Finance and Stochastics

, Volume 17, Issue 3, pp 447–475 | Cite as

Duality and convergence for binomial markets with friction



We prove limit theorems for the super-replication cost of European options in a binomial model with friction. Examples covered are markets with proportional transaction costs and illiquid markets. A dual representation for the super-replication cost in these models is obtained and used to prove the limit theorems. In particular, the existence of a liquidity premium for the continuous-time limit of the model proposed in Çetin et al. (Finance Stoch. 8:311–341, 2004) is proved. Hence, this paper extends the previous convergence result of Gökay and Soner (Math Finance 22:250–276, 2012) to the general non-Markovian case. Moreover, the special case of small transaction costs yields, in the continuous limit, the G-expectation of Peng as earlier proved by Kusuoka (Ann. Appl. Probab. 5:198–221, 1995).


Super-replication Liquidity Binomial model Limit theorems G-Expectation 

Mathematics Subject Classification (2010)

91G10 60F05 60H30 

JEL Classification

G11 G13 D52 



Research supported by the European Research Council Grant 228053-FiRM, the Swiss Finance Institute and the ETH Foundation. The authors would like to thank Prof. Kusuoka and Marcel Nutz for insightful discussions.


  1. 1.
    Bank, P., Baum, D.: Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14, 1–18 (2004) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Barles, G., Soner, H.M.: Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance Stoch. 2, 369–397 (1998) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) MATHGoogle Scholar
  4. 4.
    Çetin, U., Rogers, L.C.G.: Modelling liquidity effects in discrete time. Math. Finance 17, 15–29 (2007) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Çetin, U., Jarrow, R., Protter, P.: Liquidity risk and arbitrage pricing theory. Finance Stoch. 8, 311–341 (2004) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Çetin, U., Soner, H.M., Touzi, N.: Option hedging under liquidity costs. Finance Stoch. 14, 317–341 (2010) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Chalasani, P., Jha, S.: Randomized stopping times and American option pricing with transaction costs. Math. Finance 11, 33–77 (2001) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dolinsky, Y.: Hedging of Game Options with the Presence of Transaction Costs (2012), submitted. arXiv:1103.1165,v3
  9. 9.
    Dolinsky, Y., Nutz, M., Soner, H.M.: Weak approximation of G-expectations. Stoch. Process. Appl. 122, 664–675 (2012) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Dudley, R.M.: Distances of probability measures and random variables. Ann. Math. Stat. 39, 1563–1572 (1968) MathSciNetMATHGoogle Scholar
  12. 12.
    Duffie, D., Protter, P.: From discrete to continuous time finance: weak convergence of the financial gain process. Math. Finance 2, 1–15 (1992) MATHCrossRefGoogle Scholar
  13. 13.
    Gökay, S., Soner, H.M.: Liquidity in a binomial market. Math. Finance 22, 250–276 (2012) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Guasoni, P., Rasonyi, M., Schachermayer, W.: Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Probab. 18, 491–520 (2008) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991) MATHGoogle Scholar
  16. 16.
    Kusuoka, S.: Limit theorem on option replication cost with transaction costs. Ann. Appl. Probab. 5, 198–221 (1995) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Levental, S., Skorohod, A.V.: On the possibility of hedging options in the presence of transaction costs. Ann. Appl. Probab. 7, 410–443 (1997) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Peng, S.: Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stoch. Process. Appl. 118, 2223–2253 (2008) MATHCrossRefGoogle Scholar
  19. 19.
    Pennanen, T., Penner, I.: Hedging of claims with physical delivery under convex transaction costs. SIAM J. Financ. Math. 1, 158–178 (2010) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  21. 21.
    Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance 14, 19–48 (2004) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Soner, H.M., Shreve, S.E., Cvitanić, J.: There is no nontrivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5, 327–355 (1995) MATHCrossRefGoogle Scholar
  23. 23.
    Soner, H.M., Touzi, N.: The dynamic programming equation for second order stochastic target problems. SIAM J. Control Optim. 48, 2344–2365 (2009) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Soner, H.M., Touzi, N., Zhang, J.: Dual formulation of second order target problems. Ann. Appl. Prob. (2012), in press. arXiv:1003.6050

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Dept. of MathematicsETH ZurichZurichSwitzerland
  2. 2.Swiss Finance InstituteZurichSwitzerland

Personalised recommendations