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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

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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).

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  • $\begingroup$ Thank you, very good explanation. Just to be sure, is it a better approximation to use a lognormal distribution for the returns instead of this normal distribution? And in that case, how can we handle the fact that a lognormal distribution is always positive, but the returns can be negative? $\endgroup$
    – Sithered
    May 12, 2015 at 8:33
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    $\begingroup$ Using a log-normal distribution to model the returns is makes no sense since as you said returns can be negative. $\endgroup$
    – AFK
    May 12, 2015 at 18:46
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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

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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.

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