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Option pricing with linear market impact and non-linear Black and Scholes equations

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  • Gregoire Loeper

Abstract

We consider a model of linear market impact, and address the problem of replicating a contingent claim in this framework. We derive a non-linear Black-Scholes Equation that provides an exact replication strategy. This equation is fully non-linear and singular, but we show that it is well posed, and we prove existence of smooth solutions for a large class of final payoffs, both for constant and local volatility. To obtain regularity of the solutions, we develop an original method based on Legendre transforms. The close connections with the problem of hedging with it gamma constraints studied by Cheridito, Soner and Touzi and with the problem of hedging under it liquidity costs are discussed. We also derive a modified Black-Scholes formula valid for asymptotically small impact parameter, and finally provide numerical simulations as an illustration.

Suggested Citation

  • Gregoire Loeper, 2013. "Option pricing with linear market impact and non-linear Black and Scholes equations," Papers 1301.6252, arXiv.org, revised Aug 2016.
    • Handle: RePEc:arx:papers:1301.6252
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    References listed on IDEAS

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    1. 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..
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    3. Soner, H. Mete & Cetin, Umut & Touzi, Nizar, 2010. "Option hedging for small investors under liquidity costs," LSE Research Online Documents on Economics 28992, London School of Economics and Political Science, LSE Library.
    4. H. Mete Soner & Nizar Touzi, 2007. "Hedging Under Gamma Constraints By Optimal Stopping And Face‐Lifting," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 59-79, January.
    5. B. Tóth & Z. Eisler & F. Lillo & J. Kockelkoren & J.-P. Bouchaud & J.D. Farmer, 2012. "How does the market react to your order flow?," Quantitative Finance, Taylor & Francis Journals, vol. 12(7), pages 1015-1024, May.
    6. Umut Çetin & H. Soner & Nizar Touzi, 2010. "Option hedging for small investors under liquidity costs," Finance and Stochastics, Springer, vol. 14(3), pages 317-341, September.
    7. Rüdiger Frey & Alexander Stremme, 1997. "Market Volatility and Feedback Effects from Dynamic Hedging," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 351-374, October.
    8. Liu, Hong & Yong, Jiongmin, 2005. "Option pricing with an illiquid underlying asset market," Journal of Economic Dynamics and Control, Elsevier, vol. 29(12), pages 2125-2156, December.
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    Cited by:

    1. Frédéric Abergel & Grégoire Loeper, 2013. "Pricing and hedging contingent claims with liquidity costs and market impact," Working Papers hal-00802402, HAL.

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