Entropy Coherent and Entropy Convex Measures of Risk - Eurandom
Entropy Coherent and Entropy Convex Measures of Risk - Eurandom
Entropy Coherent and Entropy Convex Measures of Risk - Eurandom
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Measuring ‘<strong>Risk</strong>’ (in the broad sense)<br />
◮ To measure the ‘risk’ related to a financial position X, the theories <strong>of</strong><br />
variational <strong>and</strong> homothetic preferences sketched above would lead to the<br />
definition <strong>of</strong> a loss functional L(X) = −U(X), satisfying<br />
L(X) = sup {EQ [φ(−X)] − α(Q)} <strong>and</strong><br />
Q∈Q<br />
L(X) = sup {β(Q)EQ [φ(−X)]} ,<br />
Q∈Q<br />
respectively, where φ(x) = −u(−x).<br />
◮ One could, then, look at the amount <strong>of</strong> capital one needs to hold in<br />
response to the position X, i.e., the negative certainty equivalent <strong>of</strong> X,<br />
denoted by mX, satisfying L(−mX) = φ(mX) = L(X), or equivalently,<br />
mX = φ −1sup {EQ [φ(−X)] − α(Q)}<strong>and</strong><br />
Q∈Q<br />
mX = φ −1sup {β(Q)EQ [φ(−X)]}.<br />
Q∈Q<br />
<strong>Entropy</strong> <strong>Coherent</strong> <strong>and</strong> <strong>Entropy</strong> <strong>Convex</strong> <strong>Measures</strong> <strong>of</strong> <strong>Risk</strong> Advances in Financial Mathematics, Eur<strong>and</strong>om, Eindhoven 8/40