Journal of Computational Finance

Penalty methods for bilateral XVA pricing in European and American contingent claims by a partial differential equation model

Yuwei Chen and Christina C. Christara

  • A fast iterative (penalty) method for handling the nonlinearity in the partial differential equation (PDE) problem arising from XVA pricing of European contingent claims
  • The convergence analysis of the penalty method
  • The extension of the penalty method to American-style derivatives' XVA pricing, resulting in a double-penalty method
  • Formulation of boundary conditions for the XVA pricing PDE for call options, put options and forward contracts
  • Numerical experiments demonstrating the accuracy and efficiency of the method

Accounting for default risk in the valuation of financial derivatives has become increasingly important, especially since the 2007–8 financial crisis. Under some assumptions, the valuation of financial derivatives, including a value adjustment to account for default risk (the so-called XVA), gives rise to a nonlinear partial differential equation (PDE). We propose numerical methods for handling the nonlinearity in this PDE, the most efficient of which are the discrete penalty iteration methods. We first formulate a penalty iteration method for the case of European contingent claims and study its convergence. We then extend the method to the case of American contingent claims, which results in a double-penalty iteration. We also propose boundary conditions and their discretization for the XVA PDE problem in the case of a call option, a put option and a forward contract. Numerical results demonstrate the effectiveness of our methods.

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