Geodesic
Symmetry analysis and exact solutions for a nonlinear model of the financial markets theory
Matematičeskie zametki SVFU, Tome 23 (2016) no. 1, pp. 28-45 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Group classification is obtained for the Sircar-Papanicolaou equations family with a free parameter that contains the Black-Scholes equation as the simplest partial case. The five-dimensional group of equivalence transformations is calculated and three-dimensional kernel of principal Lie algebras and four-dimensional principal Lie algebras in cases of two free element specifications are found. Optimal subalgebras systems and corresponding invariant solutions or invariant submodels are calculated for every Lie algebra. Invariant solutions are included in more general multiparameter solutions families that are invariant with respect to the whole Lie algebra.
Keywords: nonlinear partial differential equation, Black-Scholes equation, Sircar-Papanicolaou model, pricing options, group analysis, invariant sub-model, dynamic hedging, feedback effects of hedging.
Mots-clés : invariant solution 
@article{SVFU_2016_23_1_a3,
     author = {M. M. Dyshaev and V. E. Fedorov},
     title = {Symmetry analysis and exact solutions for a nonlinear model of the financial markets theory},
     journal = {Matemati\v{c}eskie zametki SVFU},
     pages = {28--45},
     year = {2016},
     volume = {23},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVFU_2016_23_1_a3/}
}
TY  - JOUR
AU  - M. M. Dyshaev
AU  - V. E. Fedorov
TI  - Symmetry analysis and exact solutions for a nonlinear model of the financial markets theory
JO  - Matematičeskie zametki SVFU
PY  - 2016
SP  - 28
EP  - 45
VL  - 23
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SVFU_2016_23_1_a3/
LA  - ru
ID  - SVFU_2016_23_1_a3
ER  - 
%0 Journal Article
%A M. M. Dyshaev
%A V. E. Fedorov
%T Symmetry analysis and exact solutions for a nonlinear model of the financial markets theory
%J Matematičeskie zametki SVFU
%D 2016
%P 28-45
%V 23
%N 1
%U http://geodesic.mathdoc.fr/item/SVFU_2016_23_1_a3/
%G ru
%F SVFU_2016_23_1_a3
M. M. Dyshaev; V. E. Fedorov. Symmetry analysis and exact solutions for a nonlinear model of the financial markets theory. Matematičeskie zametki SVFU, Tome 23 (2016) no. 1, pp. 28-45. http://geodesic.mathdoc.fr/item/SVFU_2016_23_1_a3/

[1] Black F., Scholes M., “The pricing of options and corporate liabilities”, J. Political Econ., 81 (1973), 637–659 | DOI | MR

[2] Derman E., Taleb N., “The illusions of dynamic replication”, Quant. Finance, 5:4 (2005), 323–326 | DOI | MR | Zbl

[3] Haug E. G., Taleb N. N., “Option traders use (very) sophisticated heuristics, never the Black-Scholes-Merton formula”, J. Econ. Behavior Organization, 77:2 (2011), 97–106 | DOI | MR

[4] Sircar K. R., Papanicolaou G., “General Black-Scholes models accounting for increased market volatility from hedging strategies”, Appl. Math. Finance, 5 (1998), 45–82 | DOI | Zbl

[5] Frey R., Stremme A., “Market volatility and feedback effects from dynamic hedging”, Math. Finance, 7:4 (1997), 351–374 | DOI | MR | Zbl

[6] Schönbucher P., Wilmott P., “The feedback effect of hedging in illiquid markets”, SIAM J. Appl. Math., 2000, 232–272 | DOI | MR | Zbl

[7] Brandimarte P., Numerical methods in finance economics, John Wiley Sons Publ., Hoboken, NJ, 2004 | MR

[8] Morelli M. J., Montagna G., Nicrosini O., Treccani M., Farina M., Amato P., “Pricing financial derivatives with neural networks”, Phys. A., 338 (2004), 160–165 | DOI

[9] Ovsiannikov L. V., Group Analysis of Differential Equations, Nauka, Moscow, 1978 | MR

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

[11] Chirkunov Yu. A. and Khabirov S. V., Elements of symmetry analysis of differential equations of continuum mechanics, NGTU, Novosibirsk, 2012

[12] Bordag L. A., Chmakova A. Y., “Explicit solutions for a nonlinear model of financial derivatives”, Int. J. Theor. Appl. Finance, 10:1 (2007), 1–21 | DOI | MR | Zbl

[13] Bordag L. A., Frey R., “Pricing options in illiquid markets: symmetry reductions and exact solutions”, Nonlinear models in mathematical finance: New research trends in option pricing, ed. M. Ehrhardt, Nova Sci. Publ., Inc., New York, 2008, 103–130 | MR

[14] Bordag L. A., “On option-valuation in illiquid markets: invariant solutions to a nonlinear model”, Mathematical control theory and finance, ed. A. Sarychev, A. Shiryaev, M. Guerra, and M. R. Grossinho, Springer-Verl., Berlin; Heidelberg, 2008, 71–94 | DOI | MR | Zbl

[15] Mikaelyan A., Analytical study of the Schönbucher-Wilmott model of the feedback effect in illiquid markets, Halmstad Univ., 2009

[16] Bordag L. A., Mikaelyan A., “Models of self-financing hedging strategies in illiquid markets: symmetry reductions and exact solutions”, J. Lett. Math. Phys, 96:1-3 (2011), 191–207 | DOI | MR | Zbl