Return Serial Correlation, Variance Scaling, and Quadratic Variation
Summary
The document asks how return serial correlation relates to the claim that return variance grows linearly with the length of the observation interval, a scaling used in stochastic volatility modeling and quadratic variation arguments. It also asks how to interpret the claim that this variance scaling is equivalent to returns having no serial correlation. The response frames the usual intuition through geometric Brownian motion: price changes combine a drift component with a random Brownian increment, whose independent increments support the standard square-root-of-time scaling for return volatility.
The answer characterizes Brownian increments as white noise and points readers toward stochastic calculus texts for details on quadratic variation and Itô’s lemma. It does not prove the proposed equivalence or define serial correlation at specific return lags. The equivalence requires assumptions about stationarity and dependence structure; zero autocorrelation alone need not imply independent increments or guarantee the scaling in every process. The discussion also notes that realized volatility estimates can depend on the time scale used when returns are serially correlated.
Key ideas
- Under geometric Brownian motion, price changes combine drift with random Brownian increments.
- Independent Brownian increments underpin the familiar linear variance and square-root volatility scaling with time.
- The question concerns how return autocorrelation affects variance scaling and quadratic variation.
- The response offers intuition and references but does not prove the claimed equivalence.
- Variance scaling cannot generally be inferred from zero autocorrelation without additional assumptions.
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Full text
# Serial correlation, quadratic variation and variance of returns
# 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 ?
## Answer by mark leeds (score 1)
https://quant.stackexchange.com/a/54912
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.
For a gentle intro, Mikosch's "Elementary Stochastic Calculus With Finance in View" is good but then there are more advanced ones like Karatzas and Schreve or Protter etc. It depends on your background and what you want to know. The levels can vary a lot.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.