We discuss the valuation problem for a broad spectrum of plain vanilla and path-dependent options in a general modeling framework, and specifically for L\'evy driven models. Among the derivatives which we consider are digitals, double digitals, asset-or-nothing options, self-quantos, lookback and one-touch options. Extensions to the multivariate case, i.e. basket options and options on the minimum or maximum of several assets are considered as well. As a link to credit risk we derive a valuation formula for equity default swaps. The key idea of this Fourier, resp. Laplace transform based approach is the clear separation of two ingredients, namely the payoff function and the underlying process . The latter can be the driving process itself, but also the supremum or infimum or another process derived from it. For L\'evy processes the analytically extended characteristic functions of the supremum and the infimum process are derived .
The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid in a general framework; i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on the payoff function and the process arises naturally. We also extend these results to the multi-dimensional case, and discuss the calculation of Greeks by Fourier transform methods. As an application, we price options on the minimum of two assets in L\'evy and stochastic volatility models .
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We discuss the valuation problem for a broad spectrum of plain vanilla and path-dependent options
The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid
R
modeling framework, and specifically for L\'evy driven models. Among the derivatives which we consider are digitals, double digitals, asset-or-nothing options, self-quantos, lookback and one-touch options. Extensions to the multivariate case,
framework;
R
basket options and options on the minimum or maximum of several assets are considered as well. As a link to credit risk we derive a valuation formula for equity default swaps. The key idea of this Fourier, resp. Laplace transform based approach is the clear separation of two ingredients, namely
when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on
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underlying process . The latter can be the driving process itself, but also the supremum or infimum or another process derived from it. For L\'evy processes the analytically extended characteristic functions of the supremum and the infimum process are derived
process arises naturally. We also extend these results to the multi-dimensional case, and discuss the calculation of Greeks by Fourier transform methods. As an application, we price options on the minimum of two assets in L\'evy and stochastic volatility models