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There are many ways answering this, here is one: We assume the asset price at $t=T$, $S_T = S_{T-1} \times (S_T / S_{T-1})$. Assuming continuous compounding, we can write, $S_T = S_{T-1} \times \exp(R_{T-1})$. Working the same way for the previous period, we get $S_{T} = S_{T-2} \times \exp(R_{T-1}+R_T)$. Working all the way back to the initial value of ...


I wrote this paper a couple of years ago where we discuss this kind of topic. On page 6, you see a formula that comes from a paper from Acerbi available in Szego's book: $$\sigma^2(ES^{(N)}_\alpha(X)) \overset{N>>1}{=} \frac{1}{N(1-\alpha)^2} \int_0^{F^{-1}(1-\alpha)} dx \int_0^{F^{-1}(1-\alpha)} dy \{ \min( F(x), F(y) ) - F(x)F(y) \}$$ This should ...

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