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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**

[1] H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäuser, Basel, (1995).

[2] R. Andreev, Space-time wavelet FEM for parabolic equations, SAM, ETH Zurich, (2010), SAM-Report 20.

[3] S. Beuchler, R. Schneider and Ch. Schwab, Multiresolution weighted norm equivalences and applications, Numerische Mathematik 98 (2004), pp. 67-97.

[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.