Valuation of Structured Financial Products by Adaptive Multiwavelet Methods in High Dimensions

We introduce a new numerical approach to value structured financial products. These financial products typically feature a large number of underlying assets and require the explicit modeling of the dependence structure of these assets. We follow the approach of Kraft and Steffensen (Rev Finance 11:209–252, 2006), who explicitly describe the possible value combinations of the assets via a Markov chain with a portfolio state space. As the number of states increases exponentially with the number of assets in the portfolio, this model so far has been – despite its theoretical appeal – not computationally tractable. The price of a structured financial product in this model is determined by a coupled system of parabolic PDEs, describing the value of the portfolio for each state of the Markov chain depending on the time and macroeconomic state variables. A typical portfolio of n assets leads to a system of N = 2 n coupled parabolic partial differential equations. It is shown that this high number of PDEs can be solved by combining an adaptive multiwavelet method with the Hierarchical Tucker Format. We present numerical results for n = 128.

[1]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[2]  Matemática,et al.  Society for Industrial and Applied Mathematics , 2010 .

[3]  Janet M. Tavakoli,et al.  Structured Finance and Collateralized Debt Obligations: New Developments in Cash and Synthetic Securitization , 2003 .

[4]  R. Nochetto,et al.  Theory of adaptive finite element methods: An introduction , 2009 .

[5]  Risk Allocation, Debt Fueled Expansion and Financial Crisis , 2009 .

[6]  Wolfgang Dahmen,et al.  Adaptive Wavelet Schemes for Nonlinear Variational Problems , 2003, SIAM J. Numer. Anal..

[7]  Silvia Bertoluzza,et al.  Adaptive Wavelet Methods , 2011 .

[8]  Karsten Urban,et al.  A new error bound for reduced basis approximation of parabolic partial differential equations , 2012 .

[9]  Ronald A. DeVore,et al.  Multiscale, Nonlinear and Adaptive Approximation , 2009 .

[10]  Michael Christian Lehn,et al.  FLENS - a flexible library for efficient numerical solutions , 2008 .

[11]  Karsten Urban,et al.  An efficient space-time adaptive wavelet Galerkin method for time-periodic parabolic partial differential equations , 2014, Math. Comput..

[12]  Dirk Pflüger,et al.  Spatially Adaptive Sparse Grids for High-Dimensional Problems , 2010 .

[13]  Sebastian Kestler.  On the adaptive tensor product wavelet Galerkin method with applications in finance , 2013 .

[14]  W. Hackbusch,et al.  Black Box Low Tensor-Rank Approximation Using Fiber-Crosses , 2009 .

[15]  Jack Dongarra,et al.  LAPACK Users' Guide, 3rd ed. , 1999 .

[16]  David Lando,et al.  Credit Risk Modeling , 2009 .

[17]  Daniel Kressner,et al.  Algorithm 941 , 2014 .

[18]  D. Hardin,et al.  Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets , 1999 .

[19]  F. Loonstra.  III – Linear Algebra , 1969 .

[20]  Karsten Urban.  Adaptive Wavelet Methods , 2008 .

[21]  D. Hardin,et al.  Fractal Functions and Wavelet Expansions Based on Several Scaling Functions , 1994 .

[22]  Rob Stevenson,et al.  An Adaptive Wavelet Method for Solving High-Dimensional Elliptic PDEs , 2009 .

[23]  Benjamin Peherstorfer,et al.  Spatially adaptive sparse grids for high-dimensional data-driven problems , 2010, J. Complex..

[24]  F. Douglas Swesty,et al.  A Comparison of Algorithms for the Efficient Solution of the Linear Systems Arising from Multigroup Flux-limited Diffusion Problems , 2004 .

[25]  Willi-Hans Steeb,et al.  Matrix Calculus and the Kronecker Product with Applications and C++ Programs , 1997 .

[26]  Charles L. Lawson,et al.  Basic Linear Algebra Subprograms for Fortran Usage , 1979, TOMS.

[27]  W. Hackbusch,et al.  A New Scheme for the Tensor Representation , 2009 .

[28]  C. Bluhm,et al.  Structured Credit Portfolio Analysis, Baskets and CDOs , 2006 .

[29]  Karsten Urban,et al.  Wavelet Methods for Elliptic Partial Differential Equations , 2008 .

[30]  Lars Grasedyck,et al.  Hierarchical Singular Value Decomposition of Tensors , 2010, SIAM J. Matrix Anal. Appl..

[31]  T. Bielecki,et al.  Credit Risk: Modeling, Valuation And Hedging , 2004 .

[32]  Jack J. Dongarra,et al.  An extended set of FORTRAN basic linear algebra subprograms , 1988, TOMS.

[33]  Gabriel Wittum,et al.  Efficient Hierarchical Approximation of High-Dimensional Option Pricing Problems , 2007, SIAM J. Sci. Comput..

[34]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[35]  J. Ballani,et al.  Black box approximation of tensors in hierarchical Tucker format , 2013 .

[36]  Wavelet finite element method for option pricing in highdimensional di! usion market models , 2010 .

[37]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[38]  Wolfgang Dahmen,et al.  Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations , 2013, Foundations of Computational Mathematics.

[39]  C. Loan.  The ubiquitous Kronecker product , 2000 .

[40]  Lars Grasedyck,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig a Projection Method to Solve Linear Systems in Tensor Format a Projection Method to Solve Linear Systems in Tensor Format , 2022 .

[41]  Janet M. Tavakoli,et al.  Collateralized debt obligations and structured finance : new developments in cash and synthetic securitization , 2003 .

[42]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[43]  Andreas Joachim Rupp,et al.  High dimensional wavelet methods for structured financial products , 2014 .

[44]  Jack J. Dongarra,et al.  A set of level 3 basic linear algebra subprograms , 1990, TOMS.

[45]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods II—Beyond the Elliptic Case , 2002, Found. Comput. Math..

[46]  A. Nouy.  A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations , 2010 .

[47]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[48]  J. Demmel,et al.  Sun Microsystems , 1996 .

[49]  Rob Stevenson,et al.  Fast evaluation of nonlinear functionals of tensor product wavelet expansions , 2011, Numerische Mathematik.

[50]  L. Watson,et al.  Numerical analysis 2000 Vol. IV: optimization and nonlinear equations , 2000 .

[51]  R. Jarrow.  The Role of ABS , CDS and CDOs in the Credit Crisis and the Economy , 2011 .

[52]  Mogens Steffensen,et al.  Bankruptcy, Counterparty Risk, and Contagion , 2006 .

[53]  Rob P. Stevenson,et al.  Space-time adaptive wavelet methods for parabolic evolution problems , 2009, Math. Comput..