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I'd like to understand why $\Sigma$ is the same under both measures $\mathbb{P}$ and $\mathbb{Q}$. Is it an assumption or a general fact based on theoretical concepts?

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    $\begingroup$ This is not true in real life. This is a big assumption. You can write down simple models in which there is no variance risk premium but in general these two matrices are very different. In fact, forget about matrices. The risk-neutral variance of one individual stock is different to its real-world variance (as investors are risk-averse). Under Black Scholes dynamics both variances are the same but under Heston dynamics (stochastic volatility model) you observe a variance risk premium. Its sign, magnitude and parametrization is up for debate but its existence is not really. $\endgroup$ – Alex Jul 28 at 20:26
  • $\begingroup$ what is $\mathbb{P}$ and $\mathbb{Q}$? $\endgroup$ – develarist Jul 29 at 13:27
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Just to expand on Alex answer.

Empirically it is simply not true. Focusing on the diagonal of the variance-covariance matrix, we know that there is a large variance risk premium. Take a look at table 3 from Carr and Wu (2009).

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Regarding covariances we do not have much evidence, because there are no options on every single pair of stocks. However, we do know that there is an overall market correlation risk premium (Driessen, Maenhout and Vilkov (2008))

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