ADI method of credit spread option pricing based on jump-diffusion model

Document Type : Research Article


1 Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.

2 Department of Mathematics, Faculty of Mathematics Science and Computer, Allameh Tabataba’i University, Tehran, Iran


As the main contribution of this article, we establish an option on a credit spread under a stochastic interest rate. The intense volatilities in financial markets cause interest rates to change greatly; thus, we consider a jump term in addition to a diffusion term in our interest rate model. However, this decision leads us to a partial integral differential equation. Since the integral part might bring some difficulties, we put forward a fairly new numerical scheme based on the alternating direction implicit method. In the remainder of the article, we discuss consistency, stability, and convergence of the proposed approach. As the final step, with the help of the MATLAB program, we provide numerical results of implementing our method on the governing equation.


1. Deng, S-J.,Johnson, B. and Sogomonian, A. Exotic electricity options and the valuation of electricity generation and transmission assets, De cis. Support Syst. 30 (2001) 383–392.
2. Evans, G., Blackledge, J. and Yardley, P. Numerical methods for partial differential equations, Springer-Verlag London, Ltd., London, 2000.
3. Heydari, M.H., Hooshmandasl, M.R., Loghmani, Gh. Barid, and Cattani, C. Wavelets Galerkin method for solving stochastic heat equation, Int. J. Comput. Math. 93(9), (2015) 1579–1596.
4. Heydari, M.H., Mahmoudi, M.R., Shakiba, A. and Avazzadeh, Z. Chebyshev cardinal wavelets and their application in solving nonlinear stochas tic differential equations with fractional Brownian motion, Commun.
Nonlinear Sci. Numer. Simul. 64 (2018), 98–121.
5. in ’t Hout, K.J. and Mishra, C. Stability of ADI schemes for multidimen sional diffusion equations with mixed derivative terms, Appl. Numer. Math. 74 (2013), 83–94.
6. Jeong, D. and Kim, J. A comparison study of ADI and operator splitting methods on option pricing models, J. Comput. Appl. Math. 247 (2013), 162–171.
7. LeVeque, R.J. Finite difference methods for ordinary and partial differential equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007.
8. Li, L., Jiang, Z., Yin, Z. Fourth-order compact finite difference method for solving two-dimensional convection-diffusion equation, Adv. Difference Equ. 2018, Paper No. 234, 24 pp.
9. Liao, H-L. and Sun, Z-Z. Maximum Norm Error Bounds of ADI and Compact ADI Methods for Solving Parabolic Equations, Numer. Methods Partial Differential Equations 26 (2010), 37–60.
10. Longstaff, F. A., Schwartz, E.S. Valuing credit derivatives, J. Fixed Income, 6 (1995) 6–12.
11. Marchuk, G.I. Splitting and alternating direction methods, Handbook of numerical analysis, Vol. I, 197–462, Handb. Numer. Anal., I, North Holland, Amsterdam, 1990.
12. Mohamadinejad, R., Biazar, J. and Neisy, A. Spread Option Pricing Using Two Jump-Diffusion Interest Rates, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 82 (2020) 171–182.
13. O’Brien, G.G., Hyman, M.A. and Kaplan, S. A study of the numerical solution of partial differential equations, J. Math. Phys. 29, (1951) 223–251.
14. Østerby, O. Numerical solution of parabolic equations, Department of Computer Science, Aarhus University, 2015.
15. Østerby, O. On the stability of ADI methods, DAIMI Report Series, 42(2017).
16. Safaei, M., Neisy, A. and Nematollahi, N. New Splitting Scheme for Pricing American Options Under the Heston Model, Comput. Econ. 52(2018) 405–420.
17. Strikwerda, J. Finite difference schemes and partial differential equations, SIAM, Philadelphia, PA (2007).
18. Su, X. and Wang, W.Pricing options with credit risk in reduced form model, J. Korean Statist. Soc. 41 (2012) 437–444.
19. Tchuindjo, L. Closed-form solutions for pricing credit-risky bonds and bond options, Appl. Math. Comput. 217 (2011) 6133–6143.
20. Unger, A.J.A. Pricing index-based catastrophe bonds: Part 1 Formulation and discretization issues using a numerical PDE approach, Comput. and Geosci. 36 (2010) 150–160.
21. Zhou, R., Du, S., Yu, M. and Yang, F.Pricing credit spread option with Longstaff-Schwartz and GARCH models in Chinese bond market, J. Syst. Sci. Complex, 28 (2015) 1363–1373.