Cuatro ensayos sobre valoracion de derivados y estrategias de inversion

Resumen

In this thesis, we study some models used in pricing financial derivatives. In particular, we will use two kinds of valuation techniques, Replication and Indifference Pricing. We will deal with the real-time valuation problem, option pricing with variable interest rates and optimal investment and option pricing when transaction costs are present. The analysis and design of appropriate numerical techniques will be necessary in this study. The outline of the thesis is as follows: In Chapter 1 we will propose a Reduced Bases functions method for the real-time valuation problem. Under the idea of paying computational cost just once, in a first step we will construct an interpolant polynomial in several variables that admits \textit{tensorial valuation} (computes option prices for several parameter values simultaneously) and it is computationally very efficient. Interpolant polynomials of models with several parameters suffer of the so called ``Curse of dimensionality'', which blows their storage cost. In a second step, a Reduced Bases function approach, designed and applied to the previous polynomial is presented. This approach will drastically reduce the impact of the mentioned curse, obtaining a low-storage and fast multiple-evaluation polynomial. The algorithms are described so that they can be applied to n-variable models and option types. Analysis of performance with a particular model (GARCH) and real market prices will also be presented. Chapter 2 is devoted to model extension incorporating variable interest rates. We will present a discrete GARCH model with deterministic variable interest rates and an extension of Heston's SV model with a bond that is able to incorporate a stochastic component. The existence, under certain hypothesis, of a risk free measure for the discrete model is proved and the stock's dynamics is computed. For the continuous model, a semi-explicit formula is obtained and analysis with market data are carried out. In Chapter 3, in a scenario with proportional transaction costs, a mesh-adaptative Chebyshev collocation method is developed to numerically solve the Optimal Investment problem under Potential Utility. Explicit analytical formulas can be obtained for certain interesting cases. Consequently, they are employed in the analysis of the numerical error of the method. Chapter 4 is dedicated to Option Pricing with transaction costs under Exponential Utility. After dealing with the numerical problems associated to the Exponential Utility function, a new non-linear PDE is obtained and a Fourier pseudospectral method is proposed to numerically solve it. Theoretical stability and convergence of the method is proved. We also carry out numerical experiments to test the effects in the option price of incorporating transaction costs.