Computational Methods for Quantitative Finance

The subject of mathematical finance has undergone rapid development in recent years, with mathematical descriptions of financial markets evolving both in volume and technical sophistication. Pivotal in this development have been quantitative models and computational methods for calibrating mathematical models to market data, and for obtaining option prices of concrete products from the calibrated models.

In this development, two broad classes of computational methods have emerged: statistical sampling approaches and grid-based methods. They correspond, roughly speaking, to the characterization of arbitrage-free prices as conditional expectations over all sample paths of a stochastic process model of the market behavior, or to the characterization of prices as solutions (in a suitable sense) of the corresponding Kolmogorov forward and/or backward partial differential equations, or PDEs for short, the canonical example being the Black–Scholes equation and its extensions.

Sampling methods contain, for example, Monte-Carlo and Quasi-Monte-Carlo Methods, whereas grid-based methods contain, for example, Finite Difference, Finite Element, Spectral and Fourier transformation methods (which, by the use of the Fast Fourier Transform, require approximate evaluation of Fourier integrals on grids). The present text discusses the analysis and implementation of grid-based methods.

The importance of numerical methods for the efficient valuation of derivative contracts cannot be overstated: often, the selection of mathematical models for the valuation of derivative contracts is determined by the ease and efficiency of their numerical evaluation to the extent that computational efficiency takes priority over mathematical sophistication and general applicability.

The presentation of the material is structured in two parts: Part I “Basic Methods”, and Part II “Advanced Methods”. The material in the first part of these notes has evolved over several years, in graduate courses which were taught to students in the joint ETH and Uni Zürich MSc programme in quantitative finance, whereas Part II is based on PhD research projects in computational finance.

This distinction between Parts I and II is certainly subjective, and we have seen it evolve over time, in line with the development of the field. In the formulation of the methods and in their analysis, we have tried to maintain mathematical rigor whenever possible, without compromising ease of understanding of the computational methods per se. This has, in particular in Part I, lead to an engineering style of method presentation and analysis in many places. In Part II, fewer such compromises have been made. The formulation of forward and backward equations for rather large classes of jump processes has entailed a somewhat heavy machinery of Sobolev spaces of fractional and variable, state dependent order, of Dirichlet forms, etc. There is a close correspondence of many notions to objects on the stochastic side where the stochastic processes in market models are studied through their Dirichlet forms.

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