The Binomial Pricing Model in Finance: A Formalization in Isabelle

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10395)

Abstract

The binomial pricing model is an option valuation method based on a discrete-time model of the evolution of an equity market. It allows one to determine the fair price of derivatives from the payoff they generate at their expiration date. A formalization of this model in the proof assistant Isabelle is provided. We formalize essential notions in finance such as the no-arbitrage principle and prove that, under the hypotheses of the model, the market is complete, meaning that any European derivative can be replicated by creating a portfolio that generates the same payoff regardless of the evolution of the market.

References

  1. 1.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Böhme, S., Nipkow, T.: Sledgehammer: judgement day. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS (LNAI), vol. 6173, pp. 107–121. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14203-1_9 CrossRefGoogle Scholar
  3. 3.
    Cox, J.C., Ross, S.A., Rubinstein, M.: Option pricing: a simplified approach. J. Financ. Econ. 7(3), 229–263 (1979)CrossRefMATHGoogle Scholar
  4. 4.
    Durrett, R.: Probability: Theory and Examples. The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth Inc., Duxbury Press, Belmont (1991)MATHGoogle Scholar
  5. 5.
    Hölzl, J.: Construction and stochastic applications of measure spaces in higher-order logic. Ph.D. thesis, Institut für Informatik, Technische Universität München, October 2012Google Scholar
  6. 6.
    Hölzl, J.: Markov chains and Markov decision processes in Isabelle/HOL. J. Autom. Reason., 1–43 (2016). doi:10.1007/s10817-016-9401-5
  7. 7.
    Hölzl, J., Lochbihler, A., Traytel, D.: A formalized hierarchy of probabilistic system types. In: Urban, C., Zhang, X. (eds.) ITP 2015. LNCS, vol. 9236, pp. 203–220. Springer, Cham (2015). doi:10.1007/978-3-319-22102-1_13 Google Scholar
  8. 8.
    Hull, J.: Options, Futures and Other Derivatives. Pearson/Prentice Hall, Upper Saddle River (2009)MATHGoogle Scholar
  9. 9.
    Merton, R.: The theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Nipkow, T., Wenzel, M., Paulson, L.C.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Springer, Heidelberg (2002)CrossRefMATHGoogle Scholar
  11. 11.
    Shreve, S.E.: Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. Springer Finance. Springer, New York (2003)MATHGoogle Scholar
  12. 12.
    Wenzel, M., Paulson, L.: Isabelle/Isar. In: Wiedijk, F. (ed.) The Seventeen Provers of the World. LNCS, vol. 3600, pp. 41–49. Springer, Heidelberg (2006). doi:10.1007/11542384_8 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Univ. Grenoble Alpes, CNRS, LIGGrenobleFrance

Personalised recommendations