Computational Optimization and Applications

, Volume 45, Issue 2, pp 427–462 | Cite as

Correlation stress testing for value-at-risk: an unconstrained convex optimization approach

  • Houduo Qi
  • Defeng Sun


Correlation stress testing is employed in several financial models for determining the value-at-risk (VaR) of a financial institution’s portfolio. The possible lack of mathematical consistence in the target correlation matrix, which must be positive semidefinite, often causes breakdown of these models. The target matrix is obtained by fixing some of the correlations (often contained in blocks of submatrices) in the current correlation matrix while stressing the remaining to a certain level to reflect various stressing scenarios. The combination of fixing and stressing effects often leads to mathematical inconsistence of the target matrix. It is then naturally to find the nearest correlation matrix to the target matrix with the fixed correlations unaltered. However, the number of fixed correlations could be potentially very large, posing a computational challenge to existing methods. In this paper, we propose an unconstrained convex optimization approach by solving one or a sequence of continuously differentiable (but not twice continuously differentiable) convex optimization problems, depending on different stress patterns. This research fully takes advantage of the recently developed theory of strongly semismooth matrix valued functions, which makes fast convergent numerical methods applicable to the underlying unconstrained optimization problem. Promising numerical results on practical data (RiskMetrics database) and randomly generated problems of larger sizes are reported.


Stress testing Convex optimization Newton’s method Augmented Lagrangian functions 


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  1. 1.
    Alexander, C.: Market Models: A Guide to Financial Data Analysis. Wiley, New York (2001) Google Scholar
  2. 2.
    Alizadeh, F., Haeberly, J.-P.A., Overton, M.L.: Complementarity and nondegeneracy in semidefinite programming. Math. Program. 77, 111–128 (1997) MathSciNetGoogle Scholar
  3. 3.
    Arnold, V.I.: On matrices depending on parameters. Rus. Math. Surv. 26, 29–43 (1971) CrossRefGoogle Scholar
  4. 4.
    Bai, Z.-J., Chu, D., Sun, D.F.: A dual optimization approach to inverse quadratic eigenvalue problems with partial eigenstructure. SIAM J. Sci. Comput. 29, 2531–2561 (2007) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bhansali, V., Wise, B.: Forecasting portfolio risk in normal and stressed market. J. Risk 4(1), 91–106 (2001) Google Scholar
  6. 6.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000) MATHGoogle Scholar
  7. 7.
    Boyd, S., Xiao, L.: Least-squares covariance matrix adjustment. SIAM J. Matrix Anal. Appl. 27, 532–546 (2005) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chen, X., Qi, H.D., Tseng, P.: Analysis of nonsmooth symmetric matrix valued functions with applications to semidefinite complementarity problems. SIAM J. Optim. 13, 960–985 (2003) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) MATHGoogle Scholar
  10. 10.
    Dash, J.W.: Quantitative Finance and Risk Management: A Physicist’s Approach. World Scientific, Singapore (2004) MATHGoogle Scholar
  11. 11.
    Eaves, B.C.: On the basic theorem of complementarity. Math. Program. 1, 68–75 (1971) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fender, I., Gibson, M.S., Mosser, P.C.: An international survey of stress tests. Federal Reserve Bank of New York, Current Issues in Economics and Finance, vol. 7, No. 10 (2001) Google Scholar
  13. 13.
    Finger, C.: A methodology for stress correlation. In: Risk Metrics Monitor, Fourth Quarter (1997) Google Scholar
  14. 14.
    Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409–436 (1952) MATHMathSciNetGoogle Scholar
  15. 15.
    Higham, N.J.: Computing the nearest correlation matrix—a problem from finance. IMA J. Numer. Anal. 22, 329–343 (2002) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kercheval, A.N.: On Rebonato and Jäckel’s parametrization method for finding nearest correlation matrices. Int. J. Pure Appl. Math. 45, 383–390 (2008) MATHMathSciNetGoogle Scholar
  17. 17.
    Kupiec, P.H.: Stress testing in a Value-at-Risk framework. J. Deriv. 6(1), 7–24 (1998) CrossRefGoogle Scholar
  18. 18.
    León, A., Peris, J.E., Silva, J., Subiza, B.: A note on adjusting correlation matrices. Appl. Math. Finance 9, 61–67 (2002) MATHGoogle Scholar
  19. 19.
    Malick, J.: A dual approach to semidefinite least-squares problems. SIAM J. Matrix Anal. Appl. 26, 272–284 (2004) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    J.P. Morgan/Reuters: RiskMetrics—Technical Document, 4th edn. New York (1996) Google Scholar
  21. 21.
    Pang, J.S., Sun, D.F., Sun, J.: Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems. Math. Oper. Res. 28, 39–63 (2003) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Qi, H.D., Sun, D.F.: A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM J. Matrix Anal. Appl. 28, 360–385 (2006) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Rapisarda, F., Brigo, D., Mercurio, F.: Parameterizing correlations: a geometric interpretation. IMA J. Manag. Math. 18, 55–73 (2007) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Rebonato, R., Jäckel, P.: The most general methodology for creating a valid correlation matrix for risk management and option pricing purpose. J. Risk 2(2), 17–27 (2000) Google Scholar
  26. 26.
    Rockafellar, R.T.: Conjugate Duality and Optimization. SIAM, Philadelphia (1974) MATHGoogle Scholar
  27. 27.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998) MATHCrossRefGoogle Scholar
  30. 30.
    Sun, D.F.: The strong second order sufficient condition and the constraint nondegeneracy in nonlinear semidefinite programming and their implications. Math. Oper. Res. 31, 761–776 (2006) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Sun, D.F., Sun, J.: Semismooth matrix valued functions. Math. Oper. Res. 27, 150–169 (2002) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Sun, D.F., Sun, J., Zhang, L.W.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Program. 114, 349–391 (2008) MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Toh, K.C., Tütüncü, R.H., Todd, M.J.: Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems. Pac. J. Optim. 3, 135–164 (2007) MATHMathSciNetGoogle Scholar
  34. 34.
    Turkay, S., Epperlein, E., Christofides, N.: Correlation stress testing for value-at-risk. J. Risk 5(4), 75–89 (2003) Google Scholar
  35. 35.
    Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory I and II. In: Zarantonello, E.H. (ed.) Contributions to Nonlinear Functional Analysis, pp. 237–424. Academic, New York (1971) Google Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.School of MathematicsThe University of SouthamptonHighfieldUK
  2. 2.Department of Mathematics and Risk Management InstituteNational University of SingaporeSingaporeSingapore

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