Three Essays on Contingent Claim Valuation in Incomplete Markets

The main purpose of this dissertation is to investigate the problems of contingent claim valuation in incomplete markets, especially focused on the pricing measures for Levy processes. This dissertation is constituted by three essays and each essay is self-contained. Essay 1 reviews some known results in an incomplete market in the case of exponential utility function. We also discuss the notion of utility indifference price for a contingent claim and investigate the asymptotic behavior of utility indifference price. Essay 2 uses the Esscher transform to construct a martingale measure in the framework of geometric Levy process. By means of a relation between exponential Levy process and stochastic exponential of Levy process, we have shown that a Levy process is a martingale if and only if its stochastic exponential is a martingale. Using this result, we also define a necessary condition for the Esscher measure to be the minimal entropy martingale measure. Essay 3 formulates an approach to computing the density process of the minimal entropy martingale measure for a jump-diffusion model and the stochastic volatility model by Barndorff-Nielsen and Shepherd. In addition, we also calculate the explicit forms of the minimal entropy martingale measure for those two models.Contents Introduction 1 Essay 1. Minimal entropy martingale measure and utility indifference price 4 1. Introduction 4 2. The notion of minimal entropy martingale measure 5 3. The dual formulation on an exponential utility function 11 4. Expressions for utility indifference prices 14 5. Conclusions 22 References 23 Essay 2. The pricing measure for geometric Levy processes under incomplete financial markets 25 1. Introduction 25 2. Levy processes and exponential Levy models 28 2.1 Levy process 28 2.2 Exponential Levy process and stochastic exponential of Levy process 32 3. Martingale measures for Levy processes 35 3.1 Girsanov theorem and Esscher measure 35 3.2 The Esscher measure for compound return process: Exponential Levy process 40 3.3 The Esscher measure for simple return process: Linear Levy process 44 4. Minimal entropy martingale measure for exponential Levy process 51 5. Conclusions 54 Appendix 55 References 58 Essay 3. The minimal entropy martingale measure for jump-diffusion models 60 1. Introduction 60 2. The jump-diffusion model 61 3. The minimal entropy martingale measure for jump-diffusion model 64 4. The minimal entropy martingale measure for the BN-S stochastic volatility model 72 5. Conclusions 79 Appendix 80 References 8