Can someone give me an explanation of what integrated volatility is (and possibly why it is preferred) versus a standard measure of volatility eg variance?
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In a standard approach you would think about the evolution of a return process in the following form: $$dr_t=\mu dt+\sigma dW_t,$$ where for the sake of simplicity I assumed constant volatility and drift ($\mu$ and $\sigma$ can also depend on the time parameter $t$). Often you will be interested into the variance of your stock returns (for example to hedge your risks, quantify your exposure to risk or for portfolio optimization) during a certain time period $[\tau,\tau-h]$. Standard Itô Calculus gives you that the 'aggregated' volatility over this time interval is just $$\int_\tau ^{\tau-h}\sigma^2 dt.$$ This term is called integrated variance and can be estimated via the sum of squared returns during this period (this gives you the close connection to realized volatility.
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2$\begingroup$ I would add: If you think about the BS formula, the variance $\sigma^2$ never appears by itself but always in conjunction with $T$. Those terms $\sigma^2 T$ represent in a sense the integrated volatility for the constant vol case. If you believe that vol is subject to change in a deterministic fashion, you could replace those terms with whatever integ vola you think appropriate. $\endgroup$– Alex CCommented Jan 29, 2016 at 0:09