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.

  • 1
    $\begingroup$ 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. $\endgroup$ – Quantuple Sep 11 '20 at 6:16
  • $\begingroup$ I agree with all the above when $\delta t -> 0$. But how do you relate the variance of $ln(๐‘†_{๐‘ก_๐‘›}/๐‘†_{๐‘ก_0})$ with the variance of ๐‘†_{๐‘ก_๐‘›}/๐‘†_{๐‘ก_0} - 1 ?In that case $\delta t$ is large. Taylor formula has a strong convexity term in this case... $\endgroup$ – noob-mathematician Sep 11 '20 at 9:51

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