Can anybody explain how the $\phi_i$ became $\mu$ and how $\phi_i^2$ became $\sigma^2$. Am I correct to assume that since $\phi_i$ is the outcome, $\mu$ is the average of the outcome? But I don't understand how $\phi_i^2$ is the variance $\sigma^2$.

Extract from Paul Wilmott Introduces Quantitative Finance:

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  • $\begingroup$ The average of the $\phi^2_i$ is not equal to $\sigma^2$, but it is a close approximation if $\mu<<\sigma$. THat's why they say "assuming that the mean is small". $\endgroup$ – Alex C Jun 25 '18 at 4:43

Suppose we decompose each $\phi_i$ into the mean $\mu$ plus some extra bit $\kappa_i$. Then what would the squared term look like?

$$\phi_i^2 = (\mu + \kappa_i)^2 = \mu^2 + \kappa_i^2 + 2\mu\kappa_i$$

Now we've been told that the mean is small, which is generally code for "the square can be neglected", leaving us with $\kappa^2$ and $2\mu\kappa$.

We are not, however, looking for the value of $\phi_i^2$, but for $\mathbb{E}(\phi_i^2)$. Expectations are linear, i.e. $\mathbb{E}(a+b) = \mathbb{E}(a)+\mathbb{E}(b)$, so

$$\mathbb{E}(\phi_i^2) = \mathbb{E}(\kappa_i^2) + \mathbb{E}(2\mu\kappa_i)$$

$\mathbb{E}(\kappa_i^2)$ is by definition the variance (it is the expectation of the square of the deviations), thus $=\sigma^2$.

$\mathbb{E}(2\mu\kappa_i) = 2\mu\mathbb{E}(\kappa_i)$. Remembering our decomposition, $\kappa_i$ is the extra bit, so again by definition, it has a mean 0, so $\mathbb{E}(\kappa_i) = 0$.

$$\mathbb{E}(\phi_i^2) = \sigma^2$$

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  • $\begingroup$ Thanks Phil! By the way, I'd like to have an intuitive understanding of how you explained it. Can you recommend a basic book for me to read to avoid getting stuck with equations like these? My background is electrical engineering so I have a bit of background in calculus albeit a bit rusty. $\endgroup$ – gaston Jun 25 '18 at 13:38
  • $\begingroup$ The usual recommendations are Shreve, or Baxter & Rennie. I found Baxter & Rennie good for a more visceral grasp of the idea of measures and expectations. $\endgroup$ – Phil H Jun 26 '18 at 21:08

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