Multiquadratic-Radial Basis Functions Method for Mortgage valuation under jump-diffusion model

Document Type : Original Article


1 Department of Applied Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.

2 Department of Mathematics, Faculty of Mathematical Science, Allameh Tabataba’i University (ATU), Tehran, Iran.

3 Department of Mathematics, Shahr-e-rey branch, Islamic Azad University, Tehran, Iran.


Given the significant benefits of the Radial Basis Function (RBF) approach, here in this paper, we tried to exploit and adopt this method for the Fixed-Rate Mortgage (FRM) models. In the real world, a jump occurs due to an unknown reason and perhaps better reflects the evolution of real estate prices during bubbles and crises in the real estate markets. For the house price evolution, the jump- diffusion models are used which would lead to a Partial-Integro Differential Equation (PIDE) model. The main concentration is on the difficulty of projecting the pricing of FRM that deals with contracts in where the underlying stochastic factors are the house price and the interest rate. Utilizing the stochastic house-price and stochastic interest-rate models, we were able to develop a reliable mortgage valuation. The identified Partial-Integro Differential Equation (PIDE) from the FRM pricing model, solved by RBF considering the fact that a closed-form solution is usually unavailable. Further, to display the expected behaviour of the contract, the possible applications of the suggested method applied to UK fixed-rate mortgages. Based on available resources, a set of economic parameters was determined for the mortgage to provide an instance to show the applicability of the proposed approach.


  • Azevedo-Pereira, J. A., Newton, D. P., Paxson, D. A. (2002). UK fixed rate repayment mortgage and mortgage indemnity valuation. Real Estate Economy, 30(2), 185-211.
  • Black, F., Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.
  • Briani, M., Natalini, R., Russo, G. (2007), Implicit-Explicit numerical schemes for Jump-Diffusion processes Calcolo. 44, 33-57.
  • Calvo-Garrido, M. C., Vázquez, C. (2015). Effects of jump-diffusion models for the house price dynamics in the pricing of fixed-rate mortgages, insurance and coinsurance. Applied Mathematics and Computation. 271, 730-742.
  • Chan, R. T. L., Hubbert, S. (2014). Options pricing under the one dimension Jump-diffusion model using the radial basis function interpolation scheme. Springer, 17(2), 161-189.
  • Cont, R., Tankov, P. (2004). Financial modeling with jump processes. Chapman and Hall/CRC Financial Mathematics Series, Boca Raton, Fla, London: Chapman and Hall/CRC.
  • Cox, J. C., Ingersoll, J. E., Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53(2), 385-407. 
  • D’Halluin, Y., Forsyth, P.A., Labahn, G. (2005a). A semi-Lagrangian approach for American Asian options under jump diffusion. Journal of Science and Computer, 27, 315–345.
  • D'Halluin, Y., Forsyth, P., Vetzalz, K. (2005). Robust numerical methods for contingent claims under Jump Diffusion process. Journal of Numerical Analysis. 25, 87-112.
  • Fausshauer, G. E., Khaliq, A. Q. M., Voss, D. A. (2004a). In Proceedings of A Parallel Time Stepping Approach Using Meshfree Approximations for Pricing Options with Non-smooth Payoffs. Third World Congress of the Bachelier Finance Society, Chicago.
  • Fausshauer, G. E., Khaliq, A. Q. M., Voss, D. A. (2004b). Using meshfree approximation for multi-asset American options. Chinese Institute of Engineers, 27(4), 563-571.
  • Fausshauer, G.E. (2007). Meshfree Approximation Methods with MATLAB. Interdisciplinary Mathematical Sciences, vol. 6, World Scientific Sciences, Hackensack, N.J.
  • Franke, R. (1982). Scattered Data Interpolation: Test of Some Methods. Mathematics of Computation, 38(157), 181-200.
  • Hardy, R.L. (1971). Multiquadric equations of topography and other irregular surfaces. Journal of Geophysical Research, 76, 1905-1915. 
  • Hon, Y. C., Mao, X. Z. (1999). A radial basis function method for solving options pricing models. Financial Engineering, 8(1), 31-49.
  • Itô, K. (1951). On stochastic differential equations. Memoirs of the American Mathematical Society, 4(11), 1-51.
  • Kansa, E. J. (1990). Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers and Mathematics with Applications, 19(8-9), 147-161.
  • Kau, J., Keenan, D. C., Muller, W. J., Epperson, J. F. (1995). The valuation at origination of fixed-rate mortgages with default and prepayment. Real Estate Finance and Economics, 11(1), 5-36.
  • Kou, S.G. (2002). A jump-diffusion model for option pricing. Management Science, 48, 1086–1101.
  • Liu, G. R. (2003). Mesh Free Methods: Moving beyond the Finite Element Method. CRC Press, USA,
  • Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Finance and Econonics, 3, 125–144.
  • Pena, A. (2005). Option Pricing with Radial Basis Functions: A Tutorial. Technical report, Wilmott Magazine.
  • Pettersson, U., Larsson, E., Marcusson, G., Persson, J. (2008). Improved radial basis function methods for multi-dimensional option pricing. Computational and Applied Mathematics, 222(1), 82-93.
  • Safaei, M., Neisy, A., Nematollahi, N. (2018). New splitting scheme for pricing american options under the heston model. Computational Economics, 52(2), 405-420.
  • Shampine, L.F. (2008). Vectorized adaptive quadrature in matlab. Journal of Computational and Applied Mathematics, 211, 131-140. 
  • Sharp, N. J., Newton, D. P., Duck, P. W. (2008). An improved fixed-rate mortgage valuation methodology with interacting prepayment and default options. Real Estate Finance and Economics, 36(3), 307-342.
  • Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177-188.