Nonparametric Finance

An Introduction to Machine Learning in Finance, With Mathematical Background, Data Visualization, and R Nonparametric function estimation is an important part of machine learning, which is becoming increasingly important in quantitative finance. Nonparametric Finance provides graduate students and finance professionals with a foundation in nonparametric function estimation and the underlying mathematics. Combining practical applications, mathematically rigorous presentation, and statistical data analysis into a single volume, this book presents detailed instruction in discrete chapters that allow readers to dip in as needed without reading from beginning to end. Coverage includes statistical finance, risk management, portfolio management, and securities pricing to provide a practical knowledge base, and the introductory chapter introduces basic finance concepts for readers with a strictly mathematical background. Economic significance is emphasized over statistical significance throughout, and R code is provided to help readers reproduce the research, computations, and figures being discussed. Strong graphical content clarifies the methods and demonstrates essential visualization techniques, while deep mathematical and statistical insight backs up practical applications. Written for the leading edge of finance, Nonparametric Finance: • Introduces basic statistical finance concepts, including univariate and multivariate data analysis, time series analysis, and prediction • Provides risk management guidance through volatility prediction, quantiles, and value-at-risk • Examines portfolio theory, performance measurement, Markowitz portfolios, dynamic portfolio selection, and more • Discusses fundamental theorems of asset pricing, Black-Scholes pricing and hedging, quadratic pricing and hedging, option portfolios, interest rate derivatives, and other asset pricing principles • Provides supplementary R code and numerous graphics to reinforce complex content Nonparametric function estimation has received little attention in the context of risk management and option pricing, despite its useful applications and benefits. This book provides the essential background and practical knowledge needed to take full advantage of these little-used methods, and turn them into real-world advantage. Jussi Klemelä, PhD, is Adjunct Professor at the University of Oulu. His research interests include nonparametric function estimation, density estimation, and data visualization. He is the author of Smoothing of Multivariate Data: Density Estimation and Visualization and Multivariate Nonparametric Regression and Visualization: With R and Applications to Finance.

Preface xiii  1 Introduction 1  1.1 Statistical Finance 2  1.2 Risk Management 3  1.3 Portfolio Management 5  1.4 Pricing of Securities 6  Part I Statistical Finance 11  2 Financial Instruments 13  2.1 Stocks 13  2.1.1 Stock Indexes 14  2.1.2 Stock Prices and Returns 15  2.2 Fixed Income Instruments 19  2.2.1 Bonds 19  2.2.2 Interest Rates 20  2.2.3 Bond Prices and Returns 22  2.3 Derivatives 23  2.3.1 Forwards and Futures 23  2.3.2 Options 24  2.4 Data Sets 27  2.4.1 Daily S&P 500 Data 27  2.4.2 Daily S&P 500 and Nasdaq-100 Data 28  2.4.3 Monthly S&P 500, Bond, and Bill Data 28  2.4.4 Daily US Treasury 10 Year Bond Data 29  2.4.5 Daily S&P 500 Components Data 30  3 Univariate Data Analysis 33  3.1 Univariate Statistics 34  3.1.1 The Center of a Distribution 34  3.1.2 The Variance and Moments 37  3.1.3 The Quantiles and the Expected Shortfalls 40  3.2 Univariate Graphical Tools 42  3.2.1 Empirical Distribution Function Based Tools 43  3.2.2 Density Estimation Based Tools 53  3.3 Univariate Parametric Models 55  3.3.1 The Normal and Log-normal Models 55  3.3.2 The Student Distributions 59  3.4 Tail Modeling 61  3.4.1 Modeling and Estimating Excess Distributions 62  3.4.2 Parametric Families for Excess Distributions 65  3.4.3 Fitting the Models to Return Data 74  3.5 Asymptotic Distributions 83  3.5.1 The Central Limit Theorems 84  3.5.2 The Limit Theorems for Maxima 88  3.6 Univariate Stylized Facts 91  4 Multivariate Data Analysis 95  4.1 Measures of Dependence 95  4.1.1 Correlation Coefficients 97  4.1.2 Coefficients of Tail Dependence 101  4.2 Multivariate Graphical Tools 103  4.2.1 Scatter Plots 103  4.2.2 Correlation Matrix: Multidimensional Scaling 104  4.3 Multivariate Parametric Models 107  4.3.1 Multivariate Gaussian Distributions 107  4.3.2 Multivariate Student Distributions 107  4.3.3 Normal Variance Mixture Distributions 108  4.3.4 Elliptical Distributions 110  4.4 Copulas 111  4.4.1 Standard Copulas 111  4.4.2 Nonstandard Copulas 112  4.4.3 Sampling from a Copula 113  4.4.4 Examples of Copulas 116  5 Time Series Analysis 121  5.1 Stationarity and Autocorrelation 122  5.1.1 Strict Stationarity 122  5.1.2 Covariance Stationarity and Autocorrelation 126  5.2 Model Free Estimation 128  5.2.1 Descriptive Statistics for Time Series 129  5.2.2 Markov Models 129  5.2.3 Time Varying Parameter 130  5.3 Univariate Time Series Models 135  5.3.1 Prediction and Conditional Expectation 135  5.3.2 ARMA Processes 136  5.3.3 Conditional Heteroskedasticity Models 143  5.3.4 Continuous Time Processes 154  5.4 Multivariate Time Series Models 157  5.4.1 MGARCH Models 157  5.4.2 Covariance in MGARCH Models 159   5.5 Time Series Stylized Facts 160  6 Prediction 163  6.1 Methods of Prediction 164  6.1.1 Moving Average Predictors 164  6.1.2 State Space Predictors 166  6.2 Forecast Evaluation 170  6.2.1 The Sum of Squared Prediction Errors 170  6.2.2 Testing the Prediction Accuracy 172  6.3 Predictive Variables 175  6.3.1 Risk Indicators 175  6.3.2 Interest Rate Variables 177  6.3.3 Stock Market Indicators 178  6.3.4 Sentiment Indicators 180  6.3.5 Technical Indicators 180  6.4 Asset Return Prediction 182  6.4.1 Prediction of S&P 500 Returns 184  6.4.2 Prediction of 10-Year Bond Returns 187  Part II Risk Management 193  7 Volatility Prediction 195  7.1 Applications of Volatility Prediction 197  7.1.1 Variance and Volatility Trading 197  7.1.2 Covariance Trading 197  7.1.3 Quantile Estimation 198  7.1.4 Portfolio Selection 199  7.1.5 Option Pricing 199  7.2 Performance Measures for Volatility Predictors 199  7.3 Conditional Heteroskedasticity Models 200  7.3.1 GARCH Predictor 200  7.3.2 ARCH Predictor 203  7.4 Moving Average Methods 205  7.4.1 Sequential Sample Variance 205  7.4.2 Exponentially Weighted Moving Average 207  7.5 State Space Predictors 211  7.5.1 Linear Regression Predictor 212  7.5.2 Kernel Regression Predictor 214  8 Quantiles and Value-at-Risk 219  8.1 Definitions of Quantiles 220  8.2 Applications of Quantiles 223  8.2.1 Reserve Capital 223  8.2.2 Margin Requirements 225  8.2.3 Quantiles as a Risk Measure 226  8.3 Performance Measures for Quantile Estimators 227  8.3.1 Measuring the Probability of Exceedances 228  8.3.2 A Loss Function for Quantile Estimation 231  8.4 Nonparametric Estimators of Quantiles 233  8.4.1 Empirical Quantiles 234  8.4.2 Conditional Empirical Quantiles 238  8.5 Volatility Based Quantile Estimation 240  8.5.1 Location–Scale Model 240  8.5.2 Conditional Location–Scale Model 245  8.6 Excess Distributions in Quantile Estimation 258  8.6.1 The Excess Distributions 259  8.6.2 Unconditional Quantile Estimation 261  8.6.3 Conditional Quantile Estimators 269  8.7 Extreme Value Theory in Quantile Estimation 288  8.7.1 The Block Maxima Method 288  8.7.2 Threshold Exceedances 289  8.8 Expected Shortfall 292  8.8.1 Performance of Estimators of the Expected Shortfall 292  8.8.2 Estimation of the Expected Shortfall 293  Part III Portfolio Management 297  9 Some Basic Concepts of Portfolio Theory 299  9.1 Portfolios and Their Returns 300  9.1.1 Trading Strategies 300  9.1.2 The Wealth and Return in the One- Period Model 301  9.1.3 The Wealth Process in the Multiperiod Model 304  9.1.4 Examples of Portfolios 306  9.2 Comparison of Return and Wealth Distributions 312  9.2.1 Mean–Variance Preferences 313  9.2.2 Expected Utility 316  9.2.3 Stochastic Dominance 325  9.3 Multiperiod Portfolio Selection 326  9.3.1 One-Period Optimization 328  9.3.2 The Multiperiod Optimization 329  10 Performance Measurement 337  10.1 The Sharpe Ratio 338  10.1.1 Definition of the Sharpe Ratio 338  10.1.2 Confidence Intervals for the Sharpe Ratio 340  10.1.3 Testing the Sharpe Ratio 343  10.1.4 Other Measures of Risk-Adjusted Return 345  10.2 Certainty Equivalent 346  10.3 Drawdown 347  10.4 Alpha and Conditional Alpha 348  10.4.1 Alpha 349  10.4.2 Conditional Alpha 355  10.5 Graphical Tools of Performance Measurement 356  10.5.1 Using Wealth in Evaluation 356  10.5.2 Using the Sharpe Ratio in Evaluation 359  10.5.3 Using the Certainty Equivalent in Evaluation 364  11 Markowitz Portfolios 367  11.1 Variance Penalized Expected Return 369  11.1.1 Variance Penalization with the Risk-Free Rate 369  11.1.2 Variance Penalization without the Risk-Free Rate 371  11.2 Minimizing Variance under a Sufficient Expected Return 372  11.2.1 Minimizing Variance with the Risk-Free Rate 372  11.2.2 Minimizing Variance without the Risk-Free Rate 374  11.3 Markowitz Bullets 375  11.4 Further Topics in Markowitz Portfolio Selection 380  11.4.1 Estimation 380  11.4.2 Penalizing Techniques 381  11.4.3 Principal Components Analysis 382  11.5 Examples of Markowitz Portfolio Selection 383  12 Dynamic Portfolio Selection 385  12.1 Prediction in Dynamic Portfolio Selection 387  12.1.1 Expected Returns in Dynamic Portfolio Selection 387  12.1.2 Markowitz Criterion in Dynamic Portfolio Selection 390  12.1.3 Expected Utility in Dynamic Portfolio Selection 391  12.2 Backtesting Trading Strategies 393  12.3 One Risky Asset 394  12.3.1 Using Expected Returns with One Risky Asset 394  12.3.2 Markowitz Portfolios with One Risky Asset 401  12.4 Two Risky Assets 405  12.4.1 Using Expected Returns with Two Risky Assets 405  12.4.2 Markowitz Portfolios with Two Risky Assets 409  Part IV Pricing of Securities 419  13 Principles of Asset Pricing 421  13.1 Introduction to Asset Pricing 422  13.1.1 Absolute Pricing 423  13.1.2 Relative Pricing Using Arbitrage 424  13.1.3 Relative Pricing Using Statistical Arbitrage 428  13.2 Fundamental Theorems of Asset Pricing 430  13.2.1 Discrete Time Markets 431  13.2.2 Wealth and Value Processes 432  13.2.3 Arbitrage and Martingale Measures 436  13.2.4 European Contingent Claims 448  13.2.5 Completeness 451  13.2.6 American Contingent Claims 454  13.3 Evaluation of Pricing and Hedging Methods 456  13.3.1 The Wealth of the Seller 456  13.3.2 The Wealth of the Buyer 458  14 Pricing by Arbitrage 459  14.1 Futures and the Put–Call Parity 460  14.1.1 Futures 460  14.1.2 The Put–Call Parity 464  14.1.3 American Call Options 465  14.2 Pricing in Binary Models 466  14.2.1 The One-Period Binary Model 467  14.2.2 The Multiperiod Binary Model 470  14.2.3 Asymptotics of the Multiperiod Binary Model 475  14.2.4 American Put Options 484  14.3 Black–Scholes Pricing 485  14.3.1 Call and Put Prices 485  14.3.2 Implied Volatilities 495  14.3.3 Derivations of the Black–Scholes Prices 498  14.3.4 Examples of Pricing Using the Black–Scholes Model 501  14.4 Black–Scholes Hedging 505  14.4.1 Hedging Errors: Nonsequential Volatility Estimation 506  14.4.2 Hedging Frequency 508  14.4.3 Hedging and Strike Price 511  14.4.4 Hedging and Expected Return 512  14.4.5 Hedging and Volatility 514  14.5 Black–Scholes Hedging and Volatility Estimation 515  14.5.1 Hedging Errors: Sequential Volatility Estimation 515  14.5.2 Distribution of Hedging Errors 517  15 Pricing in Incomplete Models 521  15.1 Quadratic Hedging and Pricing 522  15.2 Utility Maximization 523  15.2.1 The Exponential Utility 524  15.2.2 Other Utility Functions 525  15.2.3 Relative Entropy 526  15.2.4 Examples of Esscher Prices 527  15.2.5 Marginal Rate of Substitution 529  15.3 Absolutely Continuous Changes of Measures 530  15.3.1 Conditionally Gaussian Returns 530  15.3.2 Conditionally Gaussian Logarithmic Returns 532  15.4 GARCH Market Models 534  15.4.1 Heston–Nandi Method 535  15.4.2 The Monte Carlo Method 539  15.4.3 Comparison of Risk-Neutral Densities 541  15.5 Nonparametric Pricing Using Historical Simulation 545  15.5.1 Prices 545  15.5.2 Hedging Coefficients 548  15.6 Estimation of the Risk-Neutral Density 551  15.6.1 Deducing the Risk-Neutral Density from Market Prices 552  15.6.2 Examples of Estimation of the Risk-Neutral Density 552  15.7 Quantile Hedging 554  16 Quadratic and Local Quadratic Hedging 557  16.1 Quadratic Hedging 558  16.1.1 Definitions and Assumptions 559  16.1.2 The One Period Model 562  16.1.3 The Two Period Model 569  16.1.4 The Multiperiod Model 575  16.2 Local Quadratic Hedging 583  16.2.1 The Two Period Model 583  16.2.2 The Multiperiod Model 587  16.2.3 Local Quadratic Hedging without Self-Financing 593  16.3 Implementations of Local Quadratic Hedging 595  16.3.1 Historical Simulation 596  16.3.2 Local Quadratic Hedging Under Independence 599  16.3.3 Local Quadratic Hedging under Dependence 604  16.3.4 Evaluation of Quadratic Hedging 610  17 Option Strategies 615  17.1 Option Strategies 616  17.1.1 Calls, Puts, and Vertical Spreads 616  17.1.2 Strangles, Straddles, Butterflies, and Condors 619  17.1.3 Calendar Spreads 621  17.1.4 Combining Options with Stocks and Bonds 623  17.2 Profitability of Option Strategies 625  17.2.1 Return Functions of Option Strategies 626  17.2.2 Return Distributions of Option Strategies 634  17.2.3 Performance Measurement of Option Strategies 644  18 Interest Rate Derivatives 649  18.1 Basic Concepts of Interest Rate Derivatives 650  18.1.1 Interest Rates and a Bank Account 651  18.1.2 Zero-Coupon Bonds 653  18.1.3 Coupon-Bearing Bonds 656  18.2 InterestRateForwards 659  18.2.1 Forward Zero-Coupon Bonds 659  18.2.2 Forward Rate Agreements 661  18.2.3 Swaps 663  18.2.4 Related Fixed Income Instruments 665  18.3 Interest Rate Options 666  18.3.1 Caplets and Floorlets 666  18.3.2 Caps and Floors 668  18.3.3 Swaptions 668  18.4 Modeling Interest Rate Markets 669  18.4.1 HJM Model 670  18.4.2 Short-Rate Models 671  References 673  Index 681