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