# Implied volatility and realized volatility

There are many articles and posts here claiming that the implied volatility is the expectation of future realized volatility. I don't understand. To begin with, isn't implied volatility homogeneous to a volatility and realized volatility homogeneous to a volatility times the square root of a time? How something not in the same dimension as realized volatility be the expectation of it?

• no idea what you're talking about. Just google a primer on implied volatility. May 17 at 23:04
• @EdwardWatson For example here quant.stackexchange.com/questions/32951/… the answer claims so but I don't see how his equation helps him to say that implied volatilities can be regarded as the market’s expectation of future realized volatilities? May 17 at 23:13

The link you posted is a variance swap. In plain English, this equation means that the fair variance swap value can be shown to equal the integral of weighted prices of out-of-the-money options over all strikes, with weights inversely proportional to strike squared. That is only indirectly related to implied vol (often computed from a vol surface). However, in terms of variance swaps, you definitely have a one for one relation to realized vol (as realized variance is the squared realized vol; which is exactly how payoffs are defined). $$N_{\text{var}} (\sigma^2_{\text{realized}} - \sigma^2_{K})$$