Introduction xi
Scope of This Book xi
Some Prevalent Misconceptions xv
Worst-Case Scenarios and Strategy xvi
Mathematics Notation xviii
Synthetic Constructs in This Text xviii
Optimal Trading Quantities and Optimal f xxi
1 The Empirical Techniques 1
Deciding on Quantity 1
Basic Concepts 4
The Runs Test 5
Serial Correlation 9
Common Dependency Errors 14
Mathematical Expectation 16
To Reinvest Trading Profits or Not 20
Measuring a Good System for Reinvestment: The Geometric Mean 21
How Best to Reinvest 25
Optimal Fixed Fractional Trading 26
Kelly Formulas 27
Finding the; Optimal f by the Geometric Mean 30
To Summarize Thus Far 32
Geometric Average Trade 34
Why You Must Know Your Optimal f 35
The Severity of Drawdown 38
Modern Portfolio Theory 39
The Markowitz Model 40
The Geometric Mean Portfolio Strategy 45
Daily Procedures for Using Optimal Portfolios 46
Allocations Greater Than 100% 49
How the Dispersion of Outcomes Affects Geometric Growth
The Fundamental Equation of Trading 58
53
2 Characteristics of Fixed Fractional Trading
and Salutary Techniques
Optimal f for Small Traders Just Starting Out 63
Threshold to Geometric 65
One Combined Bankroll versus Separate Bankrolls 68
Treat Each Play As If Infinitely Repeated 71
Efficiency Loss in Simultaneous Wagering or Portfolio Trading 73
Time Required to Reach a Specified Goal and
the Trouble with Fractional f 76
Comparing Trading Systems 80
Too Much Sensitivity to the Biggest Loss 82
Equalizing Optimal f 83
Dollar Averaging and Share Averaging Ideas 89
The Arc Sine Laws and Random Walks 92
Time Spent in a Drawdown 95
63
3 Parametric Optimal f on the Normal Distribution
The Basics of Probability Distributions 98
Descriptive Measures of Distributions 200
Moments of a Distribution 103
The Normal Distribution 108
The Central Limit Theorem 109
Working with the Normal Distribution 111
Normal Probabilities 115
The Lognormal Distribution 124
The Parametric Optimal f 125
Finding the Optimal f on the Normal Distribution 132
•1 Parametric Techniques on Other DistributionsThe Kolmogorov-Smirnov (K-S) Test 149
Creating Our Own Characteristic Distribution Function
Fitting the Parameters of the Distribution 160
Using the Parameters to Find the Optimal f 168
Performing "What Ifs" 175
Equalizing f 176
Optimal f on Other Distributions and Fitted Curves 177
Scenario Planning 178
Optimal f on Binned Data 190
Which is the Best Optimal f? 192
153
5 Introduction to Multiple Simultaneous
Positions under the Parametric Approach 193
Estimating Volatility 194
Ruin, Risk, and Reality 197
Option Pricing Models 199
A European Options Pricing Model for All Distributions 208
The Single Long Option and Optimal f 213
The Single Short Option 224
The Single Position in the Underlying Instrument 225
Multiple Simultaneous Positions with a Causal Relationship 228
Multiple Simultaneous Positions with a Random Relationship 233
6 Correlative Relationships and the
Derivation of the Efficient Frontier 237
Definition of the Problem 238
Solutions of Linear Systems Using Row-Equivalent Matrices 250
Interpreting the Results 258
7 The Geometry of Portfolios
The Capital Market Lines (CMLs) 266
The Geometric Efficient Frontier 271
Unconstrained Portfolios 278
How Optimal f Fits with Optimal Portfolios 283
Threshold to the Geometric for Portfolios 287
Completing the Loop 2878 Risk Management
Asset Allocation 294
Reallocation: Four Methods 302
Why Reallocate? 311
Portfolio Insurance—The Fourth Reallocation Technique
The Margin Constraint 320
Rotating Markets 324
To Summarize 326
Application to Stock Trading 327
A Closing Comment 328