Christoph Schwab: Introduction to PDE option pricing beyond Lévy
We survey deterministic methods for the numerical solution of derivative pricing problems in financial models beyond Levy Models. To this end, we briefly recapitulate Levy Models and the derivation and well-posedness of their Forward Equations, for a single underlying.
We then extend to additive processes as well as to Feller-Levy Processes where the characteristic triplet is deterministic, but may depend on the state.
Singularity-free variational discretization of the jump measures for the resulting forward equations is presented, and efficient evaluation of the singular integrals is explained.
Treatment of barriers and early exercise is described. Regularization and stabilization techniques for stable discretization of drift-dominated models with infinite intensity and variation are discussed.
Extensions of these techniques are indicated for
- baskets with general correlation structure in the jump dependence of the underlyings,
- stochastic volatility models,
- Fractional Brownian motion.
The list of references can be found bellow.
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A.M. Matache, T. von Petersdorff and Ch. Schwab:
Fast deterministic pricing of options on Lévy driven assets. Math. Modelling and Numerical Analysis 38 (2004) 37-72.
A.M. Matache, P.A. Nitsche and Ch. Schwab:
Wavelet Galerkin Pricing of American contracts on Lévy driven assets. Quantitative Finance 5 No. 4 (2005) 403-424.
N. Hilber, A.M. Matache and Ch. Schwab:
Sparse wavelet methods for option pricing under stochastic volatility. The Journal of Computational Finance 8(4) (2005) 1-42.
E.W. Farkas, N. Reich and Ch. Schwab:
Anisotropic stable Levy copula processes - analytical and numerical aspects. Mathematical Models and Methods in the Applied Sciences. 17 (9) (2007) 1405 - 1443.
Survey over Numerical Techniques for Lévy Models:
- O. Reichmann and Ch. Schwab: Numerical analysis of additive, Lévy and Feller processes
with applications to option pricing. Chapter 3 in Lévy Matters Vol. I, Springer Lecture Notes in Mathematics Vol. 2001, pp. 137-196.
Report 2010-06, Seminar for Applied Mathematics, ETH Zurich.
Numerical Solution of Feller-Lévy Models:
- O. Reichmann, R. Schneider and Ch. Schwab:
Wavelet solution of variable order pseudodifferential equations. Calcolo 47 (2) (2010) 65-101.
- A.M. Matache, Ch. Schwav and T. Wihler:
Fast numerical solution of parabolic integrodifferential equations with applications in finance.
SIAM J. Sci. Computing 27 (2005) 369-393
- N. Hilber, N. Reich, Ch. Schwab and Ch. Winter:
Numerical methods for Lévy processes, Finance and Stochastics (2009).
Numerical Evaluation of Dirichlet Forms:
- A. Chernov, Ch. Schwab and T. von Petersdorff. Exponential Convergence of hp Quadratures for Integral Operators with Gevrey Kernels.
ESAIM: M2AN 45 (2011) 387–422.
FBM and Forward Equations:
- O. Reichmann: Optimal space-time adaptive wavelet methods for degenerate parabolic PDEs. Report 2011-03,
Seminar for Applied Mathematics, ETH Zurich.