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Group classification for a general nonlinear model of option pricing. (English) Zbl 1398.91584

Summary: We consider a family of equations with two free functional parameters containing the classical Black-Scholes model, Schonbucher-Wilmott model, Sircar-Papanicolaou equation for option pricing as partial cases. A five-dimensional group of equivalence transformations is calculated for that family. That group is applied to a search for specifications’ parameters specifications corresponding to additional symmetries of the equation. Seven pairs of specifications are found.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
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References:

[1] [1] Sircar K.R., Papanicolaou G., “General Black-Scholes models accounting for increased market volatility from hedging strategies”, Appl. Math. Finance, 5 (1998), 45-82 · Zbl 1009.91023
[2] [2] Black F., Scholes M., “The pricing of options and corporate liabilities”, J. of Political Economy, 81 (1973), 637-659 · Zbl 1092.91524
[3] [3] Frey R., Stremme A., “Market volatility and feedback effects from dynamic hedging”, Math. Finance, 7:4 (1997), 351-374 · Zbl 1020.91023
[4] [4] Frey R., “Perfect option replication for a large trader”, Finance and Stochastics, 2 (1998), 115-148 · Zbl 0894.90017
[5] [5] Jarrow R.A., “Derivative securities markets, market manipulation and option pricing theory”, J. of Financial and Quantitative Analysis, 29 (1994), 241-261
[6] [6] Schönbucher P., Wilmott P., “The feedback-effect of hedging in illiquid markets”, SIAM J. on Appl. Math., 61 (2000), 232-272 · Zbl 1136.91407
[7] [7] Ovsyannikov L.V., Group analysis of differential equations, Academic press, New York, 1982 · Zbl 0485.58002
[8] [8] Chirkunov Yu.A., Khabirov S.V., Elements of symmetry analysis for differential equations of continuum mechanics, Novosibirsk State Technical University, Novosibirsk, 2012, 659 pp. (in Russian)
[9] [9] Gazizov R.K., Ibragimov N.H., “Lie symmetry analysis of differential equations in finance”, Nonlinear Dynamics, 17 (1998), 387-407 · Zbl 0929.35006
[10] [10] Bordag L.A., Chmakova A.Y., “Explicit solutions for a nonlinear model of financial derivatives”, Int. J. of Theoretical and Applied Finance, 10:1 (2007), 1-21 · Zbl 1291.91203
[11] [11] Bordag L.A., “On option-valuation in illiquid markets: invariant solutions to a nonlinear mode”, Mathematical Control Theory and Finance, eds. A. Sarychev, A. Shiryaev, M. Guerra and M. R. Grossinho, Springer, Berlin-Heidelberg, 2008, 71-94 · Zbl 1149.91313 · doi:10.1007/978-3-540-69532-5_5
[12] [12] Mikaelyan A., Analytical study of the Schönbucher-Wilmott model of the feedback effect in illiquid markets. Master’s Thesis in Financial Mathematics, Technical report, IDE0913, Halmstad University, Halmstad, 2009, viii+67 pp.
[13] [13] Bordag L.A., Mikaelyan A., “Models of self-financing hedging strategies in illiquid markets: symmetry reductions and exact solutions”, Journal Letters in Mathematical Physics, 96:1-3 (2011), 191-207 · Zbl 1223.35025
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