LIE THEORETIC PERSPECTIVE OF
BLACK-SCHOLES EQUATION UNDER
STOCHASTIC HESTON MODEL
Maba Boniface Matadi1, Phumlani Lawrence Zondi2 1University of Zululand, Department of Mathematical Science
Private Bag X1001
KwaDlangeZwa - 3886, SOUTH AFRICA 2 University of Zululand, Department of Mathematical Science
Private Bag X1001
KwaDlangeZwa - 3886, SOUTH AFRICA
This study examines a classical Black-Scholes (BC) model for stochastic volatility with Heston process from Lie symmetry perspective. In the same way the study includes a classification of point symmetries and the corresponding modified local one-parameter transformations. Lie symmetry analysis is presented for the case where the volatility is a stochastic process. Furthermore, an invariant solutions are calculated and illustrated numerically.
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References
[1] S. Dimas, D. Tsoubelis, SYM:A new symmetry-finding package for Mathematica, Group Analysis of Differential Equations (2005), 64-70.
[2] S.L. Heston, A closed-form solution for options with stochastic volatility
with applications to bonds and currency options, The Review of Financial
Studies, 6 (1993), 327-343.
[3] P.E. Hydon, Symmetry Methods for Differential Equations - A Beginner’s
Guide, Cambridge Texts in Applied Mathematics (2000), E Book (2010).
[4] T.P. Masebe, A Lie symmetry analysis of the Black-scholes Merton finance model through modified local one-parameter transformations, Doctoral Thesis, University of South Africa (2014).
[5] M.B. Matadi, Singularity and Lie group analyses for tuberculosis with
exogenous reinfection, International Journal of Biomathematics, 8 (2015),
1-12.
[6] M.B. Matadi, Lie Symmetry Analysis Of Early Carcinogenesis Model, Applied Mathematics E-Notes 18 (2018), 238-249.
[7] M.B. Matadi, The conservative form of tuberculosis model with demography, Far East Journal of Mathematical Sciences, 102 (2017), 2403-2416.
[8] M.B. Matadi, Symmetry and conservation laws for tuberculosis model,
International Journal of Biomathematics, 10 (2017), # 1750042.
[9] J. Merger, A. Borzi, A Lie algebraic and numerical investigation of the
Black-Scholes equation with Heston volatility model, J. Generalized Lie
Theory Appl., 2 (2016), 2-7.
[10] F. Oliveri, Lie symmetries of Differential equations: Classical results and
recent contributions, Symmetry, 2 (2010), 658-706.
[11] P.J. Olver, Application of Lie Groups to Differential Equations, SpringerVerlag, New York (1986).
[12] A. Paliathanasis, K. Krishnakumar, K.M. Tamizhmani, P.G.L. Leach, Lie
Symmetry analysis of the Black-Scholes-Merton Mmdel for European options with stochastic volatility, arXiv preprint : arXiv (2015), 1508.06797.