Integrated Variance and Its Realized-Return Estimate

Summary

The document introduces integrated variance as a measure of return variation accumulated over a time interval. In a continuous-time return model with drift and volatility, the interval’s integrated variance is the time integral of instantaneous variance. When volatility is constant, this accumulation corresponds to variance building over the length of the interval; when volatility varies over time, the integral accommodates that variation.

It connects the theoretical quantity to an empirical estimate: summing squared returns over the interval gives an estimate, linking integrated variance to realized volatility. The text motivates variance measurement for tasks such as risk hedging, exposure assessment, and portfolio optimization. Its explanation is brief and assumes a standard diffusion-style return process; it does not discuss estimator bias, sampling frequency, market microstructure effects, or conditions under which the squared-return estimate is reliable.

Key ideas

• Integrated variance accumulates instantaneous variance across a chosen time interval.
• A continuous-time return model provides the basis for defining the quantity.
• A sum of squared returns can estimate integrated variance and is related to realized volatility.
• The measure can support risk assessment, hedging, and portfolio decisions.

Tags

# Integrated volatility


# Integrated volatility












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?

## Answer by Stefan Voigt (score 4, accepted)

https://quant.stackexchange.com/a/22974

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.