Publications and preprints
- N. Gabrielli, J. Teichmann : Pathwise construction of affine processes, (pdf), 2014.
- N. Gabrielli, J. Teichmann : How to visualize the affine property, in preparation, 2014.
- N. Gabrielli : Regularity results for degenerate Kolmogorov equations of affine type (pdf), 2014
- N. Gabrielli : Affine processes from the perspective of path space valued Lévy processes (link), 2014
Past and future talks
The pdf presentations, if available, can be found at the corresponding page in the blog |
PhD Thesis
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Master Thesis
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Affine processes are continuous-time Markov processes characterized by the property that their Fourier-Laplace transform of the marginal distributions depends in a exponential affine way on the initial state. Due to their analytic tractability and high flexibility to model some specific patterns of the financial markets, the application of affine processes in mathematical finance, has widely increased lately. Such applications range over a broad variety of tasks common in finance like option pricing or term structure models simulation. The aim of this thesis is to obtain new results on affine processes and on their applications to mathematical finance, with a view towards numerics. In many pricing problems, like pricing of certain exotic options, we are interested in a pathwise approximation of the process, but, for some examples of affine processes, the discrete approximation can be challenging due to the lack of Lipschitz regularity of the vector fields along the boundary of the state space. Not to mention the high-dimensionality of the problem. It is therefore desirable to develop, describe and implement high-order schemes for affine processes. In this thesis we delve into this matter by analyzing a new representation of affine processes as path-space valued Levy processes.
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Thesis topics are jump-diffusion processes and Levy processes as market model. Therefore I examine incomplete markets and give some examples of option pricing formulae obtained through complex integration along different lines as described in Lewis' article. Another analytical formula is obtained for Variance Gamma model, a process with independent and stationary increments, as introduced in Carr-Madan's article et al. and, in this case, I have explored the advantages, in terms of computational times, when the Fast Fourier Transform is used to compute Fourier integrals. On the other hand, among jump-diffusion model I concentrate my attention on Merton Model. Finally, I have implemented with MATLAB several methods for option pricing in both models. For VG model: analytical formulae, Lewis method and formulae based on Fourier transform and FFT(as described in Carr-Madan article). For Merton model: Merton series expansions and option pricing with characteristic function.
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