Handbook of Financial Risk Management (Chapman and Hall/CRC Financial Mathematics Series)

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
List of Symbols and Notations
1. Introduction
	1.1 The need for risk management
		1.1.1 Risk management and the financial system
		1.1.2 The development of financial markets
		1.1.3 Financial crises and systemic risk
	1.2 Financial regulation
		1.2.1 Banking regulation
		1.2.2 Insurance regulation
		1.2.3 Market regulation
		1.2.4 Systemic risk
	1.3 Financial regulation overview
		1.3.1 List of supervisory authorities
		1.3.2 Timeline of financial regulation
Part I: Risk Management in the Financial Sector
	2. Market Risk
		2.1 Regulatory framework
			2.1.1 The Basel I/II framework
				2.1.1.1 Standardized measurement method
				2.1.1.2 Internal model-based approach
			2.1.2 The Basel III framework
				2.1.2.1 Standardized approach
				2.1.2.2 Internal model-based approach
		2.2 Statistical estimation methods of risk measures
			2.2.1 Definition
				2.2.1.1 Coherent risk measures
				2.2.1.2 Value-at-risk
				2.2.1.3 Expected shortfall
				2.2.1.4 Estimator or estimate?
			2.2.2 Historical methods
				2.2.2.1 The order statistic approach
				2.2.2.2 The kernel approach
			2.2.3 Analytical methods
				2.2.3.1 Derivation of the closed-form formula
				2.2.3.2 Linear factor models
				2.2.3.3 Volatility forecasting
				2.2.3.4 Extension to other probability distributions
			2.2.4 Monte Carlo methods
			2.2.5 The case of options and derivatives
				2.2.5.1 Identification of risk factors
				2.2.5.2 Methods to calculate VaR and ES risk measures
				2.2.5.3 Backtesting
				2.2.5.4 Model risk
		2.3 Risk allocation
			2.3.1 Euler allocation principle
			2.3.2 Application to non-normal risk measures
				2.3.2.1 Main results
				2.3.2.2 Calculating risk contributions with historical and simulated scenarios
		2.4 Exercises
			2.4.1 Calculating regulatory capital with the Basel I standardized measurement method
			2.4.2 Covariance matrix
			2.4.3 Risk measure
			2.4.4 Value-at-risk of a long/short portfolio
			2.4.5 Value-at-risk of an equity portfolio hedged with put options
			2.4.6 Risk management of exotic options
			2.4.7 P&L approximation with Greek sensitivities
			2.4.8 Calculating the non-linear quadratic value-at-risk
			2.4.9 Risk decomposition of the expected shortfall
			2.4.10 Expected shortfall of an equity portfolio
			2.4.11 Risk measure of a long/short portfolio
			2.4.12 Kernel estimation of the expected shortfall
	3. Credit Risk
		3.1 The market of credit risk
			3.1.1 The loan market
			3.1.2 The bond market
				3.1.2.1 Statistics of the bond market
				3.1.2.2 Bond pricing
			3.1.3 Securitization and credit derivatives
				3.1.3.1 Credit securitization
				3.1.3.2 Credit default swap
				3.1.3.3 Basket default swap
				3.1.3.4 Collateralized debt obligations
		3.2 Capital requirement
			3.2.1 The Basel I framework
			3.2.2 The Basel II standardized approach
				3.2.2.1 Standardized risk weights
				3.2.2.2 Credit risk mitigation
			3.2.3 The Basel II internal ratings-based approach
				3.2.3.1 The general framework
				3.2.3.2 The credit risk model of Basel II
				3.2.3.3 The IRB formulas
			3.2.4 The Basel III revision
				3.2.4.1 The standardized approach
				3.2.4.2 The internal ratings-based approach
			3.2.5 The securitization framework
				3.2.5.1 Overview of the approaches
				3.2.5.2 Internal ratings-based approach
				3.2.5.3 External ratings-based approach
				3.2.5.4 Standardized approach
		3.3 Credit risk modeling
			3.3.1 Exposure at default
			3.3.2 Loss given default
				3.3.2.1 Definition
				3.3.2.2 Stochastic modeling
				3.3.2.3 Economic modeling
			3.3.3 Probability of default
				3.3.3.1 Survival function
				3.3.3.2 Transition probability matrix
				3.3.3.3 Structural models
			3.3.4 Default correlation
				3.3.4.1 The copula model
				3.3.4.2 The factor model
				3.3.4.3 Estimation methods
				3.3.4.4 Dependence and credit basket derivatives
			3.3.5 Granularity and concentration
				3.3.5.1 Difference between fine-grained and concentrated portfolios
				3.3.5.2 Granularity adjustment
		3.4 Exercises
			3.4.1 Single- and multi-name credit default swaps
			3.4.2 Risk contribution in the Basel II model
			3.4.3 Calibration of the piecewise exponential model
			3.4.4 Modeling loss given default
			3.4.5 Modeling default times with a Markov chain
			3.4.6 Continuous-time modeling of default risk
			3.4.7 Derivation of the original Basel granularity adjustment
			3.4.8 Variance of the conditional portfolio loss
	4. Counterparty Credit Risk and Collateral Risk
		4.1 Counterparty credit risk
			4.1.1 Definition
			4.1.2 Modeling the exposure at default
				4.1.2.1 An illustrative example
				4.1.2.2 Measuring the counterparty exposure
				4.1.2.3 Practical implementation for calculating counterparty exposure
			4.1.3 Regulatory capital
				4.1.3.1 Internal model method
				4.1.3.2 Non-internal model methods (Basel II)
				4.1.3.3 SA-CCR method (Basel III)
			4.1.4 Impact of wrong way risk
				4.1.4.1 An example
				4.1.4.2 Calibration of the α factor
		4.2 Credit valuation adjustment
			4.2.1 Definition
				4.2.1.1 Difference between CCR and CVA
				4.2.1.2 CVA, DVA and bilateral CVA
				4.2.1.3 Practical implementation for calculating CVA
			4.2.2 Regulatory capital
				4.2.2.1 The 2010 version of Basel III
				4.2.2.2 The 2017 version of Basel III
			4.2.3 CVA and wrong/right way risk
		4.3 Collateral risk
			4.3.1 Definition
			4.3.2 Capital allocation
		4.4 Exercises
			4.4.1 Impact of netting agreements in counterparty credit risk
			4.4.2 Calculation of the effective expected positive exposure
			4.4.3 Calculation of the capital charge for counterparty credit risk
			4.4.4 Calculation of CVA and DVA measures
			4.4.5 Approximation of the CVA for an interest rate swap
			4.4.6 Risk contribution of CVA with collateral
	5. Operational Risk
		5.1 Definition of operational risk
		5.2 Basel approaches for calculating the regulatory capital
			5.2.1 The basic indicator approach
			5.2.2 The standardized approach
			5.2.3 Advanced measurement approaches
			5.2.4 Basel III approach
		5.3 Loss distribution approach
			5.3.1 Definition
			5.3.2 Parametric estimation
				5.3.2.1 Estimation of the loss severity distribution
				5.3.2.2 Estimation of the loss frequency distribution
			5.3.3 Calculating the capital charge
				5.3.3.1 Monte Carlo approach
				5.3.3.2 Analytical approaches
				5.3.3.3 Aggregation issues
			5.3.4 Incorporating scenario analysis
				5.3.4.1 Probability distribution of a given scenario
				5.3.4.2 Calibration of a set of scenarios
			5.3.5 Stability issue of the LDA model
		5.4 Exercises
			5.4.1 Estimation of the loss severity distribution
			5.4.2 Estimation of the loss frequency distribution
			5.4.3 Using the method of moments in operational risk models
			5.4.4 Calculation of the Basel II required capital
			5.4.5 Parametric estimation of the loss severity distribution
			5.4.6 Mixed Poisson process
	6. Liquidity Risk
		6.1 Market liquidity
			6.1.1 Transaction cost versus volume-based measures
				6.1.1.1 Bid-ask spread
				6.1.1.2 Trading volume
				6.1.1.3 Liquidation ratio
				6.1.1.4 Liquidity ordering
			6.1.2 Other liquidity measures
			6.1.3 The liquidity-adjusted CAPM
		6.2 Funding liquidity
			6.2.1 Asset liability mismatch
			6.2.2 Relationship between market and funding liquidity risks
		6.3 Regulation of the liquidity risk
			6.3.1 Liquidity coverage ratio
				6.3.1.1 Definition
				6.3.1.2 Monitoring tools
			6.3.2 Net stable funding ratio
				6.3.2.1 Definition
				6.3.2.2 ASF and RSF factors
			6.3.3 Leverage ratio
	7. Asset Liability Management Risk
		7.1 General principles of the banking book risk management
			7.1.1 Definition
				7.1.1.1 Balance sheet and income statement
				7.1.1.2 Accounting standards
				7.1.1.3 Role and importance of the ALCO
			7.1.2 Liquidity risk
				7.1.2.1 Definition of the liquidity gap
				7.1.2.2 Asset and liability amortization
				7.1.2.3 Dynamic analysis
				7.1.2.4 Liquidity hedging
			7.1.3 Interest rate risk in the banking book
				7.1.3.1 Introduction on IRRBB
				7.1.3.2 Interest rate risk principles
				7.1.3.3 The standardized approach
			7.1.4 Other ALM risks
				7.1.4.1 Currency risk
				7.1.4.2 Credit spread risk
		7.2 Interest rate risk
			7.2.1 Duration gap analysis
				7.2.1.1 Relationship between Macaulay duration and modified duration
				7.2.1.2 Relationship between the duration gap and the equity duration
				7.2.1.3 An illustration
				7.2.1.4 Immunization of the balance sheet
			7.2.2 Earnings-at-risk
				7.2.2.1 Income gap analysis
				7.2.2.2 Net interest income
				7.2.2.3 Hedging strategies
			7.2.3 Simulation approach
			7.2.4 Funds transfer pricing
				7.2.4.1 Net interest and commercial margins
				7.2.4.2 Computing the internal transfer rates
		7.3 Behavioral options
			7.3.1 Non-maturity deposits
				7.3.1.1 Static and dynamic modeling
				7.3.1.2 Behavioral modeling
			7.3.2 Prepayment risk
				7.3.2.1 Factors of prepayment
				7.3.2.2 Structural models
				7.3.2.3 Reduced-form models
				7.3.2.4 Statistical measure of prepayment
			7.3.3 Redemption risk
				7.3.3.1 The funding risk of term deposits
				7.3.3.2 Modeling the early withdrawal risk
		7.4 Exercises
			7.4.1 Constant amortization of a loan
			7.4.2 Computation of the amortization functions S (t, u) and S* (t, u)
			7.4.3 Continuous-time analysis of the constant amortization mortgage (CAM)
			7.4.4 Valuation of non-maturity deposits
			7.4.5 Impact of prepayment on the amortization scheme of the CAM
	8. Systemic Risk and Shadow Banking System
		8.1 Defining systemic risk
			8.1.1 Systemic risk, systematic risk and idiosyncratic risk
			8.1.2 Sources of systemic risk
				8.1.2.1 Systematic shocks
				8.1.2.2 Propagation mechanisms
			8.1.3 Supervisory policy responses
				8.1.3.1 A new financial regulatory structure
				8.1.3.2 A myriad of new standards
		8.2 Systemic risk measurement
			8.2.1 The supervisory approach
				8.2.1.1 The G-SIB assessment methodology
				8.2.1.2 Identification of G-SIIs
				8.2.1.3 Extension to NBNI SIFIs
			8.2.2 The academic approach
				8.2.2.1 Marginal expected shortfall
				8.2.2.2 Delta conditional value-at-risk
				8.2.2.3 Systemic risk measure
				8.2.2.4 Network measures
		8.3 Shadow banking system
			8.3.1 Definition
			8.3.2 Measuring the shadow banking
				8.3.2.1 The broad (or MUNFI) measure
				8.3.2.2 The narrow measure
			8.3.3 Regulatory developments of shadow banking
				8.3.3.1 Data gaps
				8.3.3.2 Mitigation of interconnectedness risk
				8.3.3.3 Money market funds
				8.3.3.4 Complex shadow banking activities
Part II: Mathematical and Statistical Tools
	9. Model Risk of Exotic Derivatives
		9.1 Basics of option pricing
			9.1.1 The Black-Scholes model
				9.1.1.1 The general framework
				9.1.1.2 Application to European options
				9.1.1.3 Principle of dynamic hedging
				9.1.1.4 The implied volatility
			9.1.2 Interest rate risk modeling
				9.1.2.1 Pricing zero-coupon bonds with the Vasicek model
				9.1.2.2 The calibration issue of the yield curve
				9.1.2.3 Caps, floors and swaptions
				9.1.2.4 Change of numéraire and equivalent martingale measure
				9.1.2.5 The HJM model
				9.1.2.6 Market models
		9.2 Volatility risk
			9.2.1 The uncertain volatility model
				9.2.1.1 Formulation of the partial differential equation
				9.2.1.2 Computing lower and upper pricing bounds
				9.2.1.3 Application to ratchet options
			9.2.2 The shifted log-normal model
				9.2.2.1 The fixed-strike parametrization
				9.2.2.2 The floating-strike parametrization
				9.2.2.3 The forward parametrization
				9.2.2.4 Mixture of SLN distributions
				9.2.2.5 Application to binary, corridor and barrier options
			9.2.3 Local volatility model
				9.2.3.1 Derivation of the forward equation
				9.2.3.2 Duality between local volatility and implied volatility
				9.2.3.3 Dupire model in practice
				9.2.3.4 Application to exotic options
			9.2.4 Stochastic volatility models
				9.2.4.1 General analysis
				9.2.4.2 Heston model
				9.2.4.3 SABR model
			9.2.5 Factor models
				9.2.5.1 Linear and quadratic Gaussian models
				9.2.5.2 Dynamics of risk factors under the forward probability measure
				9.2.5.3 Pricing caplets and swaptions
				9.2.5.4 Calibration and practice of factor models
		9.3 Other model risk topics
			9.3.1 Dividend risk
				9.3.1.1 Understanding the impact of dividends on option prices
				9.3.1.2 Models of discrete dividends
			9.3.2 Correlation risk
				9.3.2.1 The two-asset case
				9.3.2.2 The multi-asset case
				9.3.2.3 The copula method
			9.3.3 Liquidity risk
		9.4 Exercises
			9.4.1 Option pricing and martingale measure
			9.4.2 The Vasicek model
			9.4.3 The Black model
			9.4.4 Change of numéraire and Girsanov theorem
			9.4.5 The HJM model and the forward probability measure
			9.4.6 Equivalent martingale measure in the Libor market model
			9.4.7 Displaced diffusion option pricing
			9.4.8 Dupire local volatility model
			9.4.9 The stochastic normal model
			9.4.10 The quadratic Gaussian model
			9.4.11 Pricing two-asset basket options
	10. Statistical Inference and Model Estimation
		10.1 Estimation methods
			10.1.1 Linear regression
				10.1.1.1 Least squares estimation
				10.1.1.2 Relationship with the conditional normal distribution
				10.1.1.3 The intercept problem
				10.1.1.4 Coefficient of determination
				10.1.1.5 Extension to weighted least squares regression
			10.1.2 Maximum likelihood estimation
				10.1.2.1 Definition of the estimator
				10.1.2.2 Asymptotic distribution
				10.1.2.3 Statistical inference
				10.1.2.4 Some examples
				10.1.2.5 EM algorithm
			10.1.3 Generalized method of moments
				10.1.3.1 Method of moments
				10.1.3.2 Extension to the GMM approach
				10.1.3.3 Simulated method of moments
			10.1.4 Non-parametric estimation
				10.1.4.1 Non-parametric density
				10.1.4.2 Non-parametric regression
		10.2 Time series modeling
			10.2.1 ARMA process
				10.2.1.1 The VAR(1) process
				10.2.1.2 Extension to ARMA models
			10.2.2 State space models
				10.2.2.1 Specification and estimation of state space models
				10.2.2.2 Some applications
			10.2.3 Cointegration and error correction models
				10.2.3.1 Nonstationarity and spurious regression
				10.2.3.2 The concept of cointegration
				10.2.3.3 Error correction model
				10.2.3.4 Estimation of cointegration relationships
			10.2.4 GARCH and stochastic volatility models
				10.2.4.1 GARCH models
				10.2.4.2 Stochastic volatility models
			10.2.5 Spectral analysis
				10.2.5.1 Fourier analysis
				10.2.5.2 Definition of the spectral density function
				10.2.5.3 Frequency domain localization
				10.2.5.4 Main properties
				10.2.5.5 Statistical estimation in the frequency domain
				10.2.5.6 Extension to multidimensional processes
				10.2.5.7 Some applications
		10.3 Exercises
			10.3.1 Probability distribution of the t-statistic in the case of the linear regression model
			10.3.2 Linear regression without a constant
			10.3.3 Linear regression with linear constraints
			10.3.4 Maximum likelihood estimation of the Poisson distribution
			10.3.5 Maximum likelihood estimation of the exponential distribution
			10.3.6 Relationship between the linear regression and the maximum likelihood method
			10.3.7 The Gaussian mixture model
			10.3.8 Parameter estimation of diffusion processes
			10.3.9 The Tobit model
			10.3.10 Derivation of Kalman filter equations
			10.3.11 Steady state of time-invariant state space model
			10.3.12 Kalman information filter versus Kalman covariance filter
			10.3.13 Granger representation theorem
			10.3.14 Probability distribution of the periodogram
			10.3.15 Spectral density function of structural time series models
			10.3.16 Spectral density function of some processes
	11. Copulas and Dependence Modeling
		11.1 Canonical representation of multivariate distributions
			11.1.1 Sklar’s theorem
			11.1.2 Expression of the copula density
			11.1.3 Fréchet classes
				11.1.3.1 The bivariate case
				11.1.3.2 The multivariate case
				11.1.3.3 Concordance ordering
		11.2 Copula functions and random vectors
			11.2.1 Countermonotonicity, comonotonicity and scale invariance property
			11.2.2 Dependence measures
				11.2.2.1 Concordance measures
				11.2.2.2 Linear correlation
				11.2.2.3 Tail dependence
		11.3 Parametric copula functions
			11.3.1 Archimedean copulas
				11.3.1.1 Definition
				11.3.1.2 Properties
				11.3.1.3 Two-parameter Archimedean copulas
				11.3.1.4 Extension to the multivariate case
			11.3.2 Normal copula
			11.3.3 Student’s t copula
		11.4 Statistical inference and estimation of copula functions
			11.4.1 The empirical copula
			11.4.2 The method of moments
			11.4.3 The method of maximum likelihood
		11.5 Exercises
			11.5.1 Gumbel logistic copula
			11.5.2 Farlie-Gumbel-Morgenstern copula
			11.5.3 Survival copula
			11.5.4 Method of moments
			11.5.5 Correlated loss given default rates
			11.5.6 Calculation of correlation bounds
			11.5.7 The bivariate Pareto copula
	12. Extreme Value Theory
		12.1 Order statistics
			12.1.1 Main properties
			12.1.2 Extreme order statistics
			12.1.3 Inference statistics
			12.1.4 Extension to dependent random variables
		12.2 Univariate extreme value theory
			12.2.1 Fisher-Tippet theorem
			12.2.2 Maximum domain of attraction
				12.2.2.1 MDA of the Gumbel distribution
				12.2.2.2 MDA of the Fréchet distribution
				12.2.2.3 MDA of the Weibull distribution
				12.2.2.4 Main results
			12.2.3 Generalized extreme value distribution
				12.2.3.1 Definition
				12.2.3.2 Estimating the value-at-risk
			12.2.4 Peak over threshold
				12.2.4.1 Definition
				12.2.4.2 Estimating the expected shortfall
		12.3 Multivariate extreme value theory
			12.3.1 Multivariate extreme value distributions
				12.3.1.1 Extreme value copulas
				12.3.1.2 Deheuvels-Pickands representation
			12.3.2 Maximum domain of attraction
			12.3.3 Tail dependence of extreme values
		12.4 Exercises
			12.4.1 Uniform order statistics
			12.4.2 Order statistics and return period
			12.4.3 Extreme order statistics of exponential random variables
			12.4.4 Extreme value theory in the bivariate case
			12.4.5 Maximum domain of attraction in the bivariate case
	13. Monte Carlo Simulation Methods
		13.1 Random variate generation
			13.1.1 Generating uniform random numbers
			13.1.2 Generating non-uniform random numbers
				13.1.2.1 Method of inversion
				13.1.2.2 Method of transformation
				13.1.2.3 Rejection sampling
				13.1.2.4 Method of mixtures
			13.1.3 Generating random vectors
				13.1.3.1 Method of conditional distributions
				13.1.3.2 Method of transformation
			13.1.4 Generating random matrices
				13.1.4.1 Orthogonal and covariance matrices
				13.1.4.2 Correlation matrices
				13.1.4.3 Wishart matrices
		13.2 Simulation of stochastic processes
			13.2.1 Discrete-time stochastic processes
				13.2.1.1 Correlated Markov chains
				13.2.1.2 Time series
			13.2.2 Univariate continuous-time processes
				13.2.2.1 Brownian motion
				13.2.2.2 Geometric Brownian motion
				13.2.2.3 Ornstein-Uhlenbeck process
				13.2.2.4 Stochastic differential equations without an explicit solution
				13.2.2.5 Poisson processes
				13.2.2.6 Jump-diffusion processes
				13.2.2.7 Processes related to Brownian motion
			13.2.3 Multivariate continuous-time processes
				13.2.3.1 Multidimensional Brownian motion
				13.2.3.2 Multidimensional geometric Brownian motion
				13.2.3.3 Euler-Maruyama and Milstein schemes
		13.3 Monte Carlo methods
			13.3.1 Computing integrals
				13.3.1.1 A basic example
				13.3.1.2 Theoretical framework
				13.3.1.3 Extension to the calculation of mathematical expectations
			13.3.2 Variance reduction
				13.3.2.1 Antithetic variates
				13.3.2.2 Control variates
				13.3.2.3 Importance sampling
				13.3.2.4 Other methods
			13.3.3 MCMC methods
				13.3.3.1 Gibbs sampling
				13.3.3.2 Metropolis-Hastings algorithm
				13.3.3.3 Sequential Monte Carlo methods and particle filters
			13.3.4 Quasi-Monte Carlo simulation methods
		13.4 Exercises
			13.4.1 Simulating random numbers using the inversion method
			13.4.2 Simulating random numbers using the transformation method
			13.4.3 Simulating random numbers using rejection sampling
			13.4.4 Simulation of Archimedean copulas
			13.4.5 Simulation of conditional random variables
			13.4.6 Simulation of the bivariate Normal copula
			13.4.7 Computing the capital charge for operational risk
			13.4.8 Simulating a Brownian bridge
			13.4.9 Optimal importance sampling
	14. Stress Testing and Scenario Analysis
		14.1 Stress test framework
			14.1.1 Definition
				14.1.1.1 General objective
				14.1.1.2 Scenario design and risk factors
				14.1.1.3 Firm-specific versus supervisory stress testing
			14.1.2 Methodologies
				14.1.2.1 Historical approach
				14.1.2.2 Macroeconomic approach
				14.1.2.3 Probabilistic approach
		14.2 Quantitative approaches
			14.2.1 Univariate stress scenarios
			14.2.2 Joint stress scenarios
				14.2.2.1 The bivariate case
				14.2.2.2 The multivariate case
			14.2.3 Conditional stress scenarios
				14.2.3.1 The conditional expectation solution
				14.2.3.2 The conditional quantile solution
			14.2.4 Reverse stress testing
				14.2.4.1 Mathematical computation of reverse stress testing
				14.2.4.2 Practical solutions
		14.3 Exercises
			14.3.1 Construction of a stress scenario with the GEV distribution
			14.3.2 Conditional expectation and linearity
			14.3.3 Conditional quantile and linearity
	15. Credit Scoring Models
		15.1 The method of scoring
			15.1.1 The emergence of credit scoring
				15.1.1.1 Judgmental credit systems versus credit scoring systems
				15.1.1.2 Scoring models for corporate bankruptcy
				15.1.1.3 New developments
			15.1.2 Variable selection
				15.1.2.1 Choice of the risk factors
				15.1.2.2 Data preparation
				15.1.2.3 Variable selection
			15.1.3 Score modeling, validation and follow-up
				15.1.3.1 Cross-validation approach
				15.1.3.2 Score modeling
				15.1.3.3 Score follow-up
		15.2 Statistical methods
			15.2.1 Unsupervised learning
				15.2.1.1 Clustering
				15.2.1.2 Dimension reduction
			15.2.2 Parametric supervised methods
				15.2.2.1 Discriminant analysis
				15.2.2.2 Binary choice models
			15.2.3 Non-parametric supervised methods
				15.2.3.1 k-nearest neighbor classifier
				15.2.3.2 Neural networks
				15.2.3.3 Support vector machines
				15.2.3.4 Model averaging
		15.3 Performance evaluation criteria and score consistency
			15.3.1 Shannon entropy
				15.3.1.1 Definition and properties
				15.3.1.2 Application to scoring
			15.3.2 Graphical methods
				15.3.2.1 Performance curve, selection curve and discriminant curve
				15.3.2.2 Some properties
			15.3.3 Statistical methods
				15.3.3.1 Kolmogorov-Smirnov test
				15.3.3.2 Gini coefficient
				15.3.3.3 Choice of the optimal cut-off
		15.4 Exercises
			15.4.1 Elastic net regression
			15.4.2 Cross-validation of the ridge linear regression
			15.4.3 K-means and the Lloyd’s algorithm
			15.4.4 Derivation of the principal component analysis
			15.4.5 Two-class separation maximization
			15.4.6 Maximum likelihood estimation of the probit model
			15.4.7 Computation of feed-forward neural networks
			15.4.8 Primal and dual problems of support vector machines
			15.4.9 Derivation of the AdaBoost algorithm as the solution of the additive logit model
			15.4.10 Weighted estimation
	Conclusion
Appendix A: Technical Appendix
	A.1 Numerical analysis
		A.1.1 Linear algebra
			A.1.1.1 Eigendecomposition
			A.1.1.2 Generalized eigendecomposition
			A.1.1.3 Schur decomposition
		A.1.2 Approximation methods
			A.1.2.1 Spline functions
			A.1.2.2 Positive definite matrix approximation
			A.1.2.3 Numerical integration
			A.1.2.4 Finite difference methods
		A.1.3 Numerical optimization
			A.1.3.1 Quadratic programming problem
			A.1.3.2 Non-linear unconstrained optimization
			A.1.3.3 Sequential quadratic programming algorithm
			A.1.3.4 Dynamic programming in discrete time with finite states
	A.2 Statistical and probability analysis
		A.2.1 Probability distributions
			A.2.1.1 The Bernoulli distribution
			A.2.1.2 The binomial distribution
			A.2.1.3 The geometric distribution
			A.2.1.4 The Poisson distribution
			A.2.1.5 The negative binomial distribution
			A.2.1.6 The gamma distribution
			A.2.1.7 The beta distribution
			A.2.1.8 The noncentral chi-squared distribution
			A.2.1.9 The exponential distribution
			A.2.1.10 The normal distribution
			A.2.1.11 The Student’s t distribution
			A.2.1.12 The log-normal distribution
			A.2.1.13 The Pareto distribution
			A.2.1.14 The generalized extreme value distribution
			A.2.1.15 The generalized Pareto distribution
			A.2.1.16 The skew normal distribution
			A.2.1.17 The skew t distribution
			A.2.1.18 The Wishart distribution
		A.2.2 Special results
			A.2.2.1 Affine transformation of random vectors
			A.2.2.2 Change of variables
			A.2.2.3 Relationship between density and quantile functions
			A.2.2.4 Conditional expectation in the case of the normal distribution
			A.2.2.5 Calculation of a useful integral function in credit risk models
	A.3 Stochastic analysis
		A.3.1 Brownian motion and Wiener process
		A.3.2 Stochastic integral
		A.3.3 Stochastic differential equation and Itô’s lemma
			A.3.3.1 Existence and uniqueness of a stochastic differential equation
			A.3.3.2 Relationship with diffusion processes
			A.3.3.3 Itô calculus
			A.3.3.4 Extension to the multidimensional case
		A.3.4 Feynman-Kac formula
		A.3.5 Girsanov theorem
		A.3.6 Fokker-Planck equation
		A.3.7 Reflection principle and stopping times
		A.3.8 Some diffusion processes
			A.3.8.1 Geometric Brownian motion
			A.3.8.2 Ornstein-Uhlenbeck process
			A.3.8.3 Cox-Ingersoll-Ross process
			A.3.8.4 Multidimensional processes
	A.4 Exercises
		A.4.1 Discrete-time random process
		A.4.2 Properties of Brownian motion
		A.4.3 Stochastic integral for random step functions
		A.4.4 Power of Brownian motion
		A.4.5 Exponential of Brownian motion
		A.4.6 Exponential martingales
		A.4.7 Existence of solutions to stochastic differential equations
		A.4.8 Itô calculus and stochastic integration
		A.4.9 Solving a PDE with the Feynman-Kac formula
		A.4.10 Fokker-Planck equation
		A.4.11 Dynamic strategy based on the current asset price
		A.4.12 Strong Markov property and maximum of Brownian motion
		A.4.13 Moments of the Cox-Ingersoll-Ross process
		A.4.14 Probability density function of Heston and SABR models
		A.4.15 Discrete dynamic programming
		A.4.16 Matrix computation
Bibliography
Subject Index
Author Index