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QUANTITATIVE FINANCEANDRISK MANAGEMENTA Physicist's ApproachSecond EditionJan W DashWorld ScientificLONDON SINGAPORE BEIJING SHANGHAI HONGKONG TAIPEf CHENNAI

Table of ContentsAcknowledgments for the 1st and 2nd EditionsxxiPART I: INTRODUCTION, OVERVIEW, AND EXERCISE13. Introduction / Outline; Note for 2nd Edition; Unresolved ProblemsWho/ How/What, "Tech. Index", Messages, Personal NoteSummary Outline: Book ContentsNote for the 2nd Edition (2015)Climate Change Risk Management - NEW TOPICEvaluation - Where do we stand? What about the future?3356772. Overview (Tech. Index 1/10)Objectives of Quantitative Finance and Risk ManagementT ools of Quantitative Finance and Risk ManagementThe Traditional Areas of Risk ManagementWhen Will We Ever See Real-Time Color Movies of Risk?Many People Participate in Risk ManagementQuants in Quantitative Finance and Risk ManagementReferences991114161618213. An Exercise (Tech. Index 1/10)Part #1: Data, Statistics, and Reporting Using a SpreadsheetPart #2: Repeat Part #1 Using ProgrammingPart #3: A Few Quick and Tricky Hypothetical QuestionsMessages and AdviceReferences232326272828PART II: RISK LAB (NUTS AND BOLTS OF RISK MANAGEMENT)294. Equity Options (Tech. Index 3/10)Pricing and Hedging One OptionAmerican OptionsBasket Options and Index OptionsOther Types of Equity Options; ExoticsPortfolio Risk (Introduction)Scenario Analysis (Introduction)Referencesvii3131343537373738

vj-Quantitative Finance and Risk Management5. FX Options (Tech. Index 4/10)FX Forwards and OptionsSome Practical Details for FX OptionsHedging FX Options with Greeks: Details and AmbiguitiesFX Volatility Skew and/or SmilePricing Barrier Options with SkewDouble Barrier Option: Practical ExampleThe "Two-Country Paradox"Quanto Options and CorrelationsFX Options in the presence of Stochastic Interest RatesNumerical Codes, Closed Form Sanity Checks, and IntuitionReferences39 9434446515254565757586. Equity Volatility Skew (Tech. Index 6/10)Put-Call Parity: Theory and ViolationsThe Volatility SurfaceDealing with SkewPerturbative Skew and Barrier OptionsStatic ReplicationStochastic VolatilityLocal Volatility and SkewThe Skew-Implied Probability DistributionLocal vs. Implied Volatility Skew; Derman's Rules of ThumbOption Replication with GadgetsIntuitive Models and Different Volatility RegimesThe Macro-Micro Model and Equity Volatility RegimesJump Diffusion ProcessesAppendix 1: Algorithm for "Perturbative Skew" ApproachAppendix 2: A Technical Issue for Stochastic 78797. Forward Curves (Tech. Index 4/10)Market Input RatesConstruction of the Forward-Rate CurveReferences818184928. Interest-Rate Swaps and Credit Default Swaps (Tech. Index 5/10)Interest Rate Swaps: Pricing and RiskInterest Rate Swaps: Pricing and Risk DetailsCross-Currency SwapsCredit Default Swaps (CDS)References9393100] ]5] \7j 229. Bonds: An Overview (Tech. Index 2/10)Types of Bonds123123

Table of ContentsixBond IssuanceBond TradingFlight to QualityBond MathReferences12712912913013410. Interest-Rate Caps (Tech. Index 4/10)introduction to CapsThe Black Caplet FormulaN'on-USD CapsRelations between Caps, Floors, and Swaps- iedging Delta and Gamma for Libor CapsMedging Volatility and Vega LaddersYlatrices of Cap Prices' rime Caps and a Vega TrapMT Rates; Volatility Dependence of CMT 111. Interest-Rate Swaptions (Tech. Index 5/10)European SwaptionsBermuda/American Swaption Pricingj elta and Vega Risk: Move Inputs or Forwards?Swaptions and Corporate Liability Risk Management: . actieal Example: A Deal Involving a SwaptionVUscellaneous Swaption Topics \eferences153153157159160162164167! 2. Portfolios and Scenarios (Tech. Index 3/10): i;roduction to Portfolio Risk Using Scenario AnalysisMinitions of Portfolios1 )cfinitions of Scenarios -lany Portfolios and ScenariosA Scenario SimulatorRisk Analyses and Presentations169169169171174175175PART III: EXOTICS, DEALS, AND GASE STÜDIES17713. A Complex CVR Option (Tech. Index 5/10)The M&A ScenarioCVR Starting Point: A Put SpreadCVR Extension Options and Other ComplicationsThe Arbs and the Mispricing of the CVR OptionA Simplified CVR: Two Put Spreads with Extension LogicNon-Academic Corporate Decision for Option ExtensionThe CVR Option Pricing179179180180182183185187

xQuantitative Finance and Risk ManagementAnalytic CVR Pricing MethodologySome Practical Aspects of CVR Pricing and HedgingThe CVR BuybackA Legal Event Related to the CVRReferences19114. Two More Case Studies (Tech. Index 5/10)Case Study: DECS and Synthetic ConvertiblesCredit Spreads, Discounting, Convertibles, and DECsD)23: The Complex DEC Synthetic ConvertibleCase Study: Equity Call with Variable Strike and ExpirationReferences20120120420 /21121815. More Exotics and Risk (Tech. Index 5/10)Contingent CapsDigital Options: Pricing and HedgingHistorical Simulations and HedgingYield-Curve Shape and Principal-Component OptionsPrincipal-Component Risk Measures (Tilt Delta etc.)Hybrid 2-Dimensional Barrier Options—ExamplesReload OptionsReferences21921922322522722822923223616. A PotPourri of Deals (Tech. Index 5/10)TIPS (Treasury Inflation Protected Securities)Municipal Derivatives, Muni Issuance, Derivative HedgingDifference Option on an Equity Index and a Basket of StocksResettable Options: CliquetsPower OptionsPath-Dependent Options and Monte Carlo SimulationPeriodic CapsARM CapsIndex-Amortizing SwapsA Hypothetical Repo Options DealConvertible Issuance 25917. Single Barrier Options (Tech. Index 6/10)Knock-Out OptionsThe Semi-Group Property including a BarrierCalculating Barrier OptionsKnock-ln OptionsUseful Integrals for Barrier OptionsSingle Barrier Rebates at Touch and at MaturityOther Topics Involving Single Barrier Options261263265266267269271272' 94198*98'9t'

Table of ContentsxiReferences27618. Double Barrier Options (Tech. Index 7/10)Double Barrier Solution with an Infinite Set of ImagesDouble Barrier Option PricingRebates for Double Barrier OptionsReferences27727828028228419. Hybrid 2-D Barrier Options (Tech. Index 7/10)Pricing the Barrier 2-Dimension Hybrid Options'Jseful Integrals for 2D Barrier OptionsReferences28528828929020. Average-Rate Options (Tech. Index 8/10)Arithmetic Average Rate Options in General Gaussian ModelsResults for Average-Rate Options in the MRG ModelSimple Harmonie Oscillator Derivation for Average OptionsThermodynamic Identity Derivation for Average OptionsAverage Options with Log-Normal Rate DynamicsGaussian into Lognormal Using a Simple TrickReferences291292296297298298299300PART IV: QUANTITATIVE RISK MANAGEMENT30121. Fat Tail Volatility (Tech. Index 5/10)Gaussian Behavior and Deviations from GaussianReview of Some Math FormalismOutliers and Fat TailsÜse of the Equivalent Gaussian Fat-Tail VolatilityPractical Considerations for the Fat-Tail ParametersOverlapping vs. Non-overlapping Windows and on Matrix Formalism and the Jsf- Sphere(Tech. Index 8/10)The Importance and Difficulty of Correlation RiskOne Correlation in Two DimensionsTwo Correlations in Three Dimensions; the Azimuthai AngleCorrelations in Four Dimensions - PictureCorrelations in Five and Higher DimensionsSpherical Representation of the Cholesky DecompositionReferences23.Stressed Correlations and Random Matrices (Tech. Index 5/10)317317318319322324326328329

xjjQuantitative Finance and Risk ManagementCorrelation Stress Scenarios Using DataStressed Random Correlation MatricesStochastic Correlation Matrices Using the-sphere24. Optimally Stressed PD Correlation Matrices (Tech. Index 7/10)Least-Squares Eitting for the Optimal PD Stressed MatrixNumerical Considerations for Optimal PD Stressed MatrixExample of Optimal PD Fit to a NPD Stressed MatrixSVD Algorithm for the Starting PD Correlation MatrixPD Stressed Correlations by "Walking through the Matrix"Nearest Neighbor Technique for PD Stressed 5035125. Models for Correlation Dynamics, Uncertainties (Tech. Index 6/10).353"Just Make the Correlations Zero" Model; Three Versions353Long-Term vs. Short-Term Correlations; Macro-Micro Model355Macro Long-term Correlation Simulation Example356Macro Moves for the Whole Correlation Matrix359Correlation Dependence on Volatility360Implied, Current, and Historical Correlations for Baskets363SSA and Noise-Reduced Correlations - Preview364Factor Models, Idiosyncratic Risk, and Correlations364Correlated Idiosyncratic Residuais and Applications366References36726. Plain-Vanilla VAR and Component VAR (Tech. Index 4/10)Historical VAR (HVAR) and Monte Carlo HVARScenariosQuadratic VAR and Component VARs (CVARs)Monte-Carlo VARBacktesting VARComponent VAR (CVAR) and CVAR Volatility from MCConfidence Levels for Individual Variables in VARReferences36937237537637837938038238427. Enhanced/Stressed VAR (Tech. Index 5/10)Improved Plain-Vanilla VAR (IPV-VAR)Regulatory Stressed VAR, Turbulent VAR, VAR UncertaintyEnhanced/Stressed VAR (ES- VAR)Illiquidity Penalty for Enhanced VARSubadditivity, Integrated VAR, Backtesting"Bayesian/Scenario VAR"References38538538939039239g401 28.VAR, CVAR, CVAR Volatility Formalism (Tech. Index 7/10)403

Table of ContentsxiiiSet-up and Overview of the Formal VAR ResultsCalculation of the Generating FunctionVAR, CVARs ( Component VARs), CVAR VolatilitiesAffective Number of SD for Underlying VariablesExtension to Multiple Time Steps using Rath Integrals40340540841141329. VAR and Component VAR for Two Variables (Tech. Index 5/10)The Component VAR (CVAR) Volatility with Two VariablesGcometry, Math: Risk Ellipse, VAR Line, CVAR, CVAR Vol41541541630. Corporate-Level VAR (Tech. Index 3/10) ggregation, Desks, Business Units, Corporate Hierarchy esk CVARs and Correlations between Desk Risks\ged Inventory and IlliquidityReferences42142142342542731. Credit Risk: Issuer, Counterparty (Tech. Index 5/10); -.suer Credit Riski'ransition/Default Probability Matrices and Issuer Riskalculation of Issuer Risk - Generic CaseMonte-Carlo Simulations of Credit Issuer Riskvniple Example of Issuer Credit Risk Calculation uer Credit Risk and Market Risk: Separation via Spreads; .parating Market and Credit Risk without Double CountingUnified Credit Market Risk Model-ninterparty Credit Risk Example: SwapsVA "Risk Neutral" and PFE "Real World" Simulations- i l:, and the Macro-Micro Modelorrelated Defaults - Analytic Results for 2D Merton Model \ lisc. Topics: WWR, FVA, Factor Models, Firmwide 4745045345445445745945932. Model Risk Overview (Tech. Index 3/10)Summary of Model RiskModel Risk and Risk ManagementTime Scales and ModelsLong-Term Macro Component with Quasi-Random BehaviorLiquidity Model LimitationsWhich Model Should We Use?Psychology and ModelsModel Risk, Model Reserves, and Bid-Offer SpreadsModel Quality AssuranceModels and 468469

.ßwmfifoöve Finance anc/ RüA: Managcmcn'33. Model Quality Assurance (Tech. Index 4/10)Model Quality Assurance Goals, Activities, and Procedures.Model QA: Sample DocumentationUser Section of Model QA DocumentationQuantitative Section of Model QA DocumentationSystems Section of Model QA DocumentationReferences47 f47!47 474747 34. Systems Risk Overview (Tech. Index 3/10)Advice and a Message to Non-Technical ManagersWhat are the "Three-Fives Systems Criteria"?What is the Fundamental Theorem of Systems Risk?What are Some Systems Traps and Risks?The Birth and Development of a SystemSystems Risk in Mergers and StartupsVendor Systems RiskNew Paradigms in Systems and Parallel ProcessingLanguages for Models: Fortran 90, C \ C, Python, OthersWhat's the "Systems Solution"?Are Software Development Risks Unique to Wall Street?References484848484848',48 48849049049249249435. Strategie Computing (Tech. Index 3/10)Introduction and BackgroundIllustration of Parallel Processing for FinanceSome Aspects of Parallel ProcessingTechnology, Strategy and ChangeReferences49549649649750050136. Data Risk: Qualitative, SSA, Generalized z-Score Polynomials (Tech.Index 5/10)503Important Qualitative Aspect of Data Risk503NEW TOPIC: SSA, MSSA, and Data Smoothing/Cleaning506NEW: Generalized z-scores: General Measure Polynomials510References51137. Correlations, Data, and Random Matrix Theory (Tech. Index 6/10) .513Fluctuations and Uncertainties in Measured Correlations513Time Windowing514Correlations, the Number of Data Points, and Variables517Intrinsic and Windowing Uncertainties: Example518The Fisher Transform for Correlations519NEW: Noise-Cleaned Correlations via SSAZZZ" 520Random Matrix Theory Benchmarks for Noise in Correlations.".!"'\ZZ'Z'.52\

Table of ContentsxvNEW: Approximate Analytic Probability Distribution for ANY Eigenvalue in theZero-Correlation Wishart Matrix522A Few Other Aspects of Data and Correlations524References52438. Wishart's Theorem and Fisher's Transform (Tech. Index 9/10)Warm Up: The Distribution for a Volatility EstimateThe Wishart DistributionThe Probability Function for One Estimated CorrelationFisher's Transform and the Correlation Probability FunctionNEW: Hedge Fund Style-Change Risk and Correlation ChangesDerivation - Fourier Transform of the Wishart DistributionResult - Wishart Distribution Fourier TransformReferences52752853053253353553553754039. Economic Capital (Tech. Index 4/10)Basic Idea of Economic CapitalThe Classification of Risk Components of Economic CapitalExposures for Economic Capital: What Should They Be?Attacks on Economic Capital at High CLAllocation: Standalone, Component VAR, or Other?The Cost of Economic CapitalAn Economic-Capital Utility FunctionFirm wide Sharpe Ratio and Economic CapitalRevisiting Expected Losses; the Importance of Time ScalesTraditional Measures of 55540. Unused-Limit Risk (Tech. Index 5/10)General Aspects of Risk LimitsThe Unused Limit Risk Model: OverviewUnused Limit Economic Capital for Issuer Credit Risk557557559565PART V: PATH INTEGRALS, GREEN FUNCTIONS, AND OPTIONS.5674LPath Integrals and Options: Overview (Tech. Index 4/10)42. Path Integrals and Options I: Introduction (Tech. Index 7/10)Introduction to Path IntegralsHeretical Remarks on Rigor and All ThatPath-lntegral Warm-up: The Black Scholes ModelConnection of Path Integral with the Stochastic EquationsDividends and Jumps with Path IntegralsDiscrete Bermuda OptionsAmerican Options569573574576577589591598606

.Quantitative Finance and Risk ManagementAppendix 1: Girsanov's Theorem and Path IntegralsAppendix 2: A Short Dictionary of Common NotationsAppendix 3: No-Arbitrage, Hedging and Path IntegralsAppendix 4: Pertubation Theory, Local Volatility, SkewFigure Captions for this ChapterReferences6066106116166'6 2643. Path Integrals and Options II: Interest Rates (Tech. Index 8/10)I. Path Integrals: ReviewII. The G

Many People Participate in Risk Management 16 Quants in Quantitative Finance and Risk Management 18 References 21 3. An Exercise (Tech. Index 1/10) 23 Part #1: Data, Statistics, and Reporting Using a Spreadsheet 23 Part #2: Repeat Part #1 Using Programming 26 Part #3: A Few Quick and Tricky Hypothetical Questions 27 Messages and Advice 28 ...