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.

Finance risk management has been playing an increasingly important role in the finance sector, to analyse finance data and to prevent any potential crisis. It has been widely recognised that Value at Risk (VaR) is an effective method for finance risk management and evaluation. This thesis conducts a comprehensive review on a number of VaR methods and discusses in depth their strengths and limitations. Among these VaR methods, Monte Carlo simulation and analysis has proven to be the most accurate VaR method in finance risk evaluation due to its strong modelling capabilities. However, one major challenge in Monte Carlo analysis is its high computing complexity of O(n²). To speed up the computation in Monte Carlo analysis, this thesis parallelises Monte Carlo using the MapReduce model, which has become a major software programming model in support of data intensive applications. MapReduce consists of two functions - Map and Reduce. The Map function segments a large data set into small data chunks and distribute these data chunks among a number of computers for processing in parallel with a Mapper processing a data chunk on a computing node. The Reduce function collects the results generated by these Map nodes (Mappers) and generates an output. The parallel Monte Carlo is evaluated initially in a small scale MapReduce experimental environment, and subsequently evaluated in a large scale simulation environment. Both experimental and simulation results show that the MapReduce based parallel Monte Carlo is greatly faster than the sequential Monte Carlo in computation, and the accuracy level is maintained as well. In data intensive applications, moving huge volumes of data among the computing nodes could incur high overhead in communication. To address this issue, this thesis further considers data locality in the MapReduce based parallel Monte Carlo, and evaluates the impacts of data locality on the performance in computation.