×

Convex duality in mean-variance hedging under convex trading constraints. (English) Zbl 1277.60079

The authors study here mean-variance hedging problems under portfolio constraints in a general semimartingale model, namely they consider the minimisation problem \[ \min_{\textstyle{(x,\theta)\in \mathbb{R}\times\Theta_{S}}}\operatorname{E}\left[\left|H-x-\int_{0}^{T}\theta_s dS_s\right|^2\right], \] where \(H\) represents a (square integrable \(\mathcal{F}_T\)-measurable) contingent claim, \(x\) the initial endowment, \(S\) any locally square-integrable semimartingale and \[ \Theta_S:=\{\theta\in\mathcal{L}(S) \mid \int\theta d S \text{ is a square-integrable semimartingale}\} \] the space of all acceptable strategies (here, \(\mathcal{L}(S)\) represents the space of \(S\)-integrable predictable process for \(S\)). They first prove that the space \(\{\int\theta dS \mid \theta\in\Theta_S\}\) is closed. Existence of a solution to the mean-variance hedging problem follows from this closedness property, and convex duality tools are used in order to study some properties of this solution. This extends the work of C. Labbé and A. J. Heunis [Adv. Appl. Probab. 39, No. 1, 77–104 (2007; Zbl 1110.93051)], where \(S\) was an Itō process.

MSC:

60G48 Generalizations of martingales
91G10 Portfolio theory
93E20 Optimal stochastic control
49N10 Linear-quadratic optimal control problems
60H05 Stochastic integrals

Citations:

Zbl 1110.93051
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis , 3rd edn. Springer, Berlin. · Zbl 1156.46001
[2] Aubin, J.-P. (2000). Applied Functional Analysis , 2nd edn. Wiley-Interscience, New York.
[3] Bielecki, T. R., Jin, H., Pliska, S. R. and Zhou, X. Y. (2005). Continuous-time mean-variance portfolio selection with bankruptcy prohibition. Math. Finance 15 , 213-244. · Zbl 1153.91466 · doi:10.1111/j.0960-1627.2005.00218.x
[4] Choulli, T., Krawczyk, L. and Stricker, C. (1998). \(\mathcal{E}\)-martingales and their applications in mathematical finance. Ann. Prob. 26 , 853-876. · Zbl 0938.60032 · doi:10.1214/aop/1022855653
[5] Cvitanić, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl. Prob. 2 , 767-818. · Zbl 0770.90002 · doi:10.1214/aoap/1177005576
[6] Czichowsky, C. and Schweizer, M. (2011). Closedness in the semimartingale topology for spaces of stochastic integrals with constrained integrands. In Séminaire de Probabilités XLIII (Lecture Notes Math. 2006 ), Springer, Berlin, pp. 413-436 · Zbl 1225.60073 · doi:10.1007/978-3-642-15217-7_18
[7] Czichowsky, C. and Schweizer, M. (2012). Cone-constrained continuous-time Markowitz problems. To appear in Ann. Appl. Prob. · Zbl 1268.91162
[8] Delbaen, F. (2006). The structure of m-stable sets and in particular of the set of risk neutral measures. In In memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX (Lecture Notes Math. 1874 ), Springer, Berlin, pp. 215-258. · Zbl 1121.60043
[9] Delbaen, F. \et (1997). Weighted norm inequalities and hedging in incomplete markets. Finance Stoch. 1 , 181-227. · Zbl 0916.90016
[10] Ekeland, I. and Temam, R. (1976). Convex Analysis and Variational Problems . North-Holland, Amsterdam. · Zbl 0322.90046
[11] Föllmer, H. and Kramkov, D. (1997). Optional decompositions under constraints. Prob. Theory Relat. Fields 109 , 1-25. · Zbl 0882.60063 · doi:10.1007/s004400050122
[12] Hou, C. and Karatzas, I. (2004). Least-squares approximation of random variables by stochastic integrals. In Stochastic Analysis and Related Topics in Kyoto (Adv. Stud. Pure Math. 41 ), Mathematical Society, Japan, Tokyo, pp. 141-166. · Zbl 1057.60066
[13] Hu, Y. and Zhou, X. Y. (2005). Constrained stochastic LQ control with random coefficients, and application to portfolio selection. SIAM J. Control Optimization 44 , 444-466. · Zbl 1210.93082 · doi:10.1137/S0363012904441969
[14] Jin, H. and Zhou, X. Y. (2007). Continuous-time Markowitz’s problems in an incomplete market, with no-shorting portfolios. In Stochastic Analysis and Applications (Abel Symp. 2 ), Springer, Berlin, pp. 435-459. · Zbl 1151.91516 · doi:10.1007/978-3-540-70847-6
[15] Karatzas, I. and Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance Stoch. 11 , 447-493. · Zbl 1144.91019 · doi:10.1007/s00780-007-0047-3
[16] Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance (Appl. Math. 39 ). Springer, New York. · Zbl 0941.91032
[17] Karatzas, I. and Žitković, G. (2003). Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Prob. 31 , 1821-1858. · Zbl 1076.91017 · doi:10.1214/aop/1068646367
[18] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Prob. 9 , 904-950. · Zbl 0967.91017 · doi:10.1214/aoap/1029962818
[19] Labbé, C. and Heunis, A. J. (2007). Convex duality in constrained mean-variance portfolio optimization. Adv. Appl. Prob. 39 , 77-104. · Zbl 1110.93051 · doi:10.1239/aap/1175266470
[20] Mémin, J. (1980). Espaces de semi martingales et changement de probabilité. Z. Wahrscheinlichkeitsth. 52 , 9-39. · Zbl 0407.60046 · doi:10.1007/BF00534184
[21] Mnif, M. and Pham, H. (2001). Stochastic optimization under constraints. Stoch. Process. Appl. 93 , 149-180. · Zbl 1070.93050 · doi:10.1016/S0304-4149(00)00089-2
[22] Pham, H. (2000). Dynamic \(L^ p\)-hedging in discrete time under cone constraints. SIAM J. Control Optimization 38 , 665-682. · Zbl 0964.91022 · doi:10.1137/S0363012998341095
[23] Pham, H. (2002). Minimizing shortfall risk and applications to finance and insurance problems. Ann. Appl. Prob. 12 , 143-172. · Zbl 1015.93071 · doi:10.1214/aoap/1015961159
[24] Protter, P. E. (2005). Stochastic Integration and Differential Equations (Stoch. Modelling Appl. Prob. 21 ). Springer, Berlin.
[25] Rockafellar, R. T. (1970). Convex Analysis . (Princeton Math. Ser. 28 ). Princeton University Press, Princeton, NJ. · Zbl 0229.90020
[26] Rockafellar, R. T. (1976). Integral functionals, normal integrands and measurable selections. In Nonlinear Operators and the Calculus of Variations (Lecture Notes Math. 543 ), Springer, Berlin, pp. 157-207 · Zbl 0374.49001 · doi:10.1007/BFb0079944
[27] Schweizer, M. (2001). A guided tour through quadratic hedging approaches. In Option Pricing, Interest Rates and Risk Management. Cambridge University Press, pp. 538-574. · Zbl 0992.91036
[28] Schweizer, M. (2010). Mean-variance hedging. In Encyclopedia of Quantitative Finance , ed. R. Cont, John Wiley, pp. 1177-1181.
[29] Sun, W. G. and Wang, C. F. (2006). The mean-variance investment problem in a constrained financial market. J. Math. Econom. 42 , 885-895. · Zbl 1153.91565 · doi:10.1016/j.jmateco.2006.04.012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.