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AN INVESTIGATION ON NONLINEAR OPTION PRICING BEHAVIOURS THROUGH A NEW FRÉCHET DERIVATIVE-BASED QUADRATURE APPROACH

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Mathematical finance General higher-order equations and systems Controllability, observability, and system structure

Published online by Cambridge University Press:  15 November 2024

S. GULEN Open the ORCID record for S. GULEN [Opens in a new window]  and
M. SARI Open the ORCID record for M. SARI [Opens in a new window]
S. GULEN*
Affiliation:
Faculty of Arts and Sciences, Department of Mathematics, Tekirdag Namik Kemal University, 59030 Tekirdag, Turkiye
M. SARI
Affiliation:
Faculty of Science and Letters, Department of Mathematics Engineering, Istanbul Technical University, 34469 Istanbul, Turkiye; e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Complicated option pricing models attract much attention in financial industries, as they produce relatively better accurate values by taking into account more realistic assumptions such as market liquidity, uncertain volatility and so forth. We propose a new hybrid method to accurately explore the behaviour of the nonlinear pricing model in illiquid markets, which is important in financial risk management. Our method is based on the Newton iteration technique and the Fréchet derivative to linearize the model. The linearized equation is then discretized by a differential quadrature method in space and a quadratic trapezoid rule in time. It is observed through computations that the accurate solutions for the model emerge using very few grid points and time elements, compared with the finite difference method in the literature. Furthermore, this method also helps to avoid consideration of the convergence issues of the Newton approach applied to the nonlinear algebraic system containing many unknowns at each time step if an implicit method is used in time discretization. It is important to note that the Fréchet derivative supports to enhance the convergence order of the proposed iterative scheme.

Keywords

nonlinear Black–Scholes equationilliquid marketsFréchet derivativelinearizationdifferential quadrature method

MSC classification

Primary: 91G20: Derivative securities
Secondary: 35G31: Initial-boundary value problems for nonlinear higher-order equations 93B18: Linearizations 65M06: Finite difference methods
Type
Research Article
Information
The ANZIAM Journal , Volume 66 , Issue 4 , October 2024 , pp. 238 - 257
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Ambrosetti, A. and Giovanni, P., A primer of nonlinear analysis (Cambridge University Press, Cambridge, 1995) 1–46.Google Scholar
Ankudinova, J. and Ehrhardt, M., “On the numerical solution of nonlinear Black–Scholes equations”, Comput. Math. Appl. 56 (2008) 799–812; doi:10.1016/j.camwa.2008.02.005.CrossRefGoogle Scholar
Arenas, A. J., Gonzales-Parra, G. and Caraballo, B. M., “A nonstandard finite difference scheme for a nonlinear Black–Scholes equation”, Math. Comput. Model. 57 (2013) 1663–1670; doi:10.1016/j.mcm.2011.11.009.CrossRefGoogle Scholar
Argvros, I. K., Convergence and applications of Newton-type iterations (Springer, New York, 2008); doi:10.1007/978-0-387-72743-1.Google Scholar
Atkinson, K. and Han, W., Theoretical numerical analysis: a functional analysis framework (Springer Verlag, New York, 2009); doi:10.1007/978-1-4419-0458-4.Google Scholar
Bakstein, D. and Howison, S., “A non-arbitrage liquidity model with observable parameters for derivatives”, Oxford Preprint, 2003. Available at https://ora.ox.ac.uk/objects/uuid:e52e1442-9a2b-4ed8-8f3b-111105231c3a.Google Scholar
Bank, P. and Baum, D., “Hedging and portfolio optimization in financial markets with a large trader”, Math. Finance 14 (2004) 1–18; doi:10.1111/j.0960-1627.2004.00179.x.CrossRefGoogle Scholar
Bellman, R., Kashef, B. G. and Casti, J., “Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations”, J. Comput. Phys. 10 (1972) 40–52; doi:10.1016/0021-9991(72)90089-7.CrossRefGoogle Scholar
Bert, C. W. and Malik, M., “Differential quadrature method in computational mechanics: a review”, Appl. Mach. Rev. 49 (1996) 1–27; doi:10.1115/1.3101882.CrossRefGoogle Scholar
Bordag, L. A., “On option-valuation in illiquid markets: invariant solutions to a nonlinear model”, in: Mathematical control theory and finance (eds Sarychev, A., Shiryaev, A., Guerra, M. and Grossinho, M. R.) (Springer, Berlin–Heidelberg, 2008); doi:10.1007/978-3-540-69532-5_5.Google Scholar
Bordag, L. A., Geometrical properties of differential equations: applications of the Lie group analysis in financial mathematics (World Scientific, Singapore, 2015); doi:10.1142/9549.CrossRefGoogle Scholar
Bordag, L. A. and Chmakova, A. Y., “Explicit solutions for a nonlinear model of financial derivatives”, J. Theor. Appl. Finance 10 (2007) 1–21; doi:10.1142/S021902490700407X.CrossRefGoogle Scholar
Bordag, L. A. and Frey, R., “Pricing options in illiquid markets: symmetry reductions and exact solutions”, in: Nonlinear models in mathematical finance: new research trends in option pricing (ed. Ehrhardt, M.) (Nova Science Publishers, New York, 2008).Google Scholar
Cetin, U., Jarrow, R. A. and Protter, P., “Liquidity risk and arbitrage pricing theory”, Finan. Stoch. 8 (2004) 311–341.CrossRefGoogle Scholar
Chang, C. T., Tsai, C. S. and Lin, T. T., “The modified differential quadrature and their applications”, Chem. Eng. Commun. 123 (1993) 135–164; doi:10.1080/00986449308936170.CrossRefGoogle Scholar
Chen, W. Q., Lv, C. F. and Bian, Z. G., “Free vibration analysis of generally laminated beams via state-space-based differential quadrature”, Compos. Struct. 64 (2004) 417–425; doi:10.1016/S0263-8223(03)00190-9.CrossRefGoogle Scholar
Company, R., Jodar, L. and Pintos, J. R., “Numerical analysis and computing for option pricing models in illiquid markets”, Math. Comput. Model. 52(10) (2010) 1066–1073; doi:10.1016/j.mcm.2010.02.037.CrossRefGoogle Scholar
Company, R., Jodar, L. and Pintos, J. R., “A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets”, Math. Comput. Simul. 82 (2012) 1972–1985; doi:10.1016/j.matcom.2010.04.026.CrossRefGoogle Scholar
Company, R., Jodar, L., Ponsoda, E. and Ballester, C., “Numerical analysis and simulation of option pricing problems modeling illiquid markets”, Comput. Math. Appl. 59 (2010), 2964–2975; doi:10.1016/j.camwa.2010.02.014.CrossRefGoogle Scholar
Cuomo, S., Sica, F. and Toraldo, G., “Greeks computation in the option pricing problem by means of RBF-PU methods”, J. Comput. Math. 376 (2020) Article ID: 112882; doi:10.1016/j.cam.2020.112882.CrossRefGoogle Scholar
Deuflhard, P., Newton methods for nonlinear problems (Springer, Berlin–Heidelberg, 2011); doi:10.1007/978-3-642-23899-4.CrossRefGoogle Scholar
During, B., Fournier, B. and Jüngel, A., “Convergence of a high order compact finite difference scheme for nonlinear Black–Scholes equation”, ESAIM Math. Model. Numer. Anal. 38 (2004) 359–369; doi: https://doi.org/10.1051/m2an:2004018.CrossRefGoogle Scholar
Duris, K., Tan, S. H., Lai, C. H. and Ševčovič, D., “Comparison of the analytical approximation formula and Newton’s method for solving a class of nonlinear Black–Scholes equations”, Comput. Methods Appl. Math. 16 (2016) 35–50; doi:10.1515/cmam-2015-0035.CrossRefGoogle Scholar
Edeki, S. O., Ugbebor, O. O. and Owoloko, E. A., “On a nonlinear transaction-cost model for stock prices in an illiquid market driven by a relaxed Black–Scholes model assumptions”, Malays. J. Math. Sci. 11 (2017) 83–96.Google Scholar
Ehrhardt, M. and Valkov, R., “An unconditionally stable explicit finite difference scheme for nonlinear European option pricing problems”, Preprint BUW-IMACM, 2013. Available at http://www-amna.math.uni-wuppertal.de/~ehrhardt/talks/ehrhardt-amna2014.pdf.Google Scholar
Erdogan, U., Sari, M. and Kocak, H., “Efficient numerical treatment of nonlinearities in the advection-diffusion-reaction equations”, Int. J. Numer. Method Heat Fluid Flow 29 (2019) 132–145; doi:10.1108/HFF-05-20170198.CrossRefGoogle Scholar
Esekon, J., Onyango, S. and Ongoti, N. O., “Analytical solution of a nonlinear Black–Scholes partial differential equation”, Int. J. Pure Appl. Math. 61 (2010) 219–230; doi:10.12732/ijpam.v82i4.4.Google Scholar
Federov, V. E. and Dyshaev, M. M., “Invariant solutions for nonlinear models of illiquid markets”, Math. Method Appl. 41 (2018) 8963–8972; doi:10.1002/mma.4772.CrossRefGoogle Scholar
Frey, R., “Perfect option replication for a large trader”, Finance Stoch. 2 (1998) 115–148. doi:10.1007/s007800050035.CrossRefGoogle Scholar
Frey, R., “Market illiquidity as a source of model risk in dynamic hedging”, Working Paper, Leipzig University, 2000.Google Scholar
Frey, R. and Patie, P., “Risk management for derivatives in illiquid markets: a simulation study” in: Advances in finance and Stochastics (eds Sandman, K. and Schönbucher, P. J.) (Springer, Berlin–Heidelberg, 2002); doi:10.1007/978-3-662-04790-3_8.Google Scholar
Goxiola, O. G., Chavez, J. R. and Santiago, J. A., “A nonlinear option pricing model through the Adomian decomposition method”, Int. J. Appl. Comput. Math. 2 (2016) 453–467; doi:10.1007/s40819-015-0070-6.CrossRefGoogle Scholar
Gulen, S. and Sari, M., “A Frechet derivative-based novel approach to option pricing models in illiquid markets”, Math. Methods Appl. Sci. 45 (2022) 899–913; doi:10.1002/mma.7821.CrossRefGoogle Scholar
Guo, J. and Wang, W., “On the numerical solution of nonlinear option pricing equation in illiquid markets”, Comput. Math. Appl. 60 (2015) 117–133; doi:10.1016/j.camwa.2014.11.015.CrossRefGoogle Scholar
Heider, P., “Numerical methods for nonlinear Black–Scholes equation”, Appl. Math. Finance 17 (2010) 59–81; doi:10.1080/13504860903075670.CrossRefGoogle Scholar
Jarrow, R. A., “Market manipulation, bubbles corners and short squeezes”, J. Financial Quant. Anal. 27 (1992) 311–336; doi:10.2307/2331322.CrossRefGoogle Scholar
Kashef, B. and Bellman, R., “Solution of the partial differential equation of the Hodgkin–Huxley model using differential quadrature”, Math. Biosci. 19 (1974) 1–8; doi:10.1016/0025-5564(74)90027-3.CrossRefGoogle Scholar
Kim, Y., Bae, H. O. and Koo, H. K., “Option pricing and Greeks via a moving least square meshfree method”, Quant. Finance 14 (2014) 1753–1764; doi:10.1080/14697688.2013.845686.CrossRefGoogle Scholar
Koleva, M. N. and Vulkov, L. G., “On splitting-based numerical methods for nonlinear models of European options”, Int. J. Comput. Math. 93 (2016) 781–796; doi:10.1080/00207160.2014.884713.CrossRefGoogle Scholar
Korkut, S. O., Kaymak, N. G. and Tanoglu, G. A., “A conserved linearization approach for solving nonlinear oscillations problems”, Appl. Math. Inf. Sci. 12 (2018) 537–543; doi:10.18576/amis/120308.CrossRefGoogle Scholar
Liu, H. and Yong, J., “Option pricing with an illiquid underlying asset market”, J. Econ. Dyn. Control. 29 (2005) 2125–2156; doi:10.1016/j.jedc.2004.11.004.CrossRefGoogle Scholar
Mansell, G., Merryfield, W. and Shizgal, B., “A comparison of differential quadrature methods for the solution of partial differential equations”, Comput. Methods Appl. Mech. Engrg. 104 (1993) 295–316; doi:10.1016/0045-7825(93)90028-V.CrossRefGoogle Scholar
O’Mahoney, D. C., “A differential quadrature solution of the two-dimensional inverse heat conduction problem”, Int. Comm. Heat Mass Transfer 30 (2003) 1061–1070; doi:10.1016/S0735-1933(03)00172-6.CrossRefGoogle Scholar
Platen, E. and Schweizer, M., “On feedback effects from hedging derivatives”, Math. Finance 8 (1998) 67–84; doi:10.1111/1467-9965.00045.CrossRefGoogle Scholar
Schönbucher, P. and Wilmott, P., “The feedback effect of hedging in illiquid markets”, SIAM J. Appl. Math. 61 (2000) 232–272; doi:10.1137/S0036139996308534.Google Scholar
Shu, C., “Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation”, Ph. D. Thesis, University of Glasgow, UK, 1991.Google Scholar
Shu, C., Differential quadrature and its application in engineering (Springer, London, 2000); doi:10.1007/978-1-4471-0407-0.CrossRefGoogle Scholar
Shu, C. and Richards, B. E., “Application of generalized differential quadrature to solve two-dimensional incompressible Navier–Stokes equations”, Int. J. Numer. Methods Fluids 15 (1992) 791–798; doi:10.1002/fld.1650150704.CrossRefGoogle Scholar
Sircar, R. and Papanicolaou, G., “General Black–Scholes models accounting for increased market volatility from hedging strategies”, Appl. Math. Finance 5 (1998) 45–82; doi:10.1080/135048698334727.CrossRefGoogle Scholar
Smith, G. D., Numerical solution of partial differential equations: finite difference methods (Oxford University Press, New York, 1985).Google Scholar
Striz, A. G., Wang, X. and Bert, C. W., “Harmonic differential quadrature method and applications to analysis of structural components”, Acta Mech. 111 (1995) 85–94; doi:10.1007/BF01187729.CrossRefGoogle Scholar
Xionghua, W. and Zhihong, D., “Differential quadrature method for pricing American options”, Numer. Methods Partial Differential Equations 18 (2002) 711–725; doi:10.1002/num.10028.CrossRefGoogle Scholar
Yan, H., “Adaptive wavelet precise integration method for nonlinear Black–Scholes model based on variational iteration method”, Abstr. Appl. Anal. 2013 (2013) Article ID: 735919; doi:10.1155/2013/735919.CrossRefGoogle Scholar
Ycel, U., “Approximation of Sturm–Liouville eigenvalues using differential quadrature (DQ) method”, J. Comput. Appl. Math. 192 (2006) 310–319; doi:10.1016/j.cam.2005.05.008.CrossRefGoogle Scholar