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A Trader's Guide To Quantitative Methods (Don’t Panic ):

Basic Maths & Stats

Table of Contents

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Table of Contents - CHAPTER Listing

 

 

0      Foreword and Highlights of Key Points for this Series
1      A Trader’s Guide to Quantitative Methods: Don’t Panic
 
PART I – Basic Maths
 
2      Basic Maths (slopes, curves and wobbly bits)
3      Slopes and Derivatives
4      Sums and integrals
5      Basic Statistics – Measuring the Uncertainty
6      Probability Basics – Expectations & Distributions
7      Matrix methods
8      Expanding the Complex - Series Representation
 
PART II – Selected Applied Maths
 
9      (Almost) everything you need to know about Curves & Surfaces
10    Root Finding
11    Valuation and Risk Under Uncertainty – Part 1
12    Stochastic Calculus (that even traders can understand)
13    Valuation Under Uncertainty – Part 2
14    Introduction to Time Series Analysis and Dynamics

15    Introduction to Optimisation
16    A First Description of Four Important Numerical Methods

17    Introduction to Principal Component Analysis

Appendices
References
Subject Index

 

 

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Table of Contents - DETAILED Listing

 

0	Foreword and Highlights of Key Points for this Series
0.1	Why the TG2 Series of Books?
0.2	What is, and is not, important, and who is this for anyway?
0.2.1	Who is this Series for?
0.2.2	Pedagogical issues
0.2.3	Notation, Grammar, Spelling
0.2.4	About Accompanying (free) software, and Commercial Software.
0.2.5	About these books and relationship to ARTSchool
0.3	Future direction & Road Map for the Series.
0.4	About the Authors
0.4.1	Invitation for Contribution

1	A Trader’s Guide to Quantitative Methods:  Don’t Panic
1.1	Objectives
1.2	Overview
1.3	Quantitative methods and the derivatives business
1.4	Various classes of methods for different jobs
1.5	Various methods within each class of methods
1.6	Modelling of the world vs. solution of the models

PART I – Basic Maths

2	Basic Maths (slopes, curves, & wobbly bits)

3	Slopes and Derivatives
3.1	Slopes – What are they and why are they so important?
3.2	Slopes and Derivatives
3.2.1	Slopes and Derivatives as  “Speeds” & “Straight-lines”
3.2.2	Types of Slopes: Chord and Tangent (derivative) Slopes
3.2.3	The Derivative: A Slope “in the limit”
3.2.4	The Mean Value Theorem: If you know all the slopes/changes, do you know the value?
3.2.5	Chord Slope vs. Tangent Slope – Which is Right?
3.2.6	Examples of Slopes: Duration, Delta/Theta, Forward Prices
3.3	Slopes of Slopes – Accelerations, Diffusions, & Curvature
3.3.1	Curvature Measurement: Chord Based vs. Tangent Based
3.3.2	Curvature Examples: Returns, Yield/Vol Arbitrage, Options Gamma
3.4	Higher Order Derivatives
3.4.1	Higher Order Derivatives and “Fundamental” Properties
3.4.2	Slopes as Predictors vs. Alternate Methods
3.5	Generalised Derivatives
3.5.1	Multi-Dimensional (partial) Derivatives
3.5.2	Total Derivatives
3.5.3	Cross Derivatives (and Correlation)
3.5.3   The Jacobian and The Hessian
3.6	Notation and Interpretation
3.6.1	Notation for “Change”
3.6.2	Notation for Slopes/Derivatives
3.6.3	Notation for Higher Order Differentials
3.6.4	Example Higher Order Interacting Differentials: Option “Greeks”
3.7	Differential Equations as “Equations made of Slopes”
3.7.1	If you know all the slopes/changes, do you know the value (again)?
3.7.2	Example: The “forward price” vs. “slope & return”
3.7.3	Example: The “market forecast & uncertainty” as “slopes”
3.7.4	Example: Black-Scholes as “slopes”
3.8	Differentiation vs. (mathematical) Derivatives
3.8.1	(Some) Rules for (Analytical) Differentiation
3.9	Approximating Slopes and Derivatives – Numerical Differentiation
3.9.1	The Most Basic Approximations
3.9.2	Who is Approximating Whom?
3.9.3	Complex Curves and Alternatives
3.10	“Requirements” for (Mathematical) Derivatives
3.10.1	Continuity in the Curve & in the Slope(s)
3.10.2	Types of Singularities
3.10.3	Multiplicity and Uniqueness
3.10.4	Lipschitz Continuity and The “usual calculus”
3.10.5	Other Types of Derivatives
3.11	Treatment of Singularities and Slopes of Discontinuities
3.11.1	Example of an “Easy Singularity” – Forward Rates from a Bootstrapped Yield Curve
3.11.2	Example of a “Intermediate Singularity” – Asian Options
3.11.3	Example of a “Hard Singularity” – Digital Options
3.11.4	Example of a “Very Hard Singularity” – The Markets

4	Sums and Integrals
4.1	Sums, Products, and Areas
4.1.1	Example Areas – Simple Interest
4.1.2	Example Areas – Probability Weighted Expectations/Average
4.1.3	Example Areas – Probability Weighted Options Expectations
4.1.4	Products
4.1.5	Example Areas/Volumes – Speed vs. Accumulation and Position Value
4.1.6	Example Areas/Volumes – Simple Interest with Reinvestment
4.1.7	Products and Sums Example: Cash, Spot & Forward Rates
4.1.8	Products and Sums Example: Bermudian Options Exercise
4.1.9	Products and Sums Example: Arithmetic vs. Geometric Returns
4.2	Integration:  The Sum in the Limit
4.2.1	Illustration via Compound Interest and Spot/Forward Arbitrage
4.2.2	The “area under the curve again” – CumNorm(x) & Cos(x)
4.3	Integration and Boundary Conditions
4.3.1	Definite vs. Indefinite Integrals
4.3.2	Green’s/Stokes’ Theorem – Integration by Parts
4.3.3	Dimension Reduction (or Increase)
4.3.4	Relationship to Differential Equations
4.3.5	Integral Transforms
4.3.6	Greens Functions and Singularities
4.4	(Some) Rules for integration
4.4.1	Relationship to Derivatives
4.4.2	Sample Rules for Specific Function Families
4.5	Interpretations of Sums & Integrals in Trading and Risk Management
4.5.1	IRR vs. Zero-Coupon Curve
4.5.2	Cumulative Probability – Error Function
4.5.3	Cumulative Probability – Option Strike Price
4.5.4	Cumulative Probability – Value-at-Risk (VaR) from Distributions
4.5.5	Expected Option Pay-out Value
4.6	Multi-dimensional Sums and Integrals
4.7	Conditions and “Types” of Integrals
4.8	Quadrature:  Integral Approximation
4.8.1	The Basic Idea
4.8.2	The “area under the curve again” – Convergence Bias
4.8.3	Basis Functions
4.8.4	Example Quadrature – Digital Option
4.8.5	Example Quadrature – Value-at-Risk (VaR) from Histograms
4.8.6	What is Acceptable Numerical Error vs. Business Error?
4.8.7	Integral (Approximation) by “Rank”

5	Basic Statistics – Measuring Uncertainty
5.1	Introduction to Characterising Trading Phenomenon
5.1.1	The Law of Large Numbers and “Chances”
5.1.2	Does the past predict the future, and data sets with “Memory”
5.2	Histograms – Width & Shape vs. Frequency & Probability
5.2.1	Histograms – The Basic Idea
5.2.2	Quantiles – “Direct” Measures
5.2.3	Histograms: Pre- and Post-Processing
5.2.4	Histograms – Trending vs. De-Trending
5.2.5	Example Histograms: British Petroleum – “Data Pre-Processing”
5.2.6	Example Histograms: S&P Index – “The Law of Large Numbers”
5.2.7	Example Histograms: US Bond Credit & Default – “Other Shapes”
5.2.8	Example Histograms: T-Bond Implied & Historical Volatility – “Statistical Smoothening”
5.3	Moments – The Traditional Statistical Measures
5.3.1	The “Usual Suspects”: Average, Variance, Skew, & Kurtosis
5.3.2	Other Types of Central Tendency Summaries
5.3.3	Other Types of Variability Summaries
5.3.4	Example Summary Measures: Credit & Default Data
5.3.5	Example Summary Measures: S&P Index Histories
5.3.6	Relationship to Quantiles, Distributions, and “Calibration”
5.3.7	Geometric Moments
5.4	Weighting and Filtering
5.4.1	Weighting and Filtering Example – VaR
5.5	(Historical) Volatility as an Empirical Statistical Measure
5.5.1	Historical Volatility – Sampling Length, Frequency, & Weighting
5.5.2	Historical Volatility – Inter-Period & Calendar Effects
5.5.3	Historical Volatility – Special Variations (Open/High/Low/Close)
5.6	Correlation and Covariance: Multi-Dimensional Statistics
5.6.1	The Basic Idea – Correlation & Covariance
5.6.2	Example Covariance: A Bond + FX Position
5.6.3	Example Covariance: 2-D Histograms & the Curse of Dimensionality
5.6.4	Independence vs. Zero-Correlation, Causality vs. Correlation, and Orthogonality
5.6.5	Auto/Serial-Correlation
5.6.6	n-Dimensional Covariance (Covariance Matrix with Equity, FX, & IR)
5.6.7	A Few Comments on Covariance Matrices in Trading and Risk Management
5.7	General Moments
5.8	Stationarity
5.8.1	Stationarity Example: FTSE Historical Volatility
5.8.2	Stationarity Example: US Corporate Default/Recovery Rates
5.9	“Moving” Moments
5.9.1	Example: Moving Averages (DJIA, EUR/USD, SIMEX)
5.9.2	Example: Histories of Historical Volatility – FTSE and S&P “vol”
5.9.3	Example: Moving Correlation in Structured Products 
                 (Quanto’s, Convertible Bonds, Hedging, Bond Spreads, Asset Allocation)
5.9.4	Example: Moving Quantiles (VaR/Economic Capital)
5.9.5	Histories of “Historicals” and Serial Correlation
5.9.6	Stationarity vs. Moving Moments in Pricing/Risk Formulations
5.10	Basic Stats – Do’s & Don'ts
 
6 Probability Basics – Expectations & Models of Uncertainty
6.1 Expectations
6.1.1 The Law of Large Numbers (again): Samples vs. Population
6.1.2 Expectations in Trading and Risk Management
6.1.3 Expectations of Expectations
6.2 Probabilities
6.2.1 Intuitive Description vs. Set Theoretic Description
6.2.2 Manipulating Probabilities and Boolean Operators
6.2.3 Probability Spaces & Algebras (Borel, Sigma, etc)
6.3 Expectations and Distributions
6.3.1 Discrete vs. Continuous Processes
6.3.2 Distributions for Random Number Generation and Predictions
6.4 Discrete Expectations and Distributions
6.4.1 Discrete Expectations
6.4.2 Discrete Distributions
6.4.3 The Four Most Important Discrete Distributions (Binomial, General, Hypergeometric, & Poisson)
6.5 Continuous Expectations and Distributions
6.5.1 Continuous Expectations
6.5.2 Continuous Distributions
6.5.3 The Five Most Important Continuous Distributions (Log/Normal, Uniform, c2, Weibull, & General)
6.6 Multi-Dimensional Distributions
6.6.1 Example: Discrete Distributions in 2- Dimensions
6.6.2 Example: Continuous Distributions in 2- Dimensions
6.6.3 Distributions in n- Dimensions
6.7 Expectations and Distributions Summary
6.8 Examples of Using Distributions in Trading and Risk Management
6.8.1 Valuation and Risk of a Bet: Coin & Dice Games
6.8.2 First Valuation of Vanilla and Digital Options
6.8.3 First Valuation of Credit Default Insurance
6.9 First Introduction to Modelling the Markets with Distributions
6.9.1 A 1st “Good” Shape for Uncertainty – Equity Markets
6.9.2 A 1st “Good” Shape for Uncertainty – IR Markets and Term-Structure/Mean Reversion
6.9.3 A 1st “Good” Shape for Uncertainty – FX Markets and Jumps/Channels
6.9.4 A 1st “Good” Shape for Uncertainty – Commodities and Skews/Jumps
6.9.5 A 1st “Good” Shape for Default Uncertainty – Stationarity, Data Quality, & Supply Push
6.9.6 Summary of Some Non-Market Modelling Considerations
6.9.7 Summary of Some Technical Modelling Considerations
6.10 Inverse Distributions
6.11 Approximating Distributions
6.11.1 Distribution Transformations
6.11.2 Fitted Distributions (Quasi-Closed Form Approximations)
6.11.3 Approximating Discrete Distributions
6.11.4 Approximating Continuous Distributions
6.11.5 Continuous Distribution Approximation vs. Discrete Distribution?
6.12 Properties and Manipulation of Distributions
6.12.1 Manipulation: Area Preservation as an Auxiliary Equation
6.12.2 Manipulation: Shifting or De-coupling the Average
6.12.3 Manipulation: Scaling, Stretching, and Area Preservation
6.12.4 Terminology: Prices based vs. Returns based Distribution
6.12.5 The Central Limit Theorem
6.12.6 Quantiles, Standard Deviation, and Multiples Rules
6.13 Moment Generating Functions and the Characteristic Equation
6.14 Statistical Inference
6.14.1 Hypothesis Testing and Statistical Significance
6.14.2 The p-value and z-test: Standard Error, Confidence Intervals, and Statistical Significance.
6.14.3 The p-value: Are Two VaR’s the Same?
6.14.4 The t-test: Are 2 Means the Same?
6.14.5 Example: Compare Two Investment’s Returns: Asset Allocation
6.14.6 The F-test: Are Two Variances the Same?
6.14.7 Example: Compare Two Investment’s Risks: Asset Allocation
6.14.8 The c2–test: Are Two Histograms the Same?
6.14.9 Example: Is the World Normally distributed?
6.14.10 The Kolmogorov-Smirnov-test: Are Two Distributions the Same?
6.14.11 Example: Does Risk/Return Analysis Work in a Non-Gaussian World?
6.15 Lies, Damn Lies, and Statistics – Simpson’s Paradox and “prop trading”
7 Basic Linear Algebra and Matrix Methods 101
7.1 BLA and Matrices: What and Why
7.1.1 Vectors, Arrays, and Matrices
7.1.2 BLA & Matrix Examples: Convenience and Representation
7.1.3 BLA & Matrix Example: Portfolio Covariance/Correlation Matrix
7.1.4 BLA & Matrix Example: Risk-adjusted Return Analysis
7.2 Linearity
7.2.1 “Really” Linear
7.2.2 Linear in Coefficients
7.2.3 Linear Approximation of General Problems
7.3 (Some) Rules of Matrix Arithmetic
7.3.1 Scalar Multiplication
7.3.2 Scalar and Vector Addition
7.3.3 Vector and Matrix Multiplication
7.3.4 Vector “Products”
7.4 Matrix Inversion
7.4.1 Inversion: Direct Methods – LU Decomposition
7.4.2 Inversion: Iterative Methods
7.4.3 Inversion: Choleski Decomposition
7.4.4 Inversion: Sparse Systems
7.5 Matrix Properties
7.5.1 Singularities
7.5.2 Determinants
7.5.3 Eigenvalues and Eigenvectors
8 Expanding the Complex
8.1 Series Expansion & Representation
8.1.1 Polynomial & Power Series Representation
8.1.2 Trigonometric Representation
8.1.3 Taylor Series
8.2 (a few) Series Expansion Notes
8.2.1 Singularities and Higher Order Terms
8.2.2 Digression: Stochastic and Aperiodic (Chaotic/fractal) Problems
8.2.3 Big O and Little O
8.2.4 “Big O”
8.2.5 “little o”
8.2.6 Analyticity
9 Software and Resources for Part I
9.1 General Comments about Software
9.2 Software for Basic Maths & Stats
9.2.1 Types of Software for Basic Maths & Stats
9.2.2 Specific Computational Considerations
9.3 Selected Books, and On-Line Resources
9.3.1 Selected Books & Example On-Line Resources
9.4 Consulting & Educational Services and Facilities
9.5 Data Quality

PART II – Selected Maths

10	(Almost) everything you need to know about Curves & Surfaces
10.1	Two classes of techniques
10.1.1	Interpolation Methods
10.1.2	Approximation Methods
10.1.3	The basic idea
10.1.4	Graphical illustration of differences
10.1.5	The basic idea
10.2	Interpolation Methods
10.2.1	interpolation preliminaries
10.2.2	example: straight line interpolation of two yields
10.2.3	what if data is “curved”
10.2.4	example: exponential interpolation of discount factors
10.2.5	Caveat – Fitting Implied Modelling Assumptions
10.3	Overview of Interpolation methods generalities
10.3.1	what does it mean to “fit” a curve?
10.3.2	What does it mean to “fit” a curve?
10.3.3	Piece-wise vs. global
10.3.4	other interpolation (basis) curves
10.3.5	“fit” a piece-wise curve
10.4	“Better” curve fitting and splines
10.4.1	Description and use of “splines”
10.4.2	What are “splines”?
10.4.3	description and use of “splines”
10.4.4	Application of splines
10.4.5	Splines imply specific assumptions about the “real” world
10.4.6	Splines can be as complex as we like
10.4.7	Calculation of “cubic” splines
10.4.8	The dangers of splines
10.5	Other types of splines
10.5.1	other types of splines - tension splines
10.5.2	other types of splines - adaptive corrective splines
10.5.3	(Bajor’s) Adaptive Corrective Splines
10.5.4	other types of splines – B-Splines
10.5.5	other types of splines – Bezier Splines
10.6	Interpolation Methods Accuracy
10.7	Interpolation Case Study: Cubic splines for swap/yield curves
10.7.1	DEM/EUR Swap curves
10.7.2	implications for market convention
10.7.3	cubic splines in practice
10.8	Approximation methods
10.8.1	When to use
10.8.2	What are approximation methods?
10.8.3	The basic idea (Least Squares)
10.8.4	Approximating in general
10.8.5	Approximating in general: Technical Description
10.8.6	Approximating in general: Goodness of Fit
10.8.7	Illustration by example: duration and volatility
10.8.8	Approximation Methods Accuracy
10.8.9	Goodness of fit
10.8.10	ANOVA
10.8.11	Non-linear Least Squares
10.8.12	Summary of (scalar) interpolation and approximation
10.9	Surfaces
10.10	Surfaces – Regular Data
10.10.1	Interpolation methods
10.10.2	Approximation methods
10.10.3	Surfaces- Tiling
10.11	Surfaces – Irregular Data
10.12	Multi-dimensional Surface Fitting
10.13	Case Studies
10.13.1	Are Bonds Cheap/Dear?
10.13.2	Volatility Cones
10.13.3	T-Structure of Swaptions Volatility
10.13.4	Options Arbitrage
10.14	Other considerations
10.14.1	Numerical implications
10.14.2	Problems with surface fitting
10.15	Surfaces - Approximation of Irregular Noisy Data
10.16	Summary of Curve and Surface Fitting

11	 The Answer is in the “Root”
11.1	Root finding is an important technique used in:
11.2	The idea
11.3	Illustration by example
–	yield from bond price
–	implied volatility
11.4	Two popular techniques
11.5	Bisection
11.5.2	Calculating the IRR from cash flows
11.6	Newton’s Method
11.7	Multi dimensional problems
11.8	Properties of the methods
11.9	Root Finding as (a kind of) optimisation
11.10	Summary of Root Finding

12	Valuation and Risk Under Uncertainty – Part 1
12.1	Overview of the underlying issues
12.1.1	Mathematical modelling of securities prices
12.1.2	Some Simplifying Assumptions for Economics and Finance
12.2	Modelling the Price/Returns processes – The Basic Idea
12.2.1	The “Goal”
12.2.2	A First Qualitative Model of Price Dynamics: Trends and Wobbles.
12.3	Developing a First Model for Valuation Under Uncertainty
12.3.1	Down to earth explanation of the terminology
12.3.2	Expectations
12.3.3	Re-coupling and De-coupling Drift
12.3.4	Expectations and Distributions Summary
12.3.5	Making a Bet or Pricing an Option? – 1st (Crude) Valuations under Uncertainty
12.3.6	A First Quasi Forecasting Model of Uncertainty
12.3.7	Forecasting Prices vs. Forecasting Returns Re-visited
12.4	Developing an Extended & Working First Model of Uncertainty
12.4.1	A First “Good” Shape of Uncertainty
12.4.2	A First Calibration of the Uncertainty
12.4.3	A First Model for the Time Evolution of Uncertainty
12.4.4	Formalising the First (proper) Model of Uncertainty
12.4.5	Calibration of Uncertainty and Annulisation
12.5	Putting it all together: A First “Complete” Model for Valuation Under Uncertainty
12.5.1	First Valuation of Digital Options
12.6	Other Models for Valuation Under Uncertainty
12.7	Summary: A First “Complete” Model for Valuation Under Uncertainty

13	 Stochastic Calculus (that even traders can understand)
13.1	Ito’s lemma
13.2	Martingale
13.3	Markov
13.4	IR/Term Structure/HJM
13.5	Girsanov/Change of Measure

14	 Valuation Under Uncertainty – Part 2
14.1	Highlights of Key Points
14.2	Representations of the Price Process
14.3	The price process in discrete time steps
14.3.1	Guide-lines for choosing a price process
14.3.2	The Usual Suspects
14.4	Gaussian processes
14.4.1	Arithmetic Brownian motion
14.4.2	Geometric Brownian motion (our old friend)
14.4.3	A mean reverting processes
14.4.4	Jump Diffusion
14.5	Generalised distributions
14.5.1	Some real world effects
14.5.2	Contract Effects
14.6	Special Cases
14.6.1	Multiple underlying indices and correlation
14.6.2	Description of the problem
14.6.3	Multi-Index/Multi-Factor derivatives
14.6.4	Spread Options
14.6.5	Stochastic Volatility
14.6.6	Step-by-Step Illustration of a 2-Factor Problem: Convertible Bonds
14.7	Term-Structure Issues
14.7.1	A simple One-Factor Model
14.7.2	A simple Two-Factor Model

15	Introduction to Time Series Analysis and Dynamics
15.1	Time Series Analyses - Econometrics
15.1.1	Traditional Time Series Analysis
15.1.2	Econometrics and Fundamental TSA
15.1.3	GARCH and Volatility Forecasting
15.2	Time Series Analyses - Dynamics
15.2.1	Periodic vs. Aperiodic Systems and State Space
15.2.2	Periodic Processes and Spectral Analysis
15.2.3	Predicting Markets with Fast Fourier Transforms
15.2.4	Aperiodic Processes, Fractals/Chaos and Non-Linear Dynamics
15.2.5	Valuation and Risk Management with Aperiodic Models
 
16	Introduction to Optimisation
16.1    Mathematically Optimal vs. P&L Optimal
16.2    Snapshot Optimal vs. Holding-Period Optimal
16.3    Linear Programming and Optimal Rebalances
16.4    Efficient Frontiers and Quadratic Optimisation
16.5    Optimal Portfolio Holding Period Risk-Adjusted P&L with PaR and Simulation

17	A First Description of Four Important Numerical Methods
17.1	Monte Carlo Methods
17.1.1	Pricing a Vanilla Option with Monte Carlo
17.1.1	Pricing a Barrier Option with Monte Carlo 
17.2	Trees and Lattices
17.2.1	Binomial Tree Implementation of Vanilla Options
17.2.2	Binomial Tree Implementation of Black-Derman-Toy Term-Structure Bonds/Bond Options
17.3	Finite Difference Methods
17.3.1	Explicit FD Options Pricing
17.3.2	Implicit FD Options Pricing
17.3.3	FD Pricing of Complex Structures/Options
17.3.4	FD and a General/Homogenous Valuation & Risk Framework
17.4	Finite Element, Variational, and Boundary Methods
17.4.1	Variational Principals
17.4.2	Galerkin Methods
17.4.3	Finite Elements and Pricing a Vanilla Option
17.4.4	Boundary Methods, and Non-linear Least Squares with Arrow-Debreu

18	Introduction to Principal Component Analysis
18.1	Eigenvalues and Coordinate Systems Revisited
18.2	PCA Example: Term-Structure Models - "how many factors are enough?" 

Appendices
Appendix A: Notation/Abbreviations
Abbreviations
Currencies
Greek Letters
Alphanumeric Letters
Mathematical Operators
Appendix B: ARTicles
B1: ARTicles: How much to pay a trader?
B2: ARTicles: Credit Default Swaps
B3: ARTicles: Term-Structure Calibration: Nonsense & Reality

References

Subject Index

 

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