Two topics in financial mathematics : Forward utility and consumption functions & Hedging with variance swaps in infinite dimensions
Financial Mathematics is often presented as being composed of two main branches: one dealing with investment and consumption, with the aim of answering the now ancient question of how people should invest and spend their money, and the other dealing with the pricing and hedging of derivative instruments. This distinction between both branches of Financial Mathematics is reflected in my thesis, which is a compilation of two very different subjects on which I have worked during the past three years. The first chapter, entitled “Forward Utility and Consumption Functions”, contributes to the investment branch of Financial Mathematics. Forward utilities have been introduced (under different names) a few years ago by Musiela and Zariphopoulou on the one hand, and by Henderson and Hobson on the other hand. Their idea is to define families (indexed by time and randomness) of utility functions which make the investment decisions of agents consistent over time. The contribution of this chapter is to extend the definition of forward utilities by adding consumption into the story and by giving explicit ways of constructing consumption functions from utilities and vice versa. The last part of this first chapter characterizes, in a Laplace integral form, the decreasing forward utilities (without consumption, and subject to some regularity conditions). The second chapter, entitled “Hedging with Variance Swaps in Infinite Dimensions”, contributes to the derivatives pricing and hedging branch of Financial Mathematics. It is at the interface between the works of Buehler, who has shown that one could apply the HJM framework to model (forward) variance swaps curves, and the works of Carmona and Tehranchi, who have proved that infinite dimensional interest rates models can display theoretically nice features which are absent from their finite dimensional counterpart, such as uniqueness and maturity-specific properties of hedging portfolios for contingent claims. After an introductory section on terminology and after explaining the Buehler-HJM framework, I give a concrete example of finite dimensional model and show its (theoretical) shortcomings. I then port some results of Carmona and Tehranchi from interest rates modelling to variance swaps modelling in infinite dimensions and finally give a concrete example of model and of classical payoffs to which the results apply. Because many results and prerequisites to this chapter are quite technical, I have added a short appendix, giving modest introductions to infinite dimensional stochastic analysis, Malliavin calculus and SPDEs in Hilbert spaces.