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Finance and Stochastics

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01-01-2013

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

Authors: Bruno Bouchard, Ngoc-Minh Dang

Published in: Finance and Stochastics | Issue 1/2013

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Abstract

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.

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VWAP means volume weighted average price; see Sect. 4 for a detailed presentation.

We should like to thank the referee for pointing out to us this technical issue.

Almgren, R.: Optimal trading in a dynamic market. Technical report (2009). http://www.math.nyu.edu/financial_mathematics/content/02_financial/2009-2.pdf

Almgren, R., Harts, B.: A dynamic algorithm for smart order routing. Technical report, white paper streambase (2008). http://www.streambase.com/wp-content/uploads/downloads/StreamBase_White_Paper_Smart_Order_Routing_low.pdf

Almgren, R.F., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3, 5–39 (2000)

Bertsekas, D.P., Shreve, S.E.: Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering. Academic Press, San Diego (1978) MATH

Bertsimas, D., Lo, A.W., Hummel, P.: Optimal control of execution costs for portfolios. Comput. Sci. Eng. 1, 40–53 (1999) CrossRef

Bouchard, B., Elie, R., Touzi, N.: Stochastic target problems with controlled loss. SIAM J. Control Optim. 48, 3123–3150 (2009) MathSciNetCrossRef

Bouchard, B., Dang, N.M., Lehalle, C.A.: Optimal control of trading algorithms: a general impulse control approach. SIAM J. Financ. Math. 2, 404–438 (2011) MathSciNetMATHCrossRef

Bouchard, B., Nutz, M.: 2011, Weak dynamic programming for generalized state constraints. Preprint. http://arxiv.org/abs/1105.0745

Bouchard, B., Touzi, N.: Explicit solution of the multivariate super-replication problem under transaction costs. Ann. Appl. Probab. 10, 685–708 (2000) MathSciNetMATH

10.

Bouchard, B., Vu, T.N.: The American version of the geometric dynamic programming principle: application to the pricing of American options under constraints. Appl. Math. Optim. 61, 235–265 (2010) MathSciNetMATHCrossRef

11.

Crandall, M., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Am. Math. Soc. 27, 1–67 (1992) MathSciNetMATHCrossRef

12.

Cheridito, P., Soner, M., Touzi, N.: The multi-dimensional super-replication problem under gamma constraints. Ann. Inst. Henri Poincaré, Sér. C: Anal. Non-Linéaire 22, 633–666 (2005) MathSciNetMATHCrossRef

13.

Cvitanić, J., Pham, H., Touzi, N.: Super-replication in stochastic volatility models with portfolio constraints. J. Appl. Probab. 36, 523–545 (1999) MathSciNetMATHCrossRef

14.

Cvitanić, J., Pham, H., Touzi, N.: A closed-form solution to the problem of super-replication under transaction costs. Finance Stoch. 3, 35–54 (1999) MATHCrossRef

15.

Dupuis, P., Ishii, H.: SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21, 554–580 (1993) MathSciNetMATHCrossRef

16.

Föllmer, H., Leukert, P.: Quantile hedging. Finance Stoch. 3, 251–273 (1999) MathSciNetMATHCrossRef

17.

Föllmer, H., Leukert, P.: Efficient hedging: cost versus shortfall risk. Finance Stoch. 4, 117–146 (2000) MathSciNetMATHCrossRef

18.

Kabanov, Y.M.: Hedging and liquidation under transaction costs in currency markets. Finance Stoch. 3, 237–248 (1999) MathSciNetMATHCrossRef

19.

Pagès, G., Laruelle, S., Lehalle, C.A.: Optimal split of orders across liquidity pools: a stochastic algorithm approach. SIAM J. Financ. Math. 2, 1042–1076 (2011) MATHCrossRef

20.

Soner, H.M.: Optimal control with state-space constraint I. SIAM J. Control Optim. 24, 552–561 (1986) MathSciNetMATHCrossRef

21.

Soner, H.M.: Optimal control with state-space constraint II. SIAM J. Control Optim. 24, 1110–1122 (1986) MathSciNetMATHCrossRef

22.

Soner, H.M., Touzi, N.: Super-replication under gamma constraints. SIAM J. Control Optim. 39, 73–96 (2000) MathSciNetMATHCrossRef

23.

Soner, H.M., Touzi, N.: Stochastic target problems, dynamic programming and viscosity solutions. SIAM J. Control Optim. 41, 404–424 (2002) MathSciNetMATHCrossRef

24.

Soner, H.M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4, 201–236 (2002) MathSciNetMATHCrossRef

25.

Soner, H.M., Touzi, N.: The problem of super-replication under constraints. In: Carmona, R.A., Çinlar, E., Ekeland, I., Jouini, E., Scheinkman, J., Touzi, N. (eds.) Paris–Princeton Lectures on Mathematical Finance. Lecture Notes in Mathematics, vol. 1814, pp. 133–172. Springer, Berlin (2002) CrossRef

26.

Touzi, N.: Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints. Stoch. Process. Appl. 88, 305–328 (2000) MathSciNetMATHCrossRef

Title: Generalized stochastic target problems for pricing and partial hedging under loss constraints—application in optimal book liquidation
Authors: Bruno Bouchard
Ngoc-Minh Dang
Publication date: 01-01-2013
Publisher: Springer-Verlag
Published in: Finance and Stochastics / Issue 1/2013
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-012-0198-8

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