Geodesic
On some option pricing models on illiquid markets
Čelâbinskij fiziko-matematičeskij žurnal, Tome 2 (2017) no. 1, pp. 18-29.

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

This paper presents an overview of some options pricing mathematical models, namely the model of Sircar and Papanicolaou, and the model of Schönbucher and Wilmott. These models are interesting in that they take into account the feedback effects from the large traders operations on an illiquid market. The paper presents the basic assumptions and restrictions of the models. For each of the model a final equation are given describing the dynamics of the prices of the derivative financial instruments under simulated conditions.
Keywords: options pricing, Black — Scholes model, Sircar — Papanicolaou model, Schönbucher — Wilmott model, illiquid market, stochastic process, partial differential equation, initial boundary value problem. 
@article{CHFMJ_2017_2_1_a2,
     author = {M. M. Dyshaev},
     title = {On some option pricing models on illiquid markets},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {18--29},
     publisher = {mathdoc},
     volume = {2},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2017_2_1_a2/}
}
TY  - JOUR
AU  - M. M. Dyshaev
TI  - On some option pricing models on illiquid markets
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2017
SP  - 18
EP  - 29
VL  - 2
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2017_2_1_a2/
LA  - ru
ID  - CHFMJ_2017_2_1_a2
ER  - 
%0 Journal Article
%A M. M. Dyshaev
%T On some option pricing models on illiquid markets
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2017
%P 18-29
%V 2
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHFMJ_2017_2_1_a2/
%G ru
%F CHFMJ_2017_2_1_a2
M. M. Dyshaev. On some option pricing models on illiquid markets. Čelâbinskij fiziko-matematičeskij žurnal, Tome 2 (2017) no. 1, pp. 18-29. http://geodesic.mathdoc.fr/item/CHFMJ_2017_2_1_a2/

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

[2] F. Black, “The pricing of commodity contracts”, J. of Financial Economics, 3 (1976), 167–179 | DOI

[3] S. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, The Rev. of Financial Studies, 6:2 (1993), 327–343 | DOI

[4] R. Sircar, G. Papanicolaou, “Generalized Black — Scholes models accounting for increased market volatility from hedging strategies”, Applied Mathematical Finance, 5:1 (1998), 45–82 | DOI | Zbl

[5] P. Schónbucher, P. Wilmott, “The feedback effects of hedging in illiquid markets”, SIAM J. on Applied Mathematics, 61 (2000), 232–272 | DOI | MR | Zbl

[6] E. Peters, Chaos and order in the capital markets: a new view of cycles, prices, and market volatility, 2 ed., John Wiley Sons, N. Y., 1996, 274 pp.

[7] H. Follmer, M. Schweizer, “A microeconomic approach to diffusion models of stock prices”, Mathematical Finance, 3:1 (1993), 1–23 | DOI | MR | Zbl

[8] M. Brennan, E. Schwartz, “Portfolio insurance and financial market equilibrium”, J. of Business, 62:4 (1989), 455–476 | DOI

[9] R. Frey, A. Stremme, “Volatility and feedback effects from dynamic hedging”, Mathematical Finance, 7 (1997), 351–374 | DOI | MR | Zbl

[10] B. Oksendal, Stochastic Differential Equations. An Introduction with Applications, 5 ed., Berlin–Heidelberg–New York–London–Paris–Tokyo–Hong Kong–Barcelona–Budapest, 2000, 332 pp. | MR

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

[12] L.V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982, 416 pp. | MR | MR | Zbl

[13] M.M. Dyshaev , V.E. Fedorov, “Symmetry analysis and exact solutions of a nonlinear model of the financial markets theory”, Mathematical Notes of North-Eastern Federal University, 23:1(89) (2016), 28–45 (In Russ.) | MR

[14] M.M. Dyshaev , V.E. Fedorov, “Symmetries and exact solutions of a nonlinear equation of options pricing”, Ufa mathematical journal, 9:1 (2017), 29–41 (In Russ.)

[15] L. A. Bordag, A. Y. Chmakova, “Explicit solutions for a nonlinear model of financial derivatives”, International J. of Theoretical and Applied Finance, 10:1 (2007), 1–21 | DOI | MR | Zbl

[16] L. A. Bordag, R. Frey, “Pricing options in illiquid markets: symmetry reductions and exact solutions”, Nonlinear Models in Mathematical Finance: Research Trends in Option Pricing, Chapter 3, ed. M. Ehrhardt, Nova Science Publ., Inc., N. Y., 2008, 83–109 | MR

[17] L. A. Bordag, “On option-valuation in illiquid markets: invariant solutions to a nonlinear model”, Mathematical Control Theory and Finance, 2008, 71–94, Springer, Berlin ; Heidelberg | DOI | MR | Zbl

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

[19] . E. Fedorov, M. M. Dyshaev, “Group classification for a general nonlinear model of option pricing”, Ural Mathematical J., 2:2 (2016), 37–44 | DOI

[20] M. M. Dyshaev, “Group analysis of a nonlinear generalization of Black–Scholes equation”, Chelyabinsk Physical and mathematical Journal, 1:3 (2016), 7–14 (In Russ.) | MR

[21] J. Board, C. Sutcliffe, The Effects of Trade Transparency in the London Stock Exchange: A Summary, Spec. Paper 67, Financial Markets Group, London School of Economics, 1995, 30 pp.