UBC Theses and Dissertations
Optimal design of over-the-counter derivatives in a principal-agent framework : an existence result and numerical implementations Moreno-Bromberg, Santiago
This work lies in the intersection of Mathematical Finance, Mathematical Economics and Convex Analysis. In terms of the latter, a new result (to the author’s knowledge) on a Lipschitz property of the derivatives of a convex function is presented in the first chapter. An important result on its own, it might also provide a stepping stone to an extension to Hubert spaces of Alexandrov’s theorem on the second derivatives of convex functions. The second chapter considers the problem of Adverse Selection and op timal derivative design within a Principal-Agent framework. The principal’s income is exposed to non-hedgeable risk factors arising, for instance, from weather or climate phenomena. She evaluates her risk using a coherent and law invariant risk measure and tries to minimize her exposure by selling derivative securities on her income to individual agents. The agents have mean-variance preferences with heterogeneous risk aversion coefficients. An agent’s degree of risk aversion is private information and the principal only knows their overall distribution. It is shown that the principal’s risk mini mization problem has a solution and, in terms of the pricing schedule, the latter is unique. Finding a solution to the principal’s problem requires solving a varia tional problem with global convexity constraints. In general, this cannot be done in closed form. To this end an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions is presented in the fourth chapter of this work. Several examples are provided.
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