CuuDuongThanCong.com Recent Developments in Computational Finance Foundations, Algorithms and Applications 8636hc_9789814436427_tp.indd CuuDuongThanCong.com 29/10/12 5:26 PM INTERDISCIPLINARY MATHEMATICAL SCIENCES* Series Editor: Jinqiao Duan (University of California, Los Angeles, USA) Editorial Board: Ludwig Arnold, Roberto Camassa, Peter Constantin, Charles Doering, Paul Fischer, Andrei V Fursikov, Xiaofan Li, Sergey V Lototsky, Fred R McMorris, Daniel Schertzer, Bjorn Schmalfuss, Yuefei Wang, Xiangdong Ye, and Jerzy Zabczyk Published Vol 4: Mathematical Theory of Adaptive Control Vladimir G Sragovich Vol 5: The Hilbert–Huang Transform and Its Applications eds Norden E Huang & Samuel S P Shen Vol 6: Meshfree Approximation Methods with MATLAB Gregory E Fasshauer Vol 7: Variational Methods for Strongly Indefinite Problems Yanheng Ding Vol 8: Recent Development in Stochastic Dynamics and Stochastic Analysis eds Jinqiao Duan, Shunlong Luo & Caishi Wang Vol 9: Perspectives in Mathematical Sciences eds Yisong Yang, Xinchu Fu & Jinqiao Duan Vol 10: Ordinal and Relational Clustering (with CD-ROM) Melvin F Janowitz Vol 11: Advances in Interdisciplinary Applied Discrete Mathematics eds Hemanshu Kaul & Henry Martyn Mulder Vol 12: New Trends in Stochastic Analysis and Related Topics: A Volume in Honour of Professor K D Elworthy eds Huaizhong Zhao & Aubrey Truman Vol 13: Stochastic Analysis and Applications to Finance: Essays in Honour of Jia-an Yan eds Tusheng Zhang & Xunyu Zhou Vol 14 Recent Developments in Computational Finance: Foundations, Algorithms and Applications eds Thomas Gerstner & Peter Kloeden *For the complete list of titles in this series, please go to http://www.worldscientific.com/series/ims RokTing - Recent Dev in Computational Finance.pmd CuuDuongThanCong.com 11/2/2012, 3:30 PM Interdisciplinary Mathematical Sciences – Vol 14 Recent Developments in Computational Finance Foundations, Algorithms and Applications Editors Thomas Gerstner Goethe University Frankfurt am Main, Germany Peter Kloeden Goethe University Frankfurt am Main, Germany World Scientific NEW JERSEY • 8636hc_9789814436427_tp.indd CuuDuongThanCong.com LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI 29/10/12 5:26 PM Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Interdisciplinary Mathematical Sciences — Vol 14 RECENT DEVELOPMENTS IN COMPUTATIONAL FINANCE Foundations, Algorithms and Applications Copyright © 2013 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN 978-981-4436-42-7 Printed in Singapore RokTing - Recent Dev in Computational Finance.pmd CuuDuongThanCong.com 11/2/2012, 3:30 PM September 27, 2012 11:6 World Scientific Review Volume - 9.75in x 6.5in Preface Mathematical finance has revolutionized the financial world in the past forty years A major reason for this success has been the parallel development of efficient computational methods as well as more sophisticated mathematical models These novel computational tools are the foundation of a new field of research called Computational Finance, whose main task is to calculate as accurately and efficiently as possible the risks that financial instruments generate This requires an interdisciplinary approach involving a variety of methods from financial mathematics, stochastics, statistics, numerics and scientific computing Major impacts on the development of the field were the publication of the monograph of Kloeden and Platen on stochastic numerics in 1992 and, later, the monograph of Glasserman on Monte Carlo methods in 2004 These books, as well as many others, provide the foundations of this rapidly developing subject The new computational tools have led to even more sophisticated mathematical models, which in turn require computational methods that work under requirements not handled in the existing textbooks For example, the theory of stochastic numerics has now been extended to handle non-standard assumptions on the coefficients of the stochastic differential equations Another significant new development is the multi-level Monte Carlo method of Michael Giles, while others include the use of inverse problem methods, wavelets and backward stochastic differential equations This volume consists of a series of cutting-edge surveys of recent developments in the field of computational finance written by leading international experts Several of the contributions in this volume are based on talks presented at the International Workshop on Numerical Algorithms in Computational Finance that was held from July 20-22 in 2011 at the House of Finance of the Goethe University in Frankfurt am Main These surveys make the subject accessible to a wide readership in academia and the financial world They may also be of interest to practitioners in many areas in engineering, technology and science beyond finance Besides reviews of existing results many new, previously unpublished, results are also presented The book consists of 13 chapters divided into the three parts: Foundations, Algorithms and Applications The first part Foundations is devoted to survey and review articles It begins with the article Multilevel Monte Carlo methods for applications in finance by Mike Giles and Lukasz Szpruch which presents a survey of recent progress regarding the v CuuDuongThanCong.com ws-rv975x65 September 27, 2012 11:6 vi World Scientific Review Volume - 9.75in x 6.5in Preface multilevel Monte Carlo method The chapter Convergence of numerical methods for SDEs in finance by Peter Kloeden and Andreas Neuenkirch deals with nonstandard assumptions on the coefficients of an SDEs and the effect on the convergence of numerical discretization schemes In Inverse problems in finance, Johann Baumeister gives an overview on inverse problems in finance, which are in general ill-posed and thus require special regularization The article Asymptotic and non asymptotic approximations for option valuation by Roman Bompis and Emmanuel Gobet reviews approximation methods for the derivation of closed-form solutions for option pricing problems The second part Algorithms covers the algorithmic and numerical aspects of the computational methods The article Discretization of backward stochastic Volterra integral equations by Christian Bender and Stanislav Pokalyuk deals with the approximation of backward SDEs, while the next chapter Semi-Lagrangian schemes for parabolic equations by Kristian Debrabant and Espen Robstad Jakobsen covers numerical schemes for nonlinear second order parabolic PDEs In Derivativefree weak approximation methods for stochastic differential equations Kristian Debraband and Andreas Rưßler consider stochastic Runge-Kutta methods for the weak approximation of SDEs and in the chapter Wavelet solution of degenerate Kolmogoroff forward equations Oleg Reichmann and Christoph Schwab review wavelet Galerkin discretizations for Kolmogoroff forward pricing equations Finally, in Randomized multilevel quasi-Monte Carlo path simulation, Thomas Gerstner and Marco Noll combine the multilevel Monte Carlo method with quasi-random number generation The third part Applications then deals with specific financial problems The article Drift-free simulation methods for pricing cross-market derivatives with LMM by José Luis Fernández Pérez, María Rodríguez Nogueiras, Marta Pou Bueno and Carlos Vázquez considers a simulation approach for libor rates, which avoids driftdependent paths In Application of simplest random walk algorithms for pricing barrier options by Maria Krivko and Michael V Tretyakov proposes a special discretization for barrier options close to the barrier In the article Coupling local currency Libor models to FX Libor models, John Schoenmakers focuses on the coupling of single currency libor models into a joint libor model The final chaper Dimensionwise decompositions and their efficient parallelization by Philipp Schröder, Peter Mlynczak and Gabriel Wittum tackles the pricing of high-dimensional basket options on parallel computers We would like to thank the referees for their valuable comments on the submissions We also thank Marco Noll for his tireless work to bring the manuscript into publishable format Finally, and most of all, we would like to thank the authors for their informative contributions Thomas Gerstner and Peter Kloeden CuuDuongThanCong.com Frankfurt am Main, 2012 ws-rv975x65 September 27, 2012 11:6 World Scientific Review Volume - 9.75in x 6.5in ws-rv975x65 Contents Preface v Foundations 1 Multilevel Monte Carlo methods for applications in finance Mike Giles and Lukasz Szpruch Convergence of numerical methods for SDEs in finance 49 Peter Kloeden and Andreas Neuenkirch Inverse problems in finance 81 J Baumeister Asymptotic and non asymptotic approximations for option valuation 159 R Bompis and E Gobet Algorithms 243 245 Discretization of backward stochastic Volterra integral equations Christian Bender and Stanislav Pokalyuk Semi-Lagrangian schemes for parabolic equations 279 Kristian Debrabant and Espen Robstad Jakobsen Derivative-free weak approximation methods for stochastic differential equations Kristian Debrabant and Andreas Rưßler vii CuuDuongThanCong.com 299 September 27, 2012 11:6 World Scientific Review Volume - 9.75in x 6.5in viii ws-rv975x65 Contents Wavelet solution of degenerate Kolmogoroff forward equations 317 Oleg Reichmann and Christoph Schwab Randomized multilevel quasi-Monte Carlo path simulation 349 Thomas Gerstner and Marco Noll Applications 371 10 Drift-Free Simulation methods for pricing cross-market derivatives with LMM 373 J.L Fernández, M.R Nogueiras, M Pou and C Vázquez 11 Application of simplest random walk algorithms for pricing barrier options 407 M Krivko and M.V Tretyakov 12 Coupling local currency Libor models to FX Libor models 429 John Schoenmakers 13 Dimension-wise decompositions and their efficient parallelization Philipp Schröder, Peter Mlynczak and Gabriel Wittum CuuDuongThanCong.com 445 September 27, 2012 11:6 World Scientific Review Volume - 9.75in x 6.5in PART Foundations CuuDuongThanCong.com ws-rv975x65 September 27, 2012 11:6 World Scientific Review Volume - 9.75in x 6.5in 458 ws-rv975x65 Philipp Schröder, Peter Mlynczak and Gabriel Wittum 4.1.2 Anchor ANOVA decomposition Due to its complexity advantages, the Anchor ANOVA decomposition is commonly used when handling high dimensional problems It can be constructed by choosing the Dirac measure µ(xi ) = δ(xi −ai )dxi as probability measure as well as Ωi = [0, 1], where a ∈ Ωd designates the so called anchor point which gives the decomposition its name The Anchor ANOVA decomposition can be constructed through the following projections: Pu f (xu ) = f (x)δ(x − a)dxD\u (56) [0,1]d−|u| = f (x)|x=a\xu (57) Here f (x)|x=a\xu means that xi = ∀i ∈ / u, (58) where fu can be understood as behavior of f (x) in the hyperplane through a In this context the decomposition is also named CUT-HDMR in [13] [7] provides a good summary of the methodology and some generalizations, as well as a link between the Anchor ANOVA decomposition and the multivariate Taylor-expansion For a better understanding of the effects of the Anchor ANOVA’s projections, we show the one- and two-dimensional projections of a simple forward contract as well as a geometric basket option The one-dimensional projections are shown in figure 5, while the two-dimensional projections f12 (x) are shown in figure 0.015 0.4 0.010 0.3 0.005 1 0.2 0.005 0.1 0.010 0.015 (a) Forward Figure 1 (b) Basket option One-dimensional projections of the Anchor ANOVA decomposition For the basket option, in both the one-dimensional and the two-dimensional case, one can clearly observe the non-smoothing effect of the Anchor ANOVA CuuDuongThanCong.com September 27, 2012 11:6 World Scientific Review Volume - 9.75in x 6.5in Dimension-wise decompositions and their efficient parallelization (a) Forward Figure ws-rv975x65 459 (b) Basket option Two-dimensional projections of the Anchor ANOVA decomposition Remark 13.2 (Anchor ANOVA with normal distribution) By choosing Ω = Rd and dµ(x) = δ(x − a)ϕd,λ (x)dx with ϕd,λ (x) as the multidimensional normal distribution d √ e− i=1 xi /2λi , (59) ϕd,λ (x) := 2πλi i we define the projections Pu f (xu ) = (f (x)ϕd−|u|,λ (xu )) x=a\xu , (60) which result in the Anchor ANOVA with the normal distribution 4.2 Taylor-like ANOVA decomposition Taking a closer look at the variance distribution of the individual anchor ANOVA projections, it becomes evident that not all the two-dimensional terms are of equal importance Therefore we propose an approximation to the Standard ANOVA decomposition Taking only some terms of the Standard ANOVA decomposition into account we define the following approximation, which we will call Taylor-like ANOVA expansion: Proposition 13.1 (Taylor-like ANOVA expansion of first order) Let vu := Ifu , (61) where fu denotes the ANOVA term resulting from an ANOVA decomposition of the payoff-function g(x) and If := f (x)dx (62) Ω denotes the integral operator Then d u(x, t) ≈ v0 + i=1 CuuDuongThanCong.com d vi + v1,i i=2 (63) September 27, 2012 11:6 460 World Scientific Review Volume - 9.75in x 6.5in ws-rv975x65 Philipp Schröder, Peter Mlynczak and Gabriel Wittum represents a Taylor-like approximation to the Standard ANOVA decomposition As we will see later this is, indeed a very good approximation to the full solution Compared to the ordinary ANOVA expansion, the Taylor-like ANOVA expansion has a clear complexity advantage since it depends only linearly on the dimension of the full problem Furthermore, as we will see later, it also has a precision advantage compared to the ordinary ANOVA expansion of similar dimension The complexity advantage is even greater if we look at equation (63) in terms of the solutions u(n,j) (x, t) and not the individual ANOVA terms vu 4.3 Taylor-like ANOVA expansion of higher order Two obvious possibilities exist to generalize the Taylor-like ANOVA expansion to expansions of higher order: i) A Taylor approximation of higher order, e.g 2nd order; ii) a Taylor approximation of first order around a higher dimensional (e.g 2D) problem Proposition 13.2 (Taylor approximation of first order around a higher dimensional problem) Beginning from d u(x, t) ≈ u(2) (x, t) + u(2,j) (x, t) − u(2) (x, t) , (64) j=3 we again substitute the corresponding ANOVA terms and obtain d d d i=1 (v2,i + v1,2,i ) v1,i + vi + u(x, t) ≈ v0 + i=2 (65) i=3 This formulation is still linear in the dimension d of the full problem The complexity advantage of a formulation in terms of the solutions u(n) (x, t) is even clearer here While in the formulation using the numerical solutions u(n) (x, t) still one n-dimensional and d − n (n+1) dimensional problems need to be solved In terms of a formulation in ANOVA terms, this corresponds to d one-dimensional, (2d − 3) two-dimensional and (d − 2) three-dimensional problems Proposition 13.3 (Taylor approximation of second order) If we take higher order Taylor approximations into consideration, in the case of a second order approximation we get d u(x, t) = u(n) (x, t) + λj j=n+1 ∂u (x, t) ∂λj (66) λ=λ(n) d ∂2u + (x, t) λi λj i,j=n+1 ∂λi ∂λj CuuDuongThanCong.com λ=λ(n) +O ||λ − λ(n) ||4 (67) September 27, 2012 11:6 World Scientific Review Volume - 9.75in x 6.5in Dimension-wise decompositions and their efficient parallelization ws-rv975x65 461 We will approximate the second derivative by difference quotients using central differences: ∂2u (x, t) ∂λi ∂λj λ=λ ≈ (n) u(n,i,j) (x, t) − u(n,i) (x, t) − u(n,j) (x, t) + u(n) (x, t) , λi λj (68) and end up with the following approximation for n=1: d (u(1,i) (x, t) − u(1) (x, t)) u(x, t) ≈ u(1) (x, t) + i=2 d + (u(1,i,j) (x, t) − u(1,i) (x, t) − u(1,j) (x, t) + u(1) (x, t)) i,j=2 d d 63 v1,i vi + = v0 + i=2 i=1 d + (u(1,i,j) (x, t) − u(1,i) (x, t) − u(1,j) (x, t) + u(1) (x, t)), i,j=2 which can be written as d d d v1,i + vi + u(x, t) ≈ v0 + i=2 i=1 (v1ij + vij ), (69) i=2 i= range[i].min()) { mpiProcess.setGroupID(i); mpiProcess.setComm(i); } } This way, e.g., eight processes may be split up into two groups using a twodimensional Array containing the intervals [0;3] and [4;7] Solving the scheduled problems also can be realized locally by looping over all solver objects Every process checks if its group ID matches the group ID of the current solver object In this case it triggers the calculation of the solution by calling the solve() method of the solver object Thus every process only solves problems assigned to its group ID For the rest of this section, we will focus on the operation principle of the MPI Scheduler Figure 12 illustrates the functioning of the MPI Scheduler In the initial situation all solver objects as well as the MPI Scheduler object are created in every process The MPI Scheduler is instantiated with references on the solver objects (figure 12a).∗ As explained in example 5.3.3, the MPI Scheduler object firstly groups the processes by local decision and determines Master processes within each group (figure 12b) Calling schedule(Method) results in a mapping of groups to solver objects For this the MPI Scheduler sets the process group ID to that solver’s group ID attribute that will calculate the solution to the problem associated with the solver object (figure 12c) Subsequently the MPI Scheduler initializes the calculations of the solutions in every group by executing solveScheduled() (figure 12d) If sparse grid parallelization is used within the groups, the sparse grids will now be assigned to the processes in the respective group and the partial differential equation will be solved (figure 12e) Finally, figure 12f depicts how the MPI Scheduler gathers the sparse grid solutions from the group Masters on the overall Master using gatherSolutions(Array).† When combining MPI Scheduler and sparse grid parallelization it might happen that there are fewer CPUs than “number of sparse grids × number of groups” In this case one can either overcommit the cores, i.e., assign more than one MPI process to one CPU, or reduce the number of processes per group As described in section 5.2 the sparse grid parallelization then assigns multiple grids to one process Generally, the latter would be faster in terms of runtime as it saves communication overhead between the processes ∗ For simplicity, we set one process equal to one CPU solution Array is created on every process and the MPI Scheduler must ensure that it copies the correct Vectors to the overall Master’s Array † The CuuDuongThanCong.com September 27, 2012 11:6 World Scientific Review Volume - 9.75in x 6.5in 470 ws-rv975x65 Philipp Schröder, Peter Mlynczak and Gabriel Wittum mpiScheduler: MPIScheduler theProblems: [*solver1, , *solver4] createGroups(simple) schedule(simple) solveScheduled() gatherSolutions(solution) solver4: SGSolver solver3: SGSolver groupID: int="4" timeStep: solver2: SGSolver = "&ts" groupID: TimeSteppingScheme int="3" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme groupID: int="2" solver1: SGSolver = "&ts" discret: Discretization = "&disc" = "&ts" solver: LinSolver timeStep: TimeSteppingScheme groupID: int=""= "&solver" gridTable: SparseGridTable = "&sgTable" discret: Discretization = "&disc" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme = "&ts" gridTable: SparseGridTable = "&sgTable" Solve() discret: = "&disc" solver: Discretization LinSolver = "&solver" getSolution() gridTable: SparseGridTable = "&sgTable" Solve() discret: Discretization = "&disc" getSolution() Solve() gridTable: SparseGridTable = "&sgTable" getSolution() solve() getSolution() CPU CPU CPU CPU CPU CPU CPU CPU (a) Initial situation mpiScheduler: MPIScheduler theProblems: [*solver1, , *solver4] createGroups(simple) schedule(simple) solveScheduled() gatherSolutions(solution) solver4: SGSolver solver3: SGSolver groupID: int="4" timeStep: solver2: SGSolver = "&ts" groupID: TimeSteppingScheme int="3" solver: LinSolver = "&solver" timeStep: groupID: TimeSteppingScheme int="2" solver1: SGSolver = "&ts" discret: Discretization = "&disc" = "&ts" solver: LinSolver timeStep: TimeSteppingScheme groupID: int=""= "&solver" gridTable: SparseGridTable = "&sgTable" discret: Discretization = "&disc" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme = "&ts" gridTable: SparseGridTable = "&sgTable" Solve() discret: Discretization = "&disc" solver: LinSolver = "&solver" getSolution() gridTable: SparseGridTable = "&sgTable" Solve() discret: Discretization = "&disc" getSolution() Solve() gridTable: SparseGridTable = "&sgTable" getSolution() solve() getSolution() Groups processes CPU CPU CPU CPU CPU Overall and group Master CPU CPU CPU CPU CPU Group Master (b) Grouping mpiScheduler: MPIScheduler theProblems: [*solver1, , *solver4] createGroups(simple) schedule(simple) solveScheduled() gatherSolutions(solution) CPU solver4: SGSolver solver3: SGSolver groupID: int="4" timeStep: solver2: SGSolver = "&ts" groupID: TimeSteppingScheme int="3" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme = "&ts" groupID: int="2" solver1: SGSolver discret: Discretization = "&disc" solver: LinSolver = "&solver" timeStep: = "&ts" groupID: TimeSteppingScheme int="1" gridTable: SparseGridTable = "&sgTable" discret: Discretization = "&disc" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme = "&ts" gridTable: SparseGridTable = "&sgTable" Solve() discret: = "&disc" solver: Discretization LinSolver = "&solver" getSolution() gridTable: SparseGridTable = "&sgTable" Solve() discret: Discretization = "&disc" getSolution() Solve() gridTable: SparseGridTable = "&sgTable" getSolution() solve() getSolution() Sets group IDs CPU CPU CPU Overall and group Master CPU CPU CPU CPU CPU CPU Group Master (c) Scheduling Figure 12 CuuDuongThanCong.com Interaction of the MPI Scheduler and the solver objects September 27, 2012 11:6 World Scientific Review Volume - 9.75in x 6.5in ws-rv975x65 Dimension-wise decompositions and their efficient parallelization mpiScheduler: MPIScheduler theProblems: [*solver1, , *solver4] createGroups(simple) schedule(simple) solveScheduled() gatherSolutions(solution) solver4: SGSolver solver3: SGSolver groupID: int="4" timeStep: solver2: SGSolver = "&ts" groupID: TimeSteppingScheme int="3" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme groupID: int="2" solver1: SGSolver = "&ts" discret: Discretization = "&disc" = "&ts" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme groupID: int="1" gridTable: SparseGridTable = "&sgTable" discret: Discretization = "&disc" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme = "&ts" gridTable: SparseGridTable = "&sgTable" Solve() discret: = "&disc" solver: Discretization LinSolver = "&solver" getSolution() gridTable: SparseGridTable = "&sgTable" Solve() discret: Discretization = "&disc" getSolution() Solve() gridTable: SparseGridTable = "&sgTable" getSolution() solve() getSolution() Calls solver CPU CPU CPU CPU CPU Overall and group Master 471 CPU CPU CPU CPU CPU Group Master (d) Initializing mpiScheduler: MPIScheduler theProblems: [*solver1, , *solver4] createGroups(simple) schedule(simple) solveScheduled() gatherSolutions(solution) solver4: SGSolver solver3: SGSolver groupID: int="4" timeStep: solver2: SGSolver = "&ts" groupID: TimeSteppingScheme int="3" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme groupID: int="2" solver1: SGSolver = "&ts" discret: Discretization = "&disc" = "&ts" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme groupID: int="1" gridTable: SparseGridTable = "&sgTable" discret: Discretization = "&disc" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme = "&ts" gridTable: SparseGridTable = "&sgTable" Solve() discret: = "&disc" solver: Discretization LinSolver = "&solver" getSolution() gridTable: SparseGridTable = "&sgTable" Solve() discret: Discretization = "&disc" getSolution() Solve() gridTable: SparseGridTable = "&sgTable" getSolution() solve() getSolution() Every process group solves the problem with their respective group ID CPU CPU CPU CPU CPU Overall and group Master Sparse grid parallelization within the groups CPU CPU CPU CPU CPU Group Master (e) Solving mpiScheduler: MPIScheduler theProblems: [*solver1, , *solver4] createGroups(simple) schedule(simple) solveScheduled() gatherSolutions(solution) solver4: SGSolver solver3: SGSolver groupID: int="4" timeStep: solver2: SGSolver = "&ts" groupID: TimeSteppingScheme int="3" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme groupID: int="2" solver1: SGSolver = "&ts" discret: Discretization = "&disc" = "&ts" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme groupID: int="1" gridTable: SparseGridTable = "&sgTable" discret: Discretization = "&disc" solver: LinSolver = "&solver" timeStep: TimeSteppingScheme = "&ts" gridTable: SparseGridTable = "&sgTable" Solve() discret: = "&disc" solver: Discretization LinSolver = "&solver" getSolution() gridTable: SparseGridTable = "&sgTable" Solve() discret: Discretization = "&disc" getSolution() Solve() gridTable: SparseGridTable = "&sgTable" getSolution() solve() getSolution() Gathers the solutions on the overall Master CPU CPU CPU CPU Overall and group Master CPU CPU CPU CPU CPU CPU Group Master (f) Gathering Figure 12 CuuDuongThanCong.com Interaction of the MPI Scheduler and the solver objects September 27, 2012 11:6 472 World Scientific Review Volume - 9.75in x 6.5in Philipp Schröder, Peter Mlynczak and Gabriel Wittum Bibliography [1] Beylkin, G and Mohlenkamp, M.J: Numerical operator calculus in higher dimensions Proceedings of the National Academy of Sciences, 99(16):10246, 2002 [2] Beylkin, G and Mohlenkamp, M.J.: Algorithms for numerical analysis in high dimensions SIAM Journal on Scientific Computing, 26(6):2133–2159, 2005 [3] Blackham, J.: Sparse grid solutions to the libor market model Master’s thesis, University of Oxford, 2004 [4] Brace, A., Gatarek, D and Musiela, M.: The market model of interest rate dynamics Mathematical finance, 7(2):127–155, 1997 [5] Gerstner, T and Griebel, M.: Dimension–adaptive tensor–product quadrature Computing, 71(1):65–87, 2003 [6] Goethe Center for Scientific Computing (G-CSC) Numerical Simulation Software Library SG http://sg2.sourceforge.net/ [7] Griebel, M.: Sparse grids and related approximation schemes for higher dimensional problems Foundations of Computational Mathematics (FoCM05), 2005 [8] Griebel, M., Schneider, M and Zenger, C.: A combination technique for the solution of sparse grid problems Elsevier, 1992 [9] Hackbusch, W.: A sparse matrix arithmetic based on h-matrices part i: Introduction to-matrices Computing, 62(2):89–108, 1999 [10] Hackbusch, W., Khoromskij, B.N and Tyrtyšnikov, E.E.: Hierarchical kronecker tensor product approximations J Numer Math, 13:119–156, 2005 [11] Hackbusch, W.: Iterative Lưsung grer schwachbesetzter Gleichungssysteme Teubner Verlag, 1993 [12] Message Passing Interface Forum MPI: A Message-Passing Interface Standard Version 2.2, Message Passing Interface Forum, September 2009 [13] Rabitz, H and Aliş, O.F.: General foundations of high-dimensional model representations Journal of Mathematical Chemistry, 25(2):197–233, 1999 [14] Reisinger, C.: Analysis of linear difference schemes in the sparse grid combination technique Arxiv preprint arXiv:0710.0491, 2007 [15] Reisinger, C.: Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben PhD thesis, Universität Heidelberg, 2004 [16] Sergeev, V.: Sparse grid method in the libor market model Master’s thesis, Universiteit van Amsterdam, 2006 [17] SA Smolyak: Quadrature and interpolation formulas for tensor products of certain classes of functions In Dokl Akad Nauk SSSR, volume 4, page 123, 1963 CuuDuongThanCong.com ws-rv975x65 ... http://www.worldscientific.com/series/ims RokTing - Recent Dev in Computational Finance. pmd CuuDuongThanCong.com 11/2/ 2012 , 3:30 PM Interdisciplinary Mathematical Sciences – Vol 14 Recent Developments in Computational Finance Foundations,. .. Library Interdisciplinary Mathematical Sciences — Vol 14 RECENT DEVELOPMENTS IN COMPUTATIONAL FINANCE Foundations, Algorithms and Applications Copyright © 2013 by World Scientific Publishing Co.. .Recent Developments in Computational Finance Foundations, Algorithms and Applications 8636hc_9789814436427_tp.indd CuuDuongThanCong.com 29/10/12 5:26 PM INTERDISCIPLINARY MATHEMATICAL