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

Authors: Binder, Andreas; Jadhav, Onkar; Mehrmann, Volker;

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

Abstract

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.

Comment: 39 pages, 10 figures

Country
Germany
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Dewey Decimal Classification: ddc:510

Keywords

convection-diffusion reaction equation, financial risk analysis, parametric model order reduction, short-rate models, 510 Mathematik, Computational Finance (q-fin.CP), packaged retail investment and insurance-based products, Numerical Analysis (math.NA), 35L10, 65M06, 91G30, 91G60, 91G80, PRIIPs, adaptive greedy sampling, FOS: Economics and business, Quantitative Finance - Computational Finance, proper orthogonal decomposition, Financial risk analysis, short-rate models, convection-diffusion reaction equation, finite differencemethod, parametric model order reduction, proper orthogonal decomposition, adaptive greedy sampling, Packaged retail investment and insurance-based products (PRIIPs)., FOS: Mathematics, Mathematics - Numerical Analysis, finite difference method

50 references, page 1 of 5

2. Model Hierarchy 3 2.1. Bank Account and Short-Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Vasicek and Cox-Ingersoll-Ross Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3. Hull-White Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. Yield Curve Simulation and Parameter Calibration 7 3.1. Yield Curve Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2. Parameter Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4. Numerical Methods 12 4.1. Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2. Parametric Model Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3. Greedy Sampling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4. Adaptive Greedy Sampling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5. Numerical Example 26 5.1. Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2. Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3. Model Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.4. Computational cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

[1] M. Aichinger and A. Binder. A Workout in Computational Finance. John Wiley and Sons Inc., West Sussex, UK, 1 edition, 2013. [OpenAIRE]

[2] H. Albrecher, A. Binder, V. Lautscham, and P. Mayer. Introduction to Quantitative Methods for Financial Markets. Springer-Verlag, Berlin, 1 edition, 2010.

[3] D. Amsallem, M. Zahr, and Y. Choi. Design optimization using hyper-reduced-order models. Struct. Multidisc. Optim., 51(4):919-940, 2015.

[4] D. Anderson, J.C. Tannehill, and R.H. Pletcher. Computational Fluid Mechanics and Heat Transfer. CRC Press, London, 3 edition, 2013.

[5] G. Berkooz, P. Holmes, and J.L. Lumley. The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech., 25(1):539-575, 1993. [OpenAIRE]

[6] J. Bicksler and A. Chen. An economic analysis of interest rate swaps. J. Finance, 3:645-655, 1986. [OpenAIRE]

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