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From the information I've gathered the volatility smile concept did not exist prior to 1987. Since then it can be seen in foreign exchange markets and various other investments. Equity derivatives show volatility pairs and the smile tends to seen quite easily here.


The Sharpe Ratio and the T-Statistic for the hypothesis that returns are equal to the risk free rate, are closely related (occasionally some people mistakenly think they are the same). In fact: "The t-statistic will equal the Sharpe Ratio times the square root of N (the number of returns used for the calculation)." 1 So it makes sense to show both. Then, ...


If I'm correct Backtesting VaR usually boils down to two conditions: The unconditional coverage hypothesis : the probability of an ex-post violation must be equal to the coverage rate. (ie : if 0.01 confidence level, you should get 1% violation). You can test it with the Kupiec Test . The independence hypothesis, your VaR violations should be independent. ...


You can check the Euler-based risk attribution/ risk allocation, for example here:

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