Abstract
Following the framework of Çetin et al. (Finance Stoch. 8:311–341, 2004), we study the problem of super-replication in the presence of liquidity costs under additional restrictions on the gamma of the hedging strategies in a generalized Black–Scholes economy. We find that the minimal super-replication price is different from the one suggested by the Black–Scholes formula and is the unique viscosity solution of the associated dynamic programming equation. This is in contrast with the results of Çetin et al. (Finance Stoch. 8:311–341, 2004), who find that the arbitrage-free price of a contingent claim coincides with the Black–Scholes price. However, in Çetin et al. (Finance Stoch. 8:311–341, 2004) a larger class of admissible portfolio processes is used, and the replication is achieved in the L 2 approximating sense.
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This research is part of the Chair Financial Risks of the Risk Foundation sponsored by Société Générale, the Chair Derivatives of the Future sponsored by the Fédération Bancaire Française, and the Chair Finance and Sustainable Development sponsored by EDF and Calyon. Also the first two authors thank the European Science Foundation for its support through the AMaMeF program.
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Çetin, U., Soner, H.M. & Touzi, N. Option hedging for small investors under liquidity costs. Finance Stoch 14, 317–341 (2010). https://doi.org/10.1007/s00780-009-0116-x
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DOI: https://doi.org/10.1007/s00780-009-0116-x