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Stochastic Finance, 4th Edition by Alexander Schied, Hans Föllmer

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4Monetary measures of risk

In this chapter, we discuss the problem of quantifying the risk of a financial position. As in Chapter 2, such a position will be described by the corresponding payoff profile, that is, by a real-valued function X on some set of possible scenarios. In a probabilistic model, specified by a probability measure on scenarios, we could focus on the resulting distribution of X and try to measure the risk in terms of moments or quantiles. Note that a classical measure of risk such as the variance does not capture a basic asymmetry in the financial interpretation of X: Here it is the downside risk that matters. This asymmetry is taken into account by measures such as Value at Risk which are based on quantiles for the lower ...

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