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Gaussian Process Models for Quantitative Finance.pdf
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量化金融的高斯过程模型
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Gaussian Process Models for Quantitative Finance.pdf This text rigorously integrates Gaussian process (GP) modeling into the fabric of quantitative finance. Spurred by the transformative influence of machine learning frameworks, the volume provides a detailed exposition on the theory and application of GPs. The initial three chapters are tailored to acquaint readers with GP methodology, equipping them with the necessary foundation to navigate the more advanced applications that follow. Targeting both researchers and practitioners, this work is positioned as a reference textbook, offering a broad spectrum of GP model applications that demonstrate their versatility and relevance in quantitative finance. Our exposition begins with Chaps. 1 and 2 that form the bedrock of GP theory. The introductory Sect. 1.1 sets the context for GP models within computational finance, highlighting the convergence of machine learning and financial analysis and acquainting the reader with examples that are tackled in later chapters. The rest of Chap. 1 is dedicated to the core methodologies, equipping the reader with the theoretical framework for GP modeling. Chapter 2 goes into details of GP kernels (covariance functions), drawing connections to other mathematical sub-fields, such as reproducing kernel Hilbert spaces and stochastic differential equations. Chapter 3 pivots toward more specialized GP approaches, sampling among non-Gaussian likelihoods, multi-output modeling, and heteroskedastic modeling. This chapter’s relevance shifts depending on the specific applications the reader pursues. The remaining four chapters can be explored independently of one another, each delving into a distinct application of GPs in the financial realm. Specifically, Chapter 4 discusses the application of GPs in option pricing and sensitivity analysis. Chapter 5 examines Regression Monte Carlo methods, illustrating the use of GPs as functional surrogates for the continuation value in optimal stopping problems. Chapter 6 presents the use of GPs for non-parametric modeling of financial objects, such as yield curves, implied volatility surfaces and mortality rate surfaces. Chapter 7 addresses the application of GP surrogates in stochastic control. The brief Appendix offers a review on prerequisite mathematics, like the theory of multivariate Gaussian distributions, linear algebra, and function spaces.
1 Gaussian Process Preliminaries ............................................ 1
1.1 Introduction ............................................................. 1
1.2 Fundamentals ........................................................... 4
1.3 Gaussian Process Regression .......................................... 6
1.4 GP Likelihood and Hyperparameters.................................. 11
1.4.1 Estimation and Likelihood .................................... 12
1.4.2 Examples and Discussion ..................................... 16
1.4.3 Universal Kriging and Varying Prior Means ................. 19
1.5 GPs as Kernel Smoothing and Kernel Ridge Regression ............. 21
1.6 Closing Notes........................................................... 23
2 Covariance Kernels .......................................................... 27
2.1 First Examples and Smoothness ....................................... 27
2.2 Classes and Properties of Kernels ..................................... 29
2.2.1 Stationary Kernels ............................................. 30
2.2.2 Nonstationary Kernels......................................... 37
2.3 Kernel Composition and Engineering ................................. 39
2.4 Model Selection ........................................................ 40
2.4.1 Example: Kernel Fitness ...................................... 42
2.5 Convergence and Universal Approximation .......................... 43
2.6 Connections with SDEs and Other Processes ......................... 44
3 Advanced GP Modeling Topics............................................. 49
3.1 Heteroskedastic GPs.................................................... 49
3.2 Alternative Likelihood Functions...................................... 52
3.2.1 Gaussian Process Regression with Student t-Noise ......... 53
3.2.2 Student-t Process Regression with Student-t Noise ......... 54
3.2.3 Gaussian Process GLM ....................................... 55
3.3 Multi-Output GPs ...................................................... 55
3.4 Localization............................................................. 57
3.4.1 Inducing Points ................................................ 59
3.5 Updating Equations for GPs ........................................... 61
4 Option Pricing and Sensitivities ............................................ 63
4.1 Learning to Price Options.............................................. 63
4.2 Surrogate Ingredients................................................... 66
4.3 Option Sensitivities..................................................... 70
4.3.1 GP Gradients................................................... 70
4.3.2 Illustration: Estimating Greeks in the
Black-Scholes Model.......................................... 72
4.4 Constrained GPs and No-Arbitrage.................................... 75
4.5 Portfolio Modeling and Credit Valuation Adjustments............... 78
5 Optimal Stopping ............................................................ 81
5.1 Regression Monte Carlo ............................................... 83
5.1.1 RMC GP Features ............................................. 85
5.1.2 Training Designs............................................... 86
5.1.3 Illustration: Bermudan Options ............................... 88
5.2 Active Learning and Adaptive Batching .............................. 90
6 Non-Parametric Modeling of Financial Structures ...................... 95
6.1 Modeling Term Structure .............................................. 95
6.1.1 Kriging of Commodity Curves................................ 99
6.2 Modeling Implied Volatility ........................................... 102
6.3 Swaption Cubes ........................................................ 105
6.4 Mortality Rate Surfaces ................................................ 106
6.4.1 Illustration: Danish Mortality ................................. 108
6.5 Valuation of Variable Annuities........................................ 111
7 Stochastic Control ........................................................... 115
7.1 Switching Control ...................................................... 115
7.2 Continuous Control..................................................... 117
7.3 Impulse Control ........................................................ 119
A Mathematical Background.................................................. 125
A.1 Matrices and Linear Algebra........................................... 125
A.2 Multivariate Normal Distributions..................................... 126
A.3 Differentiability, Smoothness, and Function Spaces.................. 127
A.3.1 Sobolev Spaces ................................................ 127
References......................................................................... 129
Index ............................................................................... 137
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