0 Foreword and Highlights for the
TG2 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 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 Foreword and Highlights for
TG2QM: Monte Carlo Methods
1.1
Objectives
1.2
Quantitative methods and the trading business
1.3 Modelling
of the World vs. Solution of the Models
PART I – Basic Maths
2 Slopes and Derivatives
2.1 Slopes – What
are they and why are they so important?
2.2
Slopes and Derivatives
2.2.1 Slopes and Derivatives as
“Speeds” & “Straight-lines”
2.2.2 Types of Slopes: Chord and
Tangent (derivative) Slopes
2.2.3 The Derivative: A Slope “in the
limit”
2.2.4 The Mean Value Theorem: If you
know the slopes, you know the value?
2.2.5 Chord Slope vs. Tangent Slope –
Which is Right?
2.2.6 Examples of Slopes:
Duration, Delta/Theta, Forward Prices
2.3 Slopes of
Slopes – Accelerations, Diffusions, & Curvature
2.3.1 Curvature Measurement:
Chord Based vs. Tangent Based
2.3.2 Curvature Examples:
Returns, Yield/Vol Arbitrage, Options Gamma
2.4
Higher Order Derivatives
2.4.1 Higher Order Derivatives and
“Fundamental” Properties
2.4.2 Slopes as Predictors vs.
Alternate Methods
2.5
Generalised Derivatives
2.5.1 Multi-Dimensional (partial)
Derivatives
2.5.2 Total Derivatives
2.5.3 The Jacobian and The Hessian
2.5.4 Cross Derivatives (and
Correlation)
2.6 Notation
and Interpretation
2.6.1 Notation for “Change”
2.6.2 Notation for Slopes/Derivatives
2.6.3 Notation for Higher Order
Differentials
2.6.4 Example Higher Order
Interacting Differentials: Option “Greeks”
2.7
Differential Equations as “Equations made of Slopes”
2.7.1 If you know all slopes/changes,
do you know the value (again)?
2.7.2 Example: The “forward price”
vs. “slope & return”
2.7.3 Example: The “market forecast &
uncertainty” as “slopes”
2.7.4 Example: Black-Scholes as
“slopes”
2.8
Differentiation vs. (mathematical) Derivatives
2.8.1 (Some) Rules for (Analytical)
Differentiation
2.9 Approximating Slopes and
Derivatives – Numerical Differentiation
2.9.1 The Most Basic Approximations
2.9.2 Who is Approximating Whom?
2.9.3 Complex Curves and Alternatives
2.10
“Requirements” for (Mathematical) Derivatives
2.10.1 Continuity in the
Curve & in the Slope(s)
2.10.2 Types of Singularities
2.10.3 Multiplicity and
Uniqueness
2.10.4 Lipschitz Continuity and The
“usual calculus”
2.10.5 Other Types of Derivatives
2.11
Treatment of Singularities and Slopes of Discontinuities
2.11.1 Example of an “Easy Singularity”
– Forward Rates from a Bootstrapped Yield Curve
2.11.2 Example of a “Intermediate
Singularity” – Asian Options
2.11.3 Example of a “Hard Singularity”
– Digital Options
2.11.4 Example of a “Very Hard
Singularity” – The Markets
3 Sums and Integrals
3.1 Sums,
Products, and Areas
3.1.1 Example Areas – Simple Interest
3.1.2 Example Areas – Probability
Weighted Expectations/Average
3.1.3 Example Areas – Probability
Weighted Options Expectations
3.1.4 Products
3.1.5 Example Areas/Volumes – Speed
vs. Accumulation and Position Value
3.1.6 Example Areas/Volumes – Simple
Interest with Reinvestment
3.1.7 Products and Sums Example:
Cash, Spot & Forward Rates
3.1.8 Products and Sums Example:
Bermudian Options Exercise
3.1.9 Products and Sums Example:
Arithmetic vs. Geometric Returns
3.2
Integration: The Sum in the Limit
3.2.1 Illustration via
Compound Interest and Spot/Forward Arbitrage
3.2.2 The “area under the curve again” – CumNorm(x) & Cos(x)
3.3 Integration
and Boundary Conditions
3.3.1
Definite vs. Indefinite Integrals
3.3.2 Green’s/Stokes’ Theorem
– Integration by Parts
3.3.3 Dimension Reduction (or
Increase)
3.3.4 Relationship to
Differential Equations
3.3.5
Integral Transforms
3.3.6
Greens Functions and Singularities
3.4 (Some) Rules
for Integration
3.4.1 Relationship to
Derivatives
3.4.2
Sample Rules for Specific Function Families
3.5 Interpretations of Sums & Integrals in Trading and Risk
Management
3.5.1
IRR vs. Zero-Coupon Curve
3.5.2 Cumulative Probability
– Error Function
3.5.3 Cumulative Probability
– Option Strike Price
3.5.4 Cumulative Probability
– Value-at-Risk (VaR) from Distributions
3.5.5 Expected Option Pay-out Value
3.6
Multi-dimensional Sums and Integrals
3.7
Conditions and “Types” of Integrals
3.8
Quadrature: Integral Approximation
3.8.1 The Basic Idea
3.8.2 The “area under the curve
again” – Convergence Bias
3.8.3 Basis Functions
3.8.4 Example Quadrature – Digital Option
3.8.5 Example Quadrature –
Value-at-Risk (VaR) from Histograms
3.8.6 What is Acceptable Numerical
Error vs. Business Error?
3.8.7 Integral (Approximation) by
“Rank”
4 Basic Statistics – Measuring
Uncertainty
4.1
Introduction to Characterising Trading Phenomenon
4.1.1 The Law of Large Numbers and
“Chances”
4.1.2 Does the past predict the
future, and data sets with “Memory”
4.2
Histograms – Width & Shape vs. Frequency & Probability
4.2.1 Histograms – The Basic
Idea
4.2.2 Quantiles – “Direct”
Measures
4.2.3 Histograms: Pre- and
Post-Processing
4.2.4 Histograms – Trending
vs. De-Trending
4.2.5 Example Histograms: British Petroleum – “Data Pre-Processing”
4.2.6 Example Histograms: S&P Index –
“The Law of Large Numbers”
4.2.7 Example Histograms: US Bond
Credit & Default – “Other Shapes”
4.2.8 Example Histograms: T-Bond Implied & Historical Volatility –
“Statistical Smoothening”
4.3 Moments –
The Traditional Statistical Measures
4.3.1 The “Usual Suspects”: Average,
Variance, Skew, & Kurtosis
4.3.2 Other Types of Central Tendency
Summaries
4.3.3 Other Types of Variability
Summaries
4.3.4 Example Summary Measures:
Credit & Default Data
4.3.5 Example Summary Measures: S&P
Index Histories
4.3.6 Relationship to
Quantiles, Distributions, and “Calibration”
4.3.7 Geometric Moments
4.4 Weighting
and Filtering
4.4.1 Weighting and Filtering
Example – VaR
4.5
(Historical) Volatility as an Empirical Statistical Measure
4.5.1 Historical Volatility –
Sampling Length, Frequency, & Weighting
4.5.2 Historical Volatility –
Inter-Period & Calendar Effects
4.5.3 Historical Volatility –
Special Variations (Open/High/Low/Close)
4.6
Correlation and Covariance: Multi-Dimensional Statistics
4.6.1 The Basic Idea – Correlation &
Covariance
4.6.2 Example Covariance: A Bond + FX
Position
4.6.3 Example Covariance: 2-D
Histograms & the Curse of Dimensionality
4.6.4 Independence vs.
Zero-Correlation, Causality vs. Correlation, and Orthogonality
4.6.5 Auto/Serial-Correlation
4.6.6 n-Dimensional Covariance
(Covariance Matrix with Equity, FX, & IR)
4.6.7 A Few Comments on Covariance
Matrices in Trading and Risk Management
4.7 General
Moments
4.8
Stationarity
4.8.1 Stationarity Example:
FTSE Historical Volatility
4.8.2 Stationarity Example:
US Corporate Default/Recovery Rates
4.9 “Moving”
Moments
4.9.1 Example: Moving Averages (DJIA,
EUR/USD, SIMEX)
4.9.2 Example: Histories of
Historical Volatility – FTSE and S&P “vol”
4.9.3 Example: Moving Correlation in Structured Products (Quanto’s,
Convertible Bonds, Hedging, Bond Spreads, Asset Allocation)
4.9.4 Example: Moving Quantiles
(VaR/Economic Capital)
4.9.5 Histories of “Historicals” and
Serial Correlation
4.9.6 Stationarity vs. Moving
Moments in Pricing/Risk Formulations
4.10 Basic Stats –
Do’s & Don’t’s
5 Probability Basics –
Expectations & Models of Uncertainty
5.1
Expectations
5.1.1 The Law of Large Numbers
(again): Samples vs. Population
5.1.2 Expectations in Trading
and Risk Management
5.1.3 Expectations of
Expectations
5.2 Probabilities
5.2.1 Intuitive Description vs. Set
Theoretic Description
5.2.2 Manipulating
Probabilities and Boolean Operators
5.2.3 Probability Spaces & Algebras (Borel, Sigma, etc)
5.3 Expectations
and Distributions
5.3.1 Discrete vs. Continuous
Processes
5.3.2 Distributions for
Random Number Generation and Predictions
5.4 Discrete
Expectations and Distributions
5.4.1 Discrete Expectations
5.4.2 Discrete Distributions
5.4.3 The Four Most Important
Discrete Distributions (Binomial, General, Hypergeometric, & Poisson)
5.5 Continuous
Expectations and Distributions
5.5.1 Continuous Expectations
5.5.2 Continuous
Distributions
5.5.3 The Five Most Important
Continuous Distributions (Log/Normal, Uniform, c2, Weibull, & General)
5.6
Multi-Dimensional Distributions
5.6.1 Example: Discrete Distributions
in 2- Dimensions
5.6.2 Example: Continuous
Distributions in 2- Dimensions
5.6.3 Distributions in n-
Dimensions
5.7
Expectations and Distributions Summary
5.8 Examples
of Using Distributions in Trading and Risk Management
5.8.1 Valuation and Risk of a
Bet: Coin & Dice Games
5.8.2 First Valuation of Vanilla and
Digital Options
5.8.3 First Valuation of Credit
Default Insurance
5.9 First
Introduction to Modelling the Markets with Distributions
5.9.1 A 1st “Good” Shape for
Uncertainty – Equity Markets
5.9.2 A 1st “Good” Shape for
Uncertainty – IR Markets and Term-Structure/Mean Reversion
5.9.3 A 1st “Good” Shape for
Uncertainty – FX Markets and Jumps/Channels
5.9.4 A 1st “Good” Shape for Uncertainty – Commodities and
Skews/Jumps
5.9.5 A 1st “Good” Shape for Default
Uncertainty – Stationarity, Data Quality, & Supply Push
5.9.6 Summary of Some
Non-Market Modelling Considerations
5.9.7 Summary of Some
Technical Modelling Considerations
5.10
Inverse Distributions
5.11 Approximating
Distributions
5.11.1 Distribution
Transformations
5.11.2
Fitted Distributions (Quasi-Closed Form Approximations)
5.11.3 Approximating Discrete
Distributions
5.11.4 Approximating
Continuous Distributions
5.11.5 Continuous
Distribution Approximation vs. Discrete Distribution?
5.12 Properties and Manipulation of Distributions
5.12.1 Manipulation: Area
Preservation as an Auxiliary Equation
5.12.2 Manipulation: Shifting
or De-coupling the Average
5.12.3 Manipulation: Scaling,
Stretching, and Area Preservation
5.12.4 Terminology: Prices
based vs. Returns based Distribution
5.12.5
The Central Limit Theorem
5.12.6 Quantiles, Standard
Deviation, and Multiples Rules
5.13 Moment Generating Functions and the Characteristic Equation
5.14 Statistical
Inference
5.14.1 Hypothesis Testing and
Statistical Significance
5.14.2 The p-value and z-test: Standard
Error, Confidence Intervals, and Statistical Significance.
5.14.3 The p-value: Are Two VaR’s the
Same?
5.14.4 The t-test: Are 2 Means the
Same?
5.14.5 Example: Compare Two
Investment’s Returns – Asset Allocation
5.14.6 The F-test: Are Two Variances
the Same?
5.14.7 Example: Compare Two
Investment’s Risks: Asset Allocation
5.14.8 The c2–test: Are Two Histograms
the Same?
5.14.9 Example: Is the World Normally
distributed?
5.14.10 The Kolmogorov-Smirnov-test: Are
Two Distributions the Same?
5.14.11 Example: Does Risk/Return
Analysis Work in a Non-Gaussian World?
5.15 Lies, Damn
Lies, and Statistics – Simpson’s Paradox and “prop trading”
6 Modelling Price/Returns Processes
& Valuation Under Uncertainty
6.1 A Basic Model
of Forward Prices/Returns
6.2 A Basic
Valuation Model with Time Evolution of Uncertainty
6.2.1 Forecasting vs.
Valuation
6.3
Calibration of Models
6.4 The
Black-Scholes Risk Neutral Arbitrage Free Framework
6.5
Extensions of the Basic Valuation Model
6.5.1 Arithmetic Brownian
Motion
6.5.2 Ornstein-Uhlenbeck:
Mean Reverting & CEV
6.5.3 Jump Diffusion
6.5.4 Multi-Index Contracts: 2-Factor
Valuation
6.6 Summary
PART II – Monte Carlo
Methods
7 Monte Carlo Preliminaries
7.1
Forecasting and Valuation Models as PDEs
7.2
Differential Equations/Numerical Methods Preliminaries
7.2.2 (Some) Differential Equations
Terminology
7.2.3 (Some) Numerical Methods
Terminology
8
Monte Carlo Methods – The Basics
8.1
Monte Carlo Basics
8.1.1 The Idea
8.2 MC
Basics: Forecasting Forward Prices
8.3 MC Basics:
Approximation Error
8.4 MC Basics:
Forward Prices vs. Pay-Out Functions
8.5 MC Basics:
Additional Considerations for Valuation/Risk
8.6 MC Basics:
Path Dependence
8.7 MC Basics:
The Story so Far
9 Applying the Basics to Pricing an
Option
9.1 MC
Forecasting vs. MC Black-Scholes Model Equations
9.1.1 Continuous vs. Discrete
Processes
9.1.2 Pay-out and Distribution
9.1.3 The
Valuation Formula and Black-Scholes
9.2
Simple MC Call Option Valuation
9.3 Spreadsheet
Methods
9.3.1 Spreadsheet Methods: In
Situ
9.3.2 In situ – 1 (Central Limit
Theorem)
9.3.3 In situ – 2 (Random function + Iterator function)
9.3.4 In situ – 3 (columns vs. “looping sheet” with TG2MCX®)
9.3.5 In situ – 4 (columns vs.
“looping sheet” with @Risk ®)
9.4 Some Comments
and an Interim Reality Check
9.4.1 Some Comments on the
Calculation of Histograms etc
9.4.2 Some Comments on the
Calculation of Quantiles etc
9.4.3 MC vs. Black-Scholes & Risk
Neutral vs. Real Trading
9.5 MC
VBA Code: Vanilla Options
9.5.1 MC VBA Code: Vanilla Call
Option
9.5.2 MC VBA Code: General (Call or
Put) Vanilla Option
9.6 Error
Analysis
9.6.1 Sources of Errors
9.6.2 Sensitivity to
Simulation Parameters
9.6.3 Sensitivity to Contract
Parameters
9.7 Basic MC
(vanilla) Option Pricing Summary
10 Pricing Exotic Options with MC
10.1 Pricing a
Barrier Option with Monte Carlo Simulation
10.1.1 Closed From Solution: Barrier
Option: Pricing
10.1.2 Close Form Barrier Option:
Pricing Formula
10.1.3 Closed Form Solution Knock-Out
Call Example
10.1.4 Knock-Out Call Option
Orientation
10.1.5
MC Knock-Out Call: In situ Spreadsheet Implementation
10.1.6 MC VBA Code: Knock-Out Call/Put
10.1.7 MC and Path (in-)Dependence,
Continuums, and Scaling
10.1.8 MC VBA Code: Knock-In Call/Put
10.1.9 MC Barrier Error Analysis and
Usage Considerations
10.1.10 MC Barrier Options Summary
10.2 Pricing Asian
Options with Monte Carlo Simulations
10.2.1 (Quasi) Closed Form Solutions:
Arithmetic Asian Options
10.2.2 Asian Options: Contract
Specification Issues
10.2.3 Asian Options: Pay-Out Profiles
10.2.4
MC Asian Options: In-situ Spreadsheet Implementation
10.2.5 MC Asian Options: VBA Code
10.2.6 Asian Options: Geometric Average
Price
10.2.7
MC Asian Options: Error Analyses and Usage
10.2.8 MC Asian Options Summary
10.3 Pricing
Compound Options with Monte Carlo Simulations
10.3.1 Compound Option Structure
10.3.2 Compound Option: Call on Call Pricing Formula
10.3.3 MC Compound Option Strategy
10.3.4
MC Compound Options – In situ spreadsheet Implementation
10.3.5 MC Compound Options: VBA Code
10.3.6 MC Compound Options: Error
Analysis and Usage
10.3.7 Summary of MC Compound Options
10.4 Pricing Early
Exercise Options with Monte Carlo Simulations
10.4.1 Early Exercise Options and
Structures
10.4.2 American Options: Traditional
Pricing
10.4.3 Early Exercise with MC Methods
10.4.4
MC American Options: VBA Code
10.4.5
MC Early Exercise: Error Analysis and Usage
10.4.6 Summary: MC Early Exercise
10.5 Summary of
Pricing Exotic Options with MC
11 Special Cases
11.1 All Markets
are Mean-Reverting
11.1.1 MC Mean-Reverting Valuation
11.1.2
MC Mean-Reverting Valuation: Spreadsheet Implementation
11.2 Multi-Factor
Valuations
11.2.1 Multiple underlying indices and
correlation
11.2.2
A 2-Factor MC Iterator
11.2.3 Exchange of Asset and Spread
Options
11.2.4 Stochastic Volatility (2-Factor
+ “To Mean-Revert or Not?”)
11.2.5 Convertible Bonds (2-Factor + Other Extensions)
11.3 Term-Structure
Valuation (for any Asset Class)
11.4 Summary: MC
Special Cases
12 (Traditional) Position Keeping,
Hedging, and Risk with MC
12.1 Traditional
Hedging Objectives & Methods
12.1.1 (Traditional) Hedging
Objectives
12.1.2 Risk Measures
and Hedging Techniques
12.2
Traditional Sensitivity Risk Measures
12.3
Sensitivity-Risk Usage Review
12.4
Monte Carlo Sensitivity-based Risk Measures
12.4.1 Brute Force MC Sensitivity Risk
Measures
12.4.2 “Clever” MC Sensitivity Risk
Measures
12.5 Monte Carlo
and VaR/Economic Capital
12.5.1 The Basic VaR Idea
12.5.2 Just a Few Practical VaR
Considerations
12.5.3 Illustration of MCVaR
12.5.4 MC and Economic Capital
13 PaR and MC “Complete Strategy”
Based Methods
13.1 The PaR
Methodology
13.1.1 Trading, Hedging, Selling,
Managing, and Real World P&L
13.1.2 Trading, Hedging, Selling,
Managing, and Simulated P&L
13.1.3 Forward Testing vs. Backward
Testing
13.1.4 Basic Elements of a (Simple) PaR
Calculator
13.1.5 A Spreadsheet Example (Suitable
for simple positions)
13.1.6 Why Monte Carlo?
13.2
Sensitivity Hedge PaR Example: Options Arbitrage vs. Hedge Efficiency
13.2.1 A Super-Simple Hedging/Trading
MC-PaR Implementation
13.2.2 MC-PaR Verification and Usage
13.2.3 MC-PaR “Optimal Risk-Adjusted
Trading Strategy”
13.2.4 More Reality: Transactions
Costs, Negative Skew, and “Policy”
13.3 Profile
Matching PaR Example: Digital Options/Ratio Call-Spreads
13.3.1 The Basic Digital Option Problem
13.3.2 A Short Description of Digital
Option Profile Matching
13.3.3 A (Static) Digital Option
Profile Matching Example
13.3.4 Dynamic vs. Static Strategies
13.3.5 MCPaR Analysis of Digital Option
Profile Matching
13.4 A Simple
Forward/Backward PaR Example: Bonds/Bond Futures
13.4.1 Forward Testing (with a 1-factor
Model)
13.4.2 Back Testing the Hedging
Strategy and the (1-factor) model
13.5
Investment/Structuring/Asset Allocation PaR Example: Convertible Bonds,
n-Factors, and Beyond
13.5.1 Implementation and
Usage of (a few) “extra” features
13.5.2 (A few) Example Applications of
MCPaR to CBs Trading
13.6 Holding
Period P&L Optimal Investing, Trading, & Hedging
13.7 MCPaR – The
Story so Far
14 PaR
Portfolio Simulation: 2 (PaR for portfolios)
14.1 Portfolio and
Management Issues – the Big Picture
14.1.1 What you do for a living?
14.1.2 Complex Market and Auxiliary
Effects
14.1.3 Portfolio Composition Effects
14.1.4 Operations/Mandate
Effects
14.2 Real World
Simulation of Portfolios
14.2.1 A General MCPaR Simulator
Example
14.2.2 Generating Forward
Market Scenarios
14.2.3 Some Example Issues in a General
MCPaR Portfolio Implementation and Simulation
14.2.4 Back Testing with Historical
Market Data
14.3 Portfolio
Strategy Efficient Frontiers
15
Properties of the MC Method
15.1 Accuracy,
Convergence, Stability, and Cost
15.1.1
The MC Integration Strategy
15.1.2 Numerical Accuracy
15.1.3 Numerical Convergence
15.1.4 Numerical Stability
15.1.5 Computational Effort
15.2 Random Number
Generation
15.2.1
Random Number Generation Basics – LCGs
15.2.2
Seeding
15.2.3
Intrinsic Random Number Generators
15.2.4
Random Number/Distribution Transformation
15.2.5
Verification of Random Number Generators
15.3 MC
Computational Variability in Trading and Risk Management
15.3.1 Performance Assessment
15.4 Reducing
Computational Cost – Random Number Methods
15.4.1
Cheaper Random Numbers
15.4.2 Sampling Methods
15.4.3 Sampling Methods:
Stratification
15.4.4 Sampling Methods:
Quasi Random Sequences
15.4.5
Quasi Random Sequence – Sample Performance
15.4.6 Sampling Methods vs.
Quadrature vs. Variance Reduction
15.4.7 Summary: Sampling
Methods
15.5 Reducing
Computational Cost – Variance Reduction Methods
15.5.1 Antithetic Variance
Reduction
15.5.2
Sample Performance
15.5.3
Control Variate Methods
15.5.4
Sample Performance
15.6 Variance
Reduction Summary
16 Implementing MC methods in the
Real World
16.1
Trading/Mandate Specific Issues
16.2 MC/Simulation
Objectives & Goals
16.3 Tools and
Consideration for Implementing MC
16.3.1 Reality Impact
16.3.2 Spreadsheets, Macros, and
Add-ins
16.3.3 "Code"
16.3.4 Cost of coding an MC application
17 Pros and Cons of MC Methods
17.1 Monte Carlo
Simulation: Pros/Cons
17.2 Alternative
Solutions
17.2.1 Analytical solutions
17.2.2
MC vs. Tree Based Methods
17.2.3
MC vs. Finite Difference
18 Software and Resources
18.1 General
MC Resources
18.2
Distributions/Random Numbers
18.3
Actual MC Code
18.4
MC “packages” and Spreadsheets
18.5 Advanced
Implementations
Appendices
Appendix A:
Notation/Abbreviations
Abbreviations
Currencies
Greek Letters
Alphanumeric Letters
Mathematical Operators
Appendix B: ARTicles
B1: ARTicles: Term-Structure Calibration:
Nonsense & Reality
B2: ARTicles: Credit Default Swaps
B3: ARTicles: How much to pay a trader?
References
Subject Index
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