NUMERICAL MODELLING: APPLICATIONS TO MARINE SYSTEMS
CONTENTS GETTING STARTED ALTERNATIVE PERSPECTIVES PRICING OR RISK-MANAGEMENT? MINDING OUR P’S AND Q’S INSTRUMENTS OR PORTFOLIOS? THE TIME DIMENSION TYPE OF CREDIT-RISK MODEL CLARIFYING OUR PERSPECTIVE A USEFUL DICHOTOMY MODELLING IMPLICATIONS RARE EVENTS SEEING THE FOREST MODELLING FRAMEWORKS DIAGNOSTIC TOOLS ESTIMATION TECHNIQUES THE PUNCHLINE PREREQUISITES OUR SAMPLE PORTFOLIO A QUICK PRE-SCREENING A CLOSER LOOK AT OUR PORTFOLIO THE DEFAULT-LOSS DISTRIBUTION TAIL PROBABILITIES AND RISK MEASURES DECOMPOSING RISK SUMMING UP FINAL THOUGHTS REFERENCES PART I MODELLING FRAMEWORKS REFERENCE A NATURAL FIRST STEP MOTIVATING A DEFAULT MODEL TWO INSTRUMENTS MULTIPLE INSTRUMENTS DEPENDENCE ADDING FORMALITY AN IMPORTANT ASIDE A NUMERICAL SOLUTION AN ANALYTICAL APPROACH CONVERGENCE PROPERTIES ANOTHER ENTRY POINT MIXTURE OR ACTUARIAL MODELS BINOMIAL-MIXTURE MODELS THE BETA-BINOMIAL MIXTURE MODEL THE LOGIT- AND PROBIT-NORMAL MIXTURE MODELS POISSON-MIXTURE MODELS THE POISSON-GAMMA APPROACH OTHER POISSON-MIXTURE APPROACHES POISSON-MIXTURE COMPARISON CREDITRISK+ A ONE-FACTOR IMPLEMENTATION A MULTI-FACTOR CREDITRISK+ EXAMPLE THRESHOLD MODELS THE GAUSSIAN MODEL THE LATENT VARIABLE INTRODUCING DEPENDENCE THE DEFAULT TRIGGER CONDITIONALITY DEFAULT CORRELATION CALIBRATION GAUSSIAN MODEL RESULTS THE LIMIT-LOSS DISTRIBUTION THE LIMIT-LOSS DENSITY ANALYTIC GAUSSIAN RESULTS TAIL DEPENDENCE THE TAIL-DEPENDENCE COEF?CIENT GAUSSIAN COPULA TAIL-DEPENDENCE T-COPULA TAIL-DEPENDENCE THE T-DISTRIBUTED APPROACH A REVISED LATENT-VARIABLE DE?NITION BACK TO DEFAULT CORRELATION THE CALIBRATION QUESTION IMPLEMENTING THE T-THRESHOLD MODEL PAUSING FOR A BREATHER NORMAL-VARIANCE MIXTURE MODELS COMPUTING DEFAULT CORRELATION HIGHER MOMENTS TWO CONCRETE CASES THE VARIANCE-GAMMA MODEL THE GENERALIZED HYPERBOLIC CASE A FLY IN THE OINTMENT CONCRETE NORMAL-VARIANCE RESULTS THE CANONICAL MULTI-FACTOR SETTING THE GAUSSIAN APPROACH THE NORMAL-VARIANCE-MIXTURE SET-UP A PRACTICAL MULTI-FACTOR EXAMPLE UNDERSTANDING THE NESTED STATE-VARIABLE DE?NITION SELECTING MODEL PARAMETERS MULTIVARIATE RISK MEASURES THE GENESIS OF CREDIT-RISK MODELLING MERTON’S IDEA INTRODUCING ASSET DYNAMICS DISTANCE TO DEFAULT INCORPORATING EQUITY INFORMATION EXPLORING GEOMETRIC BROWNIAN MOTION MULTIPLE OBLIGORS TWO CHOICES THE INDIRECT APPROACH A SURPRISING SIMPLI?CATION INFERRING KEY INPUTS SIMULATING THE INDIRECT APPROACH THE DIRECT APPROACH EXPECTED VALUE OF AN,T VARIANCE AND VOLATILITY OF AN,T COVARIANCE AND CORRELATION OF AN,T AND AM,T DEFAULT CORRELATION BETWEEN FIRMS N AND M COLLECTING THE RESULTS THE TASK OF CALIBRATION A DIRECT-APPROACH INVENTORY A SMALL PRACTICAL EXAMPLE PART II DIAGNOSTIC TOOLS REFERENCES A REGULATORY PERSPECTIVE THE BASEL ACCORDS BASEL IRB THE BASIC STRUCTURE A NUMBER OF IMPORTANT DETAILS THE FULL STORY IRB IN ACTION SOME FORESHADOWING THE GRANULARITY ADJUSTMENT A FIRST TRY A COMPLICATED ADD-ON THE ONE-FACTOR GAUSSIAN CASE GETTING A BIT MORE CONCRETE THE CREDITRISK+ CASE A FINAL EXPERIMENT RISK ATTRIBUTION THE MAIN IDEA A SURPRISING RELATIONSHIP THE JUSTI?CATION A DIRECT ALGORITHM SOME ILLUSTRATIVE RESULTS A SHREWD SUGGESTION THE NORMAL APPROXIMATION INTRODUCING THE SADDLEPOINT APPROXIMATION THE INTUITION THE DENSITY APPROXIMATION THE TAIL PROBABILITY APPROXIMATION EXPECTED SHORTFALL A BIT OF ORGANIZATION CONCRETE SADDLEPOINT DETAILS THE SADDLEPOINT DENSITY TAIL PROBABILITIES AND SHORTFALL INTEGRALLS A QUICK ASIDE ILLUSTRATIVE RESULTS OBLIGOR-LEVEL RISK CONTRIBUTIONS THE VAR CONTRIBUTIONS SHORTFALL CONTRIBUTIONS THE CONDITIONALLY INDEPENDENT SADDLEPOINT APPROXIMATION IMPLEMENTATION A MULTI-MODEL EXAMPLE COMPUTATIONAL BURDEN AN INTERESTING CONNECTION MONTE CARLO METHODS BRAINS OR BRAWN? A SILLY, BUT INFORMATIVE PROBLEM THE MONTE CARLO METHOD MONTE CARLO IN FINANCE DEALING WITH SLOWNESS INTERVAL ESTIMATION A ROUGH, BUT WORKABLE SOLUTION AN EXAMPLE OF CONVERGENCE ANALYSIS TAKING STOCK VARIANCE-REDUCTION TECHNIQUES INTRODUCING IMPORTANCE SAMPLING SETTING UP THE PROBLEM THE ESSCHER TRANSFORM FINDING ? IMPLEMENTING THE TWIST SHIFTING THE MEAN YET ANOTHER TWIST TYING UP LOOSE ENDS DOES IT WORK? PART III PARAMETER ESTIMATION DEFAULT PROBABILITIES SOME PRELIMINARY MOTIVATION A MORE NUANCED PERSPECTIVE ESTIMATION A USEFUL MATHEMATICAL OBJECT APPLYING THIS IDEA COHORT APPROACH HAZARD-RATE APPROACH GETTING MORE PRACTICAL GENERATING MARKOV-CHAIN OUTCOMES POINT ESTIMATES AND TRANSITION STATISTICS DESCRIBING UNCERTAINTY INTERVAL ESTIMATES RISK-METRIC IMPLICATIONS RISK-NEUTRAL DEFAULT PROBABILITIES BASIC CASH-FLOW ANALYSIS INTRODUCING DEFAULT RISK INCORPORATING DEFAULT RISK INFERRING HAZARD RATES A CONCRETE EXAMPLE BACK TO OUR P’S AND Q’S DEFAULT AND ASSET CORRELATION REVISITING DEFAULT CORRELATION SIMULATING A DATASET A FAMILIAR SETTING THE ACTUAL RESULTS THE METHOD OF MOMENTS THE THRESHOLD CASE LIKELIHOOD APPROACH THE BASIC INSIGHT A ONE-PARAMETER EXAMPLE ANOTHER EXAMPLE A MORE COMPLICATED SITUATION TRANSITION LIKELIHOOD APPROACH THE ELLIPTICAL COPULA THE LOG-LIKELIHOOD KERNEL INFERRING THE STATE VARIABLES A FINAL EXAMPLE