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

Monte Carlo Methods

Table of Contents

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Table of Contents - CHAPTER Listing

 

0          Foreword and Highlights for the TG2 Series

1          Foreword and Highlights for TG2QM: Monte Carlo Methods

 

PART I – Basic Maths

 

2          Slopes and Derivatives

3          Sums and Integrals

4          Basic Statistics – Measuring Uncertainty

5          Probability Basics – Expectations & Models of Uncertainty

6          Modelling Price/Returns Processes & Valuation Under Uncertainty

 

PART II – Monte Carlo Methods

 

7          Monte Carlo Preliminaries

8          Monte Carlo Methods – The Basics

9          Applying the Basics to Pricing an Option

10        Pricing Exotic Options with MC

11        Special Cases

12        (Traditional) Position Keeping, Hedging, and Risk with MC

13        PaR and MC “Complete Strategy” Based Methods

14        PaR Portfolio Simulation: 2 (PaR for portfolios)

15        Properties of the MC Method

16        Implementing MC methods in the Real World

17        Pros and Cons of MC Methods

18        Software and Resources

 

Appendices

References

Subject Index

  

Table of Contents - DETAILED Listing

 

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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