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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

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There was some confusion from my side. You ask about autocorrelation in volatility (which is a stylized fact in financial time series) and not about autocorrelation in returns thermselves-right? My answer was about autocorrelation in returns. I would alter/delete my answer. –  Richard May 12 at 14:26

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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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Thank you a lot for explanation. I have a question. Let say I use Merton default probability and estimate asset value and asset volatility using two equations approach. And as you know in asset volatility equation we use equity volatility as an input parameter. I came up with new regression model which assumes that the annualized equity volatility of the following month can be explained by two factors by today annualized equity volatility and some other factor. However, I think my model does not make sense as I scaled my first factor using square-root-of-time rule. –  user3618375 May 11 at 7:23
    
What do you thin about it? –  user3618375 May 11 at 7:23
    
Merton model calculates default probability today for time T given the debt face value. You could use ARMA/GARCH(1,1) model to get the volatility today and predict for the time T and use the volatility at time T for input the formula. GARCH parameters are estimated using MLE which is comparable to OLS for time series. I agree regression without sound theoretical basis will give spurious results. –  user12348 May 11 at 7:48
    
Thank you for help! –  user3618375 May 11 at 15:25
    
If this answers your question, please click the answer button. –  user12348 May 11 at 17:39

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