SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Efficient numerical methods for option pricing in time-inhomogeneous models

O. Reichmann, V. Kazeev and C. Schwab

Abstract
Lévy processes have, since their initial use in the early 1990ies by D. Madan and his collaborators, become a standard tool in financial modeling. Time-inhomogeneity severely hampers their efficient performance in pricing derivatives across multiple strikes and maturities in markets which are intrinsically time inhomogeneous. We present a class of processes beyond Lévy whose time-inhomogeneous parabolic partial integrodifferential equations (PIDEs) exhibit strong degeneracies in time. The arising PIDE reads as follows: \begin{eqnarray} \label{eq:intro1}
tial_t u-t^\gamma \mathcal{A}(t) u&=&f\text{ on } I\times D, u(0)&=&g,\label{eq:intro2} \end{eqnarray} where $ (\mathcal{A}(t))_{t\geq 0}$ is an appropriate family of operators, $g$ the sufficiently smooth initial data, $\gamma$ a constant with $\gamma\in(-1,1)$, $I=(0,T)$ and a Lipschitz domain $D\subset \mathbb{R}^d$ for $d\geq1$. Note that negative exponents $\gamma$ lead to an explosion at $t=0$.

Bibliography
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[2] R. Andreev, Space-time wavelet FEM for parabolic equations, SAM, ETH Zurich, (2010), SAM-Report 20.
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[4] P. Carr, H. Geman, D. Madan and M. Yor, Self-decomposability and option pricing, Mathematical Finance 17 (2007), pp. 31-57.
[5] V. Kazeev, O. Reichmann and Ch. Schwab, Low-rank tensor structure of linear diffusion operators in the TT and QTT formats, Linear Algebra and its Applications 438 (2013), 4204-4221.

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