A boundary element PDE approach to corporate debt

A Boundary Element Method (BEM) is proposed for the analytic PDE-based evaluation of a defaultable zero-coupon bond in a two-factor firstpassage- time model. The default triggering barrier and the recovery payoff associated with the default event can be specified in a large variety of ways according to the real-life bond covenants. The advantages of BEM, when compared to domain methods, such as Finite Element Methods or Finite Difference Methods, are well known: only the boundary of the domain needs to be discretized; particularly, exterior problems with unbounded domains but bounded boundaries are handled as easily as interior problems; the solution in the interior of the domain is approximated with a rather high convergence rate and can be evaluated in particular points of the domain and not necessarily everywhere; infinity conditions are implicitly satisfied. The method is based on the explicit knowledge of a fundamental solution for the pricing PDE. However, when only the characteristic function of the underlying process is known, it is possible to recover the transition density, i.e. the fundamental solution, by Fourier inverse transform.