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Large liquidity expansion of super-hedging costs

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  • Dylan Possamai
  • Nizar Touzi
  • H. Mete Soner

Abstract

We consider a financial market with liquidity cost as in \c{C}etin, Jarrow and Protter [2004], where the supply function $S^{\epsilon}(s,\nu)$ depends on a parameter $\epsilon\geq 0$ with $S^0(s,\nu)=s$ corresponding to the perfect liquid situation. Using the PDE characterization of \c{C}etin, Soner and Touzi [2010] of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of $\epsilon$. In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option.

Suggested Citation

  • Dylan Possamai & Nizar Touzi & H. Mete Soner, 2012. "Large liquidity expansion of super-hedging costs," Papers 1208.3785, arXiv.org, revised Apr 2015.
    • Handle: RePEc:arx:papers:1208.3785
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    References listed on IDEAS

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    1. U. Çetin & R. Jarrow & P. Protter & M. Warachka, 2008. "Pricing Options in an Extended Black Scholes Economy with Illiquidity: Theory and Empirical Evidence," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 9, pages 185-221, World Scientific Publishing Co. Pte. Ltd..
    2. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(1), pages 193-194, February.
    3. Umut Çetin & Robert A. Jarrow & Philip Protter, 2008. "Liquidity risk and arbitrage pricing theory," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 8, pages 153-183, World Scientific Publishing Co. Pte. Ltd..
    4. S. Sethi & H. M. Soner & Q. Zhang & H. Jiang, 1992. "Turnpike Sets and Their Analysis in Stochastic Production Planning Problems," Mathematics of Operations Research, INFORMS, vol. 17(4), pages 932-950, November.
    5. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(2), pages 541-545, April.
    6. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(4), pages 1007-1017, August.
    7. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(6), pages 1461-1465, December.
    8. John Lehoczky & Suresh P. Sethi & H. M. Soner & Michael I. Taksar, 1991. "An Asymptotic Analysis of Hierarchical Control of Manufacturing Systems Under Uncertainty," Mathematics of Operations Research, INFORMS, vol. 16(3), pages 596-608, August.
    9. Umut Çetin & L. C. G. Rogers, 2007. "Modeling Liquidity Effects In Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 15-29, January.
    10. Halil Mete Soner & Guy Barles, 1998. "Option pricing with transaction costs and a nonlinear Black-Scholes equation," Finance and Stochastics, Springer, vol. 2(4), pages 369-397.
    11. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(5), pages 1273-1289, October.
    12. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(3), pages 819-821, June.
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    Cited by:

    1. Peter Bank & Selim Gokay, 2013. "Superreplication when trading at market indifference prices," Papers 1310.3113, arXiv.org.
    2. Bruno Bouchard & Ludovic Moreau & Mete H. Soner, 2013. "Hedging under an expected loss constraint with small transaction costs," Papers 1309.4916, arXiv.org, revised Sep 2014.

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