Testing VIX-Implied Bands Against Historical Equity Moves
Summary
The document examines whether realized SPY price moves fall outside bands implied by the VIX. It converts annualized implied volatility into a monthly estimate using a square-root-of-time adjustment, then applies one- through four-standard-deviation thresholds to historical observations. The author compares observed band breaches with counts expected under a normal distribution and questions whether the results support normally distributed equity prices.
A response points out that the analysis should use log returns rather than raw price changes, since the usual normal-return model applies to logarithmic returns. It also notes that volatility should be scaled to the relevant horizon and that drift may be ignored or estimated. The posted counts are descriptive, but the document does not provide a full methodology for aligning option-implied volatility with realized forward returns, handling overlapping horizons, or testing statistical uncertainty. Its normality conclusion is therefore tentative, and the reply does not independently validate the backtest.
Key ideas
- Annualized implied volatility can be converted to a shorter-horizon estimate with square-root-of-time scaling.
- The original test compares forward equity moves with standard-deviation bands derived from VIX.
- Log returns are the appropriate quantities for evaluating a lognormal price model.
- Drift and the alignment of implied volatility with the tested return horizon affect the comparison.
- The reported breach counts alone do not establish that equity returns are normally distributed.
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Full text
# Is the market really Normal. Is Implied Volatility Historically Correct?
# Is the market really Normal. Is Implied Volatility Historically Correct?
Ok. So as of 6/10/2014's market close the SPY was 195.6 and the VIX closed at a ridiculous recent low of 10.99. Now because the VIX (IV) is the implied volatility of 1 month contracts on the SPX and because this 1 month figure is quoted annually, we take the current value of the VIX and multiply it by sqrt(20/250) to convert it to a true monthly figure.
So given a VIX of 10.99, the SPX a 1stdv move to the upside and downside will be 195.6(+/-)10.99*sqrt(20/250)*195.6, ie, 201 and 189 on the SPY, 1 month from now. If it goes above 201 or below 189, the move is greater than a 1stdv move.
So I ran a historical test to see how close the SPX has abided by the VIX, 1 month prior. So I ran a historical test from 1/2/2001 to 6/10/2014, on daily data of the SPY and the VIX. Because I lose one of month of VIX data because due to the forward lag, we have a sample size of 3358 days.
Running through how many days we broke above:
```
1stdev 234 329 1074.56
2stdev 0 47 167.9
3stdv 0 9 10.074
4stdev 0 1
```
The 1st number is the days above the upper stdev band The 2nd number is the days below the lower stdev band The 3rd number is the total # of days beyond nth stdev by the normal distribution.
From the results it seems that the market is more normal that we actually though. But other empirical evidence has shown otherwise. From the results I am seeing, It seems to suggest equity prices are normally distributed. We don't see more fat tailed events more than the normal distribution suggests. Can someone shed light if I ran the analysis incorrectly??
## Answer by user12348 (score 1)
https://quant.stackexchange.com/a/12985
You need to use log of prices, because log of returns are normally distributed. So or where x is return- $$ x=-\frac{1}{\tau} ln(\frac{S_{t+\tau}}{S_{t}}) $$ The annualized standard deviation can be scaled as +/-$ n\frac{\sigma}{\sqrt{\tau}} $ where n is your multiple. You can either ignore or estimate drift. or look at it another way, S refers to the index $$ln (S_t) = ln(S_{t-1}) + \left( \left(\mu - \frac{\sigma^2}{2} \right)dt + \sigma dW_t\right)$$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.