NN-OPT: Neural Network for Option Pricing Using Multinomial Tree

  • Hung-Ching (Justin) Chen
  • Malik Magdon-Ismail
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4234)


We provide a framework for learning to price complex options by learning risk-neutral measures (Martingale measures). In a simple geometric Brownian motion model, the price volatility, fixed interest rate and a no-arbitrage condition suffice to determine a unique risk-neutral measure. On the other hand, in our framework, we relax some of these assumptions to obtain a class of allowable risk-neutral measures. We then propose a framework for learning the appropriate risk-neural measure. In particular, we provide an efficient algorithm for backpropagating gradients through multinomial pricing trees. Since the risk-neutral measure prices all options simultaneously, we can use all the option contracts on a particular stock for learning. We demonstrate the performance of these models on historical data. Finally, we illustrate the power of such a framework by developing a real time trading system based upon these pricing methods.


Option Price Trading Cost Martingale Measure Forward Propagation Term Interest Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Black, F., Scholes, M.S.: The pricing of options and corporate liabilities. Journal of Political Economy 3, 637–654 (1973)CrossRefGoogle Scholar
  2. 2.
    Magdon-Ismail, M.: The Equivalent Martingale Measure: An Introduction to Pricing Using Expectations. IEEE Transactions on Neural Netork 12(4), 684–693 (2001)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Cox, J.C., Ross, S.A.: The valuation of options for alternative stochastic processes. Journal of Financial Economics, 145–166 (1976)Google Scholar
  4. 4.
    Harrison, J.M., Pliska, S.R.: A stochastic calculus model of continuous trading: Complete markets. Stochastic Processes and their Applications, 313–316 (1983)Google Scholar
  5. 5.
    Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modeling. Applications of Mathematics, 36. Springer, New York (1977)Google Scholar
  6. 6.
    Paul, R., Lajbcygier, J.T.C.: Improve option pricing using artificial neural networks and bootstrap methods. International Journal of Neural System 8(4), 457–471 (1997)CrossRefGoogle Scholar
  7. 7.
    Amari, S.I., Xu, L.C.L.: Option pricing with neural networks. Progress in Neural Information Processing 2, 760–765 (1996)Google Scholar
  8. 8.
    Moody, J., Saffell, M.: Learning to trade via direct reinforcement. IEEE Transactions on Neural Networks 12(4), 875–889 (2001)CrossRefGoogle Scholar
  9. 9.
    Cox, J.C., Ross, S.A., Rubinstein, M.: Option pricing: A simplified approach. Journal of Financial Economics, 229–263 (1979)Google Scholar
  10. 10.
    Ross, S.M.: An Elementary Introduction to Mathematical Finance, 2nd edn. Cambridge University Press, Cambridge (2003)MATHGoogle Scholar
  11. 11.
    Baxter, M., Prnnie, A.: Financial Calculus: An Introduction to Derivative Pricing. Cambridge University Press, Cambridge (1996)MATHGoogle Scholar
  12. 12.
    Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327 (1986)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press, Oxford (1995)Google Scholar
  14. 14.
    Haykin, S.: Neural Networks: A Comprehensive Foundation, 2nd edn. Prentice-Hall, Englewood Cliffs (1998)Google Scholar
  15. 15.
    Harrison, J.M., Pliska, S.R.: Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11, 215–260 (1981)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hung-Ching (Justin) Chen
    • 1
  • Malik Magdon-Ismail
    • 1
  1. 1.Dept. of Computer ScienceRensselaer Polytechnic InstituteTroyUSA

Personalised recommendations