Geodesic
On the usage of the lie group symmetries for term structure models with nonlinear drift and squared volatility functions
Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2018), pp. 34-46.

Voir la notice de l'article provenant de la source Math-Net.Ru

One of the central tasks of financial analysis is the study of the behavior of the dynamics of interest rates. The most well-known affine models are not able to describe real yield curves with the necessary accuracy, so more and more often researchers are trying to build more complex and, it is believed, likelihood non-affinity models of the term structure of interest rate yields. One of the main problems of constructing such models is the solution of a parabolic differential equation in partial derivatives, which sets the cost of a zero-coupon bond - in order to study the properties of models it is convenient to have such a solution in an analytical form. In this paper, we consider a generalized model with nonlinear drift and squared volatility functions, which includes most of the already known models. To solve a parabolic equation associated with such a model, we use the theory of Lie groups, which makes it possible to systematize and completely algorithmize the approach to constructing solutions. On the basis of this approach, solutions are found for some particular cases of models, both new ones that have not been previously encountered by the author, and those that already known. Also for the non-affine Ana - Gao model, a more general solution is found in comparison with the original one. In the end, a numerical experiment was carried out using real data from the European Central Bank.
Keywords: Lie group symmetries, infinitesimal generator, interest rates, yield curve, forward rate, zero-coupon bonds. 
@article{BGUMI_2018_2_a4,
     author = {D. A. Pauliu},
     title = {On the usage of the lie group symmetries for term structure models with nonlinear drift and squared volatility functions},
     journal = {Journal of the Belarusian State University. Mathematics and Informatics},
     pages = {34--46},
     publisher = {mathdoc},
     volume = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/BGUMI_2018_2_a4/}
}
TY  - JOUR
AU  - D. A. Pauliu
TI  - On the usage of the lie group symmetries for term structure models with nonlinear drift and squared volatility functions
JO  - Journal of the Belarusian State University. Mathematics and Informatics
PY  - 2018
SP  - 34
EP  - 46
VL  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BGUMI_2018_2_a4/
LA  - ru
ID  - BGUMI_2018_2_a4
ER  - 
%0 Journal Article
%A D. A. Pauliu
%T On the usage of the lie group symmetries for term structure models with nonlinear drift and squared volatility functions
%J Journal of the Belarusian State University. Mathematics and Informatics
%D 2018
%P 34-46
%V 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BGUMI_2018_2_a4/
%G ru
%F BGUMI_2018_2_a4
D. A. Pauliu. On the usage of the lie group symmetries for term structure models with nonlinear drift and squared volatility functions. Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2018), pp. 34-46. http://geodesic.mathdoc.fr/item/BGUMI_2018_2_a4/

[1] Y. Ait-Sahalia, “Testing continuous-time models of the spot interest rate”, The Review of Financial Studies, 9(2) (1996), 385–426 | DOI

[2] D. Ahn, B. Gao, “A parametric nonlinear model of term structure dynamics”, The Review of Financial Studies, 12(4) (1999), 721–762 | DOI

[3] S. Lie, “On integration of a class of linear partial differential equations by means of definite integrals”, Archiv der Mathematik, 6(3) (1881), 328–368 | MR

[4] N. H. Ibragimov, R. K. Gazizov, “Lie sdymmetry analysis of differential equations in finance”, Nonlinear Dynamics, 17(4) (1998), 387–407 | DOI | MR | Zbl

[5] W. Sinkala, PGL. Leach, J. G. O’Hara, “Zero-coupon bond prices in the Vasicek and CIR models: Their computation as group-invariant solutions”, Mathematical methods in the Applied Sciences, 31(6) (2008), 665–678 | DOI | MR | Zbl

[6] G. A. Medvedev, “O vremennoi strukture dokhodnosti. Model Vasicheka”, Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaya tekhnika i informatika, 1(18) (2012), 102–111

[7] G. A. Medvedev, “O vremennoi strukture dokhodnosti. Model Koksa – Ingersolla – Rossa”, Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaya tekhnika i informatika, 2(19) (2012), 102–111

[8] G. A. Medvedev, “Plotnosti veroyatnostei protsessov protsentnykh stavok dokhodnosti”, Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaya tekhnika i informatika, 3(36) (2016), 35–48 | MR

[9] G. A. Medvedev, “O vremennoi strukture dokhodnosti. Novaya versiya”, Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaya tekhnika i informatika, 4(25) (2013), 61–70