{"product_id":"a-workout-in-computational-finance-with-website-isbn-9781119971917","title":"A Workout in Computational Finance, with Website","description":"\u003cb\u003eA comprehensive introduction to various numerical methods used in computational finance today\u003c\/b\u003e  \u003cp\u003eQuantitative skills are a prerequisite for anyone working in finance or beginning a career in the field, as well as risk managers. A thorough grounding in numerical methods is necessary, as is the ability to assess their quality, advantages, and limitations. This book offers a thorough introduction to each method, revealing the numerical traps that practitioners frequently fall into. Each method is referenced with practical, real-world examples in the areas of valuation, risk analysis, and calibration of specific financial instruments and models. It features a strong emphasis on robust schemes for the numerical treatment of problems within computational finance. Methods covered include PDE\/PIDE using finite differences or finite elements, fast and stable solvers for sparse grid systems, stabilization and regularization techniques for inverse problems resulting from the calibration of financial models to market data, Monte Carlo and Quasi Monte Carlo techniques for simulating high dimensional systems, and local and global optimization tools to solve the minimization problem.\u003c\/p\u003e \u003cp\u003eAcknowledgements xiii\u003c\/p\u003e \u003cp\u003eAbout the Authors xv\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction and Reading Guide 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Binomial Trees 7\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Equities and Basic Options 7\u003c\/p\u003e \u003cp\u003e2.2 The One Period Model 8\u003c\/p\u003e \u003cp\u003e2.3 The Multiperiod Binomial Model 9\u003c\/p\u003e \u003cp\u003e2.4 Black-Scholes and Trees 10\u003c\/p\u003e \u003cp\u003e2.5 Strengths and Weaknesses of Binomial Trees 12\u003c\/p\u003e \u003cp\u003e2.5.1 Ease of Implementation 12\u003c\/p\u003e \u003cp\u003e2.5.2 Oscillations 12\u003c\/p\u003e \u003cp\u003e2.5.3 Non-recombining Trees 14\u003c\/p\u003e \u003cp\u003e2.5.4 Exotic Options and Trees 14\u003c\/p\u003e \u003cp\u003e2.5.5 Greeks and Binomial Trees 15\u003c\/p\u003e \u003cp\u003e2.5.6 Grid Adaptivity and Trees 15\u003c\/p\u003e \u003cp\u003e2.6 Conclusion 16\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Finite Differences and the Black-Scholes PDE 17\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 A Continuous Time Model for Equity Prices 17\u003c\/p\u003e \u003cp\u003e3.2 Black-Scholes Model: From the SDE to the PDE 19\u003c\/p\u003e \u003cp\u003e3.3 Finite Differences 23\u003c\/p\u003e \u003cp\u003e3.4 Time Discretization 27\u003c\/p\u003e \u003cp\u003e3.5 Stability Considerations 30\u003c\/p\u003e \u003cp\u003e3.6 Finite Differences and the Heat Equation 30\u003c\/p\u003e \u003cp\u003e3.6.1 Numerical Results 34\u003c\/p\u003e \u003cp\u003e3.7 Appendix: Error Analysis 36\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Mean Reversion and Trinomial Trees 39\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Some Fixed Income Terms 39\u003c\/p\u003e \u003cp\u003e4.1.1 Interest Rates and Compounding 39\u003c\/p\u003e \u003cp\u003e4.1.2 Libor Rates and Vanilla Interest Rate Swaps 40\u003c\/p\u003e \u003cp\u003e4.2 Black76 for Caps and Swaptions 43\u003c\/p\u003e \u003cp\u003e4.3 One-Factor Short Rate Models 45\u003c\/p\u003e \u003cp\u003e4.3.1 Prominent Short Rate Models 45\u003c\/p\u003e \u003cp\u003e4.4 The Hull-White Model in More Detail 46\u003c\/p\u003e \u003cp\u003e4.5 Trinomial Trees 47\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Upwinding Techniques for Short Rate Models 55\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Derivation of a PDE for Short Rate Models 55\u003c\/p\u003e \u003cp\u003e5.2 Upwind Schemes 56\u003c\/p\u003e \u003cp\u003e5.2.1 Model Equation 57\u003c\/p\u003e \u003cp\u003e5.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model 63\u003c\/p\u003e \u003cp\u003e5.3.1 Bond Details 64\u003c\/p\u003e \u003cp\u003e5.3.2 Model Details 64\u003c\/p\u003e \u003cp\u003e5.3.3 Numerical Method 65\u003c\/p\u003e \u003cp\u003e5.3.4 An Algorithm in Pseudocode 68\u003c\/p\u003e \u003cp\u003e5.3.5 Results 69\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Boundary, Terminal and Interface Conditions and their Influence 71\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Terminal Conditions for Equity Options 71\u003c\/p\u003e \u003cp\u003e6.2 Terminal Conditions for Fixed Income Instruments 72\u003c\/p\u003e \u003cp\u003e6.3 Callability and Bermudan Options 74\u003c\/p\u003e \u003cp\u003e6.4 Dividends 74\u003c\/p\u003e \u003cp\u003e6.5 Snowballs and TARNs 75\u003c\/p\u003e \u003cp\u003e6.6 Boundary Conditions 77\u003c\/p\u003e \u003cp\u003e6.6.1 Double Barrier Options and Dirichlet Boundary Conditions 77\u003c\/p\u003e \u003cp\u003e6.6.2 Artificial Boundary Conditions and the Neumann Case 78\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Finite Element Methods 81\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Introduction 81\u003c\/p\u003e \u003cp\u003e7.1.1 Weighted Residual Methods 81\u003c\/p\u003e \u003cp\u003e7.1.2 Basic Steps 82\u003c\/p\u003e \u003cp\u003e7.2 Grid Generation 83\u003c\/p\u003e \u003cp\u003e7.3 Elements 85\u003c\/p\u003e \u003cp\u003e7.3.1 1D Elements 86\u003c\/p\u003e \u003cp\u003e7.3.2 2D Elements 88\u003c\/p\u003e \u003cp\u003e7.4 The Assembling Process 90\u003c\/p\u003e \u003cp\u003e7.4.1 Element Matrices 93\u003c\/p\u003e \u003cp\u003e7.4.2 Time Discretization 97\u003c\/p\u003e \u003cp\u003e7.4.3 Global Matrices 98\u003c\/p\u003e \u003cp\u003e7.4.4 Boundary Conditions 101\u003c\/p\u003e \u003cp\u003e7.4.5 Application of the Finite Element Method to Convection-Diffusion-Reaction Problems 103\u003c\/p\u003e \u003cp\u003e7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model 105\u003c\/p\u003e \u003cp\u003e7.6 Appendix: Higher Order Elements 107\u003c\/p\u003e \u003cp\u003e7.6.1 3D Elements 109\u003c\/p\u003e \u003cp\u003e7.6.2 Local and Natural Coordinates 111\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Solving Systems of Linear Equations 117\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Direct Methods 118\u003c\/p\u003e \u003cp\u003e8.1.1 Gaussian Elimination 118\u003c\/p\u003e \u003cp\u003e8.1.2 Thomas Algorithm 119\u003c\/p\u003e \u003cp\u003e8.1.3 LU Decomposition 120\u003c\/p\u003e \u003cp\u003e8.1.4 Cholesky Decomposition 121\u003c\/p\u003e \u003cp\u003e8.2 Iterative Solvers 122\u003c\/p\u003e \u003cp\u003e8.2.1 Matrix Decomposition 123\u003c\/p\u003e \u003cp\u003e8.2.2 Krylov Methods 125\u003c\/p\u003e \u003cp\u003e8.2.3 Multigrid Solvers 126\u003c\/p\u003e \u003cp\u003e8.2.4 Preconditioning 129\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Monte Carlo Simulation 133\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 The Principles of Monte Carlo Integration 133\u003c\/p\u003e \u003cp\u003e9.2 Pricing Derivatives with Monte Carlo Methods 134\u003c\/p\u003e \u003cp\u003e9.2.1 Discretizing the Stochastic Differential Equation 135\u003c\/p\u003e \u003cp\u003e9.2.2 Pricing Formalism 137\u003c\/p\u003e \u003cp\u003e9.2.3 Valuation of a Steepener under a Two Factor Hull-White Model 137\u003c\/p\u003e \u003cp\u003e9.3 An Introduction to the Libor Market Model 139\u003c\/p\u003e \u003cp\u003e9.4 Random Number Generation 146\u003c\/p\u003e \u003cp\u003e9.4.1 Properties of a Random Number Generator 147\u003c\/p\u003e \u003cp\u003e9.4.2 Uniform Variates 148\u003c\/p\u003e \u003cp\u003e9.4.3 Random Vectors 150\u003c\/p\u003e \u003cp\u003e9.4.4 Recent Developments in Random Number Generation 151\u003c\/p\u003e \u003cp\u003e9.4.5 Transforming Variables 152\u003c\/p\u003e \u003cp\u003e9.4.6 Random Number Generation for Commonly Used Distributions 155\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Advanced Monte Carlo Techniques 161\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Variance Reduction Techniques 161\u003c\/p\u003e \u003cp\u003e10.1.1 Antithetic Variates 161\u003c\/p\u003e \u003cp\u003e10.1.2 Control Variates 163\u003c\/p\u003e \u003cp\u003e10.1.3 Conditioning 166\u003c\/p\u003e \u003cp\u003e10.1.4 Additional Techniques for Variance Reduction 168\u003c\/p\u003e \u003cp\u003e10.2 Quasi Monte Carlo Method 169\u003c\/p\u003e \u003cp\u003e10.2.1 Low-Discrepancy Sequences 169\u003c\/p\u003e \u003cp\u003e10.2.2 Randomizing QMC 174\u003c\/p\u003e \u003cp\u003e10.3 Brownian Bridge Technique 175\u003c\/p\u003e \u003cp\u003e10.3.1 A Steepener under a Libor Market Model 177\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Valuation of Financial Instruments with Embedded American\/Bermudan Options within Monte Carlo Frameworks 179\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Pricing American options using the Longstaff and Schwartz algorithm 179\u003c\/p\u003e \u003cp\u003e11.2 A Modified Least Squares Monte Carlo Algorithm for Bermudan Callable Interest Rate Instruments 181\u003c\/p\u003e \u003cp\u003e11.2.1 Algorithm: Extended LSMC Method for Bermudan Options 182\u003c\/p\u003e \u003cp\u003e11.2.2 Notes on Basis Functions and Regression 185\u003c\/p\u003e \u003cp\u003e11.3 Examples 186\u003c\/p\u003e \u003cp\u003e11.3.1 A Bermudan Callable Floater under Different Short-rate Models 186\u003c\/p\u003e \u003cp\u003e11.3.2 A Bermudan Callable Steepener Swap under a Two Factor Hull-White Model 188\u003c\/p\u003e \u003cp\u003e11.3.3 A Bermudan Callable Steepener Cross Currency Swap in a 3D IR\/FX Model Framework 189\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Characteristic Function Methods for Option Pricing 193\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Equity Models 194\u003c\/p\u003e \u003cp\u003e12.1.1 Heston Model 196\u003c\/p\u003e \u003cp\u003e12.1.2 Jump Diffusion Models 198\u003c\/p\u003e \u003cp\u003e12.1.3 Infinite Activity Models 199\u003c\/p\u003e \u003cp\u003e12.1.4 Bates Model 200\u003c\/p\u003e \u003cp\u003e12.2 Fourier Techniques 201\u003c\/p\u003e \u003cp\u003e12.2.1 Fast Fourier Transform Methods 201\u003c\/p\u003e \u003cp\u003e12.2.2 Fourier-Cosine Expansion Methods 203\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Numerical Methods for the Solution of PIDEs 209\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 A PIDE for Jump Models 209\u003c\/p\u003e \u003cp\u003e13.2 Numerical Solution of the PIDE 210\u003c\/p\u003e \u003cp\u003e13.2.1 Discretization of the Spatial Domain 211\u003c\/p\u003e \u003cp\u003e13.2.2 Discretization of the Time Domain 211\u003c\/p\u003e \u003cp\u003e13.2.3 A European Option under the Kou Jump Diffusion Model 212\u003c\/p\u003e \u003cp\u003e13.3 Appendix: Numerical Integration via Newton-Cotes Formulae 214\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Copulas and the Pitfalls of Correlation 217\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Correlation 218\u003c\/p\u003e \u003cp\u003e14.1.1 Pearson’s ρ 218\u003c\/p\u003e \u003cp\u003e14.1.2 Spearman’s ρ 218\u003c\/p\u003e \u003cp\u003e14.1.3 Kendall’s τ 220\u003c\/p\u003e \u003cp\u003e14.1.4 Other Measures 221\u003c\/p\u003e \u003cp\u003e14.2 Copulas 221\u003c\/p\u003e \u003cp\u003e14.2.1 Basic Concepts 222\u003c\/p\u003e \u003cp\u003e14.2.2 Important Copula Functions 222\u003c\/p\u003e \u003cp\u003e14.2.3 Parameter estimation and sampling 229\u003c\/p\u003e \u003cp\u003e14.2.4 Default Probabilities for Credit Derivatives 234\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Parameter Calibration and Inverse Problems 239\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Implied Black-Scholes Volatilities 239\u003c\/p\u003e \u003cp\u003e15.2 Calibration Problems for Yield Curves 240\u003c\/p\u003e \u003cp\u003e15.3 Reversion Speed and Volatility 245\u003c\/p\u003e \u003cp\u003e15.4 Local Volatility 245\u003c\/p\u003e \u003cp\u003e15.4.1 Dupire’s Inversion Formula 246\u003c\/p\u003e \u003cp\u003e15.4.2 Identifying Local Volatility 246\u003c\/p\u003e \u003cp\u003e15.4.3 Results 247\u003c\/p\u003e \u003cp\u003e15.5 Identifying Parameters in Volatility Models 248\u003c\/p\u003e \u003cp\u003e15.5.1 Model Calibration for the FTSE- 100 249\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Optimization Techniques 253\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Model Calibration and Optimization 255\u003c\/p\u003e \u003cp\u003e16.1.1 Gradient-Based Algorithms for Nonlinear Least Squares Problems 256\u003c\/p\u003e \u003cp\u003e16.2 Heuristically Inspired Algorithms 258\u003c\/p\u003e \u003cp\u003e16.2.1 Simulated Annealing 259\u003c\/p\u003e \u003cp\u003e16.2.2 Differential Evolution 260\u003c\/p\u003e \u003cp\u003e16.3 A Hybrid Algorithm for Heston Model Calibration 261\u003c\/p\u003e \u003cp\u003e16.4 Portfolio Optimization 265\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Risk Management 269\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 Value at Risk and Expected Shortfall 269\u003c\/p\u003e \u003cp\u003e17.1.1 Parametric VaR 270\u003c\/p\u003e \u003cp\u003e17.1.2 Historical VaR 272\u003c\/p\u003e \u003cp\u003e17.1.3 Monte Carlo VaR 273\u003c\/p\u003e \u003cp\u003e17.1.4 Individual and Contribution VaR 274\u003c\/p\u003e \u003cp\u003e17.2 Principal Component Analysis 276\u003c\/p\u003e \u003cp\u003e17.2.1 Principal Component Analysis for Non-scalar Risk Factors 276\u003c\/p\u003e \u003cp\u003e17.2.2 Principal Components for Fast Valuation 277\u003c\/p\u003e \u003cp\u003e17.3 Extreme Value Theory 278\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 Quantitative Finance on Parallel Architectures 285\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 A Short Introduction to Parallel Computing 285\u003c\/p\u003e \u003cp\u003e18.2 Different Levels of Parallelization 288\u003c\/p\u003e \u003cp\u003e18.3 GPU Programming 288\u003c\/p\u003e \u003cp\u003e18.3.1 CUDA and OpenCL 289\u003c\/p\u003e \u003cp\u003e18.3.2 Memory 289\u003c\/p\u003e \u003cp\u003e18.4 Parallelization of Single Instrument Valuations using (Q)MC 290\u003c\/p\u003e \u003cp\u003e18.5 Parallelization of Hybrid Calibration Algorithms 291\u003c\/p\u003e \u003cp\u003e18.5.1 Implementation Details 292\u003c\/p\u003e \u003cp\u003e18.5.2 Results 295\u003c\/p\u003e \u003cp\u003e\u003cb\u003e19 Building Large Software Systems for the Financial Industry 297\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eBibliography 301\u003c\/p\u003e \u003cp\u003eIndex 307 \u003c\/p\u003e \u003cp\u003e\u003cb\u003eMICHAEL AICHINGER\u003c\/b\u003e obtained his Ph.D. in Theoretical Physics from the Johannes Kepler University Linz with a thesis on numerical methods in density functional theory and their application to 2D finite electron systems. A mobility grant led him to the Texas A\u0026amp;M University (2003) and to the Helsinki University of Technology (2004). In 2007 Michael Aichinger joined the Industrial Mathematics Competence Center where he has been working as a senior researcher and consultant in the field of quantitative finance for the last five years. He also works for the Austrian Academy of Sciences at the Radon Institute for Computational and Applied Mathematics where he is involved in several industrial mathematics and computational physics projects. Michael has (co-) authored around 20 journal articles in the fields of computational physics and quantitative finance.\u003c\/p\u003e \u003cp\u003e\u003cb\u003eANDREAS BINDER\u003c\/b\u003e obtained his Ph.D. in Industrial Mathematics from the Johannes Kepler University Linz with a thesis on continuous casting of steel. A research grant led him to the Oxford Center for Industrial and Applied Mathematics, UK, in 1991, where he got in touch with mathematical finance for the first time. After some years being an assistant professor at the Industrial Mathematics Institute, in 1996, he left university and became managing director of MathConsult GmbH, where he heads also the Computational Finance Group. Andreas has authored two introductory books on mathematical finance and 25 journal articles in the fields of industrial mathematics and of mathematical finance.\u003c\/p\u003e \u003cp\u003eQuantitative skills are a prerequisite for anyone looking to work in the finance industry today. Within the industry, any risk professional who wants to collaborate with, or work in most front office departments needs a thorough grounding in numerical methods, and the ability to assess their quality, their advantages and their limitations.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eA Workout in Computational Finance\u003c\/i\u003e delivers a profound and hands-on account of numerical methods used in modern quantitative finance, covering valuation and risk analysis of financial instruments from vanilla bonds to complex structures. The presented algorithms include, amongst others, tree methods, finite differences and finite elements, efficient Monte Carlo methods and Fourier techniques. Local and global optimisation techniques as well as stabilising regularisation methods for model calibration are thoroughly analysed.\u003c\/p\u003e \u003cp\u003eThe authors originate from the fields of theoretical physics and industrial mathematics, respectively, and have spent their professional careers creating efficient software solutions for producing industries and for financial industries. This book develops algorithms from the ground up, thus giving the reader a sound overview of their relative strengths and weaknesses. It is aimed at practitioners in the financial industry, for whom this is key knowledge in order to achieve optimal results with available data. It also enables junior quants with an IT background to implement numerical algorithms that work right away.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eA Workout in Computational Finance\u003c\/i\u003e is accompanied by a range of worked-out examples available from www.unrisk.com\/Workout.\u003c\/p\u003e  \u003cp\u003e“Mathematical Finance needs both: a well-founded theory based on stochastic calculus as well as numerical valuation schemes that work. In \u003ci\u003eA Workout in Computational Finance\u003c\/i\u003e the authors put emphasis on the numerical aspects and present an impressive range of numerical methods. All these techniques have been implemented by their group and can be used as a starting point for building a professional software system.”\u003cbr\u003e —\u003cb\u003e\u003ci\u003eWalter Schachermayer\u003c\/i\u003e, Full Professor for Mathematical Finance, University of Vienna\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e“With their strong background in numerical simulation of industrial problems, the authors succeed to develop the concepts of different numerical schemes which are useful for computational finance and essential for valuation, risk analysis and the risk management of financial instruments. Especially in times of difficult market environments, the mathematical and algorithmic foundation of software used in banking must be a solid one which avoids additional traps of poor implementation. \u003ci\u003eA Workout in Computational Finance\u003c\/i\u003e gives clear recommendations for the preferred numerical methods for various models and instruments. The book will be utmost useful for practitioners but it also will be of great interest for researchers in the field.”\u003cbr\u003e —\u003cb\u003e\u003ci\u003eGerhard Larcher\u003c\/i\u003e, Institute of Mathematical Finance, Kepler Universität\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e“The authors cover a broad range of numerical techniques for differential equations in computational finance, such as finite elements, trees, Monte Carlo, Fourier techniques and parameter calibration. Using sound, yet compact mathematical reasoning, they capture the substance of models for interest rate and equity derivatives, and provide hands-on guidance to numerics, covering all sorts of practical challenges. A vast number of numerical results illustrate potential implementation pitfalls and the mitigation techniques presented. With its strong focus on tangible usability this book is a highly valuable manual for students as well as professionals.”\u003cbr\u003e —\u003cb\u003e\u003ci\u003eRobert Maringer\u003c\/i\u003e, Head Valuation Control Switzerland, Credit Suisse\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e“For shaping your body you should go to a gym, while for building up your numerical toolkit you need a workout in computational finance. This modern treatment of numerical methods in quantitative finance addresses problems that professionals working in the field face on a daily basis. The very clear presentation of the material also makes it a perfect fit for students having a background in the theory of mathematical finance who want to gain insight on how practical problems are tackled in the industry.”\u003cbr\u003e —\u003cb\u003e\u003ci\u003ePhilipp Mayer\u003c\/i\u003e, Financial Modeling, ING Financial Markets, Brussels\u003c\/b\u003e\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47988647788773,"sku":"NP9781119971917","price":103.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119971917.jpg?v=1761781106","url":"https:\/\/k12savings.com\/es\/products\/a-workout-in-computational-finance-with-website-isbn-9781119971917","provider":"K12savings","version":"1.0","type":"link"}