State-of-the-art algorithmic deep learning and tensoring techniques for financial institutions The computational demand of risk calculations in financial institutions has ballooned and shows no sign of stopping. It is no longer viable to simply add more computing power to deal with this increased demand. The solution? Algorithmic solutions based on deep learning and Chebyshev tensors represent a practical way to reduce costs while simultaneously increasing risk calculation capabilities. Machine Learning for Risk Calculations: A Practitioner’s View provides an in-depth review of a number of algorithmic solutions and demonstrates how they can be used to overcome the massive computational burden of risk calculations in financial institutions. This book will get you started by reviewing fundamental techniques, including deep learning and Chebyshev tensors. You’ll then discover algorithmic tools that, in combination with the fundamentals, deliver actual solutions to the real problems financial institutions encounter on a regular basis. Numerical tests and examples demonstrate how these solutions can be applied to practical problems, including XVA and Counterparty Credit Risk, IMM capital, PFE, VaR, FRTB, Dynamic Initial Margin, pricing function calibration, volatility surface parametrisation, portfolio optimisation and others. Finally, you’ll uncover the benefits these techniques provide, the practicalities of implementing them, and the software which can be used. Review the fundamentals of deep learning and Chebyshev tensors Discover pioneering algorithmic techniques that can create new opportunities in complex risk calculation Learn how to apply the solutions to a wide range of real-life risk calculations. Download sample code used in the book, so you can follow along and experiment with your own calculations Realize improved risk management whilst overcoming the burden of limited computational power Quants, IT professionals, and financial risk managers will benefit from this practitioner-oriented approach to state-of-the-art risk calculation.

Table of Contents:
Acknowledgements xvii Foreword xxi Motivation and aim of this book xxiii Part One Fundamental Approximation Methods Chapter 1 Machine Learning 3 1.1 Introduction to Machine Learning 3 1.1.1 A brief history of Machine Learning Methods 4 1.1.2 Main sub-categories in Machine Learning 5 1.1.3 Applications of interest 7 1.2 The Linear Model 7 1.2.1 General concepts 8 1.2.2 The standard linear model 12 1.3 Training and predicting 15 1.3.1 The frequentist approach 18 1.3.2 The Bayesian approach 21 1.3.3 Testing—in search of consistent accurate predictions 25 1.3.4 Underfitting and overfitting 25 1.3.5 K-fold cross-validation 27 1.4 Model complexity 28 1.4.1 Regularisation 29 1.4.2 Cross-validation for regularisation 31 1.4.3 Hyper-parameter optimisation 33 Chapter 2 Deep Neural Nets 39 2.1 A brief history of Deep Neural Nets 39 2.2 The basic Deep Neural Net model 41 2.2.1 Single neuron 41 2.2.2 Artificial Neural Net 43 2.2.3 Deep Neural Net 46 2.3 Universal Approximation Theorems 48 2.4 Training of Deep Neural Nets 49 2.4.1 Backpropagation 50 2.4.2 Backpropagation example 51 2.4.3 Optimisation of cost function 55 2.4.4 Stochastic gradient descent 57 2.4.5 Extensions of stochastic gradient descent 58 2.5 More sophisticated DNNs 59 2.5.1 Convolution Neural Nets 59 2.5.2 Other famous architectures 63 2.6 Summary of chapter 64 Chapter 3 Chebyshev Tensors 65 3.1 Approximating functions with polynomials 65 3.2 Chebyshev Series 66 3.2.1 Lipschitz continuity and Chebyshev projections 67 3.2.2 Smooth functions and Chebyshev projections 70 3.2.3 Analytic functions and Chebyshev projections 70 3.3 Chebyshev Tensors and interpolants 72 3.3.1 Tensors and polynomial interpolants 72 3.3.2 Misconception over polynomial interpolation 73 3.3.3 Chebyshev points 74 3.3.4 Chebyshev interpolants 76 3.3.5 Aliasing phenomenon 77 3.3.6 Convergence rates of Chebyshev interpolants 77 3.3.7 High-dimensional Chebyshev interpolants 79 3.4 Ex ante error estimation 82 3.5 What makes Chebyshev points unique 85 3.6 Evaluation of Chebyshev interpolants 89 3.6.1 Clenshaw algorithm 90 3.6.2 Barycentric interpolation formula 91 3.6.3 Evaluating high-dimensional tensors 93 3.6.4 Example of numerical stability 94 3.7 Derivative approximation 95 3.7.1 Convergence of Chebyshev derivatives 95 3.7.2 Computation of Chebyshev derivatives 96 3.7.3 Derivatives in high dimensions 97 3.8 Chebyshev Splines 99 3.8.1 Gibbs phenomenon 99 3.8.2 Splines 100 3.8.3 Splines of Chebyshev 101 3.8.4 Chebyshev Splines in high dimensions 101 3.9 Algebraic operations with Chebyshev Tensors 101 3.10 Chebyshev Tensors and Machine Learning 103 3.11 Summary of chapter 104 Part Two The toolkit — plugging in approximation methods Chapter 4 Introduction: why is a toolkit needed 107 4.1 The pricing problem 107 4.2 Risk calculation with proxy pricing 109 4.3 The curse of dimensionality 110 4.4 The techniques in the toolkit 112 Chapter 5 Composition techniques 113 5.1 Leveraging from existing parametrisations 114 5.1.1 Risk factor generating models 114 5.1.2 Pricing functions and model risk factors 115 5.1.3 The tool obtained 116 5.2 Creating a parametrisation 117 5.2.1 Principal Component Analysis 117 5.2.2 Autoencoders 119 5.3 Summary of chapter 120 Chapter 6 Tensors in TT format and Tensor Extension Algorithms 123 6.1 Tensors in TT format 123 6.1.1 Motivating example 124 6.1.2 General case 124 6.1.3 Basic operations 126 6.1.4 Evaluation of Chebyshev Tensors in TT format 127 6.2 Tensor Extension Algorithms 129 6.3 Step 1—Optimising over tensors of fixed rank 129 6.3.1 The Fundamental Completion Algorithm 131 6.4 Step 2—Optimising over tensors of varying rank 133 6.4.1 The Rank Adaptive Algorithm 134 6.5 Step 3—Adapting the sampling set 135 6.5.1 The Sample Adaptive Algorithm 136 6.6 Summary of chapter 137 Chapter 7 Sliding Technique 139 7.1 Slide 139 7.2 Slider 140 7.3 Evaluating a slider 141 7.3.1 Relation to Taylor approximation 142 7.4 Summary of chapter 142 Chapter 8 The Jacobian projection technique 143 8.1 Setting the background 144 8.2 What we can recover 145 8.2.1 Intuition behind g and its derivative dg 146 8.2.2 Using the derivative of f 147 8.2.3 When k < n becomes a problem 149 8.3 Partial derivatives via projections onto the Jacobian 149 Part Three Hybrid solutions — approximation methods and the toolkit Chapter 9 Introduction 155 9.1 The dimensionality problem revisited 155 9.2 Exploiting the Composition Technique 156 Chapter 10 The Toolkit and Deep Neural Nets 159 10.1 Building on P using the image of g 159 10.2 Building on f 160 Chapter 11 The Toolkit and Chebyshev Tensors 161 11.1 Full Chebyshev Tensor 161 11.2 TT-format Chebyshev Tensor 162 11.3 Chebyshev Slider 162 11.4 A final note 163 Chapter 12 Hybrid Deep Neural Nets and Chebyshev Tensors Frameworks 165 12.1 The fundamental idea 165 12.1.1 Factorable Functions 167 12.2 DNN+CT with Static Training Set 168 12.3 DNN+CT with Dynamic Training Set 171 12.4 Numerical Tests 172 12.4.1 Cost Function Minimisation 172 12.4.2 Maximum Error 174 12.5 Enhanced DNN+CT architectures and further research 174 Part Four Applications Chapter 13 The aim 179 13.1 Suitability of the approximation methods 179 13.2 Understanding the variables at play 181 Chapter 14 When to use Chebyshev Tensors and when to use Deep Neural Nets 185 14.1 Speed and convergence 185 14.1.1 Speed of evaluation 186 14.1.2 Convergence 186 14.1.3 Convergence Rate in Real-Life Contexts 187 14.2 The question of dimension 190 14.2.1 Taking into account the application 192 14.3 Partial derivatives and ex ante error estimation 195 14.4 Summary of chapter 197 Chapter 15 Counterparty credit risk 199 15.1 Monte Carlo simulations for CCR 200 15.1.1 Scenario diffusion 200 15.1.2 Pricing step—computational bottleneck 200 15.2 Solution 201 15.2.1 Popular solutions 201 15.2.2 The hybrid solution 202 15.2.3 Variables at play 203 15.2.4 Optimal setup 207 15.2.5 Possible proxies 207 15.2.6 Portfolio calculations 209 15.2.7 If the model space is not available 209 15.3 Tests 211 15.3.1 Trade types, risk factors and proxies 212 15.3.2 Proxy at each time point 213 15.3.3 Proxy for all time points 223 15.3.4 Adding non-risk-driving variables 228 15.3.5 High-dimensional problems 235 15.4 Results Analysis and Conclusions 236 15.5 Summary of chapter 239 Chapter 16 Market Risk 241 16.1 VaR-like calculations 242 16.1.1 Common techniques in the computation of VaR 243 16.2 Enhanced Revaluation Grids 245 16.3 Fundamental Review of the Trading Book 246 16.3.1 Challenges 247 16.3.2 Solution 248 16.3.3 The intuition behind Chebyshev Sliders 252 16.4 Proof of concept 255 16.4.1 Proof of concept specifics 255 16.4.2 Test specifics 257 16.4.3 Results for swap 260 16.4.4 Results for swaptions 10-day liquidity horizon 262 16.4.5 Results for swaptions 60-day liquidity horizon 265 16.4.6 Daily computation and reusability 268 16.4.7 Beyond regulatory minimum calculations 271 16.5 Stability of technique 272 16.6 Results beyond vanilla portfolios—further research 272 16.7 Summary of chapter 273 Chapter 17 Dynamic sensitivities 275 17.1 Simulating sensitivities 276 17.1.1 Scenario diffusion 276 17.1.2 Computing sensitivities 276 17.1.3 Computational cost 276 17.1.4 Methods available 277 17.2 The Solution 278 17.2.1 Hybrid method 279 17.3 An important use of dynamic sensitivities 282 17.4 Numerical tests 283 17.4.1 FX Swap 283 17.4.2 European Spread Option 284 17.5 Discussion of results 291 17.6 Alternative methods 293 17.7 Summary of chapter 294 Chapter 18 Pricing model calibration 295 18.1 Introduction 295 18.1.1 Examples of pricing models 297 18.2 Solution 298 18.2.1 Variables at play 299 18.2.2 Possible proxies 299 18.2.3 Domain of approximation 300 18.3 Test description 301 18.3.1 Test setup 301 18.4 Results with Chebyshev Tensors 304 18.4.1 Rough Bergomi model with constant forward variance 304 18.4.2 Rough Bergomi model with piece-wise constant forward variance 307 18.5 Results with Deep Neural Nets 309 18.6 Comparison of results via CT and DNN 310 18.7 Summary of chapter 311 Chapter 19 Approximation of the implied volatility function 313 19.1 The computation of implied volatility 314 19.1.1 Available methods 315 19.2 Solution 316 19.2.1 Reducing the dimension of the problem 317 19.2.2 Two-dimensional CTs 318 19.2.3 Domain of approximation 321 19.2.4 Splitting the domain 323 19.2.5 Scaling the time-scaled implied volatility 325 19.2.6 Implementation 328 19.3 Results 330 19.3.1 Parameters used for CTs 330 19.3.2 Comparisons to other methods 331 19.4 Summary of chapter 334 Chapter 20 Optimisation Problems 335 20.1 Balance sheet optimisation 335 20.2 Minimisation of margin funding cost 339 20.3 Generalisation—currently “impossible” calculations 345 20.4 Summary of chapter 346 Chapter 21 Pricing Cloning 347 21.1 Pricing function cloning 347 21.1.1 Other benefits 352 21.1.2 Software vendors 352 21.2 Summary of chapter 353 Chapter 22 XVA sensitivities 355 22.1 Finite differences and proxy pricers 355 22.1.1 Multiple proxies 356 22.1.2 Single proxy 357 22.2 Proxy pricers and AAD 358 Chapter 23 Sensitivities of exotic derivatives 359 23.1 Benchmark sensitivities computation 360 23.2 Sensitivities via Chebyshev Tensors 361 Chapter 24 Software libraries relevant to the book 365 24.1 Relevant software libraries 365 24.2 The MoCaX Suite 366 24.2.1 MoCaX Library 366 24.2.2 MoCaXExtend Library 377 Appendices Appendix A Families of Orthogonal Polynomials 385 Appendix B Exponential Convergence of Chebyshev Tensors 387 Appendix C Chebyshev Splines on Functions with No Singularity Points 391 Appendix D Computational savings details for CCR 395 D.1 Barrier option 395 D.2 Cross-currency swap 395 D.3 Bermudan Swaption 397 D.3.1 Using full Chebyshev Tensors 397 D.3.2 Using Chebyshev Tensors in TT format 397 D.3.3 Using Deep Neural Nets 399 D.4 American option 399 D.4.1 Using Chebyshev Tensors in TT format 400 D.4.2 Using Deep Neural Nets 401 Appendix E Computational savings details for dynamic sensitivities 403 E.1 FX Swap 403 E.2 European Spread Option 404 Appendix F Dynamic sensitivities on the market space 407 F.1 The parametrisation 408 F.2 Numerical tests 410 F.3 Future work . . . when k > 1 412 Appendix G Dynamic sensitivities and IM via Jacobian Projection technique 415 Appendix H MVA optimisation — further computational enhancement 419 Bibliography 421 Index 425