Advertisement

Finance and Stochastics

, Volume 20, Issue 3, pp 741–771 | Cite as

Almost-sure hedging with permanent price impact

  • Bruno BouchardEmail author
  • Grégoire Loeper
  • Yiyi Zou
Article

Abstract

We consider a financial model with permanent price impact. Continuous-time trading dynamics are derived as the limit of discrete rebalancing policies. We then study the problem of superhedging a European option. Our main result is the derivation of a quasilinear pricing equation. It holds in the sense of viscosity solutions. When it admits a smooth solution, it provides a perfect hedging strategy.

Keywords

Hedging Price impact 

Mathematics Subject Classification

91G20 93E20 49L20 

JEL Classification

G13 G12 

References

  1. 1.
    Abergel, F., Loeper, G.: Pricing and hedging contingent claims with liquidity costs and market impact. Available online at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2239498 (2013)
  2. 2.
    Barles, G.: Solutions de Viscosité des Équations de Hamilton–Jacobi. Springer, Berlin (1994) zbMATHGoogle Scholar
  3. 3.
    Bertsekas, D.P., Shreve, S.E.: Stochastic Optimal Control. The Discrete-Time Case. Academic Press, New York (1978) zbMATHGoogle Scholar
  4. 4.
    Bogachev, V.I., Ruas, M.A.S.: Measure Theory, vol. 1. Springer, Berlin (2007) CrossRefGoogle Scholar
  5. 5.
    Bouchard, B., Elie, R., Touzi, N.: Stochastic target problems with controlled loss. SIAM J. Control Optim. 48, 3123–3150 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Çetin, U., Jarrow, R.A., Protter, P.: Liquidity risk and arbitrage pricing theory. Finance Stoch. 8, 311–341 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Çetin, U., Soner, H.M., Touzi, N.: Option hedging for small investors under liquidity costs. Finance Stoch. 14, 317–341 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cheridito, P., Soner, H.M., Touzi, N.: The multi-dimensional super-replication problem under gamma constraints. Ann. Inst. Henri Poincaré, C Anal. Non Linéaire 22, 633–666 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Crauel, H.: Random Probability Measures on Polish Spaces. Stochastics Monographs, vol. 11. CRC Press, Boca Raton (2003) zbMATHGoogle Scholar
  10. 10.
    Frey, R.: Perfect option hedging for a large trader. Finance Stoch. 2, 115–141 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ladyzhenskaia, O.A., Solonnikov, V., Ural’tseva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. Am. Math. Soc., Providence (1988) Google Scholar
  12. 12.
    Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996) CrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, H., Yong, J.M.: Option pricing with an illiquid underlying asset market. J. Econ. Dyn. Control 29, 2125–2156 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Loeper, G.: Option pricing with market impact and non-linear Black and Scholes PDEs. Available online at arXiv:1301.6252 (2013)
  15. 15.
    Schönbucher, P.J., Wilmott, P.: The feedback effects of hedging in illiquid markets. SIAM J. Appl. Math. 61, 232–272 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sircar, K.R., Papanicolaou, G.: Generalized Black–Scholes models accounting for increased market volatility from hedging strategies. Appl. Math. Finance 5, 45–82 (1998) CrossRefzbMATHGoogle Scholar
  17. 17.
    Soner, H.M., Touzi, N.: Superreplication under gamma constraints. SIAM J. Control Optim. 39, 73–96 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Soner, H.M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4, 201–236 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Soner, H.M., Touzi, N.: Hedging under gamma constraints by optimal stopping and face-lifting. Math. Finance 17, 59–80 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Soner, H.M., Touzi, N.: The dynamic programming equation for second order stochastic target problems. SIAM J. Control Optim. 48, 2344–2365 (2009) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.CEREMADEUniversité Paris Dauphine, and CREST-ENSAEParis cedex 16France
  2. 2.BNP-Paribas and FiQuant—Chaire de Finance QuantitativeParisFrance

Personalised recommendations