Invariant criteria for the zero-coupon bond pricing Vasicek and Cox-Ingersoll-Ross Models

Year 2017, Volume: 5 Issue: 2, 29 - 46, 30.03.2017

Ahmet Bakkaloglu Taha Aziz F. M. Mahomed

Abstract

The zero coupon bond pricing Vasicek and Cox-Ingersoll-Ross (CIR) interest rate models are solved using the invariant approach. The invariance criteria is employed on the linear (1+1) parabolic partial differential equations corresponding to the Vasicek and CIR models in order to perform reduction into one of the four Lie canonical forms. The invariant approach helps in transforming the partial differential equation representing the Vasicek model into the first Lie canonical form which is the classical heat equation. We also find that the invariant method aids in transforming the CIR model into the second Lie canonical form and with a proper parametric selection, the CIR equation can be converted to the first Lie canonical form. For both the Vasicek and CIR models, we  obtain the transformations which map these equations into the heat equation and also to the second Lie canonical form. We construct
the fundamental solutions for the Vasicek and CIR models via these transformations by utilizing the well-known fundamental solutions  of the classical heat equation as well as solution to the second Lie canonical form. Finally, the closed-form analytical solutions of the  Cauchy initial value problems of the Vasicek and CIR models with suitable choice of terminal boundary conditions are also deduced.

Keywords

Vasicek and CIR models, invariant approach, Lie symmetry, fundamental solutions

References

• R. K. Gazizov and N.H. Ibragimov, Lie Symmetry Analysis of Differential Equations in Finance, Nonlinear Dynam. 17(4) (1998) 387−407.
• J. Goard, New Solutions to the Bond-Pricing Equation via Lie’s Classical Method, Math. Comput. Model. 32 (2000) 299−313.
• C. F. Lo and C.H. Hui, Lie-Algebraic Approach for Pricing Moving Barrier Options with Time-Dependent Parameters, J. Math. Anal. Appl. 323 (2006) 1455–1464.
• F. M. Mahomed, K.S. Mahomed, R. Naz and E. Momoniat, Invariant Approaches to Equations of Finance, Math. Comput. Appl. 18(3) (2013) 244−250.
• A. Bakkalo˘glu, T. Aziz, A. Fatima, F. M. Mahomed and Chaudry M. Khalique, Invariant Approach to Optimal Investment- Consumption Problem: the constant elasticity of variance (CEV) Model, Mathematical Methods in the Applied Sciences, 2016.
• C. A. Pooe, F.M. Mahomed and C. Wafo Soh, Fundamental Solutions for Zero-Coupon Bond Pricing Models, Nonlinear Dynam. 36 (2004) 69−76.
• B. Izgi and A. Bakkalo˘glu, Fundamental Solution of Bond Pricing in the Ho-Lee Stochastic Interest Rate Model Under the Invariant Criteria, New Trends in Mathematical Sciences, 5, 1, 196-203, 2017.
• B. Izgi and A. Bakkalo˘glu, Deterministic Solutions of the Stochastic Differential Equations Using Invariant Criteria, Proceedings of ICPAS 2017, 323-326, ISBN: 978-605-9546-02-7, 2017.
• B. Izgi and A. Bakkalo˘glu, Invariant Approaches for the Analytic Solution of the Stochastic Black-Derman Toy Model ,submitted 2017.
• V. Naicker, K. Andriopoulos and P.G.L. Leach, Symmetry Reductions of a Hamilton–Jacobi–Bellman Equation Arising in Financial Mathematics, J. Nonlinear Math. Phys. 12(2) (2005) 268−283.
• N.M. Ivanova, C. Sophocleous and P.G.L. Leach, Group Classification of a Class of Equations Arising in Financial Mathematics, J. Math. Anal. Appl. 372 (2010) 273–286.
• Y. Liu and Deng-Shan Wang, Symmetry Analysis of the Option Pricing Model with Dividend yield from Financial Markets, Appl. Math. Lett. 24 (2011) 481–486.
• N.C. Caister, K.S. Govinder and J.G. O’Hara, Optimal System of Lie group Invariant Solutions for the Asian Option PDE, Math. Meth. Appl. Sci. 34 (2011) 1353–1365.
• W. Sinkala, Two ways to Solve, Using Lie Group Analysis, the Fundamental Valuation Equation in the Double-Square-Root Model of the Term Structure, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011) 56–62.
• I. Hern´andez, C. Mateos, J. N´u˜nez and ´A.F. Tenorio, Lie Theory: Applications to Problems in Mathematical Finance and Economics, Appl. Math. Comput. 208 (2009) 446−452.
• L. Bachelier, Theorie de la Speculation, Annales Scientifiques de l’Ecole Normale Superieure. 3 (1900) 21−86.
• R.C.Merton, Optimum Consumption and Portfolio Rules in a Continuous TimeModel, J. Economic Theory. 3(4) (1971) 373−413.
• F. Black and M. Scholes,The Pricing of Options and Corporate Liabilities, J. Political Economy. 81 (1973) 637−654.
• B. Izgi, Behavioral Classification of Stochastic Differential Equations in Mathematical Finance, Ph.D. Thesis, Istanbul Technical University, 2015.
• O. Vasicek, An Equilibrium Characterization of the Term Structure, J. Financial Eco. 5 (1977) 177−188.
• J. C. Cox, J.E. Ingersoll and S.A. Ross, An Intertemporal General Equilibrium Model of Asset Prices, Econometrica. 53 (1985) 363−384.
• I.K. Johnpillai and F.M.Mahomed, Singular Invariant Equation for the (1+1) Fokker- Planck Equation, J. Physics A:Mathematical and General. 34 (2001) 11033−11051.
• F. M. Mahomed, Complete Invariant Characterization of Scalar Linear (1+1) Parabolic Equations, J. Nonlinear Math. Phys. 15 (2008) 112−123.
• A.G. Johnpillai, F.M. Mahomed and S. Abbasbandy, Fundamental Solution via Invariant Approach for a Brain Tumor Model and its Extensions, Z. Naturforsch. 69 (2014) 725 – 732.
• T. Y. N. Myint-U, Partial Differential Equations of Mathematical Physics, Amercian Elsevier Publishing Company, INC. New York, (1973).
• J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press, New Haven, Connecticut, (1923).
• A.D. Polyanin and V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, Boca Raton, FL, (1995).
• M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, New York, (1965).