Optimal contingent claims.
This is an elegant piece of mathematics applied to finance: a clear conceptualisation of a natural investment question. Suppose \(X\) is a certain underlying, i.e. a random variable. Financial markets value contingent claims on \(X\) by averaging and discounting with a certain probability \(P\). But ``you know'' that the ``right'' probability is different, namely, \(Q\). You want to invest a certain amount \(\alpha\), but you are willing to admit up to a certain risk level, \(\gamma\), and, of course, you would like to obtain out of this investment the maximum average return. Which derivative on \(X\) should you choose? In other terms you want to solve the following optimisation problem: -- Constraints: \(E_P(g(X))\leq \alpha\); \(V_Q(g(X))\leq \gamma^2\), -- Maximize \(E_Q(g(X))\) over ``all'' choices of \(g\). The answer is choose: \(g(x)= a+ b[{dQ\over dP}(x)]\) with appropriate choice of the constants \(a\) and \(b\). A number of quite interesting application follows.
scientific article; zbMATH DE number 1906235