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Formalizing the Cox–Ross–Rubinstein Pricing of European Derivatives in Isabelle/HOL

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Abstract

We formalize in the proof assistant Isabelle essential basic notions and results in financial mathematics. We provide generic formal definitions of concepts such as markets, portfolios, derivative products, arbitrages or fair prices, and we show that, under the usual no-arbitrage condition, the existence of a replicating portfolio for a derivative implies that the latter admits a unique fair price. Then, we provide a formalization of the Cox–Rubinstein model and we show that the market is complete in this model, i.e., that every derivative product admits a replicating portfolio. This entails that in this model, every derivative product admits a unique fair price. In addition, we provide Isabelle functions to compute the fair price of some derivative products.

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Notes

  1. It can be argued that this assumption is incorrect because that there is always a nonzero probability that investors will not be payed what they are owed. But because these bills are backed by national governments, this probability is very close to 0.

  2. This property is also called countable additivity in the literature.

  3. The superscript may be omitted if there is no confusion.

  4. This is a simplification as in practice, two prices are associated with each asset: a bid price, which represents the price traders are willing to pay to buy the asset, and an ask price, which represents the price traders are willing to sell the asset. Bid prices are always lower than ask prices, but on markets on which high volumes of assets are traded, both prices are typically very close.

  5. Note that buying a basket option on Apple and Google is not the same as buying a call on Apple and another one on Google.

  6. For the definition of qty-single, we use the notation \(f(a{:}{=}b)\), which in Isabelle represents an update of function f so that the image of a becomes b.

  7. Closing out all positions means getting rid of all the assets in a portfolio, i.e., selling those with a long position, and buying back those with a short position.

  8. A money market account represents a deposit account on which any amount of cash can be deposited/withdrawn at each time.

  9. Recall that the model of an equity market does not model foreign-exchanges with several currencies, although more sophisticated models for this setting do exist. The latter are closer to reality, since they permit to account for, e.g., the fact that national banks may have different risk-free rates.

References

  1. Avigad, J., Hölzl, J., Serafin, L.: A formally verified proof of the Central Limit Theorem. J. Autom. Reason. 59(4), 389–423 (2017)

    Article  MathSciNet  Google Scholar 

  2. Ballarin, C.: Locales: a module system for mathematical theories. J. Autom. Reason. 52(2), 123–153 (2014)

    Article  MathSciNet  Google Scholar 

  3. Bingham, N.H., Kiesel, R.: Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives. Springer-Verlag, London (2004)

    Book  Google Scholar 

  4. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)

    Article  MathSciNet  Google Scholar 

  5. Blanchette, J.C., Hölzl, J., Lochbihler, A., Panny, L., Popescu, A., Traytel, D.: Truly modular (co)datatypes for Isabelle/HOL. In: Klein, G., Gamboa, R. (eds.) Interactive Theorem Proving—5th International Conference, ITP 2014, Held as Part of the Vienna Summer of Logic, VSL 2014, Vienna, Austria, July 14–17, 2014. Proceedings, volume 8558 of Lecture Notes in Computer Science, pp. 93–110. Springer (2014)

  6. Cox, J.C., Ross, S.A., Rubinstein, M.: Option pricing: a simplified approach. J. Fin. Econ. 7(3), 229–263 (1979)

    Article  MathSciNet  Google Scholar 

  7. Durrett, R.: Probability: Theory and Examples. The Wadsworth & Brooks/Cole Statistics/Probability Series. Dadsworth Inc. Duxbury Press, Belmont, CA (1991)

    Google Scholar 

  8. Eberl, M.: Randomized Social Choice Theory. https://www.isa-afp.org/entries/Randomised_Social_Choice.html (2016)

  9. Echenim, M., Peltier, N.: The binomial pricing model in finance: A formalization in Isabelle. In: de Moura, L. (ed.) Automated Deduction—CADE 26–26th International Conference on Automated Deduction, Gothenburg, Sweden, August 6–11, 2017, Proceedings, volume 10395 of Lecture Notes in Computer Science, pp. 546–562. Springer (2017)

  10. Kou, S.G.: Discrete barrier and lookback options. Handb. Oper. Res. Manag. Sci. 15, 343–373 (2007)

    Google Scholar 

  11. Hölzl, J.: Construction and stochastic applications of measure spaces in higher-order logic. Ph.D. thesis, Institut für Informatik, Technische Universität München (2012)

  12. Hölzl, J.: Markov chains and Markov decision processes in Isabelle/HOL. J. Autom. Reason. 59, 1–43 (2016). https://doi.org/10.1007/s10817-016-9401-5

    Article  MathSciNet  Google Scholar 

  13. Hölzl, J., Lochbihler, A., Traytel, D.: A Formalized Hierarchy of Probabilistic System Types—Proof Pearl. In: Proceedings of ITP, pp. 203–220 (2015)

  14. Hull, J.: Options, Futures and Other Derivatives. Pearson/Prentice Hall, Upper Saddle River (2009)

    MATH  Google Scholar 

  15. Cutland, N.J., Roux, A.: Derivative Pricing in Discrete Time. Springer-Verlag, London (2013)

    Book  Google Scholar 

  16. Kaliszyk, C., Parsert, J.: Formal microeconomic foundations and the first welfare theorem. In: Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2018, pp. 91–101, New York, NY, USA, 2018. ACM

  17. Merton, R.: The theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973)

    Article  MathSciNet  Google Scholar 

  18. Nipkow, T.: Social choice theory in HOL. J. Autom. Reason. 43(3), 289–304 (2009)

    Article  Google Scholar 

  19. Nipkow, T., Klein, G.: Concrete Semantics: With Isabelle/HOL. Springer Publishing Company, Incorporated, Berlin (2014)

    Book  Google Scholar 

  20. Nipkow, T., Wenzel, M., Paulson, L.C.: Isabelle/HOL: A Proof Assistant for Higher-order Logic. Springer-Verlag, Berlin, Heidelberg (2002)

    Book  Google Scholar 

  21. Shreve, S.E.: Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. Springer Finance, Berlin (2003)

    MATH  Google Scholar 

  22. Vestergaard, R.: A constructive approach to sequential Nash equilibria. Inf. Process. Lett. 97(2), 46–51 (2006)

    Article  MathSciNet  Google Scholar 

  23. Wiedijk, F.: Formalizing Arrow’s theorem. Sadhana 34(1), 193–220 (2009)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. LIG, Univ. Grenoble Alpes, CNRS, 38000, Grenoble, France

    Mnacho Echenim & Nicolas Peltier

  2. TIMC, Univ. Grenoble Alpes, CNRS, 38000, Grenoble, France

    Hervé Guiol

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  1. Mnacho Echenim
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  2. Hervé Guiol
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  3. Nicolas Peltier
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Correspondence to Mnacho Echenim.

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Echenim, M., Guiol, H. & Peltier, N. Formalizing the Cox–Ross–Rubinstein Pricing of European Derivatives in Isabelle/HOL. J Autom Reasoning 64, 737–765 (2020). https://doi.org/10.1007/s10817-019-09528-w

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  • DOI: https://doi.org/10.1007/s10817-019-09528-w

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