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We define a convex risk measure as $$ \rho( \lambda X_1 + (1-\lambda) X_2) \le \lambda \rho( X_1 ) + (1-\lambda) \rho(X_2), $$ for $\lambda \in(0,1) $. A coherent risk measure is subadditive and homogeneous thus for coherent $\rho$ we get: $$ \rho( \lambda X_1 + (1-\lambda) X_2) \le \rho( \lambda X_1) + \rho( (1-\lambda) X_2) $$ by subadditivity and $$ ...


Translation invariance of a risk measure $\rho$ is defined as $$ \rho(X+k) = \rho(X)-k, $$ where $X$ is a random variable such that $\rho(X)$ exists and $k$ is a constant. The meaning is that if I add an amount $k$ to my risky positions then the risk is reduced by this amount. For VaR we consider the case that $X$ has a continuous distribution and that it ...


As always I recommend reading Rennie and Baxter for an introduction to option pricing that's not too technical and gives intuition about how it all works.

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