On pricing contingent claims and expected utility maximization in two interest rates financial markets via completion

In this thesis, we consider the problems of hedging contingent claims and utility maximization in the framework of two interest rates financial markets. The upper and lower hedging prices are derived for European options by means of auxiliary completions of the initial market.

The well known financial markets consider a unique interest rates for both credit and deposit purposes, see for instance the Black-Scholes, Merton and Cox-Ingersoll-Ross models. The latter assumption allows to derive a unique price called the fair price on which the buyer and seller of the claim agree. Instead of a unique price, the hedging problem in the two interest rates financial markets admits an interval of initial prices on which buyer and seller can agree.

Using a similar technique of market completions, we consider the problem of an investor searching to maximize the expected utility of his terminal wealth, in the two interest rates financial markets. We show that under suitable conditions the latter problem can be reduced to a standard investment problem. This methodology is then adapted on the problem of shortfall risk minimization.

We finally consider the problem of pricing equity linked-life insurance contracts with guarantee in the two interest rates financial markets considered. To address the problem, we adapt a technique used by Melnikov to price equity linked-life insurance contract, in one interest rate jump-diffusion financial market.

On one hand, the motivation of these problems lies on the realities of financial markets where, the credit rate is always higher than the deposit rate. Taking into account such a constraint brings new difficulties in the problems of pricing contingent claims and utility maximization (investment).

On the other hand, the previous works in this topic were mostly made in a Black-Scholes and Cox-Ross-Rubinstein (binomial) frameworks. Yet, it is common knowledge in the area of finance that stock prices involve jumps, in order to take into account new extreme events. Therefore, besides a Black-Scholes model, we have considered a pure Merton (or pure jump) model and a two factors jump-diffusion model.

The thesis is organized as follows. In chapter 2, we consider a Black-Scholes and Merton model. We solve the hedging and utility maximization problems on a two interest rates financial market. Further, we give the solution of the shortfall risk minimization problem. In chapter 3, the same problems are considered but in a more general setting of two factors jump-diffusion model. In chapter 4, we consider the problem of pricing pure endowment life insurance contract with guarantee on the different interest rates financial markets. We give an approximation of the interval of survival probabilities and the corresponding policy-holder interval of ages. Finally, chapter 5 allows us to compare our hedging results on a jump-diffusion basis to those obtained by Bergman on a Black-Scholes model.