Matching Historical and Implied Volatility Horizons
Summary
The document explains why comparing historical volatility (HV) with implied volatility (IV) requires matching their forecast horizons. IV is inferred from option prices and is forward-looking; HV is estimated from realized underlying returns and is backward-looking. The question’s proposed interpolation across nearby strikes and expirations is framed as a way to estimate a point on the volatility surface, assuming the surface is sufficiently smooth and market prices provide enough information.
The response recommends selecting an HV lookback that can be compared with IV over the same interval, then modeling HV as a time-dependent process, such as an autoregression, to forecast future volatility. That forecast can be compared with current IV for the corresponding maturity. This is a conceptual methodology, not an empirical test or a complete pricing recipe. The comparison depends on model assumptions, return data, and whether past volatility meaningfully predicts future volatility; annualization alone does not make mismatched horizons equivalent.
Key ideas
- Implied volatility reflects option prices and looks forward, while historical volatility summarizes past realized returns.
- A smooth volatility surface can be fitted to option observations to estimate volatility at a selected strike and maturity.
- Historical volatility should be measured over intervals aligned with the implied-volatility horizon being studied.
- An autoregressive model can forecast future realized volatility for comparison with current implied volatility.
- The comparison relies on assumptions about volatility dynamics and does not follow from annualization alone.
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Full text
# Comparing historical to implied volatility
# 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?
## Answer by lehalle (score 1, accepted)
https://quant.stackexchange.com/a/40461
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?
- 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:
- 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 $$HV(t-H\rightarrow t) \sim IV(t-H\rightarrow t)$$
- model HV a path-dependent way, for instance using an AutoRegressive model: $$HV(t) = c + \sum_{\ell=1}^L A_\ell HV(t-\ell) + \epsilon$$
- 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 $IV(t\rightarrow t+H)$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.