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Variance is additive also for any other distribution. The reason that variance is additive is because of the assumption of independent increments - ie, the change in underlying value between 2 moments is assumed to be independent of the change in value between 2 later moments


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If you use log-returns, then it is true that the return over n periods is the sum of the returns over each subperiod (e.g. the 10 day return is the sum of 10 1-day returns) $$ R = \sum_{i=1}^n r_i. $$ If we now look at the variance of $R$ then we get $$ VAR(R) = VAR( \sum_{i=1}^n r_i ), $$ if we assume that the returns are uncorrelated then we get $$ VAR(R) ...


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You should clarify a bit your question. Your first computation with $$Var((1+R_t)(1+R_{t+1})-1)$$ is ambiguous. What do you mean by $Var$ here? You have time subscripts $t$ and $t + 1$ so unless you specify which filtration you compute your variance, it is unclear what you're computing. If you are computing at $t = 0$, then this is not a two period return ...


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To answer your questions: Is the trading p&l meant to be the delta-hedging p&l? Yes, in his example it concerns delta hedged pnl. how come p&l is raising steadily even when stock price is rising? the trader should be losing money on the delta hedging because he is short gamma? He is short gamma but long theta. He is initially making money ...



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