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1. Brief Description of the Dissertation

This thesis is devoted to finding robust and accurate finite difference schemes for an important problem in finance, namely the Heston and SABR models. The author produces a scheme which satisfies the above two requirements.

The dissertation consists of 9 chapters. They are logically organized in general.

Chapter 1 Introduction and Scope of Work

Chapter 2 Motivation for the PDEs for Heston and SABR models

Chapter 3 One-dimensional convection-diffusion equation and the convection equation. Stability and convergence of well-known FD schemes

Chapter 4 An analysis of some standard FD schemes and stability issues. This is an important chapter because of the use of extrapolation techniques and a sharper form of stability

Chapter 5 FDM for two-factor models, for example splitting methods and the Yanenko scheme for mixed derivatives

Chapter 6 A good analysis of FD schemes. This is based on the Lawson-Morris method.

Chapter 7 Non-uniform grids and non-smooth payoff functions. Nice to have but not the main theme of the thesis.

Chapter 8 A summary of the various PDEs and boundary conditions in the thesis.

Chapter 9 A short chapter on the results of numerical experiments. The author concludes (section 9.5) that fitting in combination with splitting and Yanenko methods is a robust scheme.

2. Analysis of Dissertation

This thesis uses some of the most robust methods for solving Heston PDEs. The author starts with one-factor models to motivate the work and then progresses to more advanced two-factor models which he analyses using techniques that he has found in the literature.