Quantitative Finance and Risk Management

The following sections are included:

• Dedication

• Table of Contents

• Acknowledgements

The following sections are included:

• Who/How/What, “Tech. Index”, Messages, Personal Note

• Summary Outline: Book Contents

The following sections are included:

• Objectives of Quantitative Finance and Risk Management
  • Why is Quantitative Finance a Science?
  • Why is Quantitative Finance not a Science?
  • Quantitative Finance is Not Science but Phenomenology
  • What About Uncertainties in the Risk Itself?

• Tools of Quantitative Finance and Risk Management
  • The Greeks
  • Hedges
  • Scenarios (“What-if”, Historical, Statistical)
  • Monte-Carlo Simulation
  • Data and Risk
  • Problematic Topics of Risk: Models, Systems, Data

• The Traditional Areas of Risk Management
  • Market Risk
  • Credit Risk
  • Unified Market and Credit Risk Management?
  • Operational Risk

• When Will We Ever See Real-Time Color Movies of Risk?

• Many People Participate in Risk Management
  • Systems Programmers
  • Traders
  • Risk Managers
  • Corporate Risk Management

• Quants in Quantitative Finance and Risk Management
  • Jobs for Quants
  • Looking for a Job
  • On the Job: What's the Product?
  • Creativity and the 80-20 Rule
  • Communication Skills
  • Message to PhD Scientists and Engineers Who Want to Become Quants
  • Message to Quants Who Want to Become Quant-Group Managers

• References

The following sections are included:

• Part #1: Data, Statistics, and Reporting Using a Spreadsheet
  • Step 1: Data Collection
  • Step 2: Statistics and Reporting
  • Step 3: Bis for Steps 1, 2
  • Step 4: Correlations
  • Step 5: Written Communication
  • Step 6: Verbal Communication
  • Step 7: Celebrate
  • Analysis of Part #1

• Part #2: Repeat Part #1 Using Programming
  • Analysis of Exercise, Part #2

• Part #3: A Few Quick and Tricky Hypothetical Questions
  • Question 1: System Risk
  • Question 2: Should we do this Deal or not?
  • Question 3: Market Risk

• Messages and Advice
  • To Computer Programmers Who Skipped Exercise Part #1
  • To Those Who Can't Program and Skipped Exercise Part #2

• References

The following sections are included:

• Pricing and Hedging One Option

• American Options

• Basket Options and Index Options
  • Basket Options in Disguise: Swaptions

• Other Types of Equity Options; Exotics

• Portfolio Risk (Introduction)

• Scenario Analysis (Introduction)

• References

The following sections are included:

• FX Forwards and Options
  • Pricing FX Forwards and FX European Options

• Some Practical Details for FX Options
  • Normalization of FX Options

• Hedging FX Options with Greeks: Details and Ambiguities
  • Delta
  • Gamma
  • Vega
  • Rho and Phi
  • Theta
  • Higher Greeks (Speed, Charm, Color)
  • Other Option Details

• FX Volatility Skew and/or Smile
  • Physical Motivation for the FX Volatility Smile
  • General Skew, Smile Behavior for FX
  • Fixed Delta Strangles, Risk Reversals, and Hedging
  • Implied Probabilities of FX Rates and Option “Predictions”

• Pricing Barrier Options with Skew

• Double Barrier Option: Practical Example
  • Rebates

• The “Two-Country Paradox”

• Quanto Options and Correlations

• FX Options in the presence of Stochastic Interest Rates

• Numerical Codes, Closed Form Sanity Checks, and Intuition

• References

The following sections are included:

• Put-Call Parity: Theory and Violations
  • Skew with Less Liquid Options

• The Volatility Surface

• Dealing with Skew

• Perturbative Skew and Barrier Options
  • Some Numerical Results for Barrier Options with Skew

• Static Replication

• Stochastic Volatility

• Local Volatility and Skew
  • Analytic Results in Perturbation Theory

• The Skew-Implied Probability Distribution

• Local vs. Implied Volatility Skew; Derman's Rules of Thumb
  • Derman's First Rule of Thumb
  • Derman's Second Rule of Thumb
  • Derman's Third Rule of Thumb

• Option Replication with Gadgets

• Intuitive Models and Different Volatility Regimes
  • Sticky Strike, Sticky Delta, and Sticky Moneyness Models

• The Macro-Micro Model and Equity Volatility Regimes

• Jump Diffusion Processes

• Appendix A: Algorithm for “Perturbative Skew” Approach

• Appendix B: A Technical Issue for Stochastic Volatility

• References

The following sections are included:

• Market Input Rates
  • Cash Rates
  • Futures
  • Libor Swaps
  • Treasuries

• Construction of the Forward-Rate Curve
  • General Forward Rates
  • The Forward Rates and ED Futures
  • The Forward vs. Future Correction
  • Use of Swaps in Generating the Curve
  • Cash Rates and the Front Part of the Curve
  • The Break or Changeover Point
  • Curve-Smoothing Algorithms
  • Example of a Forward Curve
  • Model Risk and Curve Construction
  • Rate Units and Conventions
  • Compounding Rates
  • Rate Interpolation

• References

The following sections are included:

• Swaps: Pricing and Risk
  • Looking Inside a Swap: Swaplets
  • Swaplet Composition of a Swap
  • Simple Scenario Analysis for a Swap
  • The “Math Calc” for Price Changes Using the Greeks

• Interest Rate Swaps: Pricing and Risk Details
  • Hedges
  • Hedging with Cash Instruments
  • Hedging with ED Futures
  • Hedging with FRAs
  • Hedging with Other Swaps
  • Hedging with Treasuries
  • Hedging with Treasury Futures
  • Example of Swap Hedging
  • The “Delta Ladder”, or Bucketing: Breaking Up the Hedge
  • Hedging Long-Dated Swaps
  • Back Chaining Hedging Procedure: Example
  • Algorithm
  • Changes in the Hedges with Time
  • Swap Pricing Math and Risk
  • The Swap Break-Even Rate
  • Swap Risk, the Delta Ladder, and the Gamma Matrix
  • Swap Risk
  • The Delta Ladder
  • Non-Parallel Shifts, the Gamma Matrix, and the Sick Gamma Ladder
  • An Approximate Factorization Method for Incorporating the Gamma Matrix in the Gamma Ladder
  • A Non-Amortizing Swap is an FRN and a Non-Callable Bond
  • FRNs are at or Close to Par

• Counterparty Credit Risk and Swaps
  • Illustrative Counterparty Credit Risk Example for a Swap
  • Counterparty Credit Risk for General Portfolios

• References

The following sections are included:

• Types of Bonds
  • Fixed-Rate Bonds
  • Floating-Rate Bonds
  • Zero-Coupon Bonds
  • Callable Bonds and Puttable Bonds
  • Deterministic Coupon Changes: Step-Up Bonds
  • Bonds Depending on Other Markets
  • Convertible Bonds
  • Mortgage-Backed Securities
  • Asset-Backed Securities
  • Munis
  • Non USD Bonds
  • Guaranteed bonds; Bradys
  • Loans

• Bond Issuance
  • Treasury Issuance
  • Agency Issuance
  • Muni Issuance
  • Corporate Issuance
  • Asset-Backed Issuance
  • Mortgage Issuance
  • CP Issuance

• Bond Trading
  • Trading, Flight to Quality, Diversification, Convergence, and Asteroids

• Bond Math
  • Discount Factors
  • Yields
  • Duration and Convexity
  • Spreads
  • Option-Adjusted Spreads (OAS)
  • General, Specific, and Idiosyncratic Risks
  • Matrix Pricing and Factor Models

• References

The following sections are included:

• Introduction to Caps
  • At the Money Options and the Trader on a Stick

• The Black Caplet Formula
  • Floors and Floorlets
  • Cap and Caplet Implied Volatilities
  • Analytic Formula for Cap Implied Vol

• Non-USD Caps

• Relations between Caps, Floors, and Swaps
  • Put-Call Parity for Libor
  • Limiting Relations

• Hedging Delta and Gamma for Libor Caps

• Hedging Volatility and Vega Ladders
  • Implied Vols, Realized Vols, and Hedging Strategies
  • Bid/Ask Vols and Illiquidity

• Matrices of Cap Prices

• Prime Caps and a Vega Trap

• CMT Rates and Volatility Dependence of CMT Products
  • CMS Rates
  • A CMT Swap is a Volatility Product
  • CMT Caps
  • Numerical Considerations in Calculating CMT Derivatives

• References

The following sections are included:

• European Swaptions
  • Put-Call Parity and European Swaptions
  • European Swaption Volatility
  • Hedging European Swaptions: Delta and Gamma (The Matrix)
  • A Paradox, a Paradox, a Most Ingenious Vega Paradox

• Bermuda/American Swaption Pricing
  • Determination of the Local Forward Term-Structure of Volatility
  • Bermuda/American Swaption Hedging
  • Swaption Delta Hedging and Numerical Noise

• Delta and Vega Risk: Move Inputs or Forwards?
  • Delta Risk
  • Vega Risk

• Swaptions and Corporate Liability Risk Management

• Practical Example: A Deal Involving a Swaption

• Miscellaneous Swaption Topics
  • Liquidity and Basis Risk for Swaptions
  • Fixed Maturity vs. Fixed Length Forward Swap
  • Caplets and One-Period European Swaptions
  • Advance Notice
  • Skew for Swaption Volatility
  • Model Dependence of Vega and Risk Aggregation Problems
  • Theta

• References

The following sections are included:

• Introduction to Portfolio Risk Using Scenario Analysis

• Definitions of Portfolios

• Definitions of Scenarios
  • Changes In Portfolios Under Scenario Assumptions
  • Many Scenarios with a Fixed Portfolio

• Many Portfolios and Scenarios
  • Sliding or Not Sliding Down the Forward Curve

• A Scenario Simulator

• Risk Analyses and Presentations

The following sections are included:

• The M&A Scenario

• CVR Starting Point: A Put Spread

• CVR Extension Options and Other Complications
  • CVR: Averaging Specifications

• The Arbs and the Mispricing of the CVR Option

• A Simplified CVR: Two Put Spreads with Extension Logic
  • Back-Chaining Logic with Two Times
  • Non-Uniqueness of the Indifference Point for Two Put Spreads

• Non-Academic Corporate Decision for Option Extension
  • Compromise by Using the Two Indifference Points

• The CVR Option Pricing
  • First Year Paths
  • Second Year Paths
  • Third Year Paths

• Analytic CVR Pricing Methodology
  • Payout Details for the CVR
  • Possible Irrelevance of the First Two Lower Put-Spread Strikes
  • Back-chaining Logic for the CVR
  • The Averaging Conditions

• Some Practical Aspects of CVR Pricing and Hedging
  • Standard Rational, Market Logic
  • CVR Components
  • Indifference Points
  • Non-Standard Corporate Decision to Extend
  • Maximum Condition and Averaging Contributions
  • CVR Valuation with Other Interest Rates
  • CVR Hedging and Scenario Results
  • Physical Intuition
  • Indicative Scenario Results, Comparison with the “Math Calculation”

• The CVR Buyback

• A Legal Event Related to the CVR

• References

The following sections are included:

• Case Study: DECS and Synthetic Convertibles
  • Standard DECs (These are simple)
  • Analysis of a Standard DEC
  • Credit Spreads, Discounting, Convertibles, and DECs
  • Textbook Argument: Use Libor
  • Other Arguments: Use Corporate Rate with the Credit Spread
  • Compromise Point of View: Use Both Rates
  • What's the Bottom Line?

• D₁₂₃ : The Complex DEC Synthetic Convertible
  • Pricing D₁₂₃: A Preview of Green-Function Techniques
  • The Payoffs and Calculation of the Best-of Option
  • Indicative Pricing

• Case Study: Equity Call with Variable Strike and Expiration
  • Theoretical Valuation of the Option
  • Monte-Carlo Valuation
  • “Back of the Envelope” Calculation
  • Extension Option Contribution to $δC
  • Lowered Strike Option Contribution to $δC and a Consistency Check

• References

The following sections are included:

• Contingent Caps
  • Clients and Contingent Caps
  • Structure of the Contingent Cap
  • A Simple “Contingent Cap” with Independent Caplets (Not This Product)
  • The 50–50 Contingent Cap
  • Scenario Analyses and Total Return Comparisons for Clients
  • Contingent Option as a Correlated Set of Digital Options

• Digital Options: Pricing and Hedging
  • Whipsaw Theoretical Scenario Analysis for Digitals
  • Hedging A Correlated Set of Digital Options

• Historical Simulations and Hedging
  • The Process of Actually Running the Historical Simulation

• Yield-Curve Shape and Principle-Component Options
  • Principal-Component Options
  • Example of Potential Clients for Principal-Component Options

• Principal-Component Risk Measures (Tilt Delta etc.)

• Hybrid 2-Dimensional Barrier Options—Examples
  • The Cross-Market Knock-Out (XMKO) Cap
  • Marketing XMKO Caps
  • Cross Market Knock-In Put Option and Correlation Risk
  • Results for the Illustrative XMKI Put Option
  • Standard Commodity Options

• Reload Options
  • Discounts due to Restrictions on Sale of Stock
  • Option Model for Stock Restriction Discount

• References

The following sections are included:

• TIPS (Treasury Inflation Protected Securities)
  • Description of TIPS
  • TIPS and an Important Statement for the Macro-Micro Model

• Municipal Derivatives, Muni Issuance, Derivative Hedging
  • Muni Derivatives
  • Muni Bond Futures and Options, and Comparison to the Treasury Market
  • Treasury-Bond Option Models
  • Example of Forward Muni Issuance With Derivative Hedging
  • Hedging the Issuance

• Difference Option on an Equity Index and a Basket of Stocks
  • Intuition for the Difference Option Formula

• Resettable Options: Cliquets
  • Pricing Resettable Options
  • Realistic Cliquet Deals
  • Example of a Cliquet Deal

• Power Options
  • Power-Barrier Options

• Path-Dependent Options and Monte Carlo Simulation

• Periodic Caps

• ARM Caps

• Index-Amortizing Swaps
  • Introduction to Index-Amortizing Swaps (IAS)
  • Connection of IAS with Barrier Options
  • Analysis of the IAS and its Embedded Options
  • Example of Application of the IAS : Mortgage Servicing Hedging
  • Pricing the IAS, a Volatility-Dependent Swap
  • No Simplification for IAS
  • Sociology

• A Hypothetical Repo + Options Deal
  • Definition of the Deal
  • Motivation and Risks for the Investor X
  • Motivation and Risks for the Broker-Dealer BD
  • Hedging the Deal

• Convertible Issuance Risk
  • A Risk Model
  • Preview of Coming Attractions: Math Coming Up Next

• References

The following sections are included:

• Knock-Out Options

• The Semi-Group Property including a Barrier

• Calculating Barrier Options

• Knock-In Options

• Useful Integrals for Barrier Options

• Complicated Barrier Options and Numerical Techniques

• A Useful Discrete Barrier Approximation

• “Potential Theory” for General Sets of Single Barriers
  • Step Barriers

• Barrier Options with Time-Dependent Drifts and Volatilities
  • Barrier Gaussian Rate Model with No Negative Rates?

• References

The following sections are included:

• Double Barrier Solution with an Infinite Set of Images

• Double Barrier Option Pricing

• Rebates for Double Barrier Options

• References

The following sections are included:

• Pricing the Barrier 2-Dimension Hybrid Options

• Useful Integrals for 2D Barrier Options

• References

The following sections are included:

• Arithmetic Average Rate Options in General Gaussian Models
  • Examples: Average-Rate Caplet, Digital/Step Option

• Results for Average-Rate Options in the MRG Model
  • Valuation date Before Averaging Period Starts
  • Valuation date During the Averaging Period

• Simple Harmonic Oscillator Derivation for Average Options

• Thermodynamic Identity Derivation for Average Options

• Average Options with Log-Normal Rate Dynamics
  • Gaussian into Lognormal Using a Simple Trick

• References

The following sections are included:

• Gaussian Behavior and Deviations from Gaussian

• Outliers and Fat Tails
  • Example of a Fat-Tail Event in the Real World
  • Gaussian Random Numbers for MC Simulations in Excel
  • Simple Illustrative Fat-Tail Monte-Carlo Model in Excel
  • Inversion for the Fat-Tail Volatility at Definite CL in the Model

• Use of the Equivalent Gaussian Fat-Tail Volatility

• Practical Considerations for the Fat-Tail Parameters
  • Why Use Fat-Tail Gaussians and not Other Distributions?
  • How do you know that Fat-Tail Gaussians are Really Fat?
  • How do you justify the use of the 99% CL for the Fat Tail Vol?
  • Bis: How do you justify the use of 99% CL for Fat Tail Vol?
  • What About the Central Part of the Distribution?
  • Serial Correlation Problem and a Lemma to the Data Theorem
  • Lemma to the Fundamental Black-Hole Theorem of Data
  • The Red Herring
  • What about the Difference Time and Mean Reversion?
  • What about Up/Down Data Move Asymmetry?
  • What is the Relation of the Fat Tail Vol with Huge Jumps?
  • The MaxVol

• References

The following sections are included:

• The Importance and Difficulty of Correlation Risk

• One Correlation in Two Dimensions

• Two Correlations in Three Dimensions; the Azimuthal Angle
  • The Degenerate World of FX Triangles
  • FX Option-Implied Correlations
  • This Triangle Doesn't Close

• Correlations in Four Dimensions

• Correlations in Five and Higher Dimensions
  • The ℬ Variables and a Recursion Relation

• Spherical Representation of the Cholesky Decomposition

• Numerical Considerations for the N-Sphere

• References

The following sections are included:

• Correlation Stress Scenarios Using Data
  • The Average-Target-Correlation Stress
  • Exposure-Weighted Definition
  • Subset Definitions, Arbitrary Weights
  • Applying the Subset Average-Target Correlation Stress

• Stressed Random Correlation Matrices

• Random Correlation Matrices Using Historical Data
  • Advantages and Disadvantages of Random Historical Matrices

• Stochastic Correlation Matrices Using the N-sphere
  • Review of the N - Sphere and the Variables
  • The Angle Volatilities and the Angle-Angle Correlations
  • The Random Matrices
  • Picture of Random Correlation Matrices Using the N - sphere

The following sections are included:

• Least-Squares Fitting for the Optimal PD Stressed Matrix
  • Weights in the Least-Squares Fitting

• Numerical Considerations for Optimal PD Stressed Matrix

• Example of Optimal PD Fit to a NPD Stressed Matrix
  • Stressed Correlations for Equity, Currency, or Rate Baskets

• SVD Algorithm for the Starting PD Correlation Matrix
  • Summary of the Starting Algorithm (SVD + Renormalization)
  • The Singular Value Decomposition (SVD)
  • Two Important Technical SVD Points
  • The Renormalization of the Eigenvalues and Eigenfunctions
  • The Square-Root Correlation Matrix Using SVD or Cholesky

• PD Stressed Correlations by Walking through the Matrix

• References

The following sections are included:

• “Just Make the Correlations Zero” Model; Three Versions
  • Zero Correlations, Version A: “No Reason, No Correlation”
  • Zero Correlations, Version B: “We Need Long Time Scales”
  • Zero Correlations, Version C: “Its Too Hard Otherwise”

• The Macro-Micro Model for Quasi-Random Correlations
  • The Macro Quasi-Random Slope for Correlation
  • The Time Intervals for Macro Correlation Slope Changes
  • Example of Quasi-Random Macro Correlation Slope Changes
  • Details of the Numerical Results

• Correlation Dependence on Volatility
  • Numerical Example of Correlation Dependence on Volatility
  • Windowed Correlations with a Macro Quasi-Random Slope
  • Should We Forget About Correlations?
  • Can Stochastic Volatility Explain Correlation Instabilities?

• Implied, Current, and Historical Correlations for Baskets

The following sections are included:

• Quadratic Plain-Vanilla VAR and CVARs

• Monte-Carlo VAR
  • The States and Portfolio Changes of the MC Simulation

• Backtesting

• Monte-Carlo CVARs and the CVAR Volatility
  • Non-Uniqueness of the CVARs
  • Summary for MC Simulation so far
  • Connection of the Monte-Carlo and Quadratic CVARs
  • The CVAR Volatility
  • Obtaining CVAR Volatilities in Practice; “Ergodic Trick”

• Confidence Levels for Individual Variables in VAR
  • Individual Variables generally do NOT contribute at High CL

• References

The following sections are included:

• Improved Plain-Vanilla VAR (IPV-VAR)
  • Convexity and the Grid
  • Example for the Grid
  • Caveats
  • Time Scale dt, Liquidity, and Product Types
  • Cutoffs for Underlying Variable Moves
  • Data Time Periods and a Measure of VAR Uncertainty

• Enhanced/Stressed VAR (ES-VAR)
  • Higher VAR CLs for Stressed VAR and Economic Capital
  • Fat-Tail Vols for Stressed VAR
  • Stressed Correlations for Stressed VAR
  • Idiosyncratic Risk Inclusion for Enhanced VAR
  • Illiquidity Penalty for Enhanced VAR
  • Starting Time for Exposure Reduction, Asteroids, Decisions
  • Starting Exposure Level; Corrections to the Expected Level
  • The Ending Exposure Level is Not Zero
  • A Component Model for Risk Exposures
  • Exposure: Reduction vs. Double-Up in Turbulent Markets
  • Increasing the Number of Time Steps for VAR-Type Simulation

• Other VAR Topics
  • VAR and Subadditivity
  • Heath's Example of Subadditivity Problem
  • Practical Resolution of VAR and Subadditivity
  • Confidence-Level Integrated VAR Measures
  • Example of CL-Integrated VAR

• References

The following sections are included:

• Set-up and Overview of the Formal VAR Results

• Calculation of the Generating Function
  • Calculation of the Exposure*Underlying Change Moments

• VAR, the CVARs, and the CVAR Volatilities
  • The CVAR Volatility
  • The Total VAR has Zero Volatility
  • The CVAR Volatility Triangle
  • The CVAR Volatilities for the Nonlinear Risk Case

• Effective Number of SD for Underlying Variables
  • Fat-Tail Vol Consistency with the Effective Number of SD
  • Calculation of in the Linear Risk Case

• Extension to Multiple Time Steps using Path Integrals

The following sections are included:

• The CVAR Volatility with Two Variables

• Geometry for Risk Ellipse, VAR Line, CVAR, CVAR Vol
  • The VAR Line for Two Variables
  • The Risk Ellipse E for Two Variables
  • Relation of VAR and the CVARs to the Risk Ellipse
  • The CVAR Volatility for Two Variables (bis)

The following sections are included:

• Aggregation, Desks, and Business Units
  • Corporate Structure and Practical Aggregation Difficulties
  • VAR Aggregation

• Desk CVARs and Correlations between Desk Risks
  • CVARs and Stand-Alone Risks at the Desk Level
  • Correlations between Desk Risks and P&L Correlations

• Aged Inventory and Illiquidity
  • Variables for Aged Inventory
  • Aged Inventory Reports
  • Quantitative Analysis of Aged Inventory

The following sections are included:

• Transition/Default Probability Matrices
  • Examples (Bad Year, Average, Worst Cases)
  • Models for Stressed Transition/Default Probability Matrices
  • Examples of Model Transition/Default Matrices
  • Stochastic Transition/Default Matrices

• Calculation of Issuer Risk—Generic Case
  • Bond Price Changes and Spreads; the Spread DV01
  • Defaulting Bond Price Changes
  • Monte-Carlo Simulations of Credit Issuer Risk at a given CL
  • Transition/Default Probability Uncertainties
  • Recovery Rate Uncertainty
  • Spread Uncertainty for a Given Credit
  • Short Positions: Why They Don't Alleviate the Risk at High CL
  • Concentration (Large Position) Risk
  • Dependence of the Results on Portfolio Definitions
  • Credit Derivatives and Issuer Risk
  • Credit Correlations
  • Other Refinements (Exposure Changes, Liquidity, Etc.)

• Example of Issuer Credit Risk Calculation
  • Credit Factors

• Issuer Credit Risk and Market Risk: Separation via Spreads
  • Spread Uncertainty at Fixed Credit and Market Risk
  • Spread Gaps Between Different Credits and Credit Risk
  • Separation of Market and Credit Risk

• Separating Market and Credit Risk without Double Counting
  • Example for High CL Credit and Market Risk
  • Advantages of Separating Market and Credit Issuer Risk
  • Disadvantages of Separating Credit and Market Risks

• A Unified Credit + Market Risk Model
  • Geometry for Credit + Market Risk
  • Effective Correlation between Market and Credit Issuer Risks

• References

The following sections are included:

• Summary of Model Risk

• Model Risk and Risk Management

• Time Scales and Models
  • Mean Reversion
  • Jumps, Gaps, and Nonlinear Diffusion

• Long-Term Macro Component with Quasi-Random Behavior

• Liquidity Model Limitations

• Which Model Should We Use?
  • Psychological Attitudes towards Models

• Model Risk, Model Reserves, and Bid-Offer Spreads

• Model Quality Assurance

• Models and Parameters

• References

The following sections are included:

• Model Quality Assurance Goals, Activities, and Procedures
  • Desk-Level Model QA
  • Independent Model QA Group
  • Model Risk Testing Using Various Independent Models
  • Test Suites
  • Documentation, Personnel Risk, and All That
  • Independent Reproduction of the Same Model
  • Judgment on Assumptions, Methodology, Algorithms
  • Which Models need Model QA?
  • Model Testing Environment
  • Feedback to the Business Units

• Model QA: Sample Documentation

• User Section of Model QA Documentation

• Quantitative Section of Model QA Documentation

• Systems Section of Model QA Documentation

• References

The following sections are included:

• Advice and a Message to Non-Technical Managers

• What are the “Three-Fives Systems Criteria”?
  • Calculators are not Systems

• What is the Fundamental Theorem of Systems?

• What are Some Systems Traps and Risks?
  • Specific Communication Issues

• The Birth and Development of a System
  • Programming Difficulties
  • Prototyping
  • Who Controls the Systems?

• Systems in Mergers and Startups

• Vendor Systems
  • Evaluating Vendor Systems
  • In-House Developers and Vendor Systems
  • Do you REALLY want the Source Code? Maybe Think Again

• New Paradigms in Systems and Parallel Processing

• Languages for Models: Fortran 90, C⁺⁺, C, and Others
  • Do We Really Need to Write in C⁺⁺ for Model Code?
  • Fortran 90
  • C
  • Other Languages
  • The Most Common Prototype Platform

• What's the “Systems Solution”?

• Are Software Development Problems Unique to Wall Street?

• References

The following sections are included:

• Introduction and Background

• Illustration of Parallel Processing for Finance

• Some Aspects of Parallel Processing
  • Parallel Processing Computers
  • Distributed Parallel Workstation Networks
  • Nodes
  • Trivial or Linear Parallelization
  • Vectorization and Parallelization
  • Parallel Languages and High Perj4orrnance Fortran
  • Parallel Communication and the MPI
  • Parallel Computing and Data

• Technology, Strategy and Change
  • Systems Groups, End Users, and New Technology

• References

The following sections are included:

• Data Consistency

• Data Reliability

• Data Completeness

• Data Vendors

• Historical Data Problems and Data Groups

• Preparation of the Data

• Bad Data Points and Other Data Traps

The following sections are included:

• Fluctuations and Uncertainties in Measured Correlations

• Time Windowing
  • Window Size
  • Overlapping and Non-Overlapping Windows
  • Example of the Effects of Windowing
  • Example for Effect of Jumps on Correlations

• Correlations, the Number of Data Points, and Variables
  • Example with Different Numbers of Observation and Variables
  • Bottom Line: In Practice, N > p is a Red Herring

• Intrinsic and Windowing Uncertainties: Example
  • The Fisher Transform for Correlations
  • Data Uncertainties Compared with Fisher Uncertainty
  • The Importance of Intrinsic Correlation Uncertainties

• Two Miscellaneous Aspects of Data and Correlations
  • The Importance of Conventions
  • Data Problems and Positive-Definite Correlations

• References

The following sections are included:

• Warm Up: The Distribution for a Volatility Estimate

• The Wishart Distribution

• The Probability Function for One Estimated Correlation

• Fisher's Transform and the Correlation Probability Function
  • WKB Approximation and Fisher's Gaussian Approximation
  • Fisher's Volatility: Statistical Noise of Estimated Correlation

• The Wishart Distribution Using Fourier Transforms
  • Outline of the Derivation, and the Dynamic Components in Wishart's Distribution
  • Further Details; the Wishart Fourier Generating Function
  • Performing the Multiple Fourier Integrals
  • Multiple Cauchy Theorem
  • Limitations of the Proof

• References

The following sections are included:

• Basic Idea of Economic Capital
  • Adverse Conditions for EC, and an Insurance Analogy
  • Example of the Insurance Analogy
  • Stress Testing and EC
  • What-if Scenarios (WIS) as Indicators
  • Numerical What-if Scenarios
  • Underlying Economic or Political What-if Scenarios
  • Probabilities, Entropy, and What-if Scenarios
  • Historical Scenarios (HS) as Indicators
  • Statistical Measures (SM) as Indicators

• The Classification of Risk Components of Economic Capital
  • Consistent vs. Inconsistent Calculations of EC Components

• Exposures for Economic Capital: What Should They Be?

• Attacks on Economic Capital at High CL
  • First Attack Misses: Lascaux Cave Paintings and an Ergodic Statement
  • Second Attack is Closer: Not Enough Companies
  • Third Attack Hits the Target: Is EC Related to Default?
  • Given All That, What do we Actually do for Economic Capital?

• Allocation: Standalone, CVAR, or Other?
  • Standalone Risk for Allocation?
  • CVARs for Allocation?
  • Compromise Recipe for Allocation?

• The Cost of Economic Capital

• An Economic-Capital Utility Function

• Sharpe Ratios

• Revisiting Expected Losses; the Importance of Time Scales
  • Standard Assumption: No Uncertainty for Expected Losses
  • More Realistic: Expected Losses Do Have Some Uncertainty
  • Time Scales Are Again the Issue
  • Summary for Time Scales and EC

• Cost Cutting and Economic Capital

• Traditional Measures of Capital, Sharpe Ratios, Allocation
  • Traditional Capital is Not Economic Capital

• References

The following sections are included:

• General Aspects of Risk Limits
  • Types of Limits
  • Setting and Monitoring Limits
  • Example of a Conundrum with Detailed Limits

• The Unused Limit Risk Model: Overview
  • Exposure Fractions: Time Dependent Decomposition
  • The Two Components of the Unused Limit Risk Model
  • Basket Approach to Multiple Exposures and Limits
  • Illustrative Example for Unused Limit Economic Capital
  • EC Can be Reduced if Exposures are Expected to Decrease
  • Exposure Scenarios: Comparison to VAR Exposure Reduction

• Unused Limit Economic Capital for Issuer Credit Risk

The following sections are included:

• Path Integrals and Physics
• Path Integrals and Finance
• A Few Basic Details of Path Integrals
• Summary of the Chapters on Path Integrals

The following sections are included:

• Summary of this Chapter

• Introduction to Path Integrals
  • Basic Idea of Path Integrals
  • The Propagator or Green Function
  • The Path Integral is Just a Convolution of Ordinary Integrals
  • Approximations to Path Integrals
  • Heretical Remarks on Rigor and All That
  • The Rest of this Chapter

• Path-Integral Warm-up: The Black Scholes Model
  • Textbook Discussion in a Path Integral Framework
  • The Option Diffusion Equation
  • Solution of the Option Diffusion Equation
  • The Free Green Function or Propagator
  • The Classic Black-Scholes Formula
  • Handy Integrals
  • The Semi-Group Property
  • Building up the Path Integral from the Semi-Group Equation
  • Formal Continuous Limit for the Path Integral
  • More General Parameters and the Path Integral
  • The Path Integral Satisfies the Diffusion Equation
  • Dirac Delta Functions
  • Details: How the Green function satisfies the Diffusion Equation
  • The No-Arbitrage World and the Fictitious Stock Prices
  • Notation: dx(t), d_tx(t), η(t) and Brownian Motion

• Connection of Path Integral with the Stochastic Equations
  • No Arbitrage: An Equivalent Approach

• Dividends and Jumps with Path Integrals
  • Proof of the Path Integral Satisfying the Semi-Group Property

• Discrete Bermuda Options
  • The Atlantic or Critical Path for Bermudas
  • Path Classes
  • Contibutions from Path Classes for Bermuda Options
  • Semigroup Properties for the Call and Put Green Functions for Bermudas
  • Atlantic Critical Path Algorithm
  • Expression for a Bermuda Put Option

• American Options

• Appendix A: Girsanov's Theorem and Path Integrals
  • First Derivation of Girsanov's Theorem
  • A Second, Quicker Derivation of Girsanov's Theorem

• Appendix B: No-Arbitrage, Hedging and Path Integrals
  • The Path Integral and the Green Function G_0*
  • The Elimination of the Drift μ^(S) for the No Arbitrage Drift μ
  • No-Arbitrage, Hedging and Path Integrals: Result

• Appendix C: Perturbation Theory, Local Volatility, Skew

• Figure Captions for this Chapter

• References

The following sections are included:

• Summary of this Chapter
• Introduction to this Chapter
• Other Term-Structure Interest-Rate Models

• Path Integrals: Review
  • The Rest of this Chapter

• The Green Function; Discretized Gaussian Models

• The Continuous-Time Gaussian Limit
  • European Bond Option
  • Embedded Call Options with Schedules

• Mean-Reverting Gaussian Models
  • Some Useful Formulae
  • The Green Function
  • Evaluation of the Green Function
  • Simple Harmonic Oscillators and the MRG Model
  • The Zero-Coupon Bond
  • The Drift Function
  • The Classical Path of Forward Rates with a Correction Term
  • European Bond Option in the MRG Model
  • Caplet in the MRG Model
  • Negative Rates in Gaussian Models

• The Most General Model with Memory
  • Applications of Models with Memory
  • Special Cases
  • Connection of the Path Integral with the Stochastic Equation
  • Stochastic Equations with Memory
  • Physical Intuition for Memory Effects

• Wrap-up for this Chapter

• Appendix A: MRG Formalism, Stochastic Equations, Etc.
  • Relation of the MRG Path Integral to Stochastic Equations
  • The MRG Diffusion Equation
  • Inclusion of Coupons for Bond Options in the MRG Model

• Appendix B: Rate-Dependent Volatility Models
  • The Green Function for General Volatilities as a Path Integral

• Appendix C: The General Gaussian Model With Memory

• Figure Captions for This Chapter

• References

The following sections are included:

• Summary of this Chapter
• Introduction to this Chapter

• Path Integrals and Common Numerical Methods
  • Path Integrals, MC Simulations, and the Binomial Approximation
  • The Binomial Approximation to a Monte Carlo Simulation

• Basic Numerical Procedure using Path Integrals
  • Bins and Path Integrals

• The Castresana-Hogan Path-Integral Discretization
  • The Castresana-Hogan Uniform Nominal Interest Rate Grid
  • The Reason for the Uniform Grid
  • The Transition Probabilities on the Uniform Grid
  • The Number of Points at Fixed Time
  • The Rectangular Grid and “Pruning the Tree”
  • Minimum and Maximum Nominal Rates
  • The First Time Interval
  • The Physical Interest Rates and the Term Structure Constraints
  • Determination of the Alpha Parameters via Term-Structure Constraints
  • Path Integral Discretization and Stochastic Equations

• Some Numerical Topics Related to Path Integrals
  • The Path Integral, MC Simulation, and Lattice Approximations
  • Monte Carlo Simulation Using Path Integrals
  • Applications of the Hybrid Monte-Carlo/Lattice Approach
  • The Standard Back-Chaining Bermuda Algorithm
  • Do We Need the Critical Path?
  • Steps in the Back-chaining Algorithm
  • Comments for the Back-chaining Bermuda Algorithm
  • American Option Backchaining
  • Backchaining for Bonds with Both Puts and Calls

• A Few Aspects of Numerical Errors
  • Sociology and Numerical Errors
  • Quantification of Numerical Errors for Pricing
  • Oscillations in the Price as the Number of Nodes Increases
  • Anomalous Noisy Fluctuations in Option Model Output
  • Risk Management and Numerical Errors
  • The Unobserved P&L
  • Risk Anomalies in Interest-Rate Risk Ladders
  • A Few Numerical Methods in the Literature
  • The Makivic Path-Integral Monte Carlo Approach
  • Moment Methods for Arbitrary Processes
  • Parametric Analysis of Derivatives Pricing

• Some Miscellaneous Approximation Methods
  • A “Call-Filtering” Iterative Method
  • Quasi-European Approximation for Bermudas – Some Details
  • The Critical Atlantic Path in the Quasi-European (QE) Approximation
  • Shielded “Geometrical Volatility” for Bermudas
  • Mean-Reverting Gaussian Approximation to Lognormal Model
  • An Exponential Interpolation Approximation
  • Canonical Normal Integral Uniform Approximation
  • Possible Framework for Multivariate Integral Approximations

• References

The following sections are included:

• Calculating Options with Multidimensional Path Integrals

• Principal-Component Path Integrals

• References

The following sections are included:

• Introduction to the Reggeon Field Theory (RFT)
  • Chaos

• Summary of the RFT in Physics
  • The RFT Lagrangian
  • Non-interacting Free Diffusion Limit
  • Calculations in the RFT
  • Results of the RFT in Physics
  • RFT Scaling Expressed in More Standard Language
  • What should we remember about all this?

• Aspects of Applications of the RFT to Finance
  • Mapping of the RFT to Finance
  • Jumps and Fat Tails
  • Lessons from Critical Exponents

• References

The following sections are included:

• Explicit Time Scales Separating Dynamical Regions

• The Macro-Micro Yield-Curve Model
  • Ch. 48. The Multifactor Lognormal Model and Yield-Curve Kinks
  • Ch. 49. The Strong Mean-Reverting Yield-Curve Multifactor Model
  • Ch. 50. The Macro-Micro Yield-Curve Model
  • Interpretation of the Macro Component
  • Interpretation of the Micro Component

• Further Developments in the Macro-Micro Model
  • The Micro-Macro Model Applied to the FX and Equity Markets
  • Models in the Economics Literature Resembling the Macro-Micro Model
  • Formal Developments in the Macro-Micro Model
  • No Arbitrage and the Macro Micro Model
  • Other Topics

• A Function Toolkit
  • Possible Additional Macro Component for Cycles
  • The Micro Component (Trading)
  • Technical Analysis and the Macro-Micro Model

• References

The following sections are included:

• Summary of this Chapter

• The Problem of Kinks in Yield Curves for Models

• Introduction to this Chapter
  • Statistical Tests: Brief Description
  • Data Used in the Analysis
  • The Historical Quasi-Equilibrium Yield-Curve Path
  • Summary of the Results
  • The Rest of the Chapter

• Statistical Probes, Data, Quasi-Equilibrium Drift
  • Yield-Curve Data
  • Input Volatilities and Yield-Curve Correlations
  • The Input Historical “Quasi-Equilibrium” Yield-Curve Path for the Model

• Yield-Curve Kinks: Bête Noire of Yield Curve Models
  • Example of Consequence of Kinks for Risk Management
  • Kinks and Arbitrage
  • Histogram for Fluctuations of a Given Rate
  • Attempted Modifications Without Strong Mean Reversion

• EOF / Principal Component Analysis
  • Principal Components of the Yield-Curve Data
  • A Principal-Component Simulator

• Simpler Lognormal Model with Three Variates
  • Kinks in the 3-Variate LN Model

• Wrap-up and Preview of the Next Chapters
  • First Improvement: Using Strong Mean Reversion (Ch. 49)
  • Further Improvement: The Macro-Micro Model (Ch. 50, 51)

• Appendix A: Definitions and Stochastic Equations
  • Correlation Formalism in Two and Three Dimensions
  • Means and Volatilities
  • Spread Shifts between Rates of Adjacent Maturities
  • The Kink Definition

• Appendix B: EOF or Principal-Component Formalism
  • Simpler Three Principal Component Example
  • Real-Life Principal Components
  • The First 3 Principal Components for the Data
  • Anomalous Principal Component Behavior

• Figures: Multivariate Lognormal Yield-Curve Model

• References

The following sections are included:

• Summary of this Chapter

• Introduction to this Chapter
  • What Should a Realistic Multifactor Yield-Curve Model Accomplish?
  • The Cluster-Decomposition Analysis (CDA)
  • Problematic Yield-Curve Features Described by the SMRG Model
  • Generalization of the SMRG Model to the Macro-Micro Model
  • WKB: Small Fluctuations from a Quasi-Equilibrium Yield-Curve Path
  • Yield-Curve Data Used for the Analysis
  • Results of the Numerical Analysis
  • Short and Long Time Scales
  • Remainder of this Chapter

• Cluster Decomposition Analysis and the SMRG Model
  • Third-Order Correlation Functions
  • How the Third-Order Correlation Function is Used in Practice
  • Mathematical Example
  • LN Model Analysis Using 3rd-Order or Three-Point Correlation Function
  • The Historical Quasi-Equilibrium Data Yield Curve and Mean Reversion
  • Details: The Historical Quasi-Equilibrium Yield Curve Path for Data
  • SMRG Model Analysis Using 3rd-Order Correlation Function
  • Large Value of the Mean Reversion
  • Strong Mean Reversion with Other Models
  • The SMRG Model is Quite Different from the Vasicek Model
  • Fat Tails Producing Jumps/Gaps are an Extra Component
  • Troublesome Aspects of Standard Approaches to Interest Rate Dynamics
  • Physical Picture of the Way Interest Rates Really Seem to Behave

• Other Statistical Tests and the SMRG Model
  • Repeat: Problems with Kinks in Models without Strong Mean Reversion
  • Results for the SMRG Model Description of Yield Curve Data
  • Results Including a Fat Tail Gaussian Component
  • How Many Free Parameters Are We Using to Describe the Yield Curve Fluctuations?
  • Maturity and Time Independence of the Mean-Reversion Parameter
  • Other Attempts to Fit the Data without Strong Mean Reversion

• Principal Components (EOF) and the SMRG Model
  • Numerical Results

• Wrap-up for this Chapter
  • Summary of the Physical Behavior of Interest Rates over Long Times
  • Options Pricing Needs Generalization to the Macro-Micro Model

• Appendix A: Definitions and Stochastic Equations

• Appendix B: The Cluster-Decomposition Analysis (CDA)
  • Using Time-Differenced Series in the CDA
  • Checking Model Hypotheses with the CDA
  • Formal Proof of C₃ = 0 for Gaussian Processes
  • More About the Cluster Decomposition Analysis

• Figures: Strong Mean-Reverting Multifactor Yield-Curve Model

• References

The following sections are included:

• Summary of this Chapter

• Introduction to this Chapter
  • Spectral Decomposition and Time Scales
  • The Macro Component and Macroeconomics
  • The Micro Component and Short-Term Trading Activity

• Prototype: Prime (Macro) and Libor (Macro + Micro)
  • Remainder of this Chapter

• Details of the Macro-Micro Yield-Curve Model
  • The Importance of the Time Step in the Macro Simulator
  • Model for the Random Macro Time Step Dynamics
  • Dynamical Model for the Random Macro Slope or Drift
  • The Macro Volatilities for Times and Slopes
  • Weak Mean Reversion and the Connection to Standard Models
  • The Overall Average Macro Drift and No Arbitrage
  • Option Pricing Using the Macro-Micro Model
  • Developments in the Macro-Micro Model

• Wrap-up of this Chapter

• Appendix A. No Arbitrage and Yield-Curve Dynamics
  • Scenario Analysis

• Figures: Macro-Micro Model

• References

The following sections are included:

• Summary of This Chapter
  • The Micro-Macro Model Applied to the FX and Equity Markets
  • Models in the Economics Literature Resembling the Macro-Micro Model
  • Formal Developments in the Macro-Micro Model
  • No Arbitrage and the Macro Micro Model
  • Other Topics

• The Macro-Micro Model for the FX and Equity Markets
  • Review: The CDA Test and the Yield Curve
  • Equities, FX, Strong Mean Reversion, and the CDA Test
  • Probability Analyses for FX and the Macro-Micro Model
  • Long-Term FX Data: Probability Standard Model Holds is 10^(−8)

• Macro-Micro-Related Models in the Economics Literature

• Related Models for Interest Rates in the Literature
  • The Fed and the Macro Component
  • TIPS: Bertonazzi and Maloney

• Related Models for FX in the Literature
  • Feiger and Jacquillat
  • Blomberg
  • Chin

• Formal Developments in the Macro-Micro Model
  • The Green Function with a Specific Quasi-Random Drift
  • Averaging the Green Function over Macro Parameters {λ_β}
  • Options and the Macro Parameter - Averaged Green Function

• No Arbitrage and the Macro-Micro Model: Formal Aspects
  • The (Drift, Green Function) Correlation 〈μ_aG_ab〉_λ;C
  • The Macro Parameter - Averaged Diffusion Equation

• Hedging, Forward Prices, No Arbitrage, Options (Equities)
  • Hedging at Fixed {λ_β} Parameters
  • Hedging Including Averaging over the {λ_β} Parameters
  • The Forward Stock Price is the Standard Expression
  • Modified No Arbitrage Condition for Equity Options
  • The Modified Black-Scholes Equity Option Formula

• Satisfying the Interest-Rate Term-Structure Constraints

• Other Developments in the Macro-Micro Model

• Derman's Equity Regimes and the Macro-Micro Model

• Seigel's Nonequilibrium Dynamics and the MM Model

• Macroeconomics and Fat Tails (Currency Crises)

• Some Remarks on Chaos and the Macro-Micro Model
  • Data, the Macro-Micro Model, and Chaos
  • The Econometric Model for FX Crises and Chaos
  • Anomalous Dimensions, Chaos, and the Macro-Micro Model

• Technical Analysis and the Macro-Micro Model
  • Connection of Technical Analysis with the Macro-Micro Model

• The Macro-Micro Model and Interest-Rate Data 1950-1996

• Data, Models, and Rate Distribution Histograms

• Negative Forwards in Multivariate Zero-Rate Simulations

• References

The following sections are included:

• Time Thresholds; Time and Frequency; Oscillations

• Summary of Desirable Properties of Toolkit Functions
  • What about Gaps or Jumps?

• Construction of the Toolkit Functions
  • Time Thresholds
  • Power-Law Takeoff
  • Damping
  • Oscillations—Standard and Nonstandard
  • Magnitude and Phase
  • Example of a Function Toolkit Application
  • Laplace Transform of a Function f(t) in the Toolkit

• Relation of the Function Toolkit to Other Approaches
  • Function Toolkit Relation to Fourier Transfoms
  • Function Toolkit Relation to Prony Analysis
  • Function Toolkit Relation to Wavelets
  • Function Toolkit Relation to ARIMA(p,d,q) Models

• Example of Standard Micro “Noise” Plus Macro “Signal”
  • Connected 3-Point Green Function / Auto-Correlation Analysis

• The Total Macro: Quasi-Random Trends + Toolkit Cycles

• Short-Time Micro Regime, Trading, and the Function Toolkit
  • Trading and the f(t) Function Toolkit

• Appendix: Wavelets, Completeness, and the Function Toolkit
  • Examples of Wavelet Variables
  • Completeness and a Plea to the Mathematicians

• References

The following sections are included:

• Index