Slide ρ to see how correlation reshapes the joint behavior. Both surfaces describe the same statistical fact — but the copula is shape-agnostic.拖動 ρ 觀察相關性如何改變聯合行為。兩個曲面描述相同的統計事實——但 Copula 與形狀無關。
The bivariate normal density lives on (−∞, +∞)². Its shape changes when you swap normal marginals for t-distributions or lognormals.
The copula density lives on the unit square [0,1]². Its shape depends only on the dependence parameter ρ — not on what the marginals look like.
Three normal densities. Shaded region under one curve = P(X ≤ x₂). Adjust threshold to see how area changes.
Every multivariate distribution can be split cleanly into its marginal distributions and a copula that captures the dependence. This is the theoretical license to do everything that follows.每個多元分布都可以被清晰地分解為邊際分布和捕捉相依性的 Copula。這是後續所有操作的理論依據。
Let H be a joint distribution function with continuous marginals FX and FY. Then there exists a unique function C: [0,1]² → [0,1] — the copula — such that for all (x,y) ∈ ℝ²:
Conversely, if C is any copula and FX, FY are univariate distribution functions, then the function H defined above is a joint distribution function with marginals FX and FY.
Sklar lets you build any joint distribution in two completely independent steps:
① Choose marginals for X and Y separately — they don't have to be from the same family. X could be t with df=8, Y could be lognormal, Z could be a generalized extreme value. Whatever fits the data.
② Choose a copula to bind them together. The copula governs the dependence structure (linear correlation, tail dependence, asymmetry) without being tied to any particular marginal shape.
This separation is what makes copulas the lingua franca of credit portfolio modeling: every CDS gives you an obligor's marginal default probability, and the copula glues 125 of them into a joint distribution.
Apply Sklar's theorem to the bivariate normal distribution. The marginals are univariate normals; what's left over after stripping them out is the copula.將 Sklar 定理應用於二元常態分布。邊際分布為單變量常態；剝除後剩餘的部分即為 Copula。
Let (X,Y) be jointly standard normal with correlation ρ. Its joint distribution function is the bivariate standard normal CDF, denoted Φρ:
The marginals are individually standard normal, FX(x) = Φ(x) and FY(y) = Φ(y).
By Sklar, there exists a unique copula CGa such that:
Now substitute u = Φ(x) and v = Φ(y), which means x = Φ⁻¹(u) and y = Φ⁻¹(v):
This gives the explicit formula for the bivariate Gaussian copula with parameter ρ:
where u, v ∈ [0,1] are uniform inputs (the marginal CDF values). This is exactly the equation used in the worked example below — Step 1 produces u and v, Step 2 inverts to Φ⁻¹(u) and Φ⁻¹(v), Step 3 evaluates the bivariate normal CDF.
The result is the same for any marginals FX, FY, not just normal ones. If you want a joint distribution with t-marginals tied together by Gaussian dependence:
This is what makes the equation so powerful: the same CGaρ dependence object can be glued onto any combination of marginals. The copula carries the correlation; the marginals carry the shape.
Differentiate twice to get the copula density. By the change-of-variables formula, with x = Φ⁻¹(u), y = Φ⁻¹(v):
which simplifies to a clean closed form:
The exploding peaks at (0,0) and (1,1) you see in the copula density chart above are the (1−ρ²)−½ factor combined with the corner behavior of Φ⁻¹.
Each asset has t-distributed returns. Scale ≠ standard deviation: σ = s · √[df/(df−2)].每個資產的報酬服從 t 分布。尺度參數 ≠ 標準差：σ = s · √[df/(df−2)]。
Probability that both assets produce a return below the threshold.兩個資產報酬均低於門檻值的機率。
For each obligor i: Ai = √ρ · M + √(1−ρ) · εi, where M and εi are independent standard normals. ρ is the asset correlation — the "copula" parameter governing how defaults cluster.
Obligor i defaults over horizon T if Ai ≤ Φ⁻¹(pi), where pi is the marginal default probability (from CDS spreads or rating agency tables).
Given a realization of M, defaults are independent. Conditional default probability:
Bad M (negative) → high p(M) → many defaults. This is the contagion mechanism.不利的 M（負值）→ 高 p(M) → 大量違約。這就是傳染機制。
100-name homogeneous basket, 5-year PD = 5%. Watch the tail explode as correlation rises.100 個同質標的的籃子，5 年違約機率 = 5%。觀察隨著相關性上升，尾部如何急劇膨脹。
Loss in % of notional. Compare independent (ρ=0) vs simulated correlated outcome.以名義本金百分比計算的損失。比較獨立情境（ρ=0）與模擬相關情境的結果。
Expected tranche loss, default probability, and implied par spread (bps/yr) under the assumed correlation regime.在假設的相關性情境下，各分段的預期損失、違約機率及隱含平價利差（基點/年）。
| Tranche 分段 | Attach 觸發點 | Detach 脫離點 | Width 寬度 | P(any loss) 損失機率 | E[loss] 預期損失 | Spread (bps) 利差 |
|---|
Same per-name PD and correlation. Smaller pools have less diversification benefit; the loss distribution stays granular and bumpier.相同的個別違約機率和相關性。較小的資產池分散效益較低，損失分布較粗糙且起伏較大。
The "correlation smile" of structured credit: equity loves low ρ; senior hates high ρ; mezzanine is roughly correlation-neutral somewhere in between.結構性信用的「相關性微笑」：股權層偏好低 ρ；優先層懼怕高 ρ；夾層在兩者之間大致相關性中性。
Loss thresholds and the probability the pool exceeds each, by correlation regime.在不同相關性情境下，資產池超過各損失門檻的機率。
| Loss threshold 損失門檻 | ρ=0.10 | ρ=0.20 | ρ=0.40 | ρ=0.70 |
|---|
Li (2000) replaced the joint default time distribution with a Gaussian copula on the marginal survival times. Tractable, fast to calibrate, easy to plug into existing pricing systems.
Single ρ assumes constant pairwise dependence; the Gaussian copula has zero tail dependence — extreme co-movements are systematically underestimated. In 2007–2008, realized correlations spiked far beyond the 0.20–0.30 calibrated by base correlation models, repricing senior tranches by orders of magnitude.
t-copulas (positive tail dependence), Clayton/Gumbel for asymmetric tails, factor-loading copulas, random-recovery models, and direct Monte Carlo with regime-switching factor M. The plumbing is the same; the dependence kernel is richer.