PDE models for American options with counterparty risk and two stochastic factors: Mathematical analysis and numerical solution

• Published in: Computers & mathematics with applications (1987) Vol. 79; no. 5; pp. 1525 - 1542
• Main Authors: Arregui, Iñigo, Salvador, Beatriz, Ševčovič, Daniel, Vázquez, Carlos
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.03.2020; Elsevier BV
• Subjects: (non)linear PDEs; American option pricing; Computer simulation; Counterparty risk; Discretization; Finite element method; Finite elements; Fixed points (mathematics); Mathematical analysis; Mathematical models; Method of characteristics; Nonlinear differential equations; Nonlinear equations; Nonlinear programming; Parabolic variational inequalities; Partial differential equations; Securities prices; Method of characteristics; Finite elements; Parabolic variational inequalities; (non)linear PDEs; American option pricing; Counterparty risk
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2019.09.014

In this article we propose new linear and nonlinear partial differential equations (PDEs) models for pricing American options and total value adjustment in the presence of counterparty risk. An innovative aspect comes from the consideration of stochastic spreads, which increases the dimension of the problem. In this setting, we pose new complementarity problems associated to linear and nonlinear PDEs. Moreover, using the mathematical tools of semilinear variational inequalities for parabolic equations, we prove the existence and uniqueness of a solution for these models. For the numerical solution, we mainly combine a semi-Lagrangian time discretization scheme, a fixed point method to cope with nonlinear terms and a finite element method for the spatial discretization, jointly with an augmented Lagrangian active set method to solve the fully discretized complementarity problem. Finally, numerical examples illustrate the expected behaviour of the option prices and the corresponding total value adjustment, as well as the performance of the proposed numerical techniques. Moreover, we compare the numerical results from the PDEs approach with those obtained by applying Monte Carlo techniques.