# Comparing historical to implied volatility

As title states, I am trying to compare historical to implied volatility of a stock.

I approximate the single implied volatility (30 days forward) of the stock by first finding 2 series that straddle the 30 days to expiration. Per serie I then take 2 ATM options that straddle the current stock price. I interpolate the IV from the 2 options per serie so that I have the IV for the exact stock price. At this point I have the stock's IV for 2 expiration dates, I then interpolate between the 2 IVs based on day difference so I have the stock's IV for exact 30 days. 1) Does this make any sense?

Calculating historical volatility should be fairly straight forward by obtaining the std dev of the stock's past price movements. However, I struggle with what time periods I should be using. 2) Does it for example make sense to compare the 90day historical volatility to the 30day forward implied volatility? I am not quite sure if IV is already annualised at this point. If it is, surely I can compare any historical period to any IV period?

To think about the correct time scales, keep in mind that

• Implied Volatility is forward looking where
• Historical Volatility is backward looking.

Implied Vol is about current market price of options (seen via the Black and Scholes model). When you select a bunch of maturities and strikes, your implied vol is now on a volatility surface. If you believe in a smooth model of this surface and you have enough market prices to fit this surface, you virtually have the IV for any strike and maturity.

That being said, what does the implied vol reflects?

1. By construction: the expected volatility by traders (i.e. investment banks) to apply from today to the given maturity.
2. In the scope of comparison with the historical volatility; IV reflects the future HV...

Historical Volatility is the observed level of uncertainty in the price formation on the underlying. It is conditioned by model assumption; if you want to stick to BS dynamics, it leads to the "naive volatility estimator" (i.e. standard deviation of returns).

To be able to compare those two volatilities, you should believe that there is autocorrelation between all these stochastic processes:

• HV for tomorrow has to be related to HV today,
• and HV for tomorrow has to be related to IV today.

As a methodology, my suggestion is thus:

1. select your time scale on HV so that current HV matches as much as possible with past IV (i.e. on the same time intervals), such that $$HV(t-H\rightarrow t) \sim IV(t-H\rightarrow t)$$
2. model HV a path-dependent way, for instance using an AutoRegressive model: $$HV(t) = c + \sum_{\ell=1}^L A_\ell HV(t-\ell) + \epsilon$$
3. Now you can have an estimate of $HV(t\rightarrow t+H)$ and compare it to the implied vol today for a given maturity $H$, that is $IV(t\rightarrow t+H)$.