In many areas of finance and of risk management it is interesting to know how to specify time-dependent correlation matrices. In this work we propose a new methodology to create valid time-dependent instantaneous correlation matrices, which we called correlation flows. In our methodology one needs only an initial correlation matrix to create these correlation flows based on isospectral flows. The tendency of the time-dependent matrices can be controlled by requirements. An application example is presented to illustrate our methodology.
Mots-clés : Time-dependent correlation matrix, isospectral flow, matrix differential equation
@article{M2AN_2020__54_2_361_0,
author = {Teng, Long and Wu, Xueran and G\"unther, Michael and Ehrhardt, Matthias},
title = {A new methodology to create valid time-dependent correlation matrices \protect\emph{via} isospectral flows},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {361--371},
publisher = {EDP-Sciences},
volume = {54},
number = {2},
year = {2020},
doi = {10.1051/m2an/2019064},
mrnumber = {4062756},
zbl = {1441.91095},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2019064/}
}
TY - JOUR AU - Teng, Long AU - Wu, Xueran AU - Günther, Michael AU - Ehrhardt, Matthias TI - A new methodology to create valid time-dependent correlation matrices via isospectral flows JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 361 EP - 371 VL - 54 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2019064/ DO - 10.1051/m2an/2019064 LA - en ID - M2AN_2020__54_2_361_0 ER -
%0 Journal Article %A Teng, Long %A Wu, Xueran %A Günther, Michael %A Ehrhardt, Matthias %T A new methodology to create valid time-dependent correlation matrices via isospectral flows %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 361-371 %V 54 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2019064/ %R 10.1051/m2an/2019064 %G en %F M2AN_2020__54_2_361_0
Teng, Long; Wu, Xueran; Günther, Michael; Ehrhardt, Matthias. A new methodology to create valid time-dependent correlation matrices via isospectral flows. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 2, pp. 361-371. doi : 10.1051/m2an/2019064. http://www.numdam.org/articles/10.1051/m2an/2019064/
[1] , Affine Diffusions and Related Processes: Simulation, Theory and Applications. Springer International Publishing, Switzerland (2015). | MR
[2] , Remarque sur une communication de Mr. H. Schwerdtfeger. Univ. e. Politec. Torino Rend. Sem. Mat. 11 (1952) 335–336. | MR | Zbl
[3] and , Forecasting portfolio risk in normal and stressed market. J. Risk 4 (2001) 91–106. | DOI
[4] and , Least-squares covariance matrix adjustment. SIAM J. Matrix Anal. Appl. 27 (2005) 532–546. | DOI | MR | Zbl
[5] , Matrix differential equations: a continuous realization process for linear algebra problems. Nonlinear Anal. Theor. Methods Appl. 18 (1992) 1125–1146. | DOI | MR | Zbl
[6] , Matrix Groups. Springer-Verlag, New York (1979). | DOI | MR | Zbl
[7] , Quantitative Finance and Risk Management: A Physicist’ Approach. World Scientific, Singapore (2004). | DOI | MR | Zbl
[8] , A methodology for stress correlation. RiskMetrics Monitor Fourth Quarter (1997) 3–11.
[9] , and , Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin Heidelberg (2006). | MR | Zbl
[10] , Differential Geometry, Lie Groups and Symmetric Spaces. Academic, New York (1978). | MR | Zbl
[11] , Computing the nearest correlation matrix – a problem from finance. IMA J. Numer. Anal. 22 (2002) 329–343. | DOI | MR | Zbl
[12] , On Rebonato and Jäckel’s parametrization method for finding nearest correlation matrices. Int. J. Pure Appl. Math. 45 (2008) 383–390. | MR | Zbl
[13] , Stress testing in a value at risk framework. J. Derivatives 6 (1998) 7–24. | DOI
[14] , , and , A note on adjusting correlation matrices. Appl. Math. Finan. 9 (2002) 61–67. | DOI | Zbl
[15] , On the exponential solution of differential equations for a linear operator. Pure Appl. Math. VII (1954) 649–673. | DOI | MR | Zbl
[16] , A dual approach to semidefinite least-squares problems. SIAM. J. Matrix Anal. Appl. 26 (2004) 272–284. | DOI | MR | Zbl
[17] , Some results on matrices which commute with their derivatives. SIAM. J. Appl. Math. 15 (1967) 1171–1183. | DOI | MR | Zbl
[18] and , A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM. J. Matrix Anal. Appl. 28 (2006) 360–385. | DOI | MR | Zbl
[19] and , Correlation stress testing for value-at-risk: an unconstrained convex optimization approach. J. Derivatives 45 (2010) 427–462. | MR | Zbl
[20] , and , Parameterizing correlations: a geometric interpretation. IMA J. Manag. Math. 18 (2007) 55–73. | MR | Zbl
[21] and , The most general methodology to create a valid correlation matrix for risk management and option pricing purposes. J. Risk 2 (2000) 17–27. | DOI
[22] , and , Option pricing with dynamically correlated stochastic interest rate Acta. Math. Uni. Comenianae LXXXIV (2015) 179–190. | MR | Zbl
[23] , and , The pricing of quanto options under dynamic correlation. J. Comput. Appl. Math. 275 (2015) 304–310. | DOI | MR | Zbl
[24] , and , The dynamic correlation model and its application to the Heston model. In: Innovations in Derivatives Markets, edited by , , and . Springer, Cham (2016). | DOI | Zbl
[25] , and , Modelling stochastic correlation. J. Math. Ind. 6 (2016) 1–18. | MR
[26] , and , On the Heston model with stochastic correlation. Int. J. Theor. Appl. Finan. 19 (2016) 16500333. | DOI | MR
[27] , , and , A versatile approach for stochastic correlation using hyperbolic functions. Int. J. Comput. Math. 93 (2016) 524–539. | DOI | MR | Zbl
[28] , and , Quanto pricing in stochastic correlation models. Int. J. Theor. Appl. Finan. 21 (2018) 1850038. | DOI | MR | Zbl
[29] , and , Correlation stress testing for value-at-risk. J. Risk 5 (2003) 75–89. | DOI
[30] , Foundations of Differential Manifolds and Lie Group. Springer-Verlag, New York (1983). | DOI | MR | Zbl
Cité par Sources :
