Chapter 25: Credit Derivatives

Credit Default Swaps (CDS)

A CDS is like insurance against the default of a reference entity. The buyer pays a regular premium (CDS spread) and receives a payoff if a credit event occurs.

Credit events include: bankruptcy, failure to pay, restructuring.

Settlement

CDS Spread

The annual premium as a percentage of notional principal. The spread is set so that the present value of premium payments equals the present value of expected loss from default:

$s=\frac{{\sum }_{i=1}^n(1-R)\cdot q_i\cdot v_i}{{\sum }_{i=1}^n{\tau }_i\cdot p_i\cdot v_i+\text{accrual term}}$

where:

• $q_i$ = default probability in period $i$
• $p_i$ = survival probability to period $i$
• ${\tau }_i$ = accrual period length
• $v_i$ = discount factor

CDS-Bond Basis

$\text{Basis}=\text{CDS spread}-\text{Bond credit spread}$

Normally should be close to zero (arbitrage). In practice, the basis can deviate due to:

• Funding costs
• Counterparty risk
• Liquidity differences
• Delivery option value in CDS
• Short-sale constraints on bonds

CDS Valuation

Given CDS spreads for different maturities, the implied hazard rate can be bootstrapped (similar to bootstrapping zero rates from bond prices). This gives the risk-neutral default probability curve.

Credit Indices

Standardized portfolios of CDS, e.g.,:

• CDX IG: 125 investment-grade North American companies
• iTraxx Europe: 125 European investment-grade companies
• CDX HY: 100 high-yield North American companies

Indices trade with a fixed coupon. The quoted price ($P$) indicates the upfront payment:

• If spread > coupon, the protection buyer pays upfront
• If spread < coupon, the protection seller pays upfront

Basket CDS and CDOs

Basket CDS

CDS on a portfolio of reference entities. Payout depends on defaults within the basket.

• First-to-default: Pays on the first default in the basket
• nth-to-default: Pays on the nth default
• Higher "n" means lower probability of payout → lower spread

Collateralized Debt Obligations (CDOs)

A CDO pools debt instruments and issues tranches with different risk levels:

Synthetic CDO Valuation

The one-factor Gaussian copula model (Li's model, 2000) is the standard approach. It maps each reference entity to a "time to default" using a latent variable:

$x_i=\sqrt{\rho }\cdot M+\sqrt{1-\rho }\cdot Z_i$

where $M$ is a common market factor, $Z_i$ are idiosyncratic factors, and $\rho$ is the correlation. Default occurs when $x_i<N^{-1}(Q_i(T))$ for the given maturity.

Conditional on $M$, defaults are independent. The conditional probability of default:

$Q_i(T\mid M)=N\left(\frac{N^{-1}(Q_i(T))-\sqrt{\rho }\cdot M}{\sqrt{1-\rho }}\right)$

This model was widely criticized during the 2007-2008 crisis for underestimating correlation during stress periods (the "correlation smile" problem).

Base Correlation vs. Compound Correlation

The standard market model uses base correlation for pricing non-standard tranches: interpolate between equity tranches to get correlations for bespoke attachment/detachment points.

Total Return Swaps (TRS)

An agreement to exchange the total economic performance of an asset (interest + capital gains/losses) for a fixed or floating rate. Unlike a CDS, TRS transfers market risk as well as credit risk. The TRS receiver gets the total return; the payer gets LIBOR/SOFR + spread.