Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs

Rafael Company; Lucas Jódar; José-Ramón Pintos

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 6, page 1045-1061
  • ISSN: 0764-583X

Abstract

top
This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical approximation of the Gamma of the option, previously obtained. Results are illustrated with numerical examples.

How to cite

top

Company, Rafael, Jódar, Lucas, and Pintos, José-Ramón. "Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs." ESAIM: Mathematical Modelling and Numerical Analysis 43.6 (2009): 1045-1061. <http://eudml.org/doc/250609>.

@article{Company2009,
abstract = { This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical approximation of the Gamma of the option, previously obtained. Results are illustrated with numerical examples. },
author = {Company, Rafael, Jódar, Lucas, Pintos, José-Ramón},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear Black-Scholes equation; option pricing; numerical analysis; transaction costs.; nonlinear Black-Scholes equation; transaction costs},
language = {eng},
month = {6},
number = {6},
pages = {1045-1061},
publisher = {EDP Sciences},
title = {Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs},
url = {http://eudml.org/doc/250609},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Company, Rafael
AU - Jódar, Lucas
AU - Pintos, José-Ramón
TI - Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/6//
PB - EDP Sciences
VL - 43
IS - 6
SP - 1045
EP - 1061
AB - This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical approximation of the Gamma of the option, previously obtained. Results are illustrated with numerical examples.
LA - eng
KW - Nonlinear Black-Scholes equation; option pricing; numerical analysis; transaction costs.; nonlinear Black-Scholes equation; transaction costs
UR - http://eudml.org/doc/250609
ER -

References

top
  1. M. Avellaneda and A. Parás, Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Appl. Math. Finance1 (1994) 165–193.  
  2. G. Barles and H.M. Soner, Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance Stochast. 2 (1998) 369–397.  Zbl0915.35051
  3. P. Boyle and T. Vorst, Option replication in discrete time with transaction costs. J. Finance47 (1973) 271–293.  
  4. R. Company, E. Navarro, J.R. Pintos and E. Ponsoda, Numerical solution of linear and nonlinear Black–Scholes option pricing equations. Comput. Math. Appl.56 (2008) 813–821.  Zbl1155.65370
  5. M. Davis, V. Panis and T. Zariphopoulou, European option pricing with transaction fees. SIAM J. Contr. Optim.31 (1993) 470–493.  Zbl0779.90011
  6. J. Dewynne, S. Howinson and P. Wilmott, Option pricing: mathematical models and computations. Oxford Financial Press, Oxford (2000).  
  7. B. Düring, M. Fournier and A. Jungel, Convergence of a high order compact finite difference scheme for a nonlinear Black–Scholes equation. ESAIM: M2AN38 (2004) 359–369.  
  8. P. Forsyth, K. Vetzal and R. Zvan, A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Appl. Math. Finance6 (1999) 87–106.  Zbl1009.91030
  9. J.M. Harrison and S.R. Pliska, Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes Appl.11 (1981) 215–260.  Zbl0482.60097
  10. S.D. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs. Review of Futures Markets8 (1989) 222–239.  
  11. T. Hoggard, A.E. Whalley and P. Wilmott, Hedging option portfolios in the presence of transaction costs. Adv. Futures Options Research7 (1994) 217–35.  
  12. R. Kangro and R. Nicolaides, Far field boundary conditions for Black–Scholes equations. SIAM J. Numer. Anal.38 (2000) 1357–1368.  Zbl0990.35013
  13. J. Leitner, Continuous time CAPM, price for risk and utility maximization, in Mathematical Finance – Workshop of the Mathematical Finance Research Project, Konstanz, Germany, M. Kohlmann and S. Tang Eds., Birkhäuser, Basel (2001).  Zbl1004.91040
  14. H.E. Leland, Option pricing and replication with transactions costs. J. Finance40 (1985) 1283–1301.  
  15. O. Pironneau and F. Hecht, Mesh adaption for the Black and Scholes equations. East-West J. Numer. Math.8 (2000) 25–35.  Zbl0995.91026
  16. A. Rigal, Numerical analisys of three-time-level finite difference schemes for unsteady diffusion-convection problems. J. Num. Meth. Engineering30 (1990) 307–330.  
  17. G.D. Smith, Numerical solution of partial differential equations: finite difference methods. Third Edition, Clarendon Press, Oxford (1985).  Zbl0576.65089
  18. H.M. Soner, S.E. Shreve and J. Cvitanic, There is no non-trivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab.5 (1995) 327–355.  Zbl0837.90012
  19. J.C. Strikwerda, Finite difference schemes and partial differential equations. Wadsworth & Brooks/Cole Mathematics Series (1989) 32–52.  Zbl0681.65064
  20. D. Tavella and C. Randall, Pricing financial instruments – The finite difference method. John Wiley & Sons, Inc., New York (2000).  
  21. A.E. Whalley and P. Wilmott, An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Math. Finance7 (1997) 307–324.  Zbl0885.90019

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.