# Serial correlation, quadratic variation and variance of returns

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 ?

• If returns have no serial correlation (i.e. no autocorrelation) then the mean and variance of a sum of those returns is given by the sum of their mean and sum of their variance, hence the linear scaling, regardless of the distribution assumptions. The fact that equity returns have no serial correlation is a well known empirical observation (see paper on stylised market facts by Rama Cont) Jun 15 '20 at 6:16
• @Quantuple Is this the paper you mentioned -- R. Cont (2000), Empirical properties of asset returns: stylized facts and statistical issues ? Do you know if other asset classes (e.g. FX) would exhibit serial correlation? Jun 15 '20 at 14:21
• It is indeed the paper I was mentioning. It depends on which factor and the time scale of the returns you are looking at. Jun 15 '20 at 14:28
• @Quantuple: Yes, the notion of serial correlation critically depends on the time scale. For example, intraday, it's quite clear that a stock's returns is serially correlated. The more difficult question is when the turning points occur. Jun 16 '20 at 3:31

Hi: That statement is made understand the assumption that a stock's return is composed of a drift term and a random component and the stock price, $$S_t$$ is then said to follow geometric brownian motion:
$$dS_t = \mu \times S_t \times dt + \sigma \times S_t \times dW_t$$
In above, $$dW_t$$ denotes the derivative of brownian motion ( note that the derivative really doesn't exist but they say that anyway ) and is often referred to as white noise.
Brownian motion has increments that are independent and it's serial correlation, $$corr(W_s,W_t) = min(s,t)$$ but only for the same stock. For different stocks, it's zero. This info will be explained with all the details ( regarding quadratic variation, Ito's Lemma etc ) in any decent probability or finance book.