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Generalized stochastic target problems for pricing and partial hedging under loss constraints—application in optimal book liquidation


  • Bruno Bouchard


  • Ngoc-Minh Dang



We consider a singular version with state constraints of the stochastic target problems studied in Soner and Touzi (SIAM J. Control Optim. 41:404–424, 2002 ; J. Eur. Math. Soc. 4:201–236, 2002 ) and more recently Bouchard et al. (SIAM J. Control Optim. 48:3123–3150, 2009 ), among others. This provides a general framework for the pricing of contingent claims under risk constraints. Our extended version perfectly fits the market models with proportional transaction costs and the order book liquidation issues. Our main result is a direct PDE characterization of the associated pricing function. As an example application, we discuss the valuation of VWAP-guaranteed-type book liquidation contracts, for a general class of risk functions. Copyright Springer-Verlag 2013

Suggested Citation

  • Bruno Bouchard & Ngoc-Minh Dang, 2013. "Generalized stochastic target problems for pricing and partial hedging under loss constraints—application in optimal book liquidation," Finance and Stochastics, Springer, vol. 17(1), pages 31-72, January.
  • Handle: RePEc:spr:finsto:v:17:y:2013:i:1:p:31-72
    DOI: 10.1007/s00780-012-0198-8

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    References listed on IDEAS

    1. Sophie Laruelle & Charles-Albert Lehalle & Gilles Pag`es, 2009. "Optimal split of orders across liquidity pools: a stochastic algorithm approach," Papers 0910.1166,, revised May 2010.
    2. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
    3. Hans FÃllmer & Peter Leukert, 2000. "Efficient hedging: Cost versus shortfall risk," Finance and Stochastics, Springer, vol. 4(2), pages 117-146.
    4. repec:dau:papers:123456789/1533 is not listed on IDEAS
    5. Touzi, Nizar, 2000. "Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints," Stochastic Processes and their Applications, Elsevier, vol. 88(2), pages 305-328, August.
    6. Hans FÃllmer & Peter Leukert, 1999. "Quantile hedging," Finance and Stochastics, Springer, vol. 3(3), pages 251-273.
    Full references (including those not matched with items on IDEAS)

    More about this item


    Stochastic target problems; State constraints; Pricing under risk constraint; Book liquidation; 49L25; 60J60; 91G80; G11; G13; C61;

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis


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