Implied Volatility Move Ranges

Article Quant Q&A · Author: thebarryman

Summary

The document explains how to translate annualized implied volatility into a one-period price move. Under the Black–Scholes interpretation, volatility is the standard deviation of log returns, so a sigma-scaled move is expressed by multiplying the current price by the exponential of the scaled volatility. Upside and downside moves therefore differ in absolute price size.

These moves describe ranges in a probability distribution, not exact forecasts. The answer cautions against compounding volatility within one period as if it were a sequence of periodic returns. It also notes that annualization conventions may use calendar days or business days, and that market option prices often imply distributions with more large moves and asymmetric probabilities than a normal model suggests. The resulting volatility smile reflects higher implied volatility for strikes farther from the current price. The explanation gives interpretation and caveats, but no empirical test or precise confidence levels.

Key ideas

Implied volatility in the Black–Scholes framework measures the standard deviation of log returns.
A sigma-scaled price move applies volatility in the exponent, so equal positive and negative sigma moves differ in absolute size.
Volatility-based move ranges describe probabilities rather than predicting an exact future price.
The annualization period depends on whether the market convention uses calendar days or business days.
Observed volatility smiles reflect departures from the simple assumption of normally distributed log returns.

Tags

Options Volatility Derivatives pricing Statistics

Full text

# "X sigma moves" - to log or not to log?


# "X sigma moves" - to log or not to log?












I've been trying to learn options math and am getting hung up on a basic misunderstanding.

From what I have gathered, to get a single day volatility from annualized volatility, you would do something like this -

```
var dailyVol = impliedVol / sqrt(1.0 / 365);
```

My assumption would then be that if I want to find an "x sigma" daily move to the upside or downside, I would do the following -

```
var newPrice = exp(log(1 + dailyVol) * nSigma) * currPrice;
```

But when reading through options trading materials, they often will state something more like this -

```
var oneSigmaMove = currPrice * dailyVol;
var newPrice = currPrice + oneSigmaMove * nSigma;
```

This has the effect that moves to the downside represent a larger percentage change than the way that I would have interpreted "daily volatility," because it uses the same absolute price change in both the up and down directions.

Additionally, a "multi-sigma" move would effectively compound with my formulation, but not in the second formulation.

Which of the above calculations does "implied volatility" actually "imply?"

## Answer by D Stanley (score 3, accepted)

https://quant.stackexchange.com/a/85613

You are right that $\sigma$ is the standard deviation of log returns ($\ln(S_t/S_0)$) implied by the option price (meaning the constant vol that would match the price using the Black-Scholes formula), but your "new price" formula should be:

```
var dailyVol = impliedVol * sqrt(1.0 / 365);
var newPrice = currPrice * exp(dailyVol * nSigma);  // not log(1 + dailyVol)
```

So a $+1\sigma$ move would be a different absolute move than a $-1\sigma$ move. But neither of them are correct, meaning neither are predicting an exact move. It's just defining the probability distribution of possible moves.

And the move does not compound within one period like periodic returns would - so using $log (1+\sigma) * n$ is not correct

A few things to note that you did not ask:

- `sqrt(1.0 / 252)` is typically used in markets that only trade on business days

- The market generally does NOT assume that log returns are normally distributed - instead it typically assumes a greater probability of "large" moves than a normal distribution would imply (and different probabilities of up and down moves). This manifests in a "volatility smile" where strikes away from the current price have a higher implied volatility than strikes near the current price.

- The volatility gives you a confidence interval of moves given the probabilities inherent in the normal distribution. It's not predicting an exact move, but a range of possibilities.

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.

Interpreting Implied Volatility as Log-Return Move Ranges

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