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FILEID="IMG_THUMBS_3582060"/><mets:fptr FILEID="IMG_MIN_3582060"/><mets:fptr FILEID="IMG_MAX_3582060"/><mets:fptr FILEID="ALTO3582060"/></mets:div><mets:div ID="phys3582061" TYPE="page" LABEL="Seite 148" ORDER="148"><mets:fptr FILEID="IMG_DEFAULT_3582061"/><mets:fptr FILEID="IMG_THUMBS_3582061"/><mets:fptr FILEID="IMG_MIN_3582061"/><mets:fptr FILEID="IMG_MAX_3582061"/><mets:fptr FILEID="ALTO3582061"/></mets:div></mets:div></mets:structMap><mets:structMap TYPE="LOGICAL"><mets:div ID="log3555185" DMDID="md3555185" ADMID="amd3555185" TYPE="document" LABEL="Analytical and numerical approximative methods for solving multifactor models for pricing of financial derivatives" ORDER="1" CONTENTIDS="urn:nbn:de:hbz:468-20170412-150623-2"><mets:fptr FILEID="PDF_3555185"/><mets:fptr FILEID="IMG_TEASER_3581818"/><mets:div ID="log3581819" TYPE="section" LABEL="Leere Seite" ORDER="1"/><mets:div ID="log3581828" TYPE="section" LABEL="Acknowledgement" ORDER="2"/><mets:div ID="log3581831" TYPE="section" LABEL="Abstract" ORDER="3"/><mets:div ID="log3581834" TYPE="section" LABEL="Contents" ORDER="4"/><mets:div ID="log3581841" TYPE="section" LABEL="Abbreviations" ORDER="5"/><mets:div ID="log3581844" TYPE="section" LABEL="Foreword" ORDER="6"/><mets:div ID="log3581847" TYPE="section" LABEL="1 Outline of the thesis and related scientific works" ORDER="7"><mets:div ID="log3581848" TYPE="section" LABEL="1.1 Outline of the first part" ORDER="1"/><mets:div ID="log3581851" TYPE="section" LABEL="1.2 Outline of the second part" ORDER="2"/></mets:div><mets:div ID="log3581854" TYPE="section" LABEL="I Analytical Approximations of Interest Short Rate Models" ORDER="8"><mets:div ID="log3581857" TYPE="section" LABEL="2 Introduction: Pricing of financial derivatives" ORDER="1"><mets:div ID="log3581860" TYPE="section" LABEL="2.1 Bond pricing in short rate models" ORDER="1"/><mets:div ID="log3581862" TYPE="section" LABEL="2.2 Short rate models" ORDER="2"><mets:div ID="log3581864" TYPE="section" LABEL="2.2.1 One-factor models" ORDER="1"/><mets:div ID="log3581867" TYPE="section" LABEL="2.2.2 Two-factor models" ORDER="2"/><mets:div ID="log3581869" TYPE="section" LABEL="2.2.3 Multi-factor short rate models" ORDER="3"/></mets:div><mets:div ID="log3581871" TYPE="section" LABEL="2.3 The calibration algorithm" ORDER="3"/></mets:div><mets:div ID="log3581874" TYPE="section" LABEL="3 Convergence model of interest rates of CKLS type" ORDER="2"><mets:div ID="log3581876" TYPE="section" LABEL="3.1 Convergence models" ORDER="1"><mets:div ID="log3581878" TYPE="section" LABEL="3.1.1 The convergence model of Vasicek type" ORDER="1"/><mets:div ID="log3581880" TYPE="section" LABEL="3.1.2 Convergence model of CIR type" ORDER="2"/><mets:div ID="log3581882" TYPE="section" LABEL="3.1.3 Convergence model of CKLS type" ORDER="3"/></mets:div><mets:div ID="log3581884" TYPE="section" LABEL="3.2 Approximation of the domestic bond price solution" ORDER="2"><mets:div ID="log3581885" TYPE="section" LABEL="3.2.1 Accuracy of the approximation for CIR model with zero correlation" ORDER="1"/></mets:div><mets:div ID="log3581887" TYPE="section" LABEL="3.3 Numerical results for CIR model with zero correlation" ORDER="3"><mets:div ID="log3581889" TYPE="section" LABEL="3.3.1 Accuracy of the approximation for general CKLS model" ORDER="1"/><mets:div ID="log3581892" TYPE="section" LABEL="3.3.2 Improvement of the approximation" ORDER="2"/></mets:div><mets:div ID="log3581894" TYPE="section" LABEL="3.4 Formulation of the optimization problems in the calibration algorithm" ORDER="4"/><mets:div ID="log3581895" TYPE="section" LABEL="3.5 The algorithm for estimating parameters in the CIR model with zero correlation" ORDER="5"><mets:div ID="log3581896" TYPE="section" LABEL="3.5.1 Simulated data" ORDER="1"/><mets:div ID="log3581898" TYPE="section" LABEL="3.5.2 Estimation of the European parameters" ORDER="2"/><mets:div ID="log3581900" TYPE="section" LABEL="3.5.3 Estimation of the domestic parameters" ORDER="3"/><mets:div ID="log3581902" TYPE="section" LABEL="3.5.4 Simulation analysis" ORDER="4"/></mets:div><mets:div ID="log3581904" TYPE="section" LABEL="3.6 Generalization for CKLS model with zero correlation and the known e, d" ORDER="6"/><mets:div ID="log3581906" TYPE="section" LABEL="3.7 Estimation of correlation  a parameters e, d" ORDER="7"/><mets:div ID="log3581907" TYPE="section" LABEL="3.8 Calibration of the model using real market data" ORDER="8"/></mets:div><mets:div ID="log3581911" TYPE="section" LABEL="4 Estimating the short rate from the term structures" ORDER="3"><mets:div ID="log3581913" TYPE="section" LABEL="4.1 Calibration procedure" ORDER="1"/><mets:div ID="log3581915" TYPE="section" LABEL="4.2 Application to simulated data" ORDER="2"/><mets:div ID="log3581918" TYPE="section" LABEL="4.3 Application to real market data" ORDER="3"><mets:div ID="log3581919" TYPE="section" LABEL="4.3.1 Comparison between estimated short rate and overnight" ORDER="1"/><mets:div ID="log3581923" TYPE="section" LABEL="4.3.2 Estimated short rates using different sets of maturities" ORDER="2"/></mets:div></mets:div><mets:div ID="log3581931" TYPE="section" LABEL="5 Short rate as a sum of two CKLS-type processes" ORDER="4"><mets:div ID="log3581932" TYPE="section" LABEL="5.1 Model" ORDER="1"/><mets:div ID="log3581934" TYPE="section" LABEL="5.2 Two-factor Vasicek model: singularity and transformation" ORDER="2"/><mets:div ID="log3581937" TYPE="section" LABEL="5.3 Application to real market data" ORDER="3"/><mets:div ID="log3581939" TYPE="section" LABEL="5.4 Robustness of the short rate estimates" ORDER="4"/><mets:div ID="log3581941" TYPE="section" LABEL="5.5 Approximation of the bond prices in the CKLS model" ORDER="5"/></mets:div><mets:div ID="log3581945" TYPE="section" LABEL="6 A three-factor convergence model of interest rates" ORDER="5"><mets:div ID="log3581947" TYPE="section" LABEL="6.1 Formulation of the model" ORDER="1"/><mets:div ID="log3581949" TYPE="section" LABEL="6.2 Bond prices" ORDER="2"><mets:div ID="log3581951" TYPE="section" LABEL="6.2.1 Vasicek and CIR type convergence models" ORDER="1"/><mets:div ID="log3581954" TYPE="section" LABEL="6.2.2 Analytical approximation formula for general convergence model" ORDER="2"/><mets:div ID="log3581956" TYPE="section" LABEL="6.2.3 Order of accuracy in the case of uncorrelated CIR model" ORDER="3"/></mets:div><mets:div ID="log3581958" TYPE="section" LABEL="6.3 Numerical experiment" ORDER="3"/></mets:div></mets:div><mets:div ID="log3581962" TYPE="section" LABEL="II Alternating direction explicit methods, Fichera theory and Trefftz methods" ORDER="9"><mets:div ID="log3581965" TYPE="section" LABEL="7 Intro to numerical solutions, ADE schemes, Fichera theory, option pricing" ORDER="1"><mets:div ID="log3581967" TYPE="section" LABEL="7.1 Proper treatment of boundary conditions, using Fichera theory" ORDER="1"/><mets:div ID="log3581969" TYPE="section" LABEL="7.2 Option pricing with Black-Scholes model" ORDER="2"><mets:div ID="log3581970" TYPE="section" LABEL="7.2.1 Multi-dimensional Black-Scholes models" ORDER="1"/></mets:div><mets:div ID="log3581972" TYPE="section" LABEL="7.3 Alternating Direction Explicit Schemes" ORDER="3"><mets:div ID="log3581974" TYPE="section" LABEL="7.3.1 The Idea of the ADE scheme" ORDER="1"/><mets:div ID="log3581977" TYPE="section" LABEL="7.3.2 Solving PDEs with the ADE method" ORDER="2"/></mets:div></mets:div><mets:div ID="log3581980" TYPE="section" LABEL="8 Fichera theory and its application to finance" ORDER="2"><mets:div ID="log3581981" TYPE="section" LABEL="8.1 The Boundary Value Problem for the Elliptic PDE" ORDER="1"/><mets:div ID="log3581983" TYPE="section" LABEL="8.2 Application to one-factor interest rate Models of CKLS type" ORDER="2"/><mets:div ID="log3581985" TYPE="section" LABEL="8.3 A two-factor interest rate Model" ORDER="3"/><mets:div ID="log3581988" TYPE="section" LABEL="8.4 Numerical Results" ORDER="4"/></mets:div><mets:div ID="log3581991" TYPE="section" LABEL="9 ADE methods for convection-diffusion-reactions Equations" ORDER="3"><mets:div ID="log3581992" TYPE="section" LABEL="9.0.1 The modified difference quotients for the ADE method" ORDER="1"/><mets:div ID="log3581995" TYPE="section" LABEL="9.1 Stability of the ADE method" ORDER="2"><mets:div ID="log3581996" TYPE="section" LABEL="9.1.1 Stability analysis using the Matrix approach" ORDER="1"/><mets:div ID="log3582001" TYPE="section" LABEL="9.1.2 Von Neumann stability analysis for the convection-diffusion-reaction equation" ORDER="2"/></mets:div><mets:div ID="log3582007" TYPE="section" LABEL="9.2 Consistency Analysis of the ADE methods" ORDER="3"><mets:div ID="log3582008" TYPE="section" LABEL="9.2.1 Consistency of the ADE scheme for convection-diffusion-reaction equations" ORDER="1"/><mets:div ID="log3582010" TYPE="section" LABEL="9.2.2 The Consistency of the ADE method for the linear BS model" ORDER="2"/><mets:div ID="log3582012" TYPE="section" LABEL="9.2.3 Application and numerical experiments with the linear model" ORDER="3"/></mets:div></mets:div><mets:div ID="log3582014" TYPE="section" LABEL="10 ADE method for higher dimensional Black-Scholes model" ORDER="4"><mets:div ID="log3582015" TYPE="section" LABEL="10.1 ADE Schemes for Multi-Dimensional Models" ORDER="1"><mets:div ID="log3582016" TYPE="section" LABEL="10.1.1 ADE Schemes for Two-dimensional Models" ORDER="1"/><mets:div ID="log3582019" TYPE="section" LABEL="10.1.2 ADE Schemes for Three and Higher Dimensional Models" ORDER="2"/><mets:div ID="log3582021" TYPE="section" LABEL="10.1.3 Boundary Conditions" ORDER="3"/></mets:div><mets:div ID="log3582023" TYPE="section" LABEL="10.2 Numerical Scheme" ORDER="2"><mets:div ID="log3582025" TYPE="section" LABEL="10.2.1 Algorithm of the Scheme" ORDER="1"/><mets:div ID="log3582027" TYPE="section" LABEL="10.2.2 Upward Finite Difference Quotients and Its Numerical Scheme" ORDER="2"/><mets:div ID="log3582029" TYPE="section" LABEL="10.2.3 Difference Quotients and Numerical Scheme for the Downward Sweep" ORDER="3"/></mets:div><mets:div ID="log3582031" TYPE="section" LABEL="10.3 Numerical Results and Experimental Study of Convergence" ORDER="3"><mets:div ID="log3582032" TYPE="section" LABEL="10.3.1 Two Dimensional Black-Scholes Model" ORDER="1"/><mets:div ID="log3582034" TYPE="section" LABEL="10.3.2 Three Dimensional Black-Scholes Model" ORDER="2"/></mets:div><mets:div ID="log3582037" TYPE="section" LABEL="10.4 Influence of dimensionality on computational complexity of the scheme" ORDER="4"/></mets:div><mets:div ID="log3582041" TYPE="section" LABEL="11 Trefftz methods for the Black-Scholes equation, FLAME" ORDER="5"><mets:div ID="log3582042" TYPE="section" LABEL="11.1 How Trefftz methods work?" ORDER="1"/><mets:div ID="log3582047" TYPE="section" LABEL="11.2 Numerical results with Six-Point FLAME Scheme" ORDER="2"/><mets:div ID="log3582049" TYPE="section" LABEL="11.3 Comparison of FLAME and Crank-Nicolson scheme" ORDER="3"/><mets:div ID="log3582050" TYPE="section" LABEL="11.4 Further potential of the Trefftz schemes" ORDER="4"/></mets:div><mets:div ID="log3582052" TYPE="section" LABEL="Conclusion and Outlook" ORDER="6"/></mets:div></mets:div></mets:structMap><mets:structLink><mets:smLink xlink:from="log3555185" xlink:to="physroot"/><mets:smLink xlink:from="log3555185" xlink:to="phys3581818"/><mets:smLink xlink:from="log3581819" xlink:to="phys3581820"/><mets:smLink xlink:from="log3581819" xlink:to="phys3581821"/><mets:smLink xlink:from="log3581819" xlink:to="phys3581822"/><mets:smLink xlink:from="log3581819" xlink:to="phys3581823"/><mets:smLink xlink:from="log3581819" xlink:to="phys3581824"/><mets:smLink xlink:from="log3581819" xlink:to="phys3581825"/><mets:smLink 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