FOUNDATIONS FOR SCIENTIFIC INVESTING : capital markets intuition and critical... thinking skills.

728 Pages·2022·12.796 MB·English

by  TIMOTHY FALCON CRACK

Note: This is a Print Replica eBook of the 11th edition of Foundations for
Scientific Investing: Capital Markets Intuition and Critical Thinking Skills(January
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TFC/OU/2022
Foundations for Scientific Investing:
Capital Markets Intuition and
Critical Thinking Skills
Foundations for Scientific Investing:
Capital Markets Intuition and
Critical Thinking Skills
Timothy Falcon Crack
PhD (MIT), MCom, PGDipCom, BSc (HONS 1st Class), IMC
©2022 TIMOTHY FALCON CRACK
Allrightsreservedworldwide. Nopartofthisbookmaybereproduced,storedinaretrieval
system, or transmitted in any form or by any means, electronic, mechanical, photocopying,
recording, or otherwise, without the prior written permission of the author.
Theauthoracceptsnoresponsibilityorliabilitywhatsoeverinanyeventincluding(without
limitation) negligence for any damages or loss of any kind, including (without limitation)
direct, indirect, incidental, special or consequential damages, expenses or losses arising out
of,orinconnectionwith,youruseorinabilitytousetheinformationoradviceinthisbook.
Tradingoptions,margintrading,ortradingfuturescontractsareleveredfinancialstrategies
that, likeshortselling, canquicklybringfinancialruinifusedforspeculativepurposes. Op-
tions, margin trading, short selling and futures contracts are not suitable for all investors.
Prior to trading any security, or entering into any trading strategy, consult your broker or
registered investment advisor, and discuss tax effects with your tax advisor.
Published by Timothy Falcon Crack, P.O. Box 6385, Dunedin North, Dunedin 9059, New
Zealand.
Eleventh edition published January 2022.
Coverimage: UnitedStatescoinimagefromtheUnitedStatesMint(usedwithpermission).
ISBN: 978-1-99115542-9
Sometimes regulators may have only the illusion of control when it comes to
large market swings. “The Securities and Exchange Commission has long bent
itseffortstowardthemaintenanceofan‘orderly’market. Thisispraiseworthy,
but they haven’t achieved much in that direction. Only the dull markets have
been orderly. I imagine that the Maritime Commission would like to maintain
an orderly ocean. But when the heavens split and the lightnings crack, one
commission is about as effective as the other, so far as maintaining orderliness
is concerned” Schwed (2006 [1940], pp. 165–166).
Thisremindsmeoftheassertionthat“socialsystemsareinherentlyinsensitive
to most policy changes that people choose in an effort to alter the behavior of
systems” (Forrester, 1971).
Typeset by the author.
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Contents
Preface xiii
Tables xv
Figures 1
Introduction 1
1 Foundations I: Quantitative 5
1.1 The Perfect-Foresight Example . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Imperfect Foresight . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Importance of Dividends . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 Importance of Transaction Costs . . . . . . . . . . . . . . . . 10
1.1.4 Probability of a Gain versus Horizon . . . . . . . . . . . . . . 14
1.1.5 What Did We Overlook?. . . . . . . . . . . . . . . . . . . . . 23
1.1.6 Perfect Foresight Conclusion . . . . . . . . . . . . . . . . . . 23
1.2 Algebra and Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.2.1 Logarithms and Exponentials . . . . . . . . . . . . . . . . . . 24
1.2.2 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.2.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.2.4 First Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.2.5 Numerical Derivatives Example . . . . . . . . . . . . . . . . . 34
1.2.6 Black-Scholes Equation (PDE) . . . . . . . . . . . . . . . . . 36
1.2.7 Continuously Compounded Returns . . . . . . . . . . . . . . 38
1.3 Statistics and Integral Calculus . . . . . . . . . . . . . . . . . . . . . 40
1.3.1 Expected Values . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.3.2 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 43
1.3.3 Standard Normal and Cumulative Standard Normal . . . . . 44
1.3.4 Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . 45
1.3.5 Conditional Expected Values . . . . . . . . . . . . . . . . . . 45
1.3.6 Numeric Integrals . . . . . . . . . . . . . . . . . . . . . . . . 48
1.3.7 Monte Carlo in Science . . . . . . . . . . . . . . . . . . . . . 53
1.3.8 From Normal to Standard Normal, Chi-Square, F, and t . . . 56
1.3.9 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
v
CONTENTS
1.3.10 Higher Moments: Skewness and Kurtosis . . . . . . . . . . . 65
1.3.11 Kurtosis in Stock Returns . . . . . . . . . . . . . . . . . . . . 68
1.3.12 Kurtosis as Peakedness: The Phoenix from the Ashes . . . . 72
1.3.13 Covariance and Correlation . . . . . . . . . . . . . . . . . . . 80
1.3.14 Intuition for General Functional Form of Z and t Tests . . . 92
1.3.15 Three Correlation Tests (Pearson or Spearman) . . . . . . . . 93
1.3.16 Building a t-Statistic to Test the Sample Mean . . . . . . . . 97
1.3.17 How Robust Is the t-Statistic for the Mean? . . . . . . . . . . 104
1.3.18 How Robust Is the t-Statistic for the Correlation? . . . . . . 108
1.3.19 How Robust Is the F-Statistic . . . . . . . . . . . . . . . . . 109
1.3.20 Three Two-Sample t-Tests That Cause Confusion . . . . . . . 110
1.3.21 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . 112
1.3.22 Parametrics and Non-Parametrics in Brief . . . . . . . . . . . 122
1.3.23 Size and Power in Brief . . . . . . . . . . . . . . . . . . . . . 123
1.3.24 Non-Central t-Statistic for the Mean under H . . . . . . . . 124
1
1.3.25 Point Nulls, Power, and Large Samples . . . . . . . . . . . . 128
1.3.26 Errors in Variables and Instrumental Variables . . . . . . . . 130
1.4 Classroom Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 131
1.4.1 The Bean Jar . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
1.4.2 Binomial Gambling . . . . . . . . . . . . . . . . . . . . . . . . 132
1.4.3 Be the Coin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2 Foundations II: Financial Economics 139
2.1 Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.1.1 Compounding and Assumed Distributions . . . . . . . . . . . 140
2.1.2 Merton and the Standard Errors of Means and Variances . . 147
2.2 Time Value of Money Revisited . . . . . . . . . . . . . . . . . . . . . 153
2.2.1 Retirement Lump Sum . . . . . . . . . . . . . . . . . . . . . . 153
2.2.2 Health Insurance: Frequency of Payments . . . . . . . . . . . 157
2.2.3 The Benefit-Adjusted Effective Annual Rate (BEAR) . . . . 158
2.2.4 The Payment-Adjusted Effective Annual Rate (PEAR) . . . 163
2.2.5 Social Security and Life Expectancies by Race/Gender . . . . 166
2.3 Fundamental Analysis and Technical Analysis . . . . . . . . . . . . . 170
2.3.1 Fundamental Is Fundamental . . . . . . . . . . . . . . . . . . 170
2.3.2 Peter Lynch’s “One Up on Wall Street” . . . . . . . . . . . . 172
2.3.3 PVGO and P/E . . . . . . . . . . . . . . . . . . . . . . . . . 174
2.3.4 DDM I: General Model . . . . . . . . . . . . . . . . . . . . . 177
2.3.5 DDM II: Simple Models . . . . . . . . . . . . . . . . . . . . . 178
2.3.6 DDM III: Deeper Models . . . . . . . . . . . . . . . . . . . . 179
2.3.7 DDM IV: DuPont Analysis and DDMs . . . . . . . . . . . . . 185
2.3.8 18 Facts about the Price-Earnings Ratio (P/E) . . . . . . . . 186
2.3.9 Enterprise Value (EV) . . . . . . . . . . . . . . . . . . . . . . 196
2.3.10 Summary of Valuation Techniques . . . . . . . . . . . . . . . 200
2.3.11 A Brief Introduction to Technical Analysis . . . . . . . . . . 202
2.4 Transaction Costs (T-Costs) . . . . . . . . . . . . . . . . . . . . . . . 203
vi
CONTENTS
2.4.1 Types of Orders . . . . . . . . . . . . . . . . . . . . . . . . . 204
2.4.2 Order Precedence/Priority. . . . . . . . . . . . . . . . . . . . 207
2.4.3 Commissions . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
2.4.4 Bid-Ask Spreads . . . . . . . . . . . . . . . . . . . . . . . . . 211
2.4.5 Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
2.4.6 Margin Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 221
2.4.7 Fees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
2.4.8 Price Impact, Sunshine Trades, and Stands in The Market . . 224
2.4.9 Price Slippage and Execution Risk . . . . . . . . . . . . . . . 225
2.4.10 Price Improvement . . . . . . . . . . . . . . . . . . . . . . . . 226
2.4.11 Block Trades and Off-Market Trades . . . . . . . . . . . . . . 229
2.4.12 Short Selling . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
2.4.13 Tax and Wash Sale . . . . . . . . . . . . . . . . . . . . . . . . 231
2.4.14 Payment for Order Flow and Internal Crossing . . . . . . . . 233
2.4.15 Fragmentation and Disintermediation . . . . . . . . . . . . . 233
2.4.16 Extended Example I: Buying a Protective Put . . . . . . . . 235
2.5 Dividends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
2.5.1 Dividend Timelines . . . . . . . . . . . . . . . . . . . . . . . . 243
2.5.2 Statistics on Dividends and Stock Prices . . . . . . . . . . . . 247
2.5.3 Dividend Imputation . . . . . . . . . . . . . . . . . . . . . . . 249
2.5.4 Six Classic Dividend Papers . . . . . . . . . . . . . . . . . . . 251
2.5.5 Six Modern Dividend Papers . . . . . . . . . . . . . . . . . . 253
2.5.6 DDM V: Inferring Expected Returns Using DDMs . . . . . . 255
2.6 Passive Return and Risk I . . . . . . . . . . . . . . . . . . . . . . . . 255
2.6.1 Parameterizing Choice Variables, Return, and Risk . . . . . . 255
2.6.2 Analytical versus Numerical Optimization . . . . . . . . . . . 258
2.6.3 Introduction to Markowitz and Short-Selling Constraints . . 259
2.6.4 Markowitz Efficient Set Mathematics . . . . . . . . . . . . . . 262
2.6.5 CAPM I: Basics . . . . . . . . . . . . . . . . . . . . . . . . . 267
2.6.6 CAPM II: CAPM Tests and the Roll Critique . . . . . . . . . 270
2.7 Active Return and Risk I: Quantitative Approach . . . . . . . . . . . 279
2.7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
2.7.2 Quantitative Active Approach . . . . . . . . . . . . . . . . . . 283
2.7.3 Turning Characteristics into Alphas . . . . . . . . . . . . . . 284
2.7.4 Strategic versus Tactical Asset Allocation . . . . . . . . . . . 286
2.7.5 Risk Aversion, Utility, and a Candidate Objective . . . . . . . 286
2.7.6 First Moment of Return Decomposition . . . . . . . . . . . . 288
2.7.7 Second Moment of Return Decomposition . . . . . . . . . . . 290
2.7.8 A New Objective . . . . . . . . . . . . . . . . . . . . . . . . . 291
2.7.9 CAPM III: Sharpe Ratios and Active Space . . . . . . . . . . 294
2.7.10 Extra Constraints, Turnover, and Neutralization . . . . . . . 301
2.7.11 Reality Check: What Is α? . . . . . . . . . . . . . . . . . . . 304
2.7.12 The Fundamental Law of Active Management (FLAM) . . . 306
2.7.13 Black-Litterman versus Grinold-Kahn . . . . . . . . . . . . . 307
vii
CONTENTS
2.8 Active Return and Risk II: Second Moment of Return Estimation,
Forecast, and Decomposition . . . . . . . . . . . . . . . . . . . . . . 313
2.8.1 Traditional Sample VCV . . . . . . . . . . . . . . . . . . . . 313
2.8.2 Sharpe’s Single-Index Model (SIM) . . . . . . . . . . . . . . . 314
2.8.3 Multiple-Factor Structural Risk Models . . . . . . . . . . . . 316
2.8.4 Constant Correlation Model . . . . . . . . . . . . . . . . . . . 319
2.8.5 Ledoit Estimators . . . . . . . . . . . . . . . . . . . . . . . . 320
2.9 Topics in Retirement . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
2.9.1 A Brief History of Retirement . . . . . . . . . . . . . . . . . . 322
2.9.2 Diversification and the Devil’s Advocate . . . . . . . . . . . . 323
2.9.3 Money Death and Deferred Annuities . . . . . . . . . . . . . 325
2.9.4 Retirement Surveys . . . . . . . . . . . . . . . . . . . . . . . 327
2.9.5 The Costs of Unplanned Early Retirement . . . . . . . . . . . 329
2.9.6 Retaining the Option to Work in Retirement . . . . . . . . . 330
2.9.7 Government Pension Eligibility . . . . . . . . . . . . . . . . . 331
2.9.8 Behavioral Finance & Retirement . . . . . . . . . . . . . . . . 332
2.9.9 Retirement Planning Rules of Thumb . . . . . . . . . . . . . 335
2.9.10 Gender and Retirement . . . . . . . . . . . . . . . . . . . . . 340
2.9.11 Marriage and Retirement . . . . . . . . . . . . . . . . . . . . 341
2.9.12 How Do I Obtain a Healthy, Wealthy Retirement? . . . . . . 342
2.10 Backtests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
2.11 A Classic Case of Data Snooping . . . . . . . . . . . . . . . . . . . . 348
2.12 A Few Derivatives Topics . . . . . . . . . . . . . . . . . . . . . . . . 349
2.12.1 Forwards, Futures, Options, and Swaps . . . . . . . . . . . . 350
2.12.2 Black-Scholes Deviations. . . . . . . . . . . . . . . . . . . . . 351
2.12.3 Option Market Maker Hedging . . . . . . . . . . . . . . . . . 353
2.13 Approaches to Present Value . . . . . . . . . . . . . . . . . . . . . . 355
2.14 Unconventional Wisdom . . . . . . . . . . . . . . . . . . . . . . . . . 358
2.14.1 Equilibrium and No-Arbitrage . . . . . . . . . . . . . . . . . 358
2.14.2 σ versus β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
2.14.3 What Is the Beta of Your Alpha? . . . . . . . . . . . . . . . . 359
2.14.4 Phase Locking and Systemic Risk . . . . . . . . . . . . . . . . 360
2.14.5 Reality Check regarding Rebalancing . . . . . . . . . . . . . . 361
2.14.6 Differences of Opinion Make a Horse Race . . . . . . . . . . . 362
2.14.7 Persistence in Mutual Fund Performance . . . . . . . . . . . 362
2.14.8 Do Skilled Asset Managers Exist? . . . . . . . . . . . . . . . 364
2.14.9 In Defense of Active Equity . . . . . . . . . . . . . . . . . . . 367
2.14.10Personal Asset Allocation . . . . . . . . . . . . . . . . . . . . 374
2.14.11Greed, Fear, Panic, and Liquidity . . . . . . . . . . . . . . . . 376
3 Financial Theories and Empirical Evidence 379
3.1 The Zero-Sum Game and Value Creation . . . . . . . . . . . . . . . 379
3.1.1 The Zero-Sum Game . . . . . . . . . . . . . . . . . . . . . . . 379
3.1.2 Value Creation . . . . . . . . . . . . . . . . . . . . . . . . . . 381
3.2 Random Walk Hypotheses (RWH) . . . . . . . . . . . . . . . . . . . 382
viii
CONTENTS
3.2.1 Marginal Distributions of Returns and a CH Model. . . . . . 383
3.2.2 Time Series Predictability of Returns . . . . . . . . . . . . . 393
3.2.3 Skewness in Stock Returns . . . . . . . . . . . . . . . . . . . 401
3.3 The Efficient Market Hypothesis (EMH) . . . . . . . . . . . . . . . . 406
3.3.1 Martingales, the RWH, and the EMH . . . . . . . . . . . . . 407
3.3.2 Does the EMH Hold? . . . . . . . . . . . . . . . . . . . . . . 409
3.3.3 Should the EMH Hold? . . . . . . . . . . . . . . . . . . . . . 409
3.3.4 Is the EMH Testable? . . . . . . . . . . . . . . . . . . . . . . 411
3.3.5 Is it a Breach of the EMH? . . . . . . . . . . . . . . . . . . . 411
3.4 Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
3.5 Deeper Coverage: Modern Technical Analysis . . . . . . . . . . . . . 414
3.5.1 Reversals and Bid-Ask Bounce . . . . . . . . . . . . . . . . . 414
3.5.2 Lead-Lag Effects, Thin Trading, and Contrarian Profits . . . 416
3.5.3 Price Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 422
3.5.4 Earnings Momentum (Post-Earnings-Announcement Drift) . 426
3.5.5 Modern Technical Analysis Summary . . . . . . . . . . . . . 428
3.6 CAPM IV: Corporate Finance Practice . . . . . . . . . . . . . . . . 429
3.7 CAPM V: Fama-French Critique . . . . . . . . . . . . . . . . . . . . 430
3.7.1 Fama-MacBeth 1973: Cross Section . . . . . . . . . . . . . . 431
3.7.2 Black, Jensen, and Scholes 1972: Time Series . . . . . . . . . 432
3.7.3 Fama-French 1992: Cross Section . . . . . . . . . . . . . . . . 432
3.7.4 Fama-French 1993: Time Series . . . . . . . . . . . . . . . . . 433
3.7.5 Fama-French 1996: Three-Factor Model . . . . . . . . . . . . 436
3.7.6 Carhart 1997: Four-Factor Model . . . . . . . . . . . . . . . . 437
3.7.7 Responses to the Fama-French Critique . . . . . . . . . . . . 437
3.7.8 Relative Importance of Debt and Equity . . . . . . . . . . . . 439
3.7.9 Ferguson and Shockley 2003: EIV . . . . . . . . . . . . . . . 440
3.7.10 Fama-French 2015, 2016, 2017: Five-Factor Model . . . . . . 442
3.8 CAPM VI: Experimental Evidence . . . . . . . . . . . . . . . . . . . 443
3.9 Arbitrage Pricing Theory . . . . . . . . . . . . . . . . . . . . . . . . 447
4 Active Investment Topics 449
4.1 Perennial Favorites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
4.1.1 Passive Return and Risk II: ETFs and Other Managed Funds 449
4.1.2 S&P 500 Indexers, Tracking Error, and Liquidity . . . . . . . 464
4.1.3 How Many Stocks and Funds? . . . . . . . . . . . . . . . . . 464
4.1.4 Behavior of Funds and Investors . . . . . . . . . . . . . . . . 465
4.1.5 Goals of Funds . . . . . . . . . . . . . . . . . . . . . . . . . . 466
4.1.6 Value, Growth, Glamour, and Income . . . . . . . . . . . . . 467
4.1.7 International Diversification . . . . . . . . . . . . . . . . . . . 470
4.1.8 Hedge Funds and Long-Short Funds . . . . . . . . . . . . . . 470
4.1.9 The Correlation between the Returns to Stocks and Bonds . 473
4.1.10 Defined Benefit (DB) and Defined Contribution (DC) Plans . 477
4.1.11 DRIPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
4.1.12 Cash Drag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
ix
CONTENTS
4.1.13 T-Costs for Different Types of Trades . . . . . . . . . . . . . 484
4.1.14 Accruals and Earnings Quality . . . . . . . . . . . . . . . . . 485
4.1.15 Active Return and Risk III: More on Alpha Scaling. . . . . . 487
4.1.16 The Diversification Puzzle . . . . . . . . . . . . . . . . . . . . 488
4.1.17 The Lottery . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
4.1.18 Insider Trading . . . . . . . . . . . . . . . . . . . . . . . . . . 491
4.2 Fads and Fashions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
4.2.1 History and Intertwined Global Regime Changes . . . . . . . 492
4.2.2 Absolute Returns . . . . . . . . . . . . . . . . . . . . . . . . . 501
4.2.3 Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
4.2.4 Bonds Do Not Always Behave like Bonds . . . . . . . . . . . 503
4.2.5 Cash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
4.2.6 Commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
4.2.7 Emerging and Frontier Markets . . . . . . . . . . . . . . . . . 511
4.2.8 Fundamental Indexation . . . . . . . . . . . . . . . . . . . . . 512
4.2.9 Low-Beta versus Low-Volatility . . . . . . . . . . . . . . . . . 514
4.2.10 Smart Beta and Factor-Based Investing . . . . . . . . . . . . 515
4.2.11 Liability-Driven Investing (LDI) . . . . . . . . . . . . . . . . 518
4.2.12 Options and Buy-Write (Covered Call) . . . . . . . . . . . . . 520
4.2.13 Portable Alpha: Removing the Passive Return . . . . . . . . 521
4.2.14 Real Estate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
4.2.15 Securities Lending . . . . . . . . . . . . . . . . . . . . . . . . 525
4.2.16 SRI, ESG, and EDI . . . . . . . . . . . . . . . . . . . . . . . 528
4.2.17 ...But It Is Different This Time . . . . . . . . . . . . . . . . . 530
4.3 Trading for Your Personal Account . . . . . . . . . . . . . . . . . . . 530
4.3.1 Who Controls the Prices? . . . . . . . . . . . . . . . . . . . . 531
4.3.2 Extended Example II: Trading Options versus Margin Trading 533
4.3.3 Exchanges and Regulations . . . . . . . . . . . . . . . . . . . 543
4.3.4 A Few Cautionary Remarks About Brokers . . . . . . . . . . 545
4.3.5 Information Sources . . . . . . . . . . . . . . . . . . . . . . . 547
4.3.6 Market Views and Opinions . . . . . . . . . . . . . . . . . . . 547
4.3.7 The Greeks in Practice . . . . . . . . . . . . . . . . . . . . . 548
4.3.8 Black-Scholes Assumptions and Violations . . . . . . . . . . . 548
4.3.9 The Deathly Slow Crawl . . . . . . . . . . . . . . . . . . . . . 549
4.3.10 Trading Experiences and Lessons . . . . . . . . . . . . . . . . 550
4.3.11 Other Trading Tips and Tools. . . . . . . . . . . . . . . . . . 551
4.3.12 Paratrading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
A Assorted Appendices 557
A.1 Central and Non-Central Moments of the Normal . . . . . . . . . . . 557
A.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
A.3 Eigen Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
A.4 Elliptic and Logistic Distributions . . . . . . . . . . . . . . . . . . . 560
A.5 Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
A.6 ARCH/GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
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