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You are looking for the distribution of $X$ conditional on $X<0$, in particular for the conditional expectation $$E[X|X<0].$$ For $X$ normal with mean $0$ and variance $\sigma^2$, that turns out to be equal to $$ E[X1_{X<0}]/P(X<0) = 2\sigma\int_{-\infty}^0 x \phi(x) dx $$ $$=2 \sigma \int_{-\infty}^0 -\phi’(x) dx = -\sigma\sqrt{2/\pi}$$ Note ...


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You’re looking for a CTE or TVaR value. Check out Wikipedia or the associated references for formulas. https://en.m.wikipedia.org/wiki/Tail_value_at_risk http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf For the record though, you are probably underestimating your risk if you use a normal distribution. You should try using ...


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The other answers have given a good qualitative description of what the VIX measures. In this answer, I will try to give a comprehensive quantitative overview of how the VIX formula works. What is the VIX? The CBOE Volatility Index (VIX) is an index which measures the stock market’s expectation of the future 30-day volatility of the S&P-500 index. This ...


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The way you have generated the paths, you expect the annualized vol to be 0.8 (since you are multiplying by $\sqrt{dt}$, and $dt$ is expressed in years. On the realized calculation, you can get that back on the raw data with np.std(np.diff(np.log(df0['values']),1))*np.sqrt(365*24) or on the resampled daily data with respectively np.std(np.diff(np.log(df_1d[&...


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