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On the numerical solution of nonlinear Black-Scholes equations. (English) Zbl 1155.65367

Summary: Nonlinear Black-Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor’s preferences or illiquid markets (which may have an impact on the stock price), the volatility, the drift and the option price itself.

In this paper we will focus on several models from the most relevant class of nonlinear Black-Scholes equations for European and American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives due to transaction costs. We will analytically approach the option price by transforming the problem for a European Call option into a convection-diffusion equation with a nonlinear term and the free boundary problem for an American Call option into a fully nonlinear nonlocal parabolic equation defined on a fixed domain following Ševčovič’s idea. Finally, we will present the results of different numerical discretization schemes for European options for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.

65M06Finite difference methods (IVP of PDE)
60H15Stochastic partial differential equations
91B28Finance etc. (MSC2000)
[1]Wilmott, P.; Howison, S.; Dewynne, J.: The mathematics of financial derivatives, (1995)
[2]Black, F.; Scholes, M.: The pricing of options and corporate liabilities, J. polit. Econ. 81, No. 3, 637-654 (1973)
[3]Merton, R. C.: Theory of rational option pricing, Bell J. Econ. 4, No. 1, 141-183 (1973)
[4]Seydel, R.: Tools for computational finance, (2004)
[5]Tavella, D.; Randall, C.: Pricing financial instruments – the finite difference method, (2000)
[6]Leland, H. E.: Option pricing and replication with transactions costs, J. finance 40, 1283-1301 (1985)
[7]Boyle, P.; Vorst, T.: Option replication in discrete time with transaction costs, J. finance 47, No. 1, 271-293 (1992)
[8]Barles, G.; Soner, H. M.: Option pricing with transaction costs and a nonlinear black–Scholes equation, Finance stoch. 2, 369-397 (1998) · Zbl 0915.35051 · doi:10.1007/s007800050046
[9]Frey, R.; Patie, P.: Risk management for derivatives in illiquid markets: A simulation study, (2002) · Zbl 1002.91031
[10]Frey, R.; Stremme, A.: Market volatility and feedback effects from dynamic hedging, Math. finance 4, 351-374 (1997) · Zbl 1020.91023 · doi:10.1111/1467-9965.00036
[11]Schönbucher, P.; Wilmott, P.: The feedback–effect of hedging in illiquid markets, SIAM J. Appl. math. 61, 232-272 (2000) · Zbl 1136.91407 · doi:10.1137/S0036139996308534
[12]Soner, H. M.; Touzi, N.: Superreplication under gamma constraints, SIAM J. Control optim. 39, No. 1, 73-96 (2001) · Zbl 0960.91036 · doi:10.1137/S0363012998348991
[13]Zhu, Y.; Wu, X.; Chern, I.: Derivative securities and difference methods, (2004)
[14]Kwok, Y.: Mathematical models of financial derivatives, (1998)
[15]Soner, H. M.; Shreve, S. E.; Cvitanič, J.: There is no nontrivial hedging portfolio for option pricing with transaction costs, Ann. appl. Probab. 5, No. 2, 327-355 (1995) · Zbl 0837.90012 · doi:10.1214/aoap/1177004767
[16]Hoggard, T.; Whalley, A. E.; Wilmott, P.: Hedging option portfolios in the presence of transaction costs, Adv. fut. Opt. res. 7, 21-35 (1994)
[17]Avellaneda, M.; Parás, A.: Dynamic hedging portfolios for derivative securities in the presence of large transaction costs, Appl. math. Finance 1, 165-193 (1994)
[18]Bensaid, B.; Lesne, J. P.; Pagès, H.; Scheinkman, J.: Derivative asset pricing with transaction costs, Math. finance 2, 63-86 (1992) · Zbl 0900.90100 · doi:10.1111/j.1467-9965.1992.tb00039.x
[19]Hodges, S.; Neuberger, A.: Optimal replication of contingent claims under transaction costs, Rev. futures markets 8, 222-239 (1989)
[20]Davis, M. H. A.; Panas, V. G.; Zariphopoulou, T.: European option pricing with transaction costs, SIAM J. Control. optim. 31, No. 2, 470-493 (1993) · Zbl 0779.90011 · doi:10.1137/0331022
[21]Fleming, W. H.; Soner, H. M.: Controlled Markov processes and viscosity solutions, (1993) · Zbl 0773.60070
[22]Jandačka, M.; Ševčovič, D.: On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile, J. appl. Math. 2005, No. 3, 235-258 (2005) · Zbl 1128.91025 · doi:10.1155/JAM.2005.235
[23]J. Ankudinova, The numerical solution of nonlinear Black–Scholes equations, Master’s Thesis, Technische Universität Berlin, Berlin, in preparation
[24]B. Düring, Black–Scholes type equations: Mathematical analysis, parameter identification & numerical solution, Dissertation, University Mainz, 2005
[25]Ševčovič, D.: An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear black–Scholes equation, Canad. appl. Math. quarterly 15, No. 1, 77-97 (2007) · Zbl 1145.35321 · doi:http://www.math.ualberta.ca/ami/CAMQ/table_of_content/vol_15/15_1f.htm
[26]M. Ehrhardt, R.E. Mickens, Discrete artificial boundary conditions for the Black–Scholes equation of American options, Int. J. Appl. Theoret. Finance (in preparation)
[27]L.H. Thomas, Elliptic problems in linear difference equations over a network, Watson Sci. Comput. Lab. Rept., Columbia University, New York, 1949
[28]Rigal, A.: High order difference schemes for unsteady one-dimensional diffusion-convection problems, J. comput. Phys. 114, No. 1, 59-76 (1994) · Zbl 0807.65096 · doi:10.1006/jcph.1994.1149
[29]Düring, B.; Fournier, M.; Jüngel, A.: High order compact finite difference schemes for a nonlinear black–Scholes equation, Int. J. Appl. theoret. Finance 7, 767-789 (2003) · Zbl 1070.91024 · doi:10.1142/S0219024903002183
[30]Strikwerda, J. C.: Finite difference schemes and partial differential equations, (1989) · Zbl 0681.65064
[31]Dormand, J. R.; Prince, P. J.: A family of embedded Runge–Kutta formulae, J. comput. Appl. math. 6, 19-26 (1980) · Zbl 0448.65045 · doi:10.1016/0771-050X(80)90013-3
[32]Zhao, J.; Davison, M.; Corless, R. M.: Compact finite difference method for American option pricing, J. comput. Appl. math. 206, No. 1, 306-321 (2007) · Zbl 1151.91552 · doi:10.1016/j.cam.2006.07.006
[33]H. Sun, J. Zhang, A high order compact boundary value method for solving one dimensional heat equations, Tech. Rep. No. 333-02, University of Kentucky, 2002
[34]Mccartin, B. J.; Labadie, S. M.: Accurate and efficient pricing of vanilla stock options via the Crandall–Douglas scheme, Appl. math. Comput. 143, No. 1, 39-60 (2003) · Zbl 1053.91062 · doi:10.1016/S0096-3003(02)00343-0