Model order reduction for parametric high dimensional models in the analysis of financial risk

This paper presents a model order reduction (MOR) approach for high dimensional problems in the analysis of financial risk. To understand the financial risks and possible outcomes, we have to perform several thousand simulations of the underlying product. These simulations are expensive and create a need for efficient computational performance. Thus, to tackle this problem, we establish a MOR approach based on a proper orthogonal decomposition (POD) method. The study involves the computations of high dimensional parametric convection-diffusion reaction partial differential equations (PDEs). POD requires to solve the high dimensional model at some parameter values to generate a reduced-order basis. We propose an adaptive greedy sampling technique based on surrogate modeling for the selection of the sample parameter set that is analyzed, implemented, and tested on the industrial data. The results obtained for the numerical example of a floater with cap and floor under the Hull-White model indicate that the MOR approach works well for short-rate models.

[1]  H. P. Lee,et al.  PROPER ORTHOGONAL DECOMPOSITION AND ITS APPLICATIONS—PART I: THEORY , 2002 .

[2]  V. Ralph Algazi,et al.  On the optimality of the Karhunen-Loève expansion (Corresp.) , 1969, IEEE Trans. Inf. Theory.

[3]  R. Pinnau.  Model Reduction via Proper Orthogonal Decomposition , 2008 .

[4]  Tinne Haentjens,et al.  Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation , 2012 .

[5]  John C. Hull,et al.  Numerical Procedures for Implementing Term Structure Models II , 1994 .

[6]  Donald R. Jones,et al.  A Taxonomy of Global Optimization Methods Based on Response Surfaces , 2001, J. Glob. Optim..

[7]  L. Caramellino,et al.  Numerical approximations for Heston-Hull-White type models , 2015 .

[8]  Karen Willcox,et al.  Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space , 2008, SIAM J. Sci. Comput..

[9]  A. Patera,et al.  Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds , 2005 .

[10]  Andrew H. Chen,et al.  An Economic Analysis of Interest Rate Swaps , 1986 .

[11]  Georges A. Darbellay,et al.  A note on the volatility term structure in short rate models , 1998 .

[12]  A. Chatterjee.  An introduction to the proper orthogonal decomposition , 2000 .

[13]  Alan G. White,et al.  One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities , 1993, Journal of Financial and Quantitative Analysis.

[14]  Reinhard Mahnke,et al.  Physics of Stochastic Processes: How Randomness Acts in Time , 2009 .

[15]  O. C. Zienkiewicz,et al.  An ‘upwind’ finite element scheme for two‐dimensional convective transport equation , 1977 .

[16]  David Amsallem,et al.  An adaptive and efficient greedy procedure for the optimal training of parametric reduced‐order models , 2015 .

[17]  Philipp Mayer,et al.  Introduction to Quantitative Methods for Financial Markets , 2013 .

[18]  Albert Corhay,et al.  STATISTICAL PROPERTIES OF DAILY RETURNS: EVIDENCE FROM EUROPEAN STOCK MARKETS , 1994 .

[19]  Muruhan Rathinam,et al.  A New Look at Proper Orthogonal Decomposition , 2003, SIAM J. Numer. Anal..

[20]  Frank J. Fabozzi,et al.  Valuation of Fixed Income Securities and Derivatives , 1995 .

[21]  D. Brigo,et al.  Interest Rate Models , 2001 .

[22]  Gene H. Golub,et al.  Singular value decomposition and least squares solutions , 1970, Milestones in Matrix Computation.

[23]  J. Hadamard.  Sur les problemes aux derive espartielles et leur signification physique , 1902 .

[24]  Calibration of Interest Rate Models , 2009 .

[25]  Albert Cohen,et al.  Approximation of high-dimensional parametric PDEs * , 2015, Acta Numerica.

[26]  Antonio Falcó,et al.  Finite Difference Methods for Hull-White Pricing of Interest Rate Derivatives with Dynamical Consistent Curves , 2014 .

[27]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[28]  L. Sirovich.  TURBULENCE AND THE DYNAMICS OF COHERENT STRUCTURES PART I : COHERENT STRUCTURES , 2016 .

[29]  Trevor Hastie,et al.  An Introduction to Statistical Learning , 2013, Springer Texts in Statistics.

[30]  Alan G. White,et al.  Pricing Interest-Rate-Derivative Securities , 1990 .

[31]  Oldrich A. Vasicek.  An equilibrium characterization of the term structure , 1977 .

[32]  Piotr Indyk,et al.  Approximate nearest neighbors: towards removing the curse of dimensionality , 1998, STOC '98.

[33]  Andreas Binder,et al.  A workout in computational finance , 2013 .

[34]  Philip S. Beran,et al.  Reduced-order modeling - New approaches for computational physics , 2001 .

[35]  M. Subrahmanyam,et al.  Pricing and Hedging Interest Rate Options: Evidence from Cap-Floor Markets , 2002 .

[36]  Seokho Lee,et al.  Principal Component Regression by Principal Component Selection , 2015 .

[37]  T. Alderweireld,et al.  A Theory for the Term Structure of Interest Rates , 2004, cond-mat/0405293.

[38]  Guilin Sun,et al.  Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method , 2006, IEEE Transactions on Microwave Theory and Techniques.

[39]  C. Farhat,et al.  Design optimization using hyper-reduced-order models , 2015 .

[40]  Peter J. Schmid,et al.  Hybrid Reduced-Order Integration with Proper Orthogonal Decomposition and Dynamic Mode Decomposition , 2013, Multiscale Model. Simul..

[41]  A. Patera,et al.  A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations , 2005 .

[42]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[43]  Per Lötstedt,et al.  Boundary Values and Finite Difference Methods for the Single Factor Term Structure Equation , 2009 .