# 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?

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).