Abstract
The robustification of trading strategies is of particular interest in financial market applications. In this paper we robustify a portfolio strategy recently introduced in the literature against model errors in the sense of a worst case design. As it turns out, the resulting optimization problem can be solved by a sequence of linear and nonlinear semidefinite programs (SDP/NSDP), where the nonlinearity is introduced by the parameters of a parabolic differential equation. The nonlinear semidefinite program naturally arises in the computation of the worst case constraint violation which is equivalent to an eigenvalue minimization problem. Further we prove convergence for the iterates generated by the sequential SDP-NSDP approach.
Similar content being viewed by others
References
Benson, S.J., Ye, Y., Zhang, X.: Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim. 10(2), 443–461 (2000)
Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)
Bowie, J., Carr, P.: Static simplicity. Risk 7, 45–49 (1994)
Burke, J.V., Lewis, A.S., Overton, M.L.: Two numerical methods for optimizing matrix stability. Linear Algebra Appl. 351–352, 117–145 (2002)
Burke, J.V., Lewis, A.S., Overton, M.L.: Optimization and pseudospectra, with applications to robust stability. SIAM J. Matrix Anal. Appl. 25, 80–104 (2003)
Carr, P., Ellis, K., Gupta, V.: Static hedging of exotic options. J. Finance 53, 1165–1190 (1998)
Derman, E., Ergener, D., Kani, I.: Static options replication. J. Deriv. 2, 78–95 (1995)
Fares, B., Noll, D., Apkarian, P.: Robust control via sequential semidefinite programming. SIAM J. Control Optim. 40(6), 1791–1820 (2002)
Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, New York (1987)
Giese, A.M., Maruhn, J.H.: Cost-optimal static super-replication of barrier options—an optimization approach. J. Comput. Finance 10(3), 71–97 (2007)
Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, New York (1998)
Goldfarb, D., Iyengar, G.: Robust portfolio selection problems. Math. Oper. Res. 28, 1–38 (2003)
Grant, M., Boyd, S., Ye, Y.: cvx users’ guide. http://www.stanford.edu/~boyd/cvx (2007)
Henrion, D.: ℋ∞ controller design on the COMP l e ib problems with the robust control toolbox for Matlab. LAAS-CNRS Research Report, Toulouse, October 2005
Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)
Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Kanzow, C., Nagel, C., Kato, H., Fukushima, M.: Successive linearization methods for nonlinear semidefinite programs. Comput. Optim. Appl. 31(3), 251–273 (2005)
Kočvara, M., Leibfritz, F., Stingl, M., Henrion, D.: A nonlinear SDP algorithm for static output feedback problems in COMP l e ib. In: Proceedings of the 16th IFAC World Congress on Automatic Control, Prague (2005)
Leibfritz, F.: A LMI-based algorithm for designing suboptimal static ℋ2/ℋ∞ output feedback controllers. SIAM J. Control Optim. 39, 1711–1735 (2001)
Leibfritz, F.: COMPl e ib: COnstrained Matrix-optimization Problem library—a collection of test examples for nonlinear semidefinite programs, control system design and related problems. In: Special issue of Eur. J. Control on Linear Matrix Inequalities in Control (2006, to appear)
Leibfritz, F.: Static output feedback ℋ∞-synthesis by solving the ℋ2/ℋ∞-NSDP. In: Proceedings in Applied Mathematics and Mechanics, PAMM (2005)
Leibfritz, F.: Nonlinear semidefinite programs: theory and applications. Oberwolfach Reports 2 (2005)
Leibfritz, F., Mostafa, E.M.E.: An interior point constrained trust region method for a special class of nonlinear semi-definite programming problems. SIAM J. Optim. 12, 1048–1074 (2002)
Leibfritz, F., Volkwein, S.: Reduced order output feedback control design for PDE systems using proper orthogonal decomposition and nonlinear semidefinite programming. Linear Algebra Appl. 415, 542–757 (2006)
Leibfritz, F., Volkwein, S.: Numerical feedback controller design for PDE systems: using model reduction: techniques and case studies. In: Biegler, L., Ghattas, O., Heinkenschloss, M., Keyes, D., van Bloemen Waanders, B. (eds.) Real-Time PDE-Constrained Optimization. SIAM, Philadelphia (2006)
Maruhn, J.H., Sachs, E.W.: Robust static super-replication of barrier options in the black scholes model. In: Proceedings of the Conference on Robust Optimization-Directed Design (RODD), Shalimar, FL (2005)
Maruhn, J.H., Sachs, E.W.: Robust static hedging of barrier options in stochastic volatility models. Technical Report No. 06-3, University of Trier (2006)
Scherer, C.: LMI relaxation in robust control. Eur. J. Control 12(1), 3–29 (2006)
Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11, 625–653 (1999)
Toh, K.C., Todd, M.J., Tütüncü, R.H.: SDPT3—a MATLAB software package for semidefinite programming, version 2.1. Optim. Methods Softw. 11, 545–581 (1999)
Tütüncü, R.H., Koenig, M.: Robust asset allocation. Ann. Oper. Res. 132, 157–187 (2004)
Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)
Wolfe, P.: A duality theorem for nonlinear programming. Q. Appl. Math. 19(3), 239–244 (1961)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Johann Radon Institute for Computational and Applied Mathematics (RICAM) during the Special Radon Semester on Computational Mechanics.
Rights and permissions
About this article
Cite this article
Leibfritz, F., Maruhn, J.H. A successive SDP-NSDP approach to a robust optimization problem in finance. Comput Optim Appl 44, 443–466 (2009). https://doi.org/10.1007/s10589-007-9163-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-007-9163-4