Comparing of some sensitivities (Greeks) for nonlinear models of option pricing with market illiquidity

  • Dyshaev Mikhail M., Mikhail.Dyshaev@gmail.com Chelyabinsk State University, Scientific Research Department, 129, Br.Kashirins str., Chelyabinsk 454021, Russia
  • Fedorov Vladimir E., kar@csu.ru Chelyabinsk State University, Mathematical Analysis Department, 129, Br.Kashirins str., Chelyabinsk 454021, Russia; South Ural State University (National Research University), Laboratory of Functional Materials, 76, Lenin Av., Chelyabinsk 454080, Russia
Keywords: options pricing, nonlinear Black-Scholes type model, illiquid market, sensitivities (Greeks), numerical solution

Abstract

We discuss the numerical solving of nonlinear options pricing models to a market with the insufficient liquidity. Also for these models, the sensitivity coefficients of the option price (Greeks) were found numerically. These nonlinear models were selected by us on the basis of our group classification of a general model and were previously obtained in the works of Frey and Stremme, Sircar and Papanicolaou, and Schönbucher  and Wilmott. The behavior of the price and its sensitivity coefficients in the nonlinear models and in the linear Black-Scholes model is compared. The results of the comparing presents in the form of graphs, a brief comparative analysis of them was made.

References


[1]
Black, F., and Scholes, M. “The pricing of options and corporate liabilities.” Journal of Political Economy 81, 3 (1973), 637–654.

[2]
Merton, R. C. “Theory of rational option pricing.” The Bell Journal of Economics and Management Science 4, 1 (1973), 141–183.

[3]
Cox, J. C., Ross, S. A., and Rubinstein, M. “Option pricing: a simplified approach.” Journal of Financial Economics 7, 3 (1979), 229–263.

[4]
Frey, R., and Stremme, A. “Market volatility and feedback effects from dynamic hedging.” Mathematical Finance 7, 4 (1997), 351–374.

[5]
Schönbucher, P. J., and Wilmott, P. “The feedback effect of hedging in illiquid markets.” SIAM J. Appl. Math. 61, 1 (2000), 232–272.

[6]
Sircar, R. K., and Papanicolaou, G. “General Black-Scholes models accounting for increased market volatility from hedging strategies.” Appl. Math. Finance 5, 1 (1998), 45–82.

[7]
Dyshaev, M. M., and Fedorov, V. E. “Symmetries and exact solutions of a nonlinear pricing options equation.” Ufa Mathematical Journal 9 (2017), 29–40.

[8]
Fedorov, V. E., and Dyshaev, M. M. “Invariant solutions for nonlinear models of illiquid markets.” Math Meth Appl Sci 41, 18 (2018), 8963–8972.

[9]
Dyshaev, M. M., and Fedorov, V. E. “Symmetry analysis and exact solutions for a nonlinear model of the financial markets theory.” Mathematical Notes of North-Eastern Federal University 23, 1 (2016), 28–45. In Russ.

[10]
Dyshaev, M. M. “Group analysis of a nonlinear generalization for Black-Scholes equation.” Chelyabinsk Physical and Mathematical Journal 1, 3 (2016), 7–14. In Russ.

[11]
Brennan, M. J., and Schwartz, E. S. “Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis.” Journal of Financial and Quantitative Analysis 13, 3 (1978), 461– 474.

[12]
Ankudinova, J., and Ehrhardt, M. “On the numerical solution of nonlinear Black-Scholes equations.” Computers & Mathematics with Applications 56, 3 (2008), 799–812.

[13]
Company, R., Jódar, L., Ponsoda, E., and Ballester, C. “Numerical analysis and simulation of option pricing problems modeling illiquid markets.” Computers & Mathematics with Applications 59, 8 (2010), 2964– 2975.

[14]
Company, R., Navarro, E., Pintos, J. R., and Ponsoda, E. “Numerical solution of linear and nonlinear Black-Scholes option pricing equations.” Computers & Mathematics with Applications 56, 3 (2008), 813–821.

[15]
Düring, B., Fournié, M., and Jüngel, A. “High order compact finite difference schemes for a nonlinear Black-Scholes equation.” International Journal of Theoretical and Applied Finance 06, 07 (2003), 767–789.

[16]
Heider, P. “Numerical methods for non-linear Black-Scholes equations.” Applied Mathematical Finance 17 (2010), 59–81.

[17]
Guo, J., and Wang, W. “On the numerical solution of nonlinear option pricing equation in illiquid markets.” Computers & Mathematics with Applications 69, 2 (2015), 117–133.

[18]
Arenas, A. J., Gonz´alez-Parra, G., and Caraballo, B. M. “A nonstandard finite difference scheme for a nonlinear Black- Scholes equation.” Mathematical and Computer Modelling 57, 7 (2013), 1663–1670.

[19]
Düring, B., Fournié, M., and Jüngel, A. “Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation.” M2AN, Math. Model. Numer. Anal. 38, 2 (2004), 359–369.

[20]
Pooley, D. M., Vetzal, K. R., and Forsyth, P. A. “Numerical convergence properties of option pricing PDEs with uncertain volatility.” IMA Journal of Numerical Analysis 23, 2 (2003), 241–267.

[21]
Haug, E. G. “ The complete guide to option pricing formulas, second ed.” McGraw-Hill, New York, 2007.

[22]
Dyshaev, M. M., and Fedorov, V. E. “The Sensitivities (Greeks) for some models of option pricing with market illiquidity.” DOI: https://doi.org/10.13140/RG.2.2.27157.58083

[23]
Kangro, R., and Nicolaides, R. “Far field boundary conditions for Black-Scholes equations.” SIAM Journal on Numerical Analysis 38, 4 (2000), 1357–1368.

[24]
Crank, J., and Nicolson, P. “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type.” Mathematical Proceedings of the Cambridge Philosophical Society 43, 1 (1947), 50–67.
How to Cite
Dyshaev, M. and Fedorov, V. (2019) “Comparing of some sensitivities (Greeks) for nonlinear models of option pricing with market illiquidity”, Mathematical notes of NEFU, 26(2), pp. 94-108. doi: https://doi.org/10.25587/SVFU.2019.102.31514.
Section
Mathematical Modeling