Why Asset Return Variance Scales with the Time Step

Summary

The document raises a question about the time scaling of asset return variance in a Gaussian framework. It asks why variance over a small interval is expected to be of order the interval length, and how that claim relates to variance remaining finite as the interval shrinks. It also proposes estimating return variance from independent observations of simple returns over intervals of the chosen length.

No answer or derivation is included, so the document does not establish the scaling result or resolve the sampling question. In particular, it does not state the process assumptions needed to connect interval returns to variance over longer horizons. It is best treated as a prompt about time aggregation and return modeling rather than as a complete explanation or empirical study.

Key ideas

• The document asks why Gaussian asset return variance is expected to scale with the length of a small time interval.
• It proposes estimating interval return variance from a sample of independent returns.
• The source provides no derivation or answer to the scaling question.
• The claim’s assumptions and connection to longer horizon variance are left unspecified.

Tags

# variance of asset returns linear for time


# variance of asset returns linear for time












I am reading Wilmott's book, "Quantitative Finance" and try to understand the derivation that the variance of asset-returns, $V[\Delta S/S]$, is a linear function of the time step $\delta t$.

If we assume that the return distribution is gaussian, then he claims that the variance of asset-returns should $\in O(\delta t)$-functions. He claims that this should be the case because as $\delta t \rightarrow 0$, the variance remains finite. Do you see why?

If we want to approximate the variance of asset returns over a period $\delta t$, I guess one can consider $N$-independent samples of $R_i = \frac{S_{i+\delta t}-S_i}{S_i}$ and approximate the variance by:

$$V[\frac{S_{t+\delta t}-S_t}{S_t}] \approx \frac{1}{N-1}\sum_{i=1}^N(R_i-\bar{R})^2$$

Anyone knows how to prove that the RHS should be linear for $\delta t$?

Thanks in advance.

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