# 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.

• IMHO the key assumption is not that indiv returns are Gaussian, but that they are independent. Rather than considering arithmetic returns like you do ($r_{a,t_i} = S_{t_i}/S_{t_{i-1}}-1$), consider geometric returns $r_i = \ln(S_{t_i}/S_{t_{i-1}}$ (knowing that for short $\delta t$ you get that $r_a \approx r_g$). Log-returns are interesting since the return over $N$ succesive periods $\ln(S_{t_n}/S_{t_0})$ can be written as $r_N = \sum_{i=1}^N r_i$. Then if the returns are independent you get that $var(r_N) = N var(r_i)$, hence variance becoming a linear function of the time period. – Quantuple Sep 11 '20 at 6:16
• I agree with all the above when $\delta t -> 0$. But how do you relate the variance of $ln(𝑆_{𝑡_𝑛}/𝑆_{𝑡_0})$ with the variance of 𝑆_{𝑡_𝑛}/𝑆_{𝑡_0} - 1 ?In that case $\delta t$ is large. Taylor formula has a strong convexity term in this case... – noob-mathematician Sep 11 '20 at 9:51