Interpreting Implied Volatility and Price Ranges
Summary
The document clarifies that volatility is defined relative to a model and, in the Black–Scholes setting, describes the standard deviation of log returns per square root of time. It derives a confidence interval for log returns using normal quantiles, then maps the endpoints through an exponential transformation to obtain a corresponding interval for the asset price. Thus volatility does not directly represent the standard deviation of the spot price, and a one-standard-deviation range is not itself a value-at-risk calculation.
The discussion also explains why implied volatility is commonly quoted for options: it offers a compact way to compare option values across instruments and pricing inputs. The Black–Scholes interval is conditional on a model that the document acknowledges is unrealistic; actual returns can be skewed and heavy-tailed, making ranges asymmetric and wider. Annualization and the day-count convention also matter when translating volatility to a short horizon.
Key ideas
- In Black–Scholes, volatility scales the standard deviation of log returns with the square root of time.
- A log-return interval must be exponentiated to express a corresponding range for spot prices.
- Implied volatility provides a standardized shorthand for comparing option values.
- Real return distributions may be skewed and heavy-tailed, limiting normal-model confidence intervals.
- Annualized volatility must be adjusted for the chosen day-count convention when estimating daily ranges.
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Full text
# Direct use of implied volatility
# Direct use of implied volatility
I am not sure to understand exactly the direct use of implied volatility. Let's take an example: if an instrument has a daily volatility of $\sigma$, there is a 68% probability that its value will be between +/- $1+\sigma * \sqrt{\frac{1}{365}} $ of today's price. Is that correct, or am I confusing with the value at risk?
Thank you for your help
## Answer by AFK (score 1, accepted)
https://quant.stackexchange.com/a/17752
This has been asked many times already. Volatility always refers to a model. And unless stated otherwise this model is the Black-Scholes model. In this model the volatility is the standard deviation of the log-returns divided by the square-root of time: $$ \log(\frac{S_{t}}{S_0}) = (r - \frac{1}{2}\sigma^2)t + \sigma W_t \sim \mathcal{N}\left( (r - \frac{1}{2}\sigma^2)t , \sigma^2t \right) $$ The standard deviation of the log-return is NOT the standard deviation of the spot.
If the model is true (it is not) the confidence interval for the log-returns would be $$ I_\alpha = \left[ (r - \frac{1}{2}\sigma^2)t - q_{1-\alpha/2}\sigma\sqrt{t} ; (r - \frac{1}{2}\sigma^2)t + q_{1-\alpha/2}\sigma\sqrt{t} \right] $$ with $q_{1 - 0.68/2} \approx 1$ and $q_{1 - 0.95/2} \approx 2$. So that would imply a confidence interval for the spot $S_t$ $$ J_\alpha = \left[ S_0 e^{(r - \frac{1}{2}\sigma^2)t - q_{1-\alpha/2}\sigma\sqrt{t}} ; S_0e^{(r - \frac{1}{2}\sigma^2)t + q_{1-\alpha/2}\sigma\sqrt{t}} \right] $$ In reality, the distribution of returns is not normal: it exhibits skewness and fat tails so your confidence interval would not be symetric and also larger.
For a very short period of 1 day, the approximation you gave is probably not too bad if you make sure that $\sigma$ is the annualized volatility not the daily volatility as you said (they differ by a factor $\sqrt{365}$ or $\sqrt{252}$ depending on your daycount convention).
## Answer by Stuart Allan (score 2)
https://quant.stackexchange.com/a/17747
The most common use for implied volatility in valuation is for asseing options or option like postions. A volatile instrument is likley to activate or put an option postion in the money just on the basis of its volatility rather than any fundamental change in the intrinsic or fair market value of the underlying. This needs to be taken into account when pricing an auction contract. Therefore, for many options or option like postions, implied vol. is used as a short hand for the price of the contract. It is useful as it serves as a bit of a standardized price. So if the contract is measured in dollars, pigs, widgets etc. implied vol. can be understood universally
## Answer by SmallChess (score 1)
https://quant.stackexchange.com/a/17749
One reason is that implied volatility measures the relative value of the option as the price of an option depends on various parameters. As everyone has its own pricing model, it's insane to quote all parameters. This little simple IV tells you everything you'd need to know for valuation.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.