IDEAS home Printed from
   My bibliography  Save this paper

Pricing and Hedging Derivative Securities with Neural Networks and a Homogeneity Hint


  • René Garcia
  • Ramazan Gençay


We estimate a generalized option pricing formula that has a functional shape similar to the usual Black-Scholes formula by a feedforward neural network model. This functional shape is obtained when the option pricing function is homogeneous of degree one with respect to the underlying asset price and the strike price. We show that pricing accuracy gains can be made by exploiting this generalized Black-Scholes shape. Instead of setting up a learning network mapping the ratio asset price/strike price and the time to maturity directly into the derivative price, we break down the pricing function into two parts, one controlled by the ratio asset price/strike price, the other one by a function of time to maturity. The results indicate that the homogeneity hint always reduces the out-of-sample mean squared prediction error compared with a feedforward neural network with no hint. Both feedforward network models, with and without the hint, provide similar delta-hedging errors that are small relative to the hedging performance of the Black-Scholes model. However, the model with hint produces a more stable hedging performance ¸ l'aide d'un modèle de réseaux de neurones, nous estimons une formule d'évaluation d'option généralisée qui a une forme fonctionnelle similaire à la formule de Black-Scholes habituelle. Cette forme fonctionnelle s'obtient lorsque le prix d'option est une fonction homogène de degré un par rapport au prix de l'actif sous-jacent et au prix d'exercice. Nous montrons que cette forme généralisée de Black-Scholes nous permet de prévoir plus précisément les prix d'options. Au lieu de construire notre réseau d'apprentissage en entrant directement le rapport prix de l'actif sous-jacent / prix d'exercice et l'échéance dans la fonction de prix, nous décomposons cette dernière en deux parties, l'une contrôlée par le rapport prix de l'actif sous-jacent / prix d'exercice l'autre par une fonction de l'échéance. Les résultats indiquent que la forme fondée sur l'homogénéité permet toujours de réduire l'erreur quadratique moyenne de prévision hors échantillon par rapport à un réseau de neurones n'utilisant pas l'homogénéité. Les deux réseaux, avec ou sans l'homogénéité, produisent des erreurs de couverture comparables qui sont petites par rapport à la performance de couverture du modèle de Black-Scholes. Toutefois, le modèle fondé sur l'homogénéité produit une performance de couverture plus stable.

Suggested Citation

  • René Garcia & Ramazan Gençay, 1998. "Pricing and Hedging Derivative Securities with Neural Networks and a Homogeneity Hint," CIRANO Working Papers 98s-35, CIRANO.
  • Handle: RePEc:cir:cirwor:98s-35

    Download full text from publisher

    File URL:
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    1. GARCIA, René & RENAULT, Éric, 1998. "Risk Aversion, Intertemporal Substitution, and Option Pricing," Cahiers de recherche 9801, Universite de Montreal, Departement de sciences economiques.
    2. Yacine Aït-Sahalia & Andrew W. Lo, 1998. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," Journal of Finance, American Finance Association, vol. 53(2), pages 499-547, April.
    3. Hutchinson, James M & Lo, Andrew W & Poggio, Tomaso, 1994. " A Nonparametric Approach to Pricing and Hedging Derivative Securities via Learning Networks," Journal of Finance, American Finance Association, vol. 49(3), pages 851-889, July.
    4. Kaushik I. Amin & Robert A. Jarrow, 2008. "Pricing Options On Risky Assets In A Stochastic Interest Rate Economy," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 15, pages 327-347 World Scientific Publishing Co. Pte. Ltd..
    5. Broadie, Mark & Detemple, Jerome & Ghysels, Eric & Torres, Olivier, 2000. "Nonparametric estimation of American options' exercise boundaries and call prices," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1829-1857, October.
    6. Jondeau, Eric & Rockinger, Michael, 2000. "Reading the smile: the message conveyed by methods which infer risk neutral densities," Journal of International Money and Finance, Elsevier, vol. 19(6), pages 885-915, December.
    7. Eric Ghysels & Valentin Patilea & Éric Renault & Olivier Torrès, 1997. "Nonparametric Methods and Option Pricing," CIRANO Working Papers 97s-19, CIRANO.
    8. Amin, Kaushik I & Ng, Victor K, 1993. " Option Valuation with Systematic Stochastic Volatility," Journal of Finance, American Finance Association, vol. 48(3), pages 881-910, July.
    9. Turnbull, Stuart M & Milne, Frank, 1991. "A Simple Approach to Interest-Rate Option Pricing," Review of Financial Studies, Society for Financial Studies, vol. 4(1), pages 87-120.
    10. Broadie, Mark & Detemple, Jerome & Ghysels, Eric & Torres, Olivier, 2000. "American options with stochastic dividends and volatility: A nonparametric investigation," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 53-92.
    11. Bailey, Warren & Stulz, René M., 1989. "The Pricing of Stock Index Options in a General Equilibrium Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(01), pages 1-12, March.
    12. repec:taf:emetrv:v:13:y:1994:i:1:p:1-91 is not listed on IDEAS
    13. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    14. Swanson, Norman R & White, Halbert, 1995. "A Model-Selection Approach to Assessing the Information in the Term Structure Using Linear Models and Artificial Neural Networks," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(3), pages 265-275, July.
    Full references (including those not matched with items on IDEAS)


    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:cir:cirwor:98s-35. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Webmaster). General contact details of provider: .

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.