Finance and Stochastics

, Volume 15, Issue 3, pp 573–605 | Cite as

Robust pricing and hedging of double no-touch options

Article

Abstract

Double no-touch options are contracts which pay out a fixed amount provided an underlying asset remains within a given interval. In this work, we establish model-independent bounds on the price of these options based on the prices of more liquidly traded options (call and digital call options). Key steps are the construction of super- and sub-hedging strategies to establish the bounds, and the use of Skorokhod embedding techniques to show the bounds are the best possible.

In addition to establishing rigorous bounds, we consider carefully what is meant by arbitrage in settings where there is no a priori known probability measure. We discuss two natural extensions of the notion of arbitrage, weak arbitrage and weak free lunch with vanishing risk, which are needed to establish equivalence between the lack of arbitrage and the existence of a market model.

Keywords

Double no-touch option Robust pricing and hedging Skorokhod embedding problem Weak arbitrage Weak free lunch with vanishing risk Model-independent arbitrage 

Mathematics Subject Classification (2000)

91B28 60G40 60G44 

JEL Classification

C60 G13 

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References

  1. 1.
    Azéma, J., Yor, M.: Une solution simple au problème de Skorokhod. In: Dellacherie, C., Meyer, P.A., Weil, M. (eds.) Séminaire de Probabilités, XIII, Univ. Strasbourg, Strasbourg, 1977/1978. Lecture Notes in Math., vol. 721, pp. 90–115. Springer, Berlin (1979) CrossRefGoogle Scholar
  2. 2.
    Biagini, S., Cont, R.: Model-free representation of pricing rules as conditional expectations. In: Akahori, J., Ogawa, S., Watanabe, S. (eds.) Stochastic Processes and Applications to Mathematical Finance, Proceedings of the 6th Ritsumeikan International Symposium, Kyoto, Japan, 6–10 March 2006, pp. 53–84. World Scientific, Hackensack (2007) Google Scholar
  3. 3.
    Breeden, D.T., Litzenberger, R.H.: Prices of state-contingent claims implicit in option prices. J. Bus. 51, 621–651 (1978) CrossRefGoogle Scholar
  4. 4.
    Brown, H., Hobson, D., Rogers, L.C.G.: Robust hedging of barrier options. Math. Finance 11, 285–314 (2001) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Buehler, H.: Expensive martingales. Quant. Finance 6, 207–218 (2006) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Carr, P., Crosby, J.: A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options. Quant. Finance 10, 1115–1136 (2010) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Carr, P., Madan, D.P.: A note on sufficient conditions for no arbitrage. Finance Res. Lett. 2, 125–130 (2005) CrossRefGoogle Scholar
  8. 8.
    Carr, P., Wu, L.: Stochastic skew in currency options. J. Financ. Econ. 86, 213–247 (2007) CrossRefGoogle Scholar
  9. 9.
    Cassese, G.: Asset pricing with no exogenous probability measure. Math. Finance 18, 23–54 (2008) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cherny, A.: General arbitrage pricing model. III. Possibility approach. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds.) Séminaire de Probabilités XL. Lecture Notes in Math., vol. 1899, pp. 463–481. Springer, Berlin (2007) CrossRefGoogle Scholar
  11. 11.
    Cont, R.: Model uncertainty and its impact on the pricing of derivative instruments. Math. Finance 16, 519–547 (2006) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cousot, L.: Conditions on option prices for absence of arbitrage and exact calibration. J. Bank. Finance 31, 3377–3397 (2007) CrossRefGoogle Scholar
  13. 13.
    Cox, A.M.G.: Skorokhod embeddings: non-centred target distributions, diffusions and minimality. http://opus.bath.ac.uk/19625/. Ph.D. thesis, University of Bath (2004)
  14. 14.
    Cox, A.M.G.: Extending Chacon–Walsh: Minimality and generalised starting distributions. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds.) Séminaire de Probabilités XLI. Lecture Notes in Math., vol. 1934, pp. 233–264. Springer, Berlin (2008) CrossRefGoogle Scholar
  15. 15.
    Cox, A.M.G., Hobson, D.G.: Skorokhod embeddings, minimality and non-centred target distributions. Probab. Theory Relat. Fields 135, 395–414 (2006) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cox, A.M.G., Obłój, J.: Robust hedging of double touch barrier options. SIAM J. Financ. Math. 2, 141–182 (2011) CrossRefMATHGoogle Scholar
  17. 17.
    Cox, A.M.G., Hobson, D.G., Obłój, J.: Pathwise inequalities for local time: applications to Skorokhod embeddings and optimal stopping. Ann. Appl. Probab. 18, 1870–1896 (2008) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Davis, M.H.A., Hobson, D.G.: The range of traded option prices. Math. Finance 17, 1–14 (2007) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 2nd edn. de Gruyter Studies in Mathematics, vol. 27. de Gruyter, Berlin (2004) CrossRefMATHGoogle Scholar
  21. 21.
    Hirsa, A., Courtadon, G., Madan, D.B.: The effect of model risk on the valuation of barrier options. J. Risk Finance 4, 47–55 (2003) CrossRefGoogle Scholar
  22. 22.
    Hobson, D.G.: Robust hedging of the lookback option. Finance Stoch. 2, 329–347 (1998) CrossRefMATHGoogle Scholar
  23. 23.
    Jacka, S.D.: Doob’s inequalities revisited: a maximal H 1-embedding. Stoch. Process. Appl. 29, 281–290 (1988) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kellerer, H.G.: Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198, 99–122 (1972) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Laurent, J.P., Leisen, D.: Building a consistent pricing model from observed option prices. In: Avellaneda, M (ed.) Quantitative Analysis in Financial Markets: Collected Papers of the New York University Mathematical Finance Seminar, vol. 2, pp. 216–238. World Scientific, Singapore (2001) CrossRefGoogle Scholar
  26. 26.
    Maruhn, J.H.: Robust Static Super-Replication of Barrier Options. Radon Series on Computational and Applied Mathematics. de Gruyter, Berlin (2009) CrossRefMATHGoogle Scholar
  27. 27.
    Mijatović, A.: Local time and the pricing of time-dependent barrier options. Finance Stoch. 14, 13–48 (2010) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Obłój, J.: The Skorokhod embedding problem and its offspring. Probab. Surv. 1, 321–390 (2004) (electronic) MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Obłój, J.: A complete characterization of local martingales which are functions of Brownian motion and its maximum. Bernoulli 12, 955–969 (2006) MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Perkins, E.: The Cereteli–Davis solution to the H 1-embedding problem and an optimal embedding in Brownian motion. In: Seminar on Stochastic Processes, Gainesville, Fla., 1985, pp. 172–223. Birkhäuser Boston, Boston (1986) Google Scholar
  31. 31.
    Skorokhod, A.V.: Studies in the Theory of Random Processes. Addison–Wesley, Reading (1965). Translated from the Russian by Scripta Technica, Inc. MATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Mathematical Institute and Oxford-Man Institute of Quantitative FinanceUniversity of OxfordOxfordUK

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