Two ways to solve, using Lie group analysis, the fundamental valuation equation in the double-square-root model of the term structure. (English) Zbl 1221.35418

Summary: Two approaches based on Lie group analysis are employed to obtain the closed-form solution of a partial differential equation derived by F. A. Longstaff [J. Fin. Econom. 23, 195–224 (1989)] for the price of a discount bond in the double-square-root model of the term structure.


35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
35A30 Geometric theory, characteristics, transformations in context of PDEs
91G80 Financial applications of other theories
Full Text: DOI


[1] Andriopoulos, K.; Leach, P.G.L., A common theme in applied mathematics: an equation connecting applications in economics, medicine and physics, South african J sci, 102, 66-72, (2006)
[2] Bluman, G.W.; Kumei, S., Symmetries and differential equations, (1989), Springer New York · Zbl 0698.35001
[3] Cantwell, B.J., Introduction to symmetry analysis, (2002), Cambridge University Press United Kingdom · Zbl 1082.34001
[4] Dresner L. Applications of Lie’s theory of ordinary and partial differential equations. Philadelphia: IOP; 1999. · Zbl 0914.34002
[5] Gazizov, R.K.; Ibragimov, N.H., Lie symmetry analysis of differential equations in finance, Nonlinear dynam, 17, 387-407, (1998) · Zbl 0929.35006
[6] 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
[7] Goard, J.; Broadbridge, P.; Raina, G., Tractable forms of the bond pricing equation, Math comput modelling, 40, 151-172, (2004) · Zbl 1112.91035
[8] Govinder, K.S., Lie subalgebras reduction of order, and group-invariant solutions, SIAM J appl math, 258, 720-732, (2001) · Zbl 0979.34032
[9] Head, A.K., LIE, a PC program for Lie analysis of differential equations, Comput phys commun, 71, 241-248, (1993) · Zbl 0854.65055
[10] Hernández, I.; Mateos, C.; Núñez, J.; Tenorio, A.F., Lie theory: applications to problems in mathematical finance and economics, Appl math comput, 208, 446-452, (2009) · Zbl 1162.91013
[11] Ibragimov NH, editor. CRC handbook of Lie group analysis of differential equations. Boca Raton (FL): CRC Press; vol. 1, 1994; vol. 2, 1995; vol. 3, 1996.
[12] Ibragimov NH. Introduction to modern group analysis. Ufa: Tay; 2000.
[13] Kallianpur G, Karandikar RL. Introduction to option pricing theory. Boston: Birkhäuser; 2000. · Zbl 0969.91003
[14] Kwok, Y.K., Mathematical models of financial derivatives, (1998), Springer Singapore · Zbl 0931.91018
[15] Leach, P.G.L.; O’Hara, J.G.; Sinkala, W., Symmetry-based solution of a model for a combination of a risky investment and a riskless investment, J math anal appl, 334, 368-381, (2007) · Zbl 1154.91027
[16] Longstaff, F.A., A nonlinear general equilibrium model of the term structure of interest rates, J financial econom, 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, 268-283, (2005) · Zbl 1080.35163
[18] Nucci MC. Interactive REDUCE programs for calculating classical, nonclassical and Lie-Bäcklund symmetries for differential equations (preprint). Georgia Institute of Technology, Math 062090-051; 1990.
[19] Nucci MC. Interactive REDUCE programs for calculating Lie point, nonclassical, Lie-Bäcklund, and approximate symmetries of differential equations: manual and floppy disk. In: Ibragimov NH, editor. CRC handbook of Lie group analysis of differential equations, vol. 3: New trends in theoretical development and computational methods. Boca Raton (FL): CRC Press; 1996. p. 415-81.
[20] Olver, P.J., Applications of Lie groups to differential equations, (1993), Springer New York · Zbl 0785.58003
[21] Ovsiannikov, L.V., Group analysis of differential equations, (1982), Academic Press New York · Zbl 0485.58002
[22] Pooe, C.A.; Mahomed, F.M.; Wafo Soh, C., Fundamental solutions for zero-coupon bond pricing models, Nonlinear dynam, 36, 69-76, (2004) · Zbl 1122.91335
[23] Schwarz, F.A., A REDUCE package for determining Lie symmetries of ordinary and partial differential equations, Comput phys commun, 27, 179-186, (1982)
[24] Schwarz, F.A., Symmetries of differential equations: from sophus Lie to computer algebra, SIAM rev, 30, 450-481, (1988) · Zbl 0664.35004
[25] Sherring, J.; Head, A.K.; Prince, G.E., Dimsym and LIE: symmetry determining packages, Math comput modelling, 25, 153-164, (1997) · Zbl 0918.34007
[26] 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 meth appl sci, 31, 665-678, (2008) · Zbl 1132.91438
[27] Sinkala, W.; Leach, P.G.L.; O’Hara, J.G., Invariance properties of a general bond-pricing equation, J differential equations, 244, 2820-2835, (2008) · Zbl 1147.91017
[28] Sinkala, W.; Leach, P.G.L.; O’Hara, J.G., Optimal system and group-invariant solutions of the cox – ingersoll – ross pricing equation, Appl math comput, 201, 95-107, (2008) · Zbl 1142.91466
[29] Sophocleous, C.; Leach, P.G.L.; Andriopoulos, K., Algebraic properties of evolution partial differential equations modelling prices of commodities, Math meth appl sci, 31, 679-694, (2008) · Zbl 1132.35491
[30] Stephani, H., Differential equations: their solution using symmetries, (1989), Cambridge University Press New York · Zbl 0704.34001
[31] Wilmott, P.; Howison, S.; Dewynne, J., The mathematics of financial derivatives: a student introduction, (1995), Cambridge University Press New York · Zbl 0842.90008
[32] Wolfram, S., Mathematica: a system for doing mathematics by computer, (1991), Addison-Wesley New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.