On p. 3 of Lorenzo Bergomi's book on Stochastic Volatility Modeling, there is the following assertion: Indeed, to a good approximation, the variance of returns scales linearly with their time scale, thus $\langle(\delta S)^2\rangle$ is of order $(\delta t)$ and $(\delta S)$ is of order $\sqrt{\delta t}$. He then continues by saying: The contributions at order one in $(\delta t)$ and order two in $(\delta S)$ are then both of order $\delta t$ while the cross term $\delta S\delta t$ and terms of higher order in $\delta S$ are of higher order in $\delta t$, thus become negligible as $\delta t \to 0$.
In my understanding, the notation means:
- $\delta S := dS$ (as in the Ito lemma)
- $\langle X \rangle = \mathbb{E}[X]$ , where $X$ is a random variable.
Since the text had not postulated any $(S_t)_{t\geq 0}$ dynamics up to this point, I seek an explanation for the footnote corresponding to the first of the sentences above: * The property that the variance of returns scales linearly with their time scale is equivalent to the property that returns have no serial correlation. Securities' returns do in fact exhibit some amount of serial correlation at varying time scales, of the order of several days down to shorter time scales and this is manifested in the existence of statistical arbitrage desks. Serial correlation itself is of no consequence for the pricing of derivatives, however the measure of realized volatility will depend on the time scale of returns used for its estimation.*
How is serial correlation defined in this context and what is its connection to quadratic variation? How can one prove that the variance of returns scales linearly with the returns' time scale $\Leftrightarrow$ the property that returns have no serial correlation ?
