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Invariance properties of a general bond-pricing equation. (English) Zbl 1147.91017
Summary: We perform the group classification of a bond-pricing partial differential equation of mathematical finance to discover the combinations of arbitrary parameters that allow the partial differential equation to admit a nontrivial symmetry Lie algebra. As a result of the group classification we propose “natural” values for the arbitrary parameters in the partial differential equation, some of which validate the choices of parameters in such classical models as that of Vasicek and Cox-Ingersoll-Ross. For each set of these natural parameter values we compute the admitted Lie point symmetries, identify the corresponding symmetry Lie algebra and solve the partial differential equation.

91B24Price theory and market structure
91B28Finance etc. (MSC2000)
35A30Geometric theory for PDE, characteristics, transformations
35K15Second order parabolic equations, initial value problems
Mathematica; LIE
Full Text: DOI
[1] , Handbook of mathematical functions with formulas, graphs, and mathematical tables (1972) · Zbl 0543.33001 · http://www.cs.bham.ac.uk/~aps/research/projects/as/
[2] Andrews, L. C.: Special functions of mathematics for engineers, (1992)
[3] Bluman, G. W.; Kumei, S.: Symmetries and differential equations, Appl. math. Sci. 81 (1989) · Zbl 0698.35001
[4] P. Carr, A. Lipton, D. Madan, The reduction method for valuing derivative securities, working paper, 2000
[5] Cox, J. C.; Ingersoll, J. E.; Ross, S. A.: An intertemporal general equilibrium model of asset prices, Econometrica 53, 363-384 (1985) · Zbl 0576.90006 · doi:10.2307/1911241
[6] Craddock, M.; Platen, E.: Symmetry group methods for fundamental solutions, J. differential equations 207, 285-302 (2004) · Zbl 1065.35016 · doi:10.1016/j.jde.2004.07.026
[7] Gazizov, R. K.; Ibragimov, N. H.: Lie symmetry analysis of differential equations in finance, Nonlinear dynam. 17, No. 4, 387-407 (1998) · Zbl 0929.35006 · doi:10.1023/A:1008304132308
[8] Goard, J. M.: New solutions to the Bond-pricing equation via Lie’s classical method, Math. comput. Modelling 32, 299-313 (2000) · Zbl 0955.91018 · doi:10.1016/S0895-7177(00)00136-9
[9] Head, A. K.: Lie, a PC program for Lie analysis of differential equations, Comput. phys. Comm. 71, 241-248 (1993) · Zbl 0854.65055 · doi:10.1016/0010-4655(93)90007-Y
[10] Hereman, W.: Symbolic software for Lie symmetry analysis, CRC handbook of Lie group analysis of differential equations, vol. 3: new trends in theoretical development and computational methods 3 (1995)
[11] , CRC handbook of Lie group analysis of differential equations, vol. 1 1 (1994) · Zbl 0864.35001
[12] Kallianpur, G.; Karandikar, R. L.: Introduction to option pricing theory, (2000) · Zbl 0969.91003
[13] Kwok, Y. K.: Mathematical models of financial derivatives, (1998) · Zbl 0931.91018
[14] Lie, S.: On integration of a class of linear partial differential equations by means of definite integrals, Arch. math. 6, No. 3, 328-368 (1881) · Zbl 13.0298.01
[15] Lo, C. F.; Hui, C. H.: Valuation of financial derivatives with time-independent parameters: Lie-algebraic approach, Quantitative finance 1, 73-78 (2001)
[16] Longstaff, F. A.: A nonlinear general equilibrium model of the term structure of interest rates, J. financ. Econ. 23, 195-224 (1989)
[17] Naicker, V.; Andriopoulos, K.; Leach, P. G. L.: Symmetry reductions of a Hamilton -- Jacobi -- Bellman equation arising in financial mathematics, J. nonlinear math. Phys. 12, No. 2, 268-283 (2005) · Zbl 1080.35163
[18] Olver, P. J.: Applications of Lie groups to differential equations, Grad texts in math. 107 (1993)
[19] Platen, E.: A minimal financial market model, Mathematical finance, 293-301 (2001) · Zbl 1004.91029
[20] Pooe, C. A.; Mahomed, F. M.; Soh, C. W.: Fundamental solutions for zero-coupon Bond pricing models, Nonlinear dynam. 36, 69-76 (2004) · Zbl 1122.91335 · doi:10.1023/B:NODY.0000034647.76381.04
[21] Schwarz, F.: Symmetries of differential equations: from sophus Lie to computer algebra, SIAM rev. 30, 450-481 (1988) · Zbl 0664.35004 · doi:10.1137/1030094
[22] Sinkala, W.; Leach, P. G. L.; O’hara, J. G.: Zero-coupon Bond prices in the vasicek and CIR models: their computation as group-invariant solutions, Math. methods appl. Sci. 31, 665-678 (2008) · Zbl 1132.91438 · doi:10.1002/mma.935
[23] Vasicek, O.: An equilibrium characterization of the term structure, J. financ. Econ. 5, 177-188 (1977)
[24] Wolfram, S.: Mathematica: A system for doing mathematics by computer, (1991) · Zbl 0671.65002