IDEAS home Printed from https://ideas.repec.org/a/taf/quantf/v7y2007i4p459-472.html
   My bibliography  Save this article

Implied non-recombining trees and calibration for the volatility smile

Author

Listed:
  • Chris Charalambous
  • Nicos Christofides
  • Eleni D. Constantinide
  • Spiros H. Martzoukos

Abstract

In this paper we capture the implied distribution from option market data using a non-recombining (binary) tree, allowing the local volatility to be a function of the underlying asset and of time. The problem under consideration is a non-convex optimization problem with linear constraints. We elaborate on the initial guess for the volatility term structure and use nonlinear constrained optimization to minimize the least squares error function on market prices. The proposed model can accommodate European options with single maturities and, with minor modifications, options with multiple maturities. It can provide a market-consistent tree for option replication with transaction costs (often this requires a non-recombining tree) and can help pricing of exotic and Over The Counter (OTC) options. We test our model using options data for the FTSE 100 index obtained from LIFFE. The results strongly support our modelling approach.

Suggested Citation

  • Chris Charalambous & Nicos Christofides & Eleni D. Constantinide & Spiros H. Martzoukos, 2007. "Implied non-recombining trees and calibration for the volatility smile," Quantitative Finance, Taylor & Francis Journals, vol. 7(4), pages 459-472.
  • Handle: RePEc:taf:quantf:v:7:y:2007:i:4:p:459-472
    DOI: 10.1080/14697680701488692
    as

    Download full text from publisher

    File URL: http://www.tandfonline.com/doi/abs/10.1080/14697680701488692
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1080/14697680701488692?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Mark Britten‐Jones & Anthony Neuberger, 2000. "Option Prices, Implied Price Processes, and Stochastic Volatility," Journal of Finance, American Finance Association, vol. 55(2), pages 839-866, April.
    2. George Skiadopoulos, 2001. "Volatility Smile Consistent Option Models: A Survey," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(03), pages 403-437.
    3. Longstaff, Francis A, 1995. "Option Pricing and the Martingale Restriction," The Review of Financial Studies, Society for Financial Studies, vol. 8(4), pages 1091-1124.
    4. Edirisinghe, Chanaka & Naik, Vasanttilak & Uppal, Raman, 1993. "Optimal Replication of Options with Transactions Costs and Trading Restrictions," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(1), pages 117-138, March.
    5. Whaley, Robert E., 1982. "Valuation of American call options on dividend-paying stocks : Empirical tests," Journal of Financial Economics, Elsevier, vol. 10(1), pages 29-58, March.
    6. Jackwerth, Jens Carsten, 1996. "Generalized Binomial Trees," MPRA Paper 11635, University Library of Munich, Germany, revised 12 May 1997.
    7. Stutzer, Michael, 1996. "A Simple Nonparametric Approach to Derivative Security Valuation," Journal of Finance, American Finance Association, vol. 51(5), pages 1633-1652, December.
    8. Jackwerth, Jens Carsten & Rubinstein, Mark, 1996. "Recovering Probability Distributions from Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1611-1632, December.
    9. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    10. Emanuel Derman & Iraj Kani, 1998. "Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 1(01), pages 61-110.
    11. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    12. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    13. repec:bla:jfinan:v:53:y:1998:i:2:p:499-547 is not listed on IDEAS
    14. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Elyas Elyasiani & Silvia Muzzioli & Alessio Ruggieri, 2016. "Forecasting and pricing powers of option-implied tree models: Tranquil and volatile market conditions," Department of Economics 0099, University of Modena and Reggio E., Faculty of Economics "Marco Biagi".
    2. Chris Charalambous & Spiros H. Martzoukos & Zenon Taoushianis, 2022. "Estimating corporate bankruptcy forecasting models by maximizing discriminatory power," Review of Quantitative Finance and Accounting, Springer, vol. 58(1), pages 297-328, January.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    2. Silvia Muzzioli, 2010. "Towards a volatility index for the Italian stock market," Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) 10091, Universita di Modena e Reggio Emilia, Dipartimento di Economia "Marco Biagi".
    3. Silvia Muzzioli, 2013. "The Information Content of Option-Based Forecasts of Volatility: Evidence from the Italian Stock Market," Quarterly Journal of Finance (QJF), World Scientific Publishing Co. Pte. Ltd., vol. 3(01), pages 1-46.
    4. Christoffersen, Peter & Jacobs, Kris & Chang, Bo Young, 2013. "Forecasting with Option-Implied Information," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 2, chapter 0, pages 581-656, Elsevier.
    5. Wael Bahsoun & Pawel Góra & Silvia Mayoral & Manuel Morales, 2006. "Random Dynamics and Finance: Constructing Implied Binomial Trees from a Predetermined Stationary Den," Faculty Working Papers 13/06, School of Economics and Business Administration, University of Navarra.
    6. Elyas Elyasiani & Silvia Muzzioli & Alessio Ruggieri, 2016. "Forecasting and pricing powers of option-implied tree models: Tranquil and volatile market conditions," Department of Economics 0099, University of Modena and Reggio E., Faculty of Economics "Marco Biagi".
    7. Moriggia, V. & Muzzioli, S. & Torricelli, C., 2009. "On the no-arbitrage condition in option implied trees," European Journal of Operational Research, Elsevier, vol. 193(1), pages 212-221, February.
    8. René Garcia & Eric Ghysels & Eric Renault, 2004. "The Econometrics of Option Pricing," CIRANO Working Papers 2004s-04, CIRANO.
    9. Hsuan-Chu Lin & Ren-Raw Chen & Oded Palmon, 2016. "Explaining the volatility smile: non-parametric versus parametric option models," Review of Quantitative Finance and Accounting, Springer, vol. 46(4), pages 907-935, May.
    10. Y. Wang & H. Yin & L. Qi, 2004. "No-Arbitrage Interpolation of the Option Price Function and Its Reformulation," Journal of Optimization Theory and Applications, Springer, vol. 120(3), pages 627-649, March.
    11. Jurczenko, Emmanuel & Maillet, Bertrand & Negrea, Bogdan, 2002. "Revisited multi-moment approximate option pricing models: a general comparison (Part 1)," LSE Research Online Documents on Economics 24950, London School of Economics and Political Science, LSE Library.
    12. Elyas Elyasiani & Luca Gambarelli & Silvia Muzzioli, 2015. "Towards a skewness index for the Italian stock market," Department of Economics 0064, University of Modena and Reggio E., Faculty of Economics "Marco Biagi".
    13. Kim, In Joon & Park, Gun Youb, 2006. "An empirical comparison of implied tree models for KOSPI 200 index options," International Review of Economics & Finance, Elsevier, vol. 15(1), pages 52-71.
    14. Jackwerth, Jens Carsten & Rubinstein, Mark, 2003. "Recovering Probabilities and Risk Aversion from Option Prices and Realized Returns," MPRA Paper 11638, University Library of Munich, Germany, revised 2004.
    15. Bondarenko, Oleg, 2003. "Estimation of risk-neutral densities using positive convolution approximation," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 85-112.
    16. Dasheng Ji & B. Brorsen, 2011. "A recombining lattice option pricing model that relaxes the assumption of lognormality," Review of Derivatives Research, Springer, vol. 14(3), pages 349-367, October.
    17. Weihan Li & Jin E. Zhang & Xinfeng Ruan & Pakorn Aschakulporn, 2024. "An empirical study on the early exercise premium of American options: Evidence from OEX and XEO options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(7), pages 1117-1153, July.
    18. Semih Yon & Cafer Erhan Bozdag, 2014. "Test of Log-Normal Process with Importance Sampling for Options Pricing," Proceedings of Economics and Finance Conferences 0401571, International Institute of Social and Economic Sciences.
    19. Monteiro, Ana Margarida & Tutuncu, Reha H. & Vicente, Luis N., 2008. "Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity," European Journal of Operational Research, Elsevier, vol. 187(2), pages 525-542, June.
    20. Shi-jie Jiang & Mujun Lei & Cheng-Huang Chung, 2018. "An Improvement of Gain-Loss Price Bounds on Options Based on Binomial Tree and Market-Implied Risk-Neutral Distribution," Sustainability, MDPI, vol. 10(6), pages 1-17, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:quantf:v:7:y:2007:i:4:p:459-472. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/RQUF20 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.