Journal of Computational Finance

Risk.net

Stability and convergence of Galerkin schemes for parabolic equations with application to Kolmogorov pricing equations in time-inhomogeneous Lévy models

Maximillian Gaß and Kathrin Glau

  • Solving partial (integro)differential equations (P(I)DEs) is one of the fundamental approaches to computing option prices. PDE approaches and their generalization to PIDE models in the presence of jumps are especially praised in industry for their favorable stability and convergence properties. In this paper we advance the theoretical insight into stability and convergence of solutions of pricing PIDEs.
  • We extend the stability and convergence analysis provided in von Petersdorff and Schwab (2003) to the time-in-homogeneous case. When adapting the proof of von Petersdorff and Schwab, a technical complication arises from the energy norm becoming time-dependent. This compels us to deviate from the concept of their proof.
  • A notable feature of our new proof is that our main results no longer require the inverse property, one of the essential assumptions in von Petersdorff and Schwab (2003).
  • Our final convergence results are expressed in terms of the projection error, which applies generally to choices of function spaces for the exact and the approximate solution.
  • We discuss all assumptions in regards to applicability in finance.
  • Examples of function spaces for the exact and the approximate solution that arise naturally for problems in finance are provided.

Two essential quantities for the analysis of approximation schemes of evolution equations are stability and convergence. We derive stability and convergence of fully discrete approximation schemes of solutions to linear parabolic evolution equations governed by time-dependent coercive operators. We consider abstract Galerkin approximations in space combined with θ-schemes in time. The level of generality of our analysis comprises both a large class of time-dependent operators and a large choice of approximating Galerkin spaces. In particular, the results apply to partial integrodifferential equations for option pricing in time-inhomogeneous Lévy models and allow for a large variety of option types and models. The derivation builds on the strong foundation laid out in a 2003 paper by von Petersdorff and Schwab, which provides the respective results for the time-homogeneous case. We discuss the assumptions in the context of option pricing.

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