A new numerical method for pricing fixed-rate mortgages with prepayment and default options

Abstract

In this paper we consider the valuation of fixed-rate mortgages including prepayment and default options, where the underlying stochastic factors are the house price and the interest rate. The mathematical model to obtain the value of the contract is posed as a free boundary problem associated to a partial differential equation (PDE) model. The equilibrium contract rate is determined by using an iterative process. Moreover, appropriate numerical methods based on a Lagrange–Galerkin discretization of the PDE, an augmented Lagrangian active set method and a Newton iteration scheme are proposed. Finally, some numerical results to illustrate the performance of the numerical schemes, as well as the qualitative and quantitative behaviour of solution and the optimal prepayment boundary are presented.

Keywords:

• fixed-rate mortgages
• option pricing
• complementarity problem
• numerical methods
• Augmented Lagrangian Active Set formulation

2010 AMS Subject Classifications:

• 91G80
• 65M25
• 65M60
• 90C33

Acknowledgements

Paper funded by Spanish MCINN (Project MTM2010–21135–C02-01) and by Xunta de Galicia (Ayuda CN2011/004, partially funded with FEDER funds).