Machine Learning for Risk Calculations: A Practitioner's View

ISBN-13: 9781119791386 / Angielski / Twarda / 2021 / 464 str.

Mariano Zeron; Ignacio Ruiz

Spis treści

Acknowledgements xviiForeword xxiMotivation and aim of this book xxiiiPart One Fundamental Approximation MethodsChapter 1 Machine Learning 31.1 Introduction to Machine Learning 31.1.1 A brief history of Machine Learning Methods 41.1.2 Main sub-categories in Machine Learning 51.1.3 Applications of interest 71.2 The Linear Model 71.2.1 General concepts 81.2.2 The standard linear model 121.3 Training and predicting 151.3.1 The frequentist approach 181.3.2 The Bayesian approach 211.3.3 Testing--in search of consistent accurate predictions 251.3.4 Underfitting and overfitting 251.3.5 K-fold cross-validation 271.4 Model complexity 281.4.1 Regularisation 291.4.2 Cross-validation for regularisation 311.4.3 Hyper-parameter optimisation 33Chapter 2 Deep Neural Nets 392.1 A brief history of Deep Neural Nets 392.2 The basic Deep Neural Net model 412.2.1 Single neuron 412.2.2 Artificial Neural Net 432.2.3 Deep Neural Net 462.3 Universal Approximation Theorems 482.4 Training of Deep Neural Nets 492.4.1 Backpropagation 502.4.2 Backpropagation example 512.4.3 Optimisation of cost function 552.4.4 Stochastic gradient descent 572.4.5 Extensions of stochastic gradient descent 582.5 More sophisticated DNNs 592.5.1 Convolution Neural Nets 592.5.2 Other famous architectures 632.6 Summary of chapter 64Chapter 3 Chebyshev Tensors 653.1 Approximating functions with polynomials 653.2 Chebyshev Series 663.2.1 Lipschitz continuity and Chebyshev projections 673.2.2 Smooth functions and Chebyshev projections 703.2.3 Analytic functions and Chebyshev projections 703.3 Chebyshev Tensors and interpolants 723.3.1 Tensors and polynomial interpolants 723.3.2 Misconception over polynomial interpolation 733.3.3 Chebyshev points 743.3.4 Chebyshev interpolants 763.3.5 Aliasing phenomenon 773.3.6 Convergence rates of Chebyshev interpolants 773.3.7 High-dimensional Chebyshev interpolants 793.4 Ex ante error estimation 823.5 What makes Chebyshev points unique 853.6 Evaluation of Chebyshev interpolants 893.6.1 Clenshaw algorithm 903.6.2 Barycentric interpolation formula 913.6.3 Evaluating high-dimensional tensors 933.6.4 Example of numerical stability 943.7 Derivative approximation 953.7.1 Convergence of Chebyshev derivatives 953.7.2 Computation of Chebyshev derivatives 963.7.3 Derivatives in high dimensions 973.8 Chebyshev Splines 993.8.1 Gibbs phenomenon 993.8.2 Splines 1003.8.3 Splines of Chebyshev 1013.8.4 Chebyshev Splines in high dimensions 1013.9 Algebraic operations with Chebyshev Tensors 1013.10 Chebyshev Tensors and Machine Learning 1033.11 Summary of chapter 104Part Two The toolkit -- plugging in approximation methodsChapter 4 Introduction: why is a toolkit needed 1074.1 The pricing problem 1074.2 Risk calculation with proxy pricing 1094.3 The curse of dimensionality 1104.4 The techniques in the toolkit 112Chapter 5 Composition techniques 1135.1 Leveraging from existing parametrisations 1145.1.1 Risk factor generating models 1145.1.2 Pricing functions and model risk factors 1155.1.3 The tool obtained 1165.2 Creating a parametrisation 1175.2.1 Principal Component Analysis 1175.2.2 Autoencoders 1195.3 Summary of chapter 120Chapter 6 Tensors in TT format and Tensor Extension Algorithms 1236.1 Tensors in TT format 1236.1.1 Motivating example 1246.1.2 General case 1246.1.3 Basic operations 1266.1.4 Evaluation of Chebyshev Tensors in TT format 1276.2 Tensor Extension Algorithms 1296.3 Step 1--Optimising over tensors of fixed rank 1296.3.1 The Fundamental Completion Algorithm 1316.4 Step 2--Optimising over tensors of varying rank 1336.4.1 The Rank Adaptive Algorithm 1346.5 Step 3--Adapting the sampling set 1356.5.1 The Sample Adaptive Algorithm 1366.6 Summary of chapter 137Chapter 7 Sliding Technique 1397.1 Slide 1397.2 Slider 1407.3 Evaluating a slider 1417.3.1 Relation to Taylor approximation 1427.4 Summary of chapter 142Chapter 8 The Jacobian projection technique 1438.1 Setting the background 1448.2 What we can recover 1458.2.1 Intuition behind g and its derivative dg 1468.2.2 Using the derivative of f 1478.2.3 When k8.3 Partial derivatives via projections onto the Jacobian 149Part Three Hybrid solutions -- approximation methods and the toolkitChapter 9 Introduction 1559.1 The dimensionality problem revisited 1559.2 Exploiting the Composition Technique 156Chapter 10 The Toolkit and Deep Neural Nets 15910.1 Building on P using the image of g 15910.2 Building on f 160Chapter 11 The Toolkit and Chebyshev Tensors 16111.1 Full Chebyshev Tensor 16111.2 TT-format Chebyshev Tensor 16211.3 Chebyshev Slider 16211.4 A final note 163Chapter 12 Hybrid Deep Neural Nets and Chebyshev Tensors Frameworks 16512.1 The fundamental idea 16512.1.1 Factorable Functions 16712.2 DNN+CT with Static Training Set 16812.3 DNN+CT with Dynamic Training Set 17112.4 Numerical Tests 17212.4.1 Cost Function Minimisation 17212.4.2 Maximum Error 17412.5 Enhanced DNN+CT architectures and further research 174Part Four ApplicationsChapter 13 The aim 17913.1 Suitability of the approximation methods 17913.2 Understanding the variables at play 181Chapter 14 When to use Chebyshev Tensors and when to use Deep Neural Nets 18514.1 Speed and convergence 18514.1.1 Speed of evaluation 18614.1.2 Convergence 18614.1.3 Convergence Rate in Real-Life Contexts 18714.2 The question of dimension 19014.2.1 Taking into account the application 19214.3 Partial derivatives and ex ante error estimation 19514.4 Summary of chapter 197Chapter 15 Counterparty credit risk 19915.1 Monte Carlo simulations for CCR 20015.1.1 Scenario diffusion 20015.1.2 Pricing step--computational bottleneck 20015.2 Solution 20115.2.1 Popular solutions 20115.2.2 The hybrid solution 20215.2.3 Variables at play 20315.2.4 Optimal setup 20715.2.5 Possible proxies 20715.2.6 Portfolio calculations 20915.2.7 If the model space is not available 20915.3 Tests 21115.3.1 Trade types, risk factors and proxies 21215.3.2 Proxy at each time point 21315.3.3 Proxy for all time points 22315.3.4 Adding non-risk-driving variables 22815.3.5 High-dimensional problems 23515.4 Results Analysis and Conclusions 23615.5 Summary of chapter 239Chapter 16 Market Risk 24116.1 VaR-like calculations 24216.1.1 Common techniques in the computation of VaR 24316.2 Enhanced Revaluation Grids 24516.3 Fundamental Review of the Trading Book 24616.3.1 Challenges 24716.3.2 Solution 24816.3.3 The intuition behind Chebyshev Sliders 25216.4 Proof of concept 25516.4.1 Proof of concept specifics 25516.4.2 Test specifics 25716.4.3 Results for swap 26016.4.4 Results for swaptions 10-day liquidity horizon 26216.4.5 Results for swaptions 60-day liquidity horizon 26516.4.6 Daily computation and reusability 26816.4.7 Beyond regulatory minimum calculations 27116.5 Stability of technique 27216.6 Results beyond vanilla portfolios--further research 27216.7 Summary of chapter 273Chapter 17 Dynamic sensitivities 27517.1 Simulating sensitivities 27617.1.1 Scenario diffusion 27617.1.2 Computing sensitivities 27617.1.3 Computational cost 27617.1.4 Methods available 27717.2 The Solution 27817.2.1 Hybrid method 27917.3 An important use of dynamic sensitivities 28217.4 Numerical tests 28317.4.1 FX Swap 28317.4.2 European Spread Option 28417.5 Discussion of results 29117.6 Alternative methods 29317.7 Summary of chapter 294Chapter 18 Pricing model calibration 29518.1 Introduction 29518.1.1 Examples of pricing models 29718.2 Solution 29818.2.1 Variables at play 29918.2.2 Possible proxies 29918.2.3 Domain of approximation 30018.3 Test description 30118.3.1 Test setup 30118.4 Results with Chebyshev Tensors 30418.4.1 Rough Bergomi model with constant forward variance 30418.4.2 Rough Bergomi model with piece-wise constant forward variance 30718.5 Results with Deep Neural Nets 30918.6 Comparison of results via CT and DNN 31018.7 Summary of chapter 311Chapter 19 Approximation of the implied volatility function 31319.1 The computation of implied volatility 31419.1.1 Available methods 31519.2 Solution 31619.2.1 Reducing the dimension of the problem 31719.2.2 Two-dimensional CTs 31819.2.3 Domain of approximation 32119.2.4 Splitting the domain 32319.2.5 Scaling the time-scaled implied volatility 32519.2.6 Implementation 32819.3 Results 33019.3.1 Parameters used for CTs 33019.3.2 Comparisons to other methods 33119.4 Summary of chapter 334Chapter 20 Optimisation Problems 33520.1 Balance sheet optimisation 33520.2 Minimisation of margin funding cost 33920.3 Generalisation--currently "impossible" calculations 34520.4 Summary of chapter 346Chapter 21 Pricing Cloning 34721.1 Pricing function cloning 34721.1.1 Other benefits 35221.1.2 Software vendors 35221.2 Summary of chapter 353Chapter 22 XVA sensitivities 35522.1 Finite differences and proxy pricers 35522.1.1 Multiple proxies 35622.1.2 Single proxy 35722.2 Proxy pricers and AAD 358Chapter 23 Sensitivities of exotic derivatives 35923.1 Benchmark sensitivities computation 36023.2 Sensitivities via Chebyshev Tensors 361Chapter 24 Software libraries relevant to the book 36524.1 Relevant software libraries 36524.2 The MoCaX Suite 36624.2.1 MoCaX Library 36624.2.2 MoCaXExtend Library 377AppendicesAppendix A Families of Orthogonal Polynomials 385Appendix B Exponential Convergence of Chebyshev Tensors 387Appendix C Chebyshev Splines on Functions with No Singularity Points 391Appendix D Computational savings details for CCR 395D.1 Barrier option 395D.2 Cross-currency swap 395D.3 Bermudan Swaption 397D.3.1 Using full Chebyshev Tensors 397D.3.2 Using Chebyshev Tensors in TT format 397D.3.3 Using Deep Neural Nets 399D.4 American option 399D.4.1 Using Chebyshev Tensors in TT format 400D.4.2 Using Deep Neural Nets 401Appendix E Computational savings details for dynamic sensitivities 403E.1 FX Swap 403E.2 European Spread Option 404Appendix F Dynamic sensitivities on the market space 407F.1 The parametrisation 408F.2 Numerical tests 410F.3 Future work . . . when k > 1 412Appendix G Dynamic sensitivities and IM via Jacobian Projection technique 415Appendix H MVA optimisation -- further computational enhancement 419Bibliography 421Index 425

O autorze

IGNACIO RUIZ, PhD, is the head of Counterparty Credit Risk Measurement and Analytics at Scotiabank. Prior to that he has been head quant for Counterparty Credit Risk Exposure Analytics at Credit Suisse, head of Equity Risk Analytics at BNP Paribas and he founded MoCaX Intelligence, from where he offered his services as an independent consultant. He holds a PhD in Physics from the University of Cambridge.MARIANO ZERON, PhD, is Head of Research and Development at MoCaX Intelligence. Prior to that he was a quant researcher at Areski Capital. He has extensive experience with Chebyshev Tensors and Deep Neural Nets applied to risk calculations. He holds a PhD in Mathematics from the University of Cambridge.