Credit Derivatives and Correlations

Credit derivatives are financial instruments that allow trading credit risk independently of the underlying bonds. CDS (Credit Default Swap) and CDO (Collateralized Debt Obligation) played a key role in the financial crisis of 2008: understanding their mechanics and the dangers of correlations is an essential skill for the modern risk manager. The main thing is not mathematics by itself, but its correct interpretation: the Gaussian Copula model turned out to be catastrophically inadequate at the moment of crisis, and the reason was a misunderstanding of its limitations.

Credit Default Swap (CDS)

Structure. A bilateral contract between a protection buyer and a protection seller.

• The buyer pays a regular spread $s$ (basis points per year of notional $N$), usually quarterly.
• When a credit event occurs (default, bankruptcy, restructuring) — the seller compensates $(1 − R)\cdot N$ (loss given default).

CDS is insurance against credit risk, available without owning the underlying bond (naked CDS). It is both a hedging instrument and a speculative tool.

Pricing (no-arbitrage). With constant $h$ and $R$, in continuous time: $ \text{PV(premium leg)} = s \cdot \int_0^T e^{-(r+h)\cdot t} dt = s\cdot\frac{1 - e^{-(r+h)\cdot T}}{r+h} $ $ \text{PV(protection leg)} = (1−R)\cdot h \cdot \int_0^T e^{−(r+h)\cdot t} dt $

From the equality: $s = h\cdot(1 − R)$.

Bootstrapping $h$ from the CDS curve. Having CDS spreads for 1, 2, 3, 5, 7, 10 years, successively reconstruct piecewise constant $h(t)$.

Types of Credit Derivatives

• Single-name CDS. On a specific issuer.
• CDS Index (CDX, iTraxx). On a portfolio of 125 issuers. Standardized tranches.
• Credit-Linked Note (CLN). A bond whose payments depend on credit events.
• Total Return Swap. Exchange total return on a bond for LIBOR + spread.
• CDO (Collateralized Debt Obligation). Structured product (see below).

CDO and Correlation Risk

CDO Structure. A pool of $n$ credits (loans, bonds, RMBS — Residential Mortgage-Backed Securities) is divided into tranches with different priority:

• Senior tranche (e.g., 30–100% losses): receives losses last, AAA rating, low coupon.
• Mezzanine tranche (e.g., 10–30%): medium priority, BBB-A rating.
• Equity tranche (e.g., 0–10%): receives losses first, unrated, high coupon (or residual interest).

Distribution of pool losses: losses $L$ first go to equity (absorbs the first 10%), then to mezzanine, finally to senior.

Gaussian Copula (Li, 2000)

The standard model of default correlations. Each borrower $i$ has a “latent” credit index:

$ X_i = \sqrt{\rho} \cdot Z + \sqrt{1 - \rho} \cdot \varepsilon_i, \quad \text{where } Z, \varepsilon_i \sim N(0, 1) \text{ iid} $

$Z$ — common market factor, $\varepsilon_i$ — idiosyncratic. Correlation $\text{Cor}(X_i, X_j) = \rho$ for all $i \neq j$.

Default: borrower $i$ defaults if $X_i < \Phi^{-1}(p_i)$, where $p_i$ is the marginal probability of default.

Conditional default probability given common factor $Z$: $ p_i(Z) = \Phi\left(\frac{\Phi^{-1}(p_i) - \sqrt{\rho}\cdot Z}{\sqrt{1-\rho}}\right) $

With bad $Z$ $(Z = −2)$: $p_i(Z = −2) \gg p_i$. Correlations amplify “fat tails”.

Numerical Example: CDO Tranches

Pool of 100 credits, $p_i = 0.05$ each (5% annual probability of default), recovery 40%, $\rho = 0.2$.

Monte Carlo simulation (10,000 scenarios $Z$ from $N(0, 1)$):

• Average number of defaults: 5.
• For $Z = −1.65$ (5th percentile): conditional $p_i \approx 0.18$, average 18 defaults.
• For $Z = −2.33$ (1st percentile): $p_i \approx 0.30$, about 30 defaults.

Distribution of aggregate losses $L$ (as % of the pool):

• $E[L] = 0.05\cdot(1 − 0.4) = 3%$.
• $\text{VaR}_{0.99}(L) \approx 18%$.
• $\text{VaR}_{0.999}(L) \approx 30%$.

Tranches:

• Equity 0–10%: average loss $\approx 100\cdot E[\min(L, 10%)/10%]$ (relative to tranche notional) $\approx 35%$. High risk.
• Mezzanine 10–30%: average loss $\approx 8%$ of notional. AAA for the lower half, BBB for the upper.
• Senior 30–100%: loss $\approx 0.3%$ of notional. AAA with almost zero probability of write-off.

Failure in the 2008 Crisis

Gaussian Copula was applied to MBS pools with historical correlation $\rho \approx 0.2–0.3$, estimated from the “fat” 2003–2006 years. In the 2007–2008 crisis $\rho \rightarrow 0.7–0.9$: mortgage defaults across the country became correlated (national decline in housing prices).

Recalculation with $\rho = 0.7$: probability of mass default (

gt;30$ out of 100) increased from 0.001% to 5%. Senior tranches, considered AAA, depreciated by 60–80%. Bear Stearns, Lehman, AIG—all either went bankrupt or were rescued by the government because of this underestimated tail risk.

Lessons.

• Marginal distributions and correlations are not enough. Need t-copulas (heavy tails), wrong-way risk, stress-testing.
• Calibrate parameters on stressed periods, not on “calm” ones.
• Liquidity risk: even with technically correct valuation it is impossible to sell AAA-tranche at fair price during panic.

Real Applications

• CDX/iTraxx trading. Volume ~$5 trillion notional. Used by hedge funds for market-neutral strategies, and banks for portfolio hedging.
• Bespoke tranches. Customized CDOs for specific institutional clients. Revived since 2017 after disappearing post-2008.
• CLO (Collateralized Loan Obligations). Equivalent of CDO for leveraged loans. Market $1+ trillion, actively used in LBO financing.
• Synthetic CDOs. Based on CDS, no real assets. Used for regulatory arbitrage, now restricted.

Assignment. Pool of 5 credits with $p_i = 0.02$ each, $\rho = 0.3$, recovery 0%. (a) In Python (numpy.random) implement Gaussian Copula: for each of 10,000 scenarios generate $(Z, \varepsilon_1, ..., \varepsilon_5)$, count the number of defaults. (b) Plot histogram of aggregate losses. (c) Calculate expected losses on tranches: equity (0–10%), mezzanine (10–30%), senior (30–100%) — what proportion of tranche notional will be written off on average? (d) Repeat with $\rho = 0.7$ — how do senior tranche losses change? (e) Bonus: use t-copula with 4 degrees of freedom instead of normal — how will tail losses change?