## Fachbereich Mathematik

This thesis covers two important fields in financial mathematics, namely the continuous time portfolio optimisation and credit risk modelling. We analyse optimisation problems of portfolios of Call and Put options on the stock and/or the zero coupon bond issued by a firm with default risk. We use the martingale approach for dynamic optimisation problems. Our findings show that the riskier the option gets, the less proportion of his wealth the investor allocates to the risky asset. Further, we analyse the Credit Default Swap (CDS) market quotes on the Eurobonds issued by Turkish sovereign for building the term structure of the sovereign credit risk. Two methods are introduced and compared for bootstrapping the risk-neutral probabilities of default (PD) in an intensity based (or reduced form) credit risk modelling approach. We compare the market-implied PDs with the actual PDs reported by credit rating agencies based on historical experience. Our results highlight the market price of the sovereign credit risk depending on the assigned rating category in the sampling period. Finally, we find an optimal leverage strategy for delivering the payments promised by a Constant Proportion Debt Obligation (CPDO). The problem is solved via the introduction and explicit solution of a stochastic control problem by transforming the related Hamilton-Jacobi-Bellman Equation into its dual. Contrary to the industry practise, the optimal leverage function we derive is a non-linear function of the CPDO asset value. The simulations show promising behaviour of the optimal leverage function compared with the one popular among practitioners.

In this thesis we consider the problem of maximizing the growth rate with proportional and fixed costs in a framework with one bond and one stock, which is modeled as a jump diffusion with compound Poisson jumps. Following the approach from [1], we prove that in this framework it is optimal for an investor to follow a CB-strategy. The boundaries depend only on the parameters of the underlying stock and bond. Now it is natural to ask for the investor who follows a CB-strategy which is given by the stopping times \((\tau_i)_{i\in\mathbb N}\) and impulses \((\eta_i)_{i\in\mathbb N}\) how often he has to rebalance. In other words we want to obtain the limit of the inter trading times
\[
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n(\tau_{i+1}-\tau_{i}).
\]
We are able to obtain this limit which is given by the expected first exit time of the risky fraction process from some interval under the invariant measure of the Markov chain \((\eta_i)_{i\in\mathbb N}\) using the Ergodic Theorem from von Neumann and Birkhoff. In general, it is difficult to obtain the expectation of the first exit time for the process with jumps. Because of the jump part, when the process crosses the boundaries of the interval an overshoot may occur which makes it difficult to obtain the distribution. Nevertheless we can obtain the first exit time if the process has only negative jumps using scale functions. The main difficulty of this approach is that the scale functions are known only up to their Laplace transforms. In [2] and [3] the closed-form expression for the scale function of the Levy process with phase-type distributed jumps is obtained. Phase-type distributions build a rich class of positive-valued distributions: the exponential, hyperexponential, Erlang, hyper-Erlang and Coxian distributions. Since the scale function is given as a function in a closed form we can differentiate to obtain the expected first exit time using the fluctuation identities explicitly.
[1] Irle, A. and Sass,J.: Optimal portfolio policies under fixed and proportional transaction costs, Advances in Applied Probability 38, 916-942.
[2] Egami, M., Yamazaki, K.: On scale functions of spectrally negative Levy processes with phase-type jumps, working paper, July 3.
[3]Egami, M., Yamazaki, K.: Precautionary measures for credit risk management in jump models, working paper, June 17.