Ayşegül İşcanoğlu Çekiç, Ph.D.

Department of Financial Mathematics

February 2011, 84 pages

*Supervisor*: Ömür Uğur (Institute of Applied Mathematics, Middle East Technical University, Ankara)*Co-supervisor*: Ralf Korn (Department of Mathematics, University of Kaiserslautern, Germany)

### Abstract

A Constant Proportion Debt Obligation is a credit derivative which has been introduced to generate a surplus return over a riskless market return. The surplus payments should be obtained by synthetically investing in a risky asset (such as a credit index) and using a linear leverage strategy which is capped for bounding the risk.

In this thesis, we investigate two approaches for investigation of constant proportion debt obligations. First, we search for an optimal leverage strategy which minimises the mean-square distance between the final payment and the final wealth of constant proportion debt obligation by the use of optimal control methods. We show that the optimal leverage function for constant proportion debt obligations in a mean-square sense coincides with the one used in practice for geometric type diffusion processes. However, the optimal strategy will lead to a shortfall for some cases.

The second approach of this thesis is to develop a pricing formula for constant proportion debt obligations. To do so, we consider both the early defaults and the default on the final payoff features of constant proportion debt obligations. We observe that a constant proportion debt obligation can be modelled as a barrier option with rebate. In this respect, given the knowledge on barrier options, the pricing equation is derived for a particular leverage strategy.

*Keywords*: credit risk, credit derivatives, stochastic control, Laplace transforms, jump diffusion processes