Models estocàstics del tipus d'interès

Abstract

The aim of this final project is to study the pricing of zero-coupon bonds of different interest rate models in a continuous-time market in the absence of arbitrage opportunities, specifically, the Vasicek model and the Cox-Ingersoll-Ross model. First, this study needs to analyze the basis of the stochastic modeling of continuous-time market which includes to study some notions about the stochastic calculus. So, first the chapters 1 and 2 have some useful concepts and results of stochastic calculus like the Brownian motions, the stochastic integrals, the Itô calculus, the stochastic differential equations... Then, in the chapter 3 some economic concepts, the model of continuous-time market and the concept of portfolio self-financing, are defined; and also, this Black-Scholes pricing are studied. Later, in the chapter 4, some common models short term interest rate models are introduced. Last, in the chapter 5, the pricing of zero-coupon bonds are studied following the two named models in the former chapter, the Vasicek model and Cox-Ingersoll-Ross model, using pricing from chapter 3. During all the project, we suppose all the affirmations about finite random variables and stochastic processes are true P almost surely. To sum up, we have used different resources but overall, we have based on the books Introduction to stochastic calculus applied to finance ([Lam]) and An elementary introduction to stochastic interest rate modeling ([Pri]).