Model Risk in Financial Markets

The following sections are included:

• Dedication
• Preface
• List of Notations
• Contents
• List of Figures
• List of Tables

Why do I consider this book to be relevant to the community of people involved in one way or another with financial modelling? Because the next big crisis already has the seeds planted and its roots are growing stronger and bigger by the day. There are already manifestations of model risk that led to substantial losses and it may sound cliché to say that this is only the tip of the iceberg. In 1987 Merrill Lynch reported losses of 300 million USD on stripped mortgage-backed securities because of an incorrect pricing model and five years later in 1992 J.P. Morgan lost about 200 million USD in the mortgage-backed securities market because of inadequate modelling of prepayments. Bank of Tokyo/Mitsubishi announced in March 1997 that its New York-subsidiary dealing with derivatives had incurred an $83 million loss because of their internal pricing model overvalued a portfolio of swaps and options on USD interest rates. [Dowd (2002)] pointed out that the loss was caused by wrongly using a one-factor Black-Derman-Toy (BDT) model to trade swaptions. The model was calibrated to market prices of ATM swaptions but used to trade out-of-the-money (OTM) Bermudan swaptions, which was not appropriate. With the benefit of hindsight it is known now that pricing OTM swaptions and Bermudan swaptions requires multi-factor models. Also in 1997, NatWest Capital Markets reported a £50 million loss because of a mispriced portfolio of German and U.K. interest rate derivatives on the book of a single derivatives trader in London who fed his own estimates of volatility into a model pricing OTC interest rate options with long maturities. The estimates were high and led to fictitious profits. It is not clear whether the trader simply inflated the volatility estimate or she/he came up with the estimate that was more “convenient” to her/him. [Elliott (1997)] pointed out that these losses were directly linked to model risk. [Williams (1999)] remarked that model risk was not included in standard risk management software and in 1999 about 5 billion USD losses were caused by model risk…

The following sections are included:

• Introduction
• Present Value
• Constant Relative Risk Aversion Utility
• Risk versus Return: The Sharpe Ratio
  • Issues related to non-normality
  • The Sharpe ratio and negative returns
• APT
• Notes and Summary

The following sections are included:

• Introduction
• Short Rate Models
• Theory of Interest Rate Term Structure
  • Expectations Hypothesis
  • A reexamination of Log EH
  • Reconciling the arguments and examples
• Yield Curve
  • Parallel shift of a flat yield curve
  • Another proof that the yield curve cannot be flat
  • Deterministic maturity independent yields
  • Consol modelling
• Interest Rate Forward Curve Modelling
• One-factor or Multi-factor models
• Notes and Summary

The following sections are included:

• Introduction
• Transaction Costs
• Arbitrage
  • Non-convergence financial gain process
  • Distortion operator with arbitrage
• Notes and Summary

The following sections are included:

• Introduction to Model Risk
  • Parameter estimation risk
  • Model selection risk
  • Model identification risk
  • Computational implementation risk
  • Model protocol risk
• Uncertain Volatility
  • An option pricing model with uncertain volatility
• Option Pricing under Uncertainty in Complete Markets
  • Parameter uncertainty
  • Model uncertainty
  • Numerical examples
  • Accounting for parameter estimation risk in the Black-Scholes model
  • Accounting for parameter estimation risk in the CEV model
• A Simple Measure of Parameter Uncertainty Risk
• Bayesian Option Pricing
  • Modelling the future asset value under physical measure
  • Modelling the current asset value under a risk neutral measure
• Measuring Model Uncertainty
  • Worst case risk measure
• Cont's Framework for Model Uncertainty
  • An axiomatic approach
  • A coherent measure of model risk
  • A convex measure of model risk
• Notes and Summary

The following sections are included:

• Introduction to Model Risk for Portfolio Analysis
• Bayesian Averaging for Portfolio Analysis
  • Empirical Bayes priors
  • Marginal likelihood calculations
• Portfolio Optimization
  • Portfolio optimisation with stochastic interest rates
  • Stochastic market price of risk
  • Stochastic volatility
• Notes and Summary

The following sections are included:

• Introduction
• Probability Distribution Functions and Density Functions
• Gaussian Distribution
• Moments
  • Mean-median-mode inequality
  • Distributions are not defined by moments
  • Conditional expectation
• Stochastic Processes
  • Infinite returns from finite variance processes
  • Martingales
• Spurious Testing
  • Spurious mean reversion
  • Spurious regression
• Dependence Measures
  • Problems with the Pearson linear correlation coefficient
  • Pitfalls in detecting breakdown of linear correlation
  • Copulas
  • More general issues
  • Dependence and Levy processes
• Notes and Summary

The following sections are included:

• Introduction
• Controlling Risk in Insurance
  • Diversification
  • Variance
• Coherent Distortion Risk Measures
• Value-at-Risk
  • General observations
  • Expected shortfall and expected tail loss
  • Violations ratio
  • Correct representation
  • VaR may not be subadditive
  • Artificial improvement of VaR
  • Problems at long horizon
• Backtesting
  • Uncertainty in risk estimates: A short overview
  • Backtesting VaR
• Asymptotic Risk of VaR
  • Normal VaR
  • More general asymptotic standard errors for VaR
  • Exact confidence intervals for VaR
  • Examples
  • VaR at different significance levels
  • Exact confidence intervals
  • Extreme losses estimation and uncertainty
  • Backtesting expected shortfall
• Notes and Summary

The following sections are included:

• Introduction
• Problems with Estimating Diffusions
  • A brief review
  • Parameter estimation for the Vasicek model
  • Parameter estimation for the CIR model
• Problems with Estimation of Jump-Diffusion Models
  • The Gaussian-Poisson jump-diffusion model
  • ML Estimation under the Merton Model
  • Inexistence of an unbiased estimator
• A Critique of Maximum Likelihood Estimation
• Bootstrapping Can Be Unreliable Too
• Notes and Summary

The following sections are included:

• Introduction
• Problems with Monte Carlo Variance Reduction Techniques
• Pitfalls in Estimating Greeks with Pathwise Monte Carlo Simulation
• Pitfall in Options Portfolio Calculation by Approximation Methods
• Transformations and Expansions
  • Edgeworth expansion
  • Computational issues for MLE
• Calculating the Implied Volatility
  • Existence and uniqueness of implied volatility under Black-Scholes
  • Approximation formulae for implied volatility
  • An interesting example
• Incorrect Implied Volatility for Merton Model
• Notes and Summary

In this chapter real data is used for a portfolio analysis problem with the declared scope of highlighting the pitfalls that may appear when using the Sharpe ratio. There is a growing literature on improving or adjusting the classical Sharpe ratio for investments analysis. Important and useful reading can be found in [Sharpe (1994)], [Dowd (2000a)], [Lo (2002)], [Goetzmann et al. (2004)] and [Cerny (2009)]…

The following sections are included:

• Introduction
• Problems in Pricing Derivatives for Assets with a Slow Business Time
• Choosing the Correct Auxiliary Values
• Empirical Exemplifications
  • A mean-reversion model with predictability in the drift
  • Data augmentation
• MCMC Inference for the IPD model
• Derivatives Pricing
• Notes and Summary

The following sections are included:

• Introduction
• A Short Example
• Further Analysis
  • Bayesian inference with Gibbs sampling
• Hierarchical Bayesian Models for Credit Risk
  • Model specification of probabilities of default
  • Model estimation
• Standard&Poor's Rating Data
  • Data description
  • Hierarchical model for aggregated data
  • Hierarchical time-series model
  • Hierarchical model for disaggregated data
• Further Credit Modelling with MCMC Callibration
• Estimating the Transition Matrix
  • MCMC estimation
  • MLE estimation
• Notes and Summary

The following sections are included:

• Yes, We Can
• No, We Cannot
• A Non-technical Template for Model Risk Control
  • Identify the type of model risk that may appear
  • A guide for senior managers
• There is Still Work to Do

Here are the notations used by [Tang and Chen (2009)]…

The following sections are included:

• Bibliography
• Index