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Homogenization and asymptotics for small transaction costs

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  • H. Mete Soner
  • Nizar Touzi
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    Abstract

    We consider the classical Merton problem of lifetime consumption-portfolio optimization problem with small proportional transaction costs. The first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in the one-dimensional case. Unlike the existing literature, we consider a general utility function and general dynamics for the underlying assets. Our arguments are based on ideas from the homogenization theory and use the convergence tools from the theory of viscosity solutions. The multidimensional case is studied in our accompanying paper using the same approach.

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    File URL: http://arxiv.org/pdf/1202.6131
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1202.6131.

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    Date of creation: Feb 2012
    Date of revision: Jun 2013
    Handle: RePEc:arx:papers:1202.6131

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    1. Freddy Delbaen & Yuri M. Kabanov & Esko Valkeila, 2002. "Hedging under Transaction Costs in Currency Markets: a Discrete-Time Model," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 12(1), pages 45-61.
    2. Karel Janeček & Steven Shreve, 2004. "Asymptotic analysis for optimal investment and consumption with transaction costs," Finance and Stochastics, Springer, Springer, vol. 8(2), pages 181-206, 05.
    3. Halil Mete Soner & Guy Barles, 1998. "Option pricing with transaction costs and a nonlinear Black-Scholes equation," Finance and Stochastics, Springer, Springer, vol. 2(4), pages 369-397.
    4. Stefan Gerhold & Johannes Muhle-Karbe & Walter Schachermayer, 2010. "Asymptotics and Duality for the Davis and Norman Problem," Papers 1010.0627, arXiv.org, revised Aug 2011.
    5. C. Atkinson & S. Mokkhavesa, 2004. "Multi-asset portfolio optimization with transaction cost," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 11(2), pages 95-123.
    6. Constantinides, George M, 1986. "Capital Market Equilibrium with Transaction Costs," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 94(4), pages 842-62, August.
    7. Magill, Michael J. P. & Constantinides, George M., 1976. "Portfolio selection with transactions costs," Journal of Economic Theory, Elsevier, Elsevier, vol. 13(2), pages 245-263, October.
    8. Dumas, Bernard & Luciano, Elisa, 1991. " An Exact Solution to a Dynamic Portfolio Choice Problem under Transactions Costs," Journal of Finance, American Finance Association, American Finance Association, vol. 46(2), pages 577-95, June.
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    Cited by:
    1. Zura Kakushadze, 2014. "Notes on Alpha Stream Optimization," Papers 1406.1249, arXiv.org.
    2. Dylan Possama\"i & Guillaume Royer, 2014. "General indifference pricing with small transaction costs," Papers 1401.3261, arXiv.org.
    3. Christoph K\"uhn & Johannes Muhle-Karbe, 2013. "Optimal Liquidity Provision in Limit Order Markets," Papers 1309.5235, arXiv.org, revised May 2014.
    4. 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.
    5. Zura Kakushadze, 2014. "Combining Alpha Streams with Costs," Papers 1405.4716, arXiv.org, revised Jul 2014.
    6. Paolo Guasoni & Johannes Muhle-Karbe, 2012. "Portfolio Choice with Transaction Costs: a User's Guide," Papers 1207.7330, arXiv.org.
    7. Dylan Possama\"i & H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs: the multidimensional case," Papers 1212.6275, arXiv.org, revised Jan 2013.
    8. Jan Kallsen & Johannes Muhle-Karbe, 2013. "The General Structure of Optimal Investment and Consumption with Small Transaction Costs," Papers 1303.3148, arXiv.org.
    9. Jose V. Alcala & Arash Fahim, 2013. "Balancing small fixed and proportional transaction cost in trading strategies," Papers 1304.7562, arXiv.org.
    10. Maxim Bichuch & Ronnie Sircar, 2014. "Optimal Investment with Transaction Costs and Stochastic Volatility," Papers 1401.0562, arXiv.org, revised Aug 2014.
    11. Jiatu Cai & Masaaki Fukasawa, 2014. "Asymptotic replication with modified volatility under small transaction costs," Papers 1408.5677, arXiv.org.
    12. Jan Kallsen & Johannes Muhle-Karbe, 2012. "Option Pricing and Hedging with Small Transaction Costs," Papers 1209.2555, arXiv.org, revised Dec 2012.
    13. Ludovic Moreau & Johannes Muhle-Karbe & H. Mete Soner, 2014. "Trading with Small Price Impact," Papers 1402.5304, arXiv.org.

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