Theory of Credit Risk Models

Define credit risk through the six dimensions, including event, duration, frequency, severity, correlation, and capital under Basel accords, emphasizing vanilla default risk, recovery, and credit spreads.

Identify sources of credit risk across secured and unsecured loans, public debt, and counterparty exposures, including collateral, credit spreads, and OTC versus exchange-traded derivatives.

Survey credit risk models from qualitative face-to-face and credit score methods to quantitative approaches like the Merton model and Euroland Turnbull with multiple states.

Explore the challenges in credit risk modelling, including data scarcity, severity estimation with low default frequency, data fragmentation, credit enhancements, and rating agency uncertainties.

Conduct a personal interview in the face-to-face model, as the bank manager assesses collateral and surety, reviews borrower job and income, and weighs loan purpose, default, recovery, and risk change.

Automates the face-to-face credit assessment to quickly filter loan applicants. Relying on data-driven rules set by a credit risk analyst, it risks blind spots and gaming.

Introduce derivatives as instruments deriving value from underlying assets to manage market risk, cover hedges and speculators, and distinguish over-the-counter and exchange-traded types, with complete-market and no-arbitrage assumptions.

Explore futures and forwards, exchange-traded and over-the-counter derivatives that lock in future prices for underlying assets. Learn long and short payoffs, no-arbitrage pricing, and time-value adjustment for fair strike prices.

Explain the long and short positions in call and put options, the rights to buy or sell at a strike price, premium payments, payoff diagrams, and Black-Scholes pricing.

Explore how the factor mix in the Black-Scholes model shapes option premiums, including share price, strike price, time to expiry, volatility, and risk-free rate, along with call and put payoffs.

The Merton model links a firm's asset value to equity and debt, with equity as a call on assets and debt as a put, deriving default probability and credit spreads.

Discuss the drawbacks of the Merton model: assumes observable assets, equity as a long call on assets, debt as a zero-coupon short put, and a frictionless market with risk-free rate.

The KMV model extends the merchant model by using distance to default to estimate the one-year probability of default from asset value, debt thresholds, volatility, and Black-Scholes relations.

Compare the kmv model to the merton model, noting kmv supports coupon paying debt and share-derived value. See how market sentiment and equity data inform credit risk.

Model credit risk with the Jarrow Turnbull framework, using Markov processes to map credit states, estimate transition probabilities, and compute default timing and investment-grade transitions.

Examine the euro Turnbull credit migration model and its drawbacks, including time-homogeneity, data limits, state granularity, and credibility concerns of credit ratings.

Explore how value at risk guides credit portfolio models, focusing on correlation and diversification. Review multivariate structural and migration approaches, credit metrics, copulas, and recovery rates to assess portfolio risk.

Explore ten practical methods to manage credit risk, from collateral and securitization to hedging with interest rate swaps and credit default swaps, plus underwriting, diversification, and soft collection strategies.

Explore stochastic processes by linking the time domain t and the state space X. Model stock prices as X evolving over time, use patterns to simulate futures and study distributions.

Introduce stationary as a property where a stochastic process's statistical features do not change over time, and highlight weak stationarity with a constant mean and lag-based covariance.

Define increments as the changes in a stochastic process over time, from single-step to multi-step intervals, and emphasize independence as a key property for later modeling.

Explore the Markov property and filtration in stochastic processes, showing that the future depends only on the present value and independent increments enable this prediction.

Explore white noise, where x_t are iid normally distributed with mean zero. Higher variance makes it noisier, and it acts as an error term while being stationary and Markov.

Explore the random walk model x_t = x_{t-1} + ε_t, where ε_t is white noise, often normal, noting its non-stationarity, growing variance, and independent increments.

Delve into the Poisson distribution, its formula and the fact that both mean and variance equal lambda, with examples and a note on Poisson and compound Poisson processes.

Explore Poisson process with rate lambda in continuous time, starting at zero with independent, stationary increments; the process is not stationary as mean and variance grow with time, modeling claims.

Analyze the compound Poisson process as total losses equal to the sum of claims. Model the number of claims with a Poisson process and claim sizes as iid.

The lecture introduces the Markov chain, a discrete-time stochastic process with a discrete state space and the Markov property, using a healthy–ill–dead example to model disease transitions.

Explore transition probabilities in a markov chain and how they describe moves between states. See how transitions among healthy, corona, and dead estimate p_ij^m(n) as N_ij divided by N_i.

Explore the Chapman-Kolmogorov equation by showing how the probability of dying from a healthy state equals staying healthy and dying plus getting sick and dying, via an interim state.

Convert a three-state markov chain—healthy, corona, and dead—into a transition matrix p, where each row sums to one for time-homogeneous, one-step transition probabilities.

Explore a transition matrix in a Markov chain, compute multi-period death probabilities from a healthy state, and implement using R code to multiply matrices and extract future probabilities.

Introduce a recovered state to restore the Markov property in a credit risk model, showing how converting non-Markov dynamics into a modified Markov chain with absorbed dead states clarifies transitions.

Explore stationary probability distributions in finite Markov chains, compute the stationary vector pi, and analyze long-run state distributions and convergence via transition matrices.

Examine irreducibility in Markov chains by analyzing transition diagrams and matrices, distinguishing irreducible and reducible cases, and noting a unique stationary distribution for finite irreducible chains.

Explore periodicity in simple irreducible Markov chains, identify state periods and the period one condition, and show convergence to a stationary distribution with implications for credit risk models.

Explore the continuous-time markov jump process with rates in a discrete state space. Examine a simple alive or death model under age-dependent mortality using forward and backward differential equations.

Explain T Q X as the chance a life at age X dies before reaching X plus T, and T P X as chance it survives to X plus T.

Explore Kolmogorov's forward differential equation for Markov jump processes by outlining three key assumptions: the Markov property, small-h transition rates, and age-specific constant mortality for t<1 year.

Explore the mathematics of Kolmogorov's forward differential equation for credit risk models, deriving survival dynamics from transition probabilities and the force of mortality.

Solve Kolmogorov's forward equation by converting to log form, integrating 0 to t, and exponentiating to obtain survival under constant mortality; relate force of mortality to deaths over waiting time.

Compare Markov chains and Markov jump processes, highlighting discrete versus continuous time, transition probabilities versus transition rates, and the generator matrix with absorbing states.

Kolmogorov's backward differential equation shows how state transition probabilities evolve over time using the generator and transitional matrix, including survival probabilities in a Markov jump process.