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Option hedging for small investors under liquidity costs

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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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References

  1. Bank, P., Baum, D.: Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14, 1–18 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barles, G.: Soner, H.M, Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance Stoch. 2, 369–397 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Broadie, M., Cvitanić, J., Soner, H.M.: Optimal replication of contingent claims under portfolio constraints. Rev. Financ. Stud. 11, 59–79 (1998)

    Article  Google Scholar 

  4. Çetin, U., Jarrow, R., Protter, P.: Liquidity risk and arbitrage pricing theory. Finance Stoch. 8, 311–341 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Çetin, U., Jarrow, R., Protter, P., Warachka, M.: Pricing options in an extended Black–Scholes economy with illiquidity: Theory and empirical evidence. Rev. Financ. Stud. 19, 493–529 (2006)

    Article  Google Scholar 

  6. Çetin, U., Rogers, L.C.G.: Modelling liquidity effects in discrete time. Math. Finance 17, 15–29 (2006)

    Article  Google Scholar 

  7. Cheridito, P., Soner, H.M., Touzi, N.: The multi-dimensional super-replication problem under gamma constraints. Ann. Henri Poincaré (C) Non Linear Anal. 22, 633–666 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheridito, P., Soner, H.M., Touzi, N.: Small time path behavior of double stochastic integrals and applications to stochastic control. Ann. Appl. Probab. 15, 2472–2495 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cheridito, P., Soner, H.M., Touzi, N., Victoir, N.: Second order backward stochastic differential equations and fully non-linear parabolic PDEs. Commun. Pure Appl. Math. 60, 1081–1110 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cvitanić, J., Karatzas, I.: Hedging and portfolio optimization under transaction costs: A martingale approach. Math. Finance 6, 133–165 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cvitanić, J., Ma, J.: Hedging options for a large investor and forward-backward SDEs. Ann. Appl. Prob. 6, 370–398 (1996)

    Article  MATH  Google Scholar 

  13. Cvitanić, J., Pham, H., Touzi, N.: Super-replication in stochastic volatility models under portfolio constraints. J. Appl. Probab. 36, 523–545 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Davis, M., Panas, V.G., Zariphopoulou, T.: European option pricing with transaction fees. SIAM J. Control. Opt. 31, 470–493 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35, 333–363 (1982)

    Article  MATH  Google Scholar 

  16. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics, vol. 25. Springer, New York (1993)

    MATH  Google Scholar 

  17. Frey, R.: Perfect option hedging for a large trader. Finance Stoch. 2, 115–141 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Frey, R.: Market illiquidity as a source of model risk in dynamic hedging. In: Gibson, R. (ed.) Model Risk, pp. 125–136. RISK Publications, London (2000)

    Google Scholar 

  19. Gökay, S., Soner, H.M.: Çetin–Jarrow–Protter model of liquidity in a binomial market and its limit. Preprint

  20. Jarrow, R.: Derivative security markets, market manipulation, and option pricing theory. J. Financ. Quant. Anal. 29, 241–261 (1994)

    Article  Google Scholar 

  21. Longstaff, F.A.: Optimal portfolio choice and the valuation of illiquid securities. Rev. Financ. Stud. 14, 407–431 (2001)

    Article  Google Scholar 

  22. Krylov, N.V., Safanov, M.V.: A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR Izv. 16, 151–164 (1981)

    Article  MATH  Google Scholar 

  23. Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order. Reidel, Dordrecht (1981)

    Google Scholar 

  24. Ladyzhenskaya, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Quasilinear Parabolic Equations. Am. Math. Soc., Providence (1968)

    Google Scholar 

  25. Platen, E., Schweizer, M.: On feedback effects from hedging derivatives. Math. Finance 8, 67–84 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rogers, L.C.G., Singh, S.: Option pricing in an illiquid market. Technical Report, University of Cambridge, Cambridge (2005)

    Google Scholar 

  27. Schönbucher, P.J., Wilmott, P.: The feedback effects of hedging in illiquid markets. SIAM J. Appl. Math. 61, 232–272 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sircar, K.R., Papanicolaou, G.: Generalized Black–Scholes models accounting for increased market volatility from hedging strategies, Appl. Math. Finance 5, 45–82 (1998)

    Article  MATH  Google Scholar 

  29. Soner, H.M., Touzi, N.: Super-replication under gamma constraints. SIAM J. Control Optim. 39, 73–96 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. Soner, H.M., Touzi, N.: Stochastic target problems, dynamic programming and viscosity solutions. SIAM J. Control Optim. 41, 404–424 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  31. Soner, H.M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4, 201–236 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Soner, H.M., Touzi, N.: The dynamic programming equation for second order stochastic target problems. SIAM J. Control Optim. 48, 2344–2365 (2009)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. London School of Economics, London, UK

    Umut Çetin

  2. Department of Mathematics, ETH Zurich, Zurich, Switzerland

    H. Mete Soner

  3. Centre de Mathématiques Appliquées, Ecole Polytechnique, Paris, France

    Nizar Touzi

Authors
  1. Umut Çetin
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  2. H. Mete Soner
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  3. Nizar Touzi
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Correspondence to Umut Çetin.

Additional information

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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