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Full Description
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.
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 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
