Polynomial processes and their applications to mathematical finance

We introduce a class of Markov processes, called m -polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Lévy-driven SDEs with aff...

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Bibliographic Details
Published in:Finance and stochastics 2012-10, Vol.16 (4), p.711-740
Main Authors: Cuchiero, Christa, Keller-Ressel, Martin, Teichmann, Josef
Format: Article
Language:English
Subjects:
Economic Theory/Quantitative Economics/Mathematical Methods
Economics
Expected values
Explicit knowledge
Finance
Fourier transforms
Generalized method of moments
Hedging
Insurance
Management
Markov analysis
Markovian processes
Mathematical analysis
Mathematical finance
Mathematics
Mathematics and Statistics
Monte Carlo simulation
Polynomials
Prices
Pricing
Pricing policies
Probability distribution
Probability Theory and Stochastic Processes
Quantitative Finance
Random variables
Statistics for Business
Studies
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Summary:We introduce a class of Markov processes, called m -polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Lévy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Lévy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.
ISSN:0949-2984
1432-1122
DOI:10.1007/s00780-012-0188-x