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

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Abstract

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.

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References

  1. Alexander, C.: Market Models: A Guide to Financial Data Analysis. Wiley, New York (2001)

    Google Scholar 

  2. Alizadeh, F., Haeberly, J.-P.A., Overton, M.L.: Complementarity and nondegeneracy in semidefinite programming. Math. Program. 77, 111–128 (1997)

    MathSciNet  Google Scholar 

  3. Arnold, V.I.: On matrices depending on parameters. Rus. Math. Surv. 26, 29–43 (1971)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bhansali, V., Wise, B.: Forecasting portfolio risk in normal and stressed market. J. Risk 4(1), 91–106 (2001)

    Google Scholar 

  6. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    MATH  Google Scholar 

  7. Boyd, S., Xiao, L.: Least-squares covariance matrix adjustment. SIAM J. Matrix Anal. Appl. 27, 532–546 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  9. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  10. Dash, J.W.: Quantitative Finance and Risk Management: A Physicist’s Approach. World Scientific, Singapore (2004)

    MATH  Google Scholar 

  11. Eaves, B.C.: On the basic theorem of complementarity. Math. Program. 1, 68–75 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  13. Finger, C.: A methodology for stress correlation. In: Risk Metrics Monitor, Fourth Quarter (1997)

  14. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409–436 (1952)

    MATH  MathSciNet  Google Scholar 

  15. Higham, N.J.: Computing the nearest correlation matrix—a problem from finance. IMA J. Numer. Anal. 22, 329–343 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  17. Kupiec, P.H.: Stress testing in a Value-at-Risk framework. J. Deriv. 6(1), 7–24 (1998)

    Article  Google Scholar 

  18. León, A., Peris, J.E., Silva, J., Subiza, B.: A note on adjusting correlation matrices. Appl. Math. Finance 9, 61–67 (2002)

    MATH  Google Scholar 

  19. Malick, J.: A dual approach to semidefinite least-squares problems. SIAM J. Matrix Anal. Appl. 26, 272–284 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. J.P. Morgan/Reuters: RiskMetrics™—Technical Document, 4th edn. New York (1996)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  23. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rapisarda, F., Brigo, D., Mercurio, F.: Parameterizing correlations: a geometric interpretation. IMA J. Manag. Math. 18, 55–73 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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. Rockafellar, R.T.: Conjugate Duality and Optimization. SIAM, Philadelphia (1974)

    MATH  Google Scholar 

  27. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  29. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sun, D.F., Sun, J.: Semismooth matrix valued functions. Math. Oper. Res. 27, 150–169 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  34. Turkay, S., Epperlein, E., Christofides, N.: Correlation stress testing for value-at-risk. J. Risk 5(4), 75–89 (2003)

    Google Scholar 

  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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Authors and Affiliations

  1. School of Mathematics, The University of Southampton, Highfield, Southampton, SO17 1BJ, UK

    Houduo Qi

  2. Department of Mathematics and Risk Management Institute, National University of Singapore, Singapore, 117543, Singapore

    Defeng Sun

Authors
  1. Houduo Qi
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  2. Defeng Sun
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Correspondence to Defeng Sun.

Additional information

Research of H. Qi was partially supported by EPSRC Grant EP/D502535/1.

Research of D. Sun was partially supported by Grant R146-000-104-112 of the National University of Singapore.

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Qi, H., Sun, D. Correlation stress testing for value-at-risk: an unconstrained convex optimization approach. Comput Optim Appl 45, 427–462 (2010). https://doi.org/10.1007/s10589-008-9231-4

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  • DOI: https://doi.org/10.1007/s10589-008-9231-4

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