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:
- compute daily log return from daily return
- compute the expected value and variance of daily log return $$\operatorname{E}[\log r_d] = \mu$$ $$\operatorname{var}(\log r_d) = \sigma^2$$
- compute the expected value and variance of t-day log return by scaling daily values by t $$\operatorname{E}[r_t] = \mu t$$ $$\operatorname{var}(\log r_t) = \sigma^2t$$
- 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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