Shape Constrained Optimization

All Rights Reserved ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Abstract In the first part of this work, we study a particular class of infinite dimensional linear programs on the value of a function at a given number of points, with the additional constraint that this function be convex. Convexity is shown to be the key ingredient making these problems tractable. We detail some applications, with a particular focus on arbitrage constraints between call options. In the second part, we use the results of chapter one to compute tractable relax-ations to some multivariate or basket option pricing problems. We then derive tight price bounds on basket options in some particular cases. Finally, part three uses some recent results in moment theory and semidefinite programming to refine the convex relaxation techniques of part two and compute tighter constraints linking the prices of basket options.