In this chapter, we present basic numerical methods for solving one-dimensional parabolic PDEs. In particular, we introduce the Finite Difference Method (FDM) as a simple technique for generating an approximate solution to the pricing PDE. A considerable number of problems arise in the case of high-dimensional PDEs (e.g., stochastic volatility models, stochastic interest rates models, basket options) and path-dependent payoffs (e.g., Asian options). A few of these issues can be tackled through analytical tools such as integral transforms (e.g., Laplace and Fourier transforms). While we defer a treatment of these and other related issues to Chapter 7, the pricing of path-dependent contracts (such as Asian and lookback options) will be considered through a number of case studies in the second part of this book. Instead, we will not consider how to deal with multidimensional PDEs, arising with stochastic volatility or with basket options.
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