Massimo Morini is Head of Credit Models and Coordinator of Model Research at IMI Bank of Intesa San Paolo. He has spent the last ten years inventing new models, implementing them, and helping practitioners in using them for buying, selling, and hedging derivatives. This has exposed him to the most practical side of model risk, and has led him to investigate model uncertainty, model robustness, and the management of the risk of model losses. Massimo is also Professor of Fixed Income at Bocconi University and was a Research Fellow at Cass Business School, City University London. He regularly delivers advanced training in London New York and worldwide on model risk management, credit modelling, interest rate models and correlation modelling, where he teaches cutting edge innovations in quantitative finance and discusses their implications with practitioners from the major institutions. He has led workshops on financial modelling and the financial crisis at major international conferences, including Global Derivatives, the Quant Congress, and the Fixed Income Conference. He has published papers in journals including Risk Magazine, Mathematical Finance, and the Journal of Derivatives.Massimo holds a PhD in Mathematics and an MSc in Economics.

• Preface xiAcknowledgements xixPart I Theory and Practice of Model Risk Management1 Understanding Model Risk 31.1 What Is Model Risk? 31.1.1 The Value Approach 41.1.2 The Price Approach 61.1.3 A Quant Story of the Crisis 91.1.4 A Synthetic View on Model Risk 171.2 Foundations of Modelling and the Reality of Markets 221.2.1 The Classic Framework 221.2.2 Uncertainty and Illiquidity 301.3 Accounting for Modellers 381.3.1 Fair Value 381.3.2 The Liquidity Bubble and the Accountancy Boards 401.3.3 Level 1, 2, 3 .go? 411.3.4 The Hidden Model Assumptions in ‘vanilla’ Derivatives 421.4 What Regulators Said After the Crisis 481.4.1 Basel New Principles: The Management Process 491.4.2 Basel New Principles: The Model, The Market and The Product 511.4.3 Basel New Principles: Operative Recommendations 521.5 Model Validation and Risk Management: Practical Steps 531.5.1 A Scheme for Model Validation 541.5.2 Special Points in Model Risk Management 591.5.3 The Importance of Understanding Models 602 Model Validation and Model Comparison: Case Studies 632.1 The Practical Steps of Model Comparison 632.2 First Example: The Models 652.2.1 The Credit Default Swap 662.2.2 Structural First-Passage Models 672.2.3 Reduced-Form Intensity Models 692.2.4 Structural vs Intensity: Information 722.3 First Example: The Payoff. Gap Risk in a Leveraged Note 742.4 The Initial Assessment 772.4.1 First Test: Calibration to Liquid Relevant Products 772.4.2 Second Test: a Minimum Level of Realism 782.5 The Core Risk in the Product 812.5.1 Structural Models: Negligible Gap Risk 822.5.2 Reduced-Form Models: Maximum Gap Risk 822.6 A Deeper Analysis: Market Consensus and Historical Evidence 852.6.1 What to Add to the Calibration Set 852.6.2 Performing Market Intelligence 862.6.3 The Lion and the Turtle. Incompleteness in Practice 862.6.4 Reality Check: Historical Evidence and Lack of it 872.7 Building a Parametric Family of Models 882.7.1 Understanding Model Implications 932.8 Managing Model Uncertainty: Reserves, Limits, Revisions 952.9 Model Comparison: Examples from Equity and Rates 992.9.1 Comparing Local and Stochastic Volatility Models in Pricing Equity Compound and Barrier Options 992.9.2 Comparing Short Rate and Market Models in Pricing Interest Rate Bermudan Options 1053 Stress Testing and the Mistakes of the Crisis 1113.1 Learning Stress Test from the Crisis 1113.1.1 The Meaning of Stress Testing 1123.1.2 Portfolio Stress Testing 1133.1.3 Model Stress Testing 1163.2 The Credit Market and the ‘Formula that Killed Wall Street’ 1183.2.1 The CDO Payoff 1183.2.2 The Copula 1193.2.3 Applying the Copula to CDOs 1223.2.4 The Market Quotation Standard 1243.3 Portfolio Stress Testing and the Correlation Mistake 1253.3.1 From Flat Correlation Towards a Realistic Approach 1263.3.2 A Correlation Parameterization to Stress the Market Skew 1313.4 Payoff Stress and the Liquidity Mistake 1363.4.1 Detecting the Problem: Losses Concentrated in Time 1373.4.2 The Problem in Practice 1393.4.3 A Solution. From Copulas to Real Models 1453.4.4 Conclusions 1503.5 Testing with Historical Scenarios and the Concentration Mistake 1513.5.1 The Mapping Methods for Bespoke Portfolios 1523.5.2 The Lehman Test 1563.5.3 Historical Scenarios to Test Mapping Methods 1573.5.4 The Limits of Mapping and the Management of Model Risk 1643.5.5 Conclusions 1684 Preparing for Model Change. Rates and Funding in the New Era 1714.1 Explaining the Puzzle in the Interest Rates Market and Models 1714.1.1 The Death of a Market Model: 9 August 2007 1734.1.2 Finding the New Market Model 1744.1.3 The Classic Risk-free Market Model 1784.1.4 A Market Model with Stable Default Risk 1824.1.5 A Market with Volatile Credit Risk 1924.1.6 Conclusions 2004.2 Rethinking the Value of Money: The Effect of Liquidity in Pricing 2014.2.1 The Setting 2044.2.2 Standard DVA: Is Something Missing? 2064.2.3 Standard DVA plus Liquidity: Is Something Duplicated? 2074.2.4 Solving the Puzzle 2074.2.5 Risky Funding for the Borrower 2084.2.6 Risky Funding for the Lender and the Conditions for Market Agreement 2094.2.7 Positive Recovery Extension 2104.2.8 Two Ways of Looking at the Problem: Default Risk or Funding Benefit? The Accountant vs the Salesman 2114.2.9 Which Direction for Future Pricing? 214Part II Snakes in the Grass: Where Model Risk Hides5 Hedging 2195.1 Model Risk and Hedging 2195.2 Hedging and Model Validation: What is Explained by P&L Explain? 2215.2.1 The Sceptical View 2225.2.2 The Fundamentalist View and Black and Scholes 2225.2.3 Back to Reality 2245.2.4 Remarks: Recalibration, Hedges and Model Instability 2265.2.5 Conclusions: from Black and Scholes to Real Hedging 2285.3 From Theory to Practice: Real Hedging 2295.3.1 Stochastic Volatility Models: SABR 2315.3.2 Test Hedging Behaviour Leaving Nothing Out 2325.3.3 Real Hedging for Local Volatility Models 2385.3.4 Conclusions: the Reality of Hedging Strategies 2416 Approximations 2436.1 Validate and Monitor the Risk of Approximations 2436.2 The Swaption Approximation in the Libor Market Model 2456.2.1 The Three Technical Problems in Interest Rate Modelling 2456.2.2 The Libor Market Model and the Swaption Market 2476.2.3 Pricing Swaptions 2506.2.4 Understanding and Deriving the Approximation 2536.2.5 Testing the Approximation 2576.3 Approximations for CMS and the Shape of the Term Structure 2646.3.1 The CMS Payoff 2656.3.2 Understanding Convexity Adjustments 2666.3.3 The Market Approximation for Convexity Adjustments 2676.3.4 A General LMM Approximation 2696.3.5 Comparing and Testing the Approximations 2716.4 Testing Approximations Against Exact. Dupire’s Idea 2766.4.1 Perfect Positive Correlation 2786.4.2 Perfect Negative Correlation 2806.5 Exercises on Risk in Computational Methods 2836.5.1 Approximation 2836.5.2 Integration 2856.5.3 Monte Carlo 2857 Extrapolations 2877.1 Using the Market to Complete Information: Asymptotic Smile 2887.1.1 The Indetermination in the Asymptotic Smile 2887.1.2 Pricing CMS with a Smile: Extrapolating to Infinity 2927.1.3 Using CMS Information to Transform Extrapolation into Interpolation and Fix the Indetermination 2937.2 Using Mathematics to Complete Information: Correlation Skew 2957.2.1 The Expected Tranched Loss 2957.2.2 Properties for Interpolation 2987.2.3 Properties for Turning Extrapolation into Interpolation 2988 Correlations 3038.1 The Technical Difficulties in Computing Correlations 3038.1.1 Correlations in Interest Rate Modelling 3058.1.2 Cross-currency Correlations 3078.1.3 Stochastic Volatility Correlations 3128.2 Fundamental Errors in Modelling Correlations 3158.2.1 The Zero-correlation Error 3168.2.2 The 1-Correlation Error 3199 Calibration 3239.1 Calibrating to Caps/Swaptions and Pricing Bermudans 3249.1.1 Calibrating Caplets 3259.1.2 Understanding the Term Structure of Volatility 3269.1.3 Different Parameterizations 3299.1.4 The Evolution of the Term Structure of Volatility 3329.1.5 The Effect on Early-Exercise Derivatives 3349.1.6 Reducing Our Indetermination in Pricing Bermudans: Liquid European Swaptions 3359.2 The Evolution of the Forward Smiles 34010 When the Payoff is Wrong 34710.1 The Link Between Model Errors and Payoff Errors 34710.2 The Right Payoff at Default: The Impact of the Closeout Convention 34810.2.1 How Much Will be Paid at Closeout, Really? 35010.2.2 What the Market Says and What the ISDA Says 35210.2.3 A Quantitative Analysis of the Closeout 35310.2.4 A Summary of the Findings and Some Conclusions on Payoff Uncertainty 36010.3 Mathematical Errors in the Payoff of Index Options 36210.3.1 Too Much Left Out 36410.3.2 Too Much Left In 36510.3.3 Empirical Results with the Armageddon Formula 36510.3.4 Payoff Errors and Armageddon Probability 36711 Model Arbitrage 37111.1 Introduction 37111.2 Capital Structure Arbitrage 37311.2.1 The Credit Model 37311.2.2 The Equity Model 37511.2.3 From Barrier Options to Equity Pricing 37711.2.4 Capital-structure Arbitrage and Uncertainty 38111.3 The Cap-Swaption Arbitrage 39111.4 Conclusion: Can We Use No-Arbitrage Models to Make Arbitrage? 39412 Appendix 39712.1 Random Variables 39712.1.1 Generating Variables from Uniform Draws 39712.1.2 Copulas 39712.1.3 Normal and Lognormal 39812.2 Stochastic Processes 39912.2.1 The Law of Iterated Expectation 39912.2.2 Diffusions, Brownian Motions and Martingales 40012.2.3 Poisson Process 40312.2.4 Time-dependent Intensity 40412.3 Useful Results from Quantitative Finance 40512.3.1 Black and Scholes (1973) and Black (1976) 40512.3.2 Change of Numeraire 407Bibliography 409Index 417