Fundamental solution of bond pricing in the Ho-Lee stochastic interest rate model under the invariant criteria

Abstract

We study the fundamental solution of bond-pricing in the Ho-Lee stochastic interest rate model under the invariant criteria. We obtain transformations between Ho-Lee model with the corresponding linear (1+1) partial differential equation and the first Lie canonical form which is identical to the classical heat equation. These transformations help us to generate the fundamental solution for the Ho-Lee model with respect to the fundamental solution of the classical heat equation sense. Moreover, as a financial application of the Ho-Lee model, we choose the drift term from power functions and perform simulations via Milstein method. Furthermore, we obtain important results for the parameter calibration of the corresponding drift term by using the simulation results.

Keywords

• Ho-Lee stochastic interest rate model,heat equation,canonical Lie forms,Lie symmetry analysis,invariant criteria

Kaynakça

1. J.C. Cox, J.E. Ingersoll and S.A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica, 53, 385-407, 1985.
2. S. Dimas, K. Andriopoulos, D. Tsoubelis and P.G.l. Leach, Complete Specification of Some Partial Differential Equations That Arise in Financial Mathematics, J. Nonlinear Math. Phys., 73-92, 2009.
3. R.K. Gazizov and N.H. Ibragimov, Lie Symmetry Analysis of Differential Equations in Finance, Nonlinear Dynam. 17(4), 387-407, 1998.
4. J. Goard, New solutions to the bond-pricing equation via Lie’s Classical Method, Math. Comput. Model., 32, 299-313, 2000.
5. T. S. Y. Ho and S.-B. Lee, Term Structure Movements and Pricing of Interest Rate Claims, Journal of Finance, 41, 1011-1029, 1986.
6. S. Lie, On integration of a Class of linear partial differential equations by means of definite integrals Archiv for Mathematik ıg Naturvidenskab, VI(3) 328-368, 1881 [in German]. Reprinted in S.Lie, Gesammelte Abhadlundgen, 3 papers, XXXV.
7. F.M. Mahomed, K.S. Mahomed, R. Naz and E. Momoniat, Invariant Approaches to Equations of Finance, Math. Comput. Appl., 18(3), 244-250, 2013.
8. F.M. Mahomed, Complete Invariant Characterization of Scalar Linear (1+1) Parabolic Equations, J. Nonlinear Math. Phys., 15, 112-123, 2008.

9. R. Merton, Option Pricing when Underlying Stock Returns are Discontinuous, J. Financial Economics, 3, 125-144, 1976.
10. G.N. Milstein, Approximate Integration of Stochastic Differential Equations, Theor. Prob. Appl. 19: 557-562, 1974.
11. P.E. Kloeden, E. Platen, and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, Berlin, 2003.
12. C.A. Pooe, F.M. Mahomed and C. Wafo Soh, Fundamental solutions for zero-coupon bond pricing models, Nonlinear Dynam., 36, 69-76, 2004.
13. W. Sinkala, P.G.L. Leach and J. G. O’Hara, Zero-coupon Bond Prices in Vasicek and CIR Models, Math. Meth. Appl. Sci., 31, 665-678, 2008.