Finance and Stochastics

, Volume 12, Issue 4, pp 441–468 | Cite as

Pricing by hedging and no-arbitrage beyond semimartingales



We show that pricing a big class of relevant options by hedging and no-arbitrage can be extended beyond semimartingale models. To this end we construct a subclass of self-financing portfolios that contains hedges for these options, but does not contain arbitrage opportunities, even if the stock price process is a non-semimartingale of some special type. Moreover, we show that the option prices depend essentially only on a path property of the stock price process, viz. on the quadratic variation. We end the paper by giving no-arbitrage results even with stopping times for our model class.


Arbitrage Pricing Quadratic variation Robust hedging 

JEL Classification

G10 G13 


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  1. 1.
    Androshchuk, T., Mishura, Yu.: Mixed Brownian-fractional Brownian model: absence of arbitrage and related topics. Stoch. Int. J. Probab. Stoch. Process. 78, 281–300 (2006) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cheridito, P.: Mixed fractional Brownian motion. Bernoulli 7, 913–934 (2001) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cheridito, P.: Arbitrage in fractional Brownian motion models. Finance Stoch. 7, 533–553 (2003) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dasgupta, A., Kallianpur, G.: Arbitrage opportunities for a class of Gladyshev processes. Appl. Math. Optim. 41, 377–385 (2000) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dupire, B.: Pricing with a smile. Risk January, 18–20 (1994) Google Scholar
  7. 7.
    Föllmer, H.: Calcul d’Itô sans probabilités. In: Séminaire de Probabilités, XV. Lecture Notes in Mathematics, vol. 850, pp. 143–150. Springer, Berlin (1981) Google Scholar
  8. 8.
    Guasoni, P.: No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16, 569–582 (2006) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Guasoni, P., Rásonyi, M., Schachermayer, W.: Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Probab. 18(2), 491–520 (2008) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes, 2nd edn. Academic Press, San Diego (1975) MATHGoogle Scholar
  11. 11.
    Lin, S.J.: Stochastic analysis of fractional Brownian motions. Stoch. Stoch. Rep. 55, 121–140 (1995) MATHGoogle Scholar
  12. 12.
    Mishura, Yu., Valkeila, E.: On arbitrage and replication in the fractional Black–Scholes pricing model. Proc. Steklov Inst. Math. 237, 215–224 (2002) MathSciNetGoogle Scholar
  13. 13.
    Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004) MATHGoogle Scholar
  14. 14.
    Russo, F., Vallois, P.: Forward, backward and symmetric stochastic integration. Probab. Theory Relat. Fields 97, 403–421 (1993) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Schoenmakers, J., Kloeden, P.: Robust option replication for a Black–Scholes model extended with nondeterministic trends. J. Appl. Math. Stoch. Anal. 12, 113–120 (1999) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sondermann, D.: Introduction to Stochastic Calculus for Finance: A New Didactic Approach. Springer, Berlin (2006) MATHGoogle Scholar
  17. 17.
    Shiryaev, A.: On arbitrage and replication for fractal models. Research Report, vol. 20. MaPhySto, Department of Mathematical Sciences, University of Aarhus (1998) Google Scholar
  18. 18.
    Shiryaev, A.: Essentials of Stochastic Finance. Facts, Models, Theory. World Scientific, Singapore (1999) Google Scholar
  19. 19.
    Wilmott, P.: Derivatives: The Theory and Practice of Financial Engineering. Wiley, New York (1998) Google Scholar
  20. 20.
    Zähle, M.: Long range dependence, no arbitrage and the Black–Scholes formula. Stoch. Dyn. 2, 265–280 (2002) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Christian Bender
    • 1
  • Tommi Sottinen
    • 2
    • 3
  • Esko Valkeila
    • 4
  1. 1.Institute for Mathematical StochasticsTU BraunschweigBraunschweigGermany
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  3. 3.School of Science and Engineering and School of BusinessReykjavík UniversityReykjavíkIceland
  4. 4.Department of Mathematics and Systems AnalysisHelsinki University of TechnologyHelsinkiFinland

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