility model. It also compresses several assumptions: autocorrelation in volatility levels is not necessarily the same as autocorrelation in returns, and a GARCH forecast does not by itself guarantee that square-root scaling is valid. Treat the advice as a prompt to model dependence and match volatility to the horizon, not as a complete annualization procedure.

Key ideas

• Autocorrelation can invalidate naive square-root-of-time volatility scaling.
• A GARCH-family model can represent time-varying volatility and dependence.
• Black–Scholes uses a volatility assumption for the option’s maturity.
• The note offers no empirical validation or detailed scaling procedure.

Tags

# Square-root-of-time and autocorrelation


# Square-root-of-time and autocorrelation












My question is that when we have autocorrelation in daily volatilities can we scale daily volatility to annual basis using square-root-of-time rule?

Does it breach the main assumption of the rule that says volatility is constant across period? Thank you

## Answer by user12348 (score 1, accepted)

https://quant.stackexchange.com/a/11241

If the time series has autocorrelation then you are right then square root of time scaling is not applicable. Normally autocorrelation is removed using GARCH framework or ARMA/GARCH framework then you get heteroskedastic volatility by definition of GARCH.

For the second part of the question, say, you are looking at Black-scholes model. For that the volatility is assumed to be constant and is the volatility at maturity. So, volatility used is the forecasted volatility at maturity. You can forecast that volatility using GARCH also, for the sake of fitting into the BS formula.

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