THE MATHEMATICS OF MONEY MANAGEMENT:
RISK ANALYSIS TECHNIQUES FOR TRADERS
by Ralph Vince

1. Investment analysis! Mathematics. 
2. Risk management! Mathematics
3. Program trading (Securities)

Index
Parametric Approach..............................................................
Introduction........................................................................................5
Estimating Volatility..........................................................
Scope of this book..........................................................................5
Ruin, Risk and Reality........................................................
Some prevalent misconceptions.......................................................6
Option pricing models........................................................
Worst-case scenarios and stategy.....................................................6
A European options pricing model for all distributions.........
Mathematics notation......................................................................7
The single long option and optimal f...................................
Synthetic constructs in this text.......................................................7
The single short option.......................................................
Optimal trading quantities and optimal f..........................................8
The single position in The Underlying Instrument...............
Chapter 1-The Empirical Techniques...................................................9
Multiple simultaneous positions with a causal relationship...
Deciding on quantity.......................................................................9
Multiple simultaneous positions with a random relationship.
Basic concepts................................................................................9
Chapter 6 - Correlative Relationships and the Derivation of th
The runs test.................................................................................10
Frontier..................................................................................
Serial correlation..........................................................................11
Definition of The Problem..................................................
Common dependency errors..........................................................12
Solutions of Linear Systems using Row-Equivalent Matrice
Mathematical Expectation.............................................................13
Interpreting The Results......................................................
To reinvest trading profits or not...................................................14
Chapter 7 - The Geometry of Portfolios...................................
Measuring a good system for reinvestment the Geometric Mean....14
The Capital Market Lines (CMLs)......................................
How best to reinvest.....................................................................15
The Geometric Efficient Frontier........................................
Optimal fixed fractional trading....................................................15
Unconstrained portfolios.....................................................
Kelly formulas..............................................................................16
How optimal f fits with optimal portfolios...........................
Finding the optimal f by the Geometric Mean................................16
Threshold to The Geometric for Portfolios..........................
To summarize thus far..................................................................17
Completing The Loop.........................................................
Geometric Average Trade.............................................................17
Chapter 8 - Risk Management.................................................
Why you must know your optimal f...............................................18
Asset Allocation.................................................................
The severity of drawdown.............................................................18
Reallocation: Four Methods................................................
Modern portfolio theory................................................................19
Why reallocate?..................................................................
The Markovitz model....................................................................19
Portfolio Insurance ¨ The Fourth Reallocation Technique....
The Geometric Mean portfolio strategy.........................................21
The Margin Constraint........................................................
Daily procedures for using optimal portfolios................................21
Rotating Markets................................................................
Allocations greater than 100%.......................................................22
To summarize....................................................................
How the dispersion of outcomes affects geometric growth.............23
Application to Stock Trading..............................................
The Fundamental Equation of trading............................................24
A Closing Comment...........................................................
Chapter 2 - Characteristics of Fixed Fractional Trading and Salutary
APPENDIX A - The Chi-Square Test......................................
Techniques.......................................................................................26
APPENDIX B - Other Common Distributions.........................
Optimal f for small traders just starting out....................................26
The Uniform Distribution...................................................
Threshold to geometric.................................................................26
The Bernouli Distribution...................................................
One combined bankroll versus separate bankrolls..........................27
The Binomial Distribution..................................................
Threat each play as if infinitely repeated........................................28
The Geometric Distribution................................................
Efficiency loss in simultaneous wagering or portfolio trading.........28
The Hypergeometric Distribution........................................
Time required to reach a specified goal and the trouble with fractional
The Poisson Distribution.....................................................
f...................................................................................................29
The Exponential Distribution..............................................
Comparing trading systems...........................................................30
The Chi-Square Distribution...............................................
Too much sensivity to the biggest loss...........................................30
The Student's Distribution...................................................
Equalizing optimal f.....................................................................31
The Multinomial Distribution.............................................
Dollar averaging and share averaging ideas...................................32
The stable Paretian Distribution..........................................
The Arc Sine Laws and random walks...........................................33
APPENDIX C - Further on Dependency: The Turning Points a
Time spent in a drawdown............................................................34
Length Tests...........................................................................
Chapter 3 - Parametric Optimal f on the Normal Distribution.............35
The basics of probability distributions...........................................35
Descriptive measures of distributions............................................35
Moments of a distribution.............................................................36
The Normal Distribution...............................................................37
The Central Limit Theorem...........................................................38
Working with the Normal Distribution..........................................38
Normal Probabilities.....................................................................39
Further Derivatives of the Normal.................................................41
The Lognormal Distribution..........................................................41
The parametric optimal f...............................................................42
The distribution of trade P&L's.....................................................43
Finding optimal f on the Normal Distribution................................44
The mechanics of the procedure....................................................45
Chapter 4 - Parametric Techniques on Other Distributions..................49
The Kolmogorov-Smirnov (K-S) Test...........................................49
Creating our own Characteristic Distribution Function...................50
Fitting the Parameters of the distribution.......................................52
Using the Parameters to find optimal f...........................................54
Performing "What Ifs"..................................................................56
Equalizing f..................................................................................56
Optimal f on other distributions and fitted curves...........................57
Scenario planning.........................................................................57
Optimal f on binned data...............................................................60
Which is the best optimal f?..........................................................60

Chapter 5 - Introduction to Multiple Simultaneous Positions under the
Index
 
Parametric Approach........................................................................62
Estimating Volatility....................................................................62
Ruin, Risk and Reality..................................................................63
Option pricing models..................................................................63
A European options pricing model for all distributions...................66
The single long option and optimal f.............................................67
The single short option.................................................................70
The single position in The Underlying Instrument.........................71
Multiple simultaneous positions with a causal relationship.............71
Multiple simultaneous positions with a random relationship...........73
Chapter 6 - Correlative Relationships and the Derivation of the Efficient
Frontier............................................................................................74
Definition of The Problem............................................................74
Solutions of Linear Systems using Row-Equivalent Matrices.........77
Interpreting The Results................................................................78
Chapter 7 - The Geometry of Portfolios.............................................81
The Capital Market Lines (CMLs)................................................81
How best to reinvest.....................................................................15
The Geometric Efficient Frontier..................................................82
Unconstrained portfolios...............................................................84
How optimal f fits with optimal portfolios.....................................85
Threshold to The Geometric for Portfolios....................................86
Completing The Loop...................................................................86
Chapter 8 - Risk Management...........................................................89 
Asset Allocation...........................................................................89
Reallocation: Four Methods..........................................................91
Why reallocate?............................................................................93
Portfolio Insurance ¨ The Fourth Reallocation Technique..............93
The Margin Constraint..................................................................96
Rotating Markets..........................................................................97
To summarize..............................................................................97
How the dispersion of outcomes affects geometric growth.............23
Application to Stock Trading........................................................98
A Closing Comment.....................................................................98
Chapter 2 - Characteristics of Fixed Fractional Trading and Salutary
APPENDIX A - The Chi-Square Test..............................................100
APPENDIX B - Other Common Distributions.................................101
The Uniform Distribution...........................................................101

The Bernouli Distribution...........................................................101
The Binomial Distribution..........................................................102
The Geometric Distribution........................................................103
The Hypergeometric Distribution................................................104
Time required to reach a specified goal and the trouble with fractional
The Poisson Distribution.............................................................104
The Exponential Distribution......................................................105
The Chi-Square Distribution.......................................................105
Too much sensivity to the biggest loss...........................................30
The Student's Distribution...........................................................106
The Multinomial Distribution.....................................................107
The stable Paretian Distribution..................................................107
APPENDIX C - Further on Dependency: The Turning Points and Phase
Length Tests...................................................................................109