Exit problems of Lévy processes with applications in finance

Abstract

In this thesis we study the pricing of options of American type in a continuous time setting. We begin with a general introduction where we briefly sketch history and different aspects of the option pricing problem. In the first chapter we consider four perpetual options of American type driven by a geometric Brownian motion: the American put and call, the Russian option and the integral option. We derive their values exploiting properties of Brownian motion and Bessel processes. From a practical point of view perpetual options do not seem of much use, since in practice the time of expiration is always finite. However, following an appealing idea of Peter Carr, we build an approximating sequence of perpetual-type options and prove this converges pointwise to the value of the corresponding finite time American option. Next we compute for the mentioned options the first approximation. The second chapter proposes the class of ``phase type Lévy processes'' as a new model for the stock price. This is a class of jump-diffusions which is dense in all Lévy processes and whose positive and negative jumps form compound Poisson processes with jump distributions of phase type. We illustrate its analytical tractability by pricing the perpetual American put and Russian option under this model. In the third chapter we study the same problems but now for the class of Lévy processes without negative jumps. We restrict ourselves to this class, since it contains already a lot of the rich structure of Lévy processes while still being analytically tractable due to many available results exploiting the fact that the jumps of the Lévy process have one sign. A recent study of Carr and Wu offers empirical evidence supporting the case of a model where the risky asset is driven by a spectrally negative Lévy process. For this class of Lévy processes, we review theory on first exit times of finite and semi-infinite intervals. Subsequently, we determine the Laplace transform of the exit time and exit position from an interval containing the origin of the process reflected at its supremum. The proof relies on Itô -excursion theory. The fourth chapter complements the study of the previous chapter. We find the Laplace transform of the first exit time of a finite interval containing the origin of the process reflected at its infimum. Then we turn our attention to these reflected processes killed upon leaving a finite interval containing zero and determine their resolvent measures. Invoking the R-theory of irreducible Markov chains developed by Tuomen and Tweedie, we are able to give a relatively complete description of the ergodic behaviour of their transition probabilities. The obtained results on Lévy processes in chapters 3 and 4 also have applications in the context of the theories of queueing, dams and insurance risk. Finally, the fifth chapter considers the utility-optimisation problem of an agent that operates in a general semimartingale market and seeks to trade so as to maximise his utility from inter-temporal consumption and final wealth. In this setting existence is established following a direct variational approach. Also a characterisation for the optimal consumption and final wealth plan is given.