t-day covariance matrix from daily return data

Question

How do I get t-day covariance matrix from daily return data?

I have an idea of how to calculate t-day variance from daily return data. From 14.6, 14.7 in Options, Futures and Other Derivatives by John Hull, I know that assuming stock price follows Ito's process, t-day log return follows a normal distribution and its expected value and variance are just those of daily log return scaled by t. Based on these and the properties of lognormal distributions, I can do the following step:

1. compute daily log return from daily return
2. compute the expected value and variance of daily log return $$\operatorname{E}[\log r_d] = \mu$$ $$\operatorname{var}(\log r_d) = \sigma^2$$
3. compute the expected value and variance of t-day log return by scaling daily values by t $$\operatorname{E}[r_t] = \mu t$$ $$\operatorname{var}(\log r_t) = \sigma^2t$$
4. compute the expected value and variance of return by the property of log normal distributions $$\operatorname{E}[r_t] = \exp(\mu t + \frac{\sigma^2}{2})$$ $$\operatorname{var}(r_t) = [\exp(\sigma^{2})-1]\exp(2\mu +\sigma ^{2})$$

If the above are valid, what are the similar reasonings for covariance matrix if I am looking at more than a single stock?

I see in practice some people seem to just scale covariance by 252 when calculating annualized covariance. Also, they don't seem to handle the difference of return and log return. But I don't understand why.

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Answer 1

Let $M_d$ be the covariance matrix for the daily log-values.

Then $M_t = t M_d$ is the covariance matrix for the log-values after $t$ days.

From this, standard formulas give the covariances of the values, as in this answer.

One way to write those formulas is that if $X$ and $Y$ are jointly normal, then $$\text{Cov}[e^X,e^Y]\, =E[e^X]\: E[e^Y]\:(e^{\text{Cov}(X,Y)}-1)$$