Skip to main content
SpringerLink
Log in

Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function

  • Published:
Asia-Pacific Financial Markets Aims and scope Submit manuscript

Abstract

We investigate qualitative and quantitative behavior of a solution of the mathematical model for pricing American style of perpetual put options. We assume the option price is a solution to the stationary generalized Black–Scholes equation in which the volatility function may depend on the second derivative of the option price itself. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit equation for the free boundary position and the closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of a constant volatility. We also present results of numerical computations of the free boundary position, option price and their dependence on model parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Amster, P., Averbuj, C. G., Mariani, M. C., & Rial, D. (2005). A Black–Scholes option pricing model with transaction costs. Journal of Mathematical Analysis and Applications, 303, 688–695.

    Article  Google Scholar 

  • Avellaneda, M., & Paras, A. (1994). Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Applied Mathematical Finance, 1, 165–193.

    Article  Google Scholar 

  • Bakstein, D., & Howison, S. (2004). A non-arbitrage liquidity model with observable parameters. Working paper. http://eprints.maths.ox.ac.uk/53/

  • Barles, G., & Soner, H. M. (1998). Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance and Stochastics, 2, 369–397.

    Article  Google Scholar 

  • Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.

    Article  Google Scholar 

  • Bordag, L. A. (2011). Study of the risk-adjusted pricing methodology model with methods of geometrical analysis. Stochastics: International Journal of Probability and Stochastic Processes, 83(4–6), 333–345.

    Google Scholar 

  • Bordag, L. A. (2015). Geometrical properties of differential equations: Applications of the Lie group analysis in financial mathematics. Singapore: World Scientific Publishing.

    Book  Google Scholar 

  • Bordag, L. A., & Chmakova, A. Y. (2007). Explicit solutions for a nonlinear model of financial derivatives. International Journal of Theoretical and Applied Finance, 10(1), 1–21.

    Article  Google Scholar 

  • Brennan, M. J., & Schwartz, E. S. (1977). The valuation of American put options. Journal of Finance, 32, 449–462.

    Article  Google Scholar 

  • Cetin, U., Jarrow, R., & Protter, P. (2004). Liquidity risk and arbitrage pricing theory. Finance and Stochastics, 8(3), 311–441.

    Article  Google Scholar 

  • Dewynne, J. N., Howison, S. D., Rupf, J., & Wilmott, P. (1993). Some mathematical results in the pricing of American options. European Journal of Applied Mathematics, 4, 381–398.

    Article  Google Scholar 

  • Evans, J. D., Kuske, R., & Keller, J. B. (2002). American options on assets with dividends near expiry. Mathematical Finance, 12(3), 219–237.

    Article  Google Scholar 

  • Fabiao, R. F., Grossinho, M. R., & Simoes, O. (2009). Positive solutions of a Dirichlet problem for a stationary nonlinear Black–Scholes equation. Nonlinear Analysis, Theory, Methods and Applications, 71, 4624–4631.

    Article  Google Scholar 

  • Frey, R. (1998). Perfect option hedging for a large trader. Finance and Stochastics, 2, 115–142.

    Article  Google Scholar 

  • Frey, R., & Patie, P. (2002). Risk management for derivatives in illiquid markets: A simulation study. Advances in finance and stochastics (pp. 137–159). Berlin: Springer.

    Google Scholar 

  • Frey, R., & Stremme, A. (1997). Market volatility and feedback effects from dynamic hedging. Mathematical Finance, 4, 351–374.

    Article  Google Scholar 

  • Grossinho, M. R., & Morais, E. (2009). A note on a stationary problem for a Black–Scholes equation with transaction costs. International Journal of Pure and Applied Mathematics, 51, 557–565.

    Google Scholar 

  • Grossinho, M., Kord Faghan, Y., & Ševčovič, D. (2017). Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black–Scholes equations. In M. Ehrhardt, M. Günther, & J. ter Maten (Eds.), Novel methods in computational finance, Mathematics in industry (Vol. 25, pp. 129–142). Berlin: Springer.

    Google Scholar 

  • Hoggard, T., Whalley, A. E., & Wilmott, P. (1994). Hedging option portfolios in the presence of transaction costs. Advances in Futures and Options Research, 7, 21–35.

    Google Scholar 

  • Hull, J. (1989). Options, futures and other derivative securities. Englewood Cliffs: Prentice Hall.

    Google Scholar 

  • Jandačka, M., & Ševčovič, D. (2005). On the risk adjusted pricing methodology based valuation of vanilla options and explanation of the volatility smile. Journal of Applied Mathematics, 3, 235–258.

    Article  Google Scholar 

  • Kratka, M. (1998). No mystery behind the smile. Risk, 9, 67–71.

    Google Scholar 

  • Kwok, Y. K. (1998). Mathematical models of financial derivatives. Berlin: Springer.

    Google Scholar 

  • Lauko, M., & Ševčovič, D. (2011). Comparison of numerical and analytical approximations of the early exercise boundary of American put options. ANZIAM Journal, 51, 430–448.

    Article  Google Scholar 

  • Leland, H. E. (1985). Option pricing and replication with transaction costs. Journal of Finance, 40, 1283–1301.

    Article  Google Scholar 

  • Merton, R. (1971). Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory, 3, 373–413.

    Article  Google Scholar 

  • Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4, 141–183.

    Article  Google Scholar 

  • Schönbucher, P., & Wilmott, P. (2000). The feedback-effect of hedging in illiquid markets. SIAM Journal of Applied Mathematics, 61, 232–272.

    Article  Google Scholar 

  • Stamicar, R., Ševčovič, D., & Chadam, J. (1999). The early exercise boundary for the American put near expiry: Numerical approximation. Canadian Applied Math Quarterly, 7, 427–444.

    Google Scholar 

  • Ševčovič, D. (2007). An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black–Scholes equation. Canadian Applied Math Quarterly, 15, 77–97.

    Google Scholar 

  • Ševčovič, D., Stehlíková, B., & Mikula, K. (2011). Analytical and numerical methods for pricing financial derivatives (p. 309). Hauppauge: Nova Science.

    Google Scholar 

  • Ševčovič, D., & Žitňanská, M. (2016). Analysis of the nonlinear option pricing model under variable transaction costs. Asia-Pacific Financial Markets, 23(2), 153–174.

    Article  Google Scholar 

  • Zhu, S.-P. (2006). A new analytical approximation formula for the optimal exercise boundary of American put options. International Journal of Theoretical and Applied Finance, 9, 1141–1177.

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by the European Union in the FP7-PEOPLE-2012-ITN Project STRIKE—Novel Methods in Computational Finance (304617), the Project CEMAPRE MULTI/00491 financed by FCT/MEC through national funds and the Slovak research Agency Project VEGA 1/0251/16.

Author information

Authors and Affiliations

  1. CEMAPRE, Department of Mathematics, ISEG, Universidade de Lisboa, Rua do Quelhas, 6, 1200-781, Lisbon, Portugal

    Maria do Rosário Grossinho & Yaser Kord Faghan

  2. Department of Applied Mathematics and Statistics, Comenius University, 842 48, Bratislava, Slovak Republic

    Daniel Ševčovič

Authors
  1. Maria do Rosário Grossinho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yaser Kord Faghan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Ševčovič
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Ševčovič.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grossinho, M.d.R., Kord Faghan, Y. & Ševčovič, D. Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function. Asia-Pac Financ Markets 24, 291–308 (2017). https://doi.org/10.1007/s10690-017-9234-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10690-017-9234-1

Keywords

Mathematics Subject Classification

Search

Navigation