The Annals of Applied Probability

Super-replication with nonlinear transaction costs and volatility uncertainty

Peter Bank, Yan Dolinsky, and Selim Gökay

Full-text: Open access


We study super-replication of contingent claims in an illiquid market with model uncertainty. Illiquidity is captured by nonlinear transaction costs in discrete time and model uncertainty arises as our only assumption on stock price returns is that they are in a range specified by fixed volatility bounds. We provide a dual characterization of super-replication prices as a supremum of penalized expectations for the contingent claim’s payoff. We also describe the scaling limit of this dual representation when the number of trading periods increases to infinity. Hence, this paper complements the results in [Finance Stoch. 17 (2013) 447–475] and [Ann. Appl. Probab. 5 (1995) 198–221] for the case of model uncertainty.

Article information

Ann. Appl. Probab., Volume 26, Number 3 (2016), 1698-1726.

Received: November 2014
Revised: June 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91G10: Portfolio theory 91G40: Credit risk

Super-replication hedging with friction volatility uncertainty limit theorems


Bank, Peter; Dolinsky, Yan; Gökay, Selim. Super-replication with nonlinear transaction costs and volatility uncertainty. Ann. Appl. Probab. 26 (2016), no. 3, 1698--1726. doi:10.1214/15-AAP1130.

Export citation


  • [1] Aldous, D. (1981). Weak convergence of stochastic processes for processes viewed in the strasbourg manner. Unpublished Manuscript, Statist. Laboratory Univ. Cambridge.
  • [2] Bank, P. and Baum, D. (2004). Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14 1–18.
  • [3] Bouchard, B. and Nutz, M. (2014). Consistent price systems under model uncertainty. Preprint.
  • [4] Bouchard, B. and Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25 823–859.
  • [5] Çetin, U., Jarrow, R. A. and Protter, P. (2004). Liquidity risk and arbitrage pricing theory. Finance Stoch. 8 311–341.
  • [6] Chentsov, N. N. (1957). Weak convergence of stochastic processes whose trajectories have no discontinuities of the second kind and the “heuristic” approach to the Kolmogorov–Smirnov tests. Theory Probab. Appl. 1 140–144.
  • [7] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520.
  • [8] Denis, L. and Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 827–852.
  • [9] Deparis, S. and Martini, C. (2004). Superheading strategies and balayage in discrete time. In Seminar on Stochastic Analysis, Random Fields and Applications IV. Progress in Probability 58 205–219. Birkhäuser, Basel.
  • [10] Dolinsky, Y. (2014). Hedging of game options under model uncertainty in discrete time. Electron. Commun. Probab. 19 no. 19, 11.
  • [11] Dolinsky, Y. and Soner, H. M. (2013). Duality and convergence for binomial markets with friction. Finance Stoch. 17 447–475.
  • [12] Dolinsky, Y. and Soner, H. M. (2014). Robust hedging with proportional transaction costs. Finance Stoch. 18 327–347.
  • [13] Duffie, D. and Protter, P. (1992). From discrete to continuous time finance: Weak convergence of the financial gain process. Math. Finance 2 1–15.
  • [14] Föllmer, H. and Schied, A. (2002). Convex measures of risk and trading constraints. Finance Stoch. 6 429–447.
  • [15] Föllmer, H. and Schied, A. (2004). Stochastic Finance: An Introduction in Discrete Time, extended ed. De Gruyter Studies in Mathematics 27. de Gruyter, Berlin.
  • [16] Frittelli, M. and Rosazza Gianin, E. (2002). Putting order in risk measures. J. Bank. Financ. 26 1473–1486.
  • [17] Gökay, S. and Soner, H. M. (2012). Liquidity in a binomial market. Math. Finance 22 250–276.
  • [18] Hobson, D. G. (1998). Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Probab. 8 193–205.
  • [19] Kusuoka, S. (1995). Limit theorem on option replication cost with transaction costs. Ann. Appl. Probab. 5 198–221.
  • [20] Levental, S. and Skorohod, A. V. (1997). On the possibility of hedging options in the presence of transaction costs. Ann. Appl. Probab. 7 410–443.
  • [21] Lyons, T. J. (1995). Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2 117–133.
  • [22] Nutz, M. and Soner, H. M. (2012). Superhedging and dynamic risk measures under volatility uncertainty. SIAM J. Control Optim. 50 2065–2089.
  • [23] Peng, S. (2008). Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation. Stochastic Process. Appl. 118 2223–2253.
  • [24] Soner, H. M., Shreve, S. E. and Cvitanić, J. (1995). There is no nontrivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5 327–355.
  • [25] Soner, H. M., Touzi, N. and Zhang, J. (2011). Martingale representation theorem for the $G$-expectation. Stochastic Process. Appl. 121 265–287.