⚡ USD/JPY Gamma Scalping Lab

01 — FOUNDATIONS

From bivariate normal to copula

A bivariate normal mixes the marginals and the dependence into one object. A copula isolates the dependence by feeding it uniform inputs F(x), F(y) ∈ [0,1].

Compare the surfaces

Slide ρ to see how correlation reshapes the joint behavior. Both surfaces describe the same statistical fact — but the copula is shape-agnostic.

Correlation ρ 0.70

View mode

Why the change of variable matters

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.

C(u,v) = Φ_ρ[ Φ⁻¹(u), Φ⁻¹(v) ]

Bivariate normal vs Gaussian copula

Bivariate Normal Distribution

3-D Surface (rotate · drag)

Contour view

Gaussian Copula

3-D Surface (rotate · drag)

Contour view

3-D surface above the contour for each. Density mode shows the bivariate normal "bell" and the copula's corner spikes; distribution mode shows the smooth CDF climbing to 1.

01b — DERIVATION

How the copula equation is built — five conceptual stages

From a single random variable's density to the multivariate dependence object: each step is a coordinate change designed to strip out marginal shape until only the pure dependence remains.

① Density f(x)

② CDF F(x)

③ Joint density f(x,y)

④ Joint CDF F(x,y)

⑤ Copula C(u,v)

Stage 1 — Probability density f(x)

Three normal densities. Shaded region under one curve = P(X ≤ x₂). Adjust threshold to see how area changes.

f(x) = (1/σ√2π) · exp[−(x−μ)²/2σ²]

Threshold x₂ −1.50

Mathematics

Sklar's Theorem (1959) — the foundation

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.

Statement

Let H be a joint distribution function with continuous marginals F_X and F_Y. Then there exists a unique function C: [0,1]² → [0,1] — the copula — such that for all (x,y) ∈ ℝ²:

H(x,y) = C( F_X(x), F_Y(y) )

Conversely, if C is any copula and F_X, F_Y are univariate distribution functions, then the function H defined above is a joint distribution function with marginals F_X and F_Y.

Why it matters — the separation principle

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.

Constructing the Gaussian copula explicitly

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.

Step 1 — Start with a bivariate standard normal

Let (X,Y) be jointly standard normal with correlation ρ. Its joint distribution function is the bivariate standard normal CDF, denoted Φ_ρ:

Φ_ρ(x,y) = P(X ≤ x, Y ≤ y) = ∫_−∞^x ∫_−∞^y φ_ρ(s,t) ds dt

The marginals are individually standard normal, F_X(x) = Φ(x) and F_Y(y) = Φ(y).

Step 2 — Apply Sklar's identity

By Sklar, there exists a unique copula C^Ga such that:

Φ_ρ(x,y) = C^Ga( Φ(x), Φ(y) )

Now substitute u = Φ(x) and v = Φ(y), which means x = Φ⁻¹(u) and y = Φ⁻¹(v):

Φ_ρ( Φ⁻¹(u), Φ⁻¹(v) ) = C^Ga(u, v)

Step 3 — Read off the Gaussian copula

This gives the explicit formula for the bivariate Gaussian copula with parameter ρ:

C^Ga_ρ(u, v) = Φ_ρ( Φ⁻¹(u), Φ⁻¹(v) )

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.

Step 4 — Why this is "the" Gaussian copula

The result is the same for any marginals F_X, F_Y, not just normal ones. If you want a joint distribution with t-marginals tied together by Gaussian dependence:

H(x, y) = C^Ga_ρ( T_{ν_x}(x), T_{ν_y}(y) ) = Φ_ρ( Φ⁻¹[T_{ν_x}(x)], Φ⁻¹[T_{ν_y}(y)] )

This is what makes the equation so powerful: the same C^Ga_ρ dependence object can be glued onto any combination of marginals. The copula carries the correlation; the marginals carry the shape.

Step 5 — The copula density (for completeness)

Differentiate twice to get the copula density. By the change-of-variables formula, with x = Φ⁻¹(u), y = Φ⁻¹(v):

c^Ga_ρ(u, v) = ∂²C/∂u∂v = φ_ρ(x, y) / [ φ(x) · φ(y) ]

which simplifies to a clean closed form:

c^Ga_ρ(u, v) = (1/√(1−ρ²)) · exp{ [2ρxy − ρ²(x² + y²)] / [2(1−ρ²)] }

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.

02 — WORKED EXAMPLE

Two-asset joint loss via Gaussian copula

t-distributed returns, ρ = 0.7. Three-step recipe: marginal CDFs → normal-equivalent quantiles → bivariate normal CDF.

Inputs — two risky assets

Each asset has t-distributed returns. Scale ≠ standard deviation: σ = s · √[df/(df−2)].

Asset X — mean μ 0.04

Asset X — scale s 0.15

Asset X — df ν 8

Asset Y — mean μ 0.07

Asset Y — scale s 0.17

Asset Y — df ν 5

Copula correlation ρ 0.70

Loss threshold (return ≤ ?) 0.000

Step-by-step calculation

Probability that both assets produce a return below the threshold.

Step 1 — Marginal probabilities under each t-distribution

z_x = (k − μ_x) / s_x = —
z_y = (k − μ_y) / s_y = —
F_x(k) = T_{ν_x}(z_x) = —
F_y(k) = T_{ν_y}(z_y) = —

Step 2 — Map to standard normal quantiles

Φ⁻¹[F_x(k)] = —
Φ⁻¹[F_y(k)] = —

Step 3 — Apply bivariate standard normal CDF

C(u,v) = Φ_ρ[ Φ⁻¹(F_x), Φ⁻¹(F_y) ]
= Φ_0.70[ —, — ]
= —

P(X≤k)—

P(Y≤k)—

P(X≤k ∩ Y≤k)—

03 — APPLICATION TO CREDIT

The one-factor Gaussian copula for credit baskets

Vasicek (1987) / Li (2000): each obligor's "asset return" decomposes into a single systematic factor M and an idiosyncratic shock. Default if asset value falls below the default threshold.

The model

Step 1 — Factor decomposition

For each obligor i: A_i = √ρ · M + √(1−ρ) · ε_i, where M and ε_i are independent standard normals. ρ is the asset correlation — the "copula" parameter governing how defaults cluster.

Step 2 — Default trigger

Obligor i defaults over horizon T if A_i ≤ Φ⁻¹(p_i), where p_i is the marginal default probability (from CDS spreads or rating agency tables).

Step 3 — Conditional independence

Given a realization of M, defaults are independent. Conditional default probability:

p(M) = Φ( [ Φ⁻¹(p) − √ρ · M ] / √(1−ρ) )

Bad M (negative) → high p(M) → many defaults. This is the contagion mechanism.

How ρ reshapes the default-count distribution

100-name homogeneous basket, 5-year PD = 5%. Watch the tail explode as correlation rises.

Distribution of the number of defaults over 5 years, by asset correlation ρ. Higher ρ ⇒ fatter right tail and bimodality emerges at extreme ρ.

04 — MONTE CARLO LABORATORY

Simulate a credit basket under the copula

Set portfolio characteristics, run draws, observe how correlation transforms loss outcomes from binomial-like to wildly skewed.

Simulation controls

Number of names N 100

5-year PD per name 5.0%

Asset correlation ρ 0.20

Recovery rate R 40%

Number of paths 10,000

Mean losses—

Std dev—

95% VaR—

99.5% VaR—

Portfolio loss distribution

Loss in % of notional. Compare independent (ρ=0) vs simulated correlated outcome.

Histogram of losses from — Monte Carlo paths under the one-factor Gaussian copula.

05 — CDO TRANCHE ANALYSIS

5-year synthetic CDO tranche pricing & risk

A CDO carves the loss distribution into vertical slices. Equity absorbs first losses, mezzanine sits in the middle, senior is "protected" — but only if correlation is well-estimated.

Pool & structure

Pool size (names)

5-yr PD per name 5.0%

Asset correlation ρ 0.20

Recovery R 40%

Tranche attachments (% of pool)

Equity 0 → 3%

Mezz Jr → 7%

Mezz Sr → 15%

Senior → 30%

Tranche risk profile

Expected tranche loss, default probability, and implied par spread (bps/yr) under the assumed correlation regime.

Tranche	Attach	Detach	Width	P(any loss)	E[loss]	Spread (bps)

Spread ≈ E[tranche loss] / (T · width) · 10,000 bps. A first-order par approximation, ignoring discounting and timing.

Pool size effect — 10 / 20 / 50 / 100 names

Same per-name PD and correlation. Smaller pools have less diversification benefit; the loss distribution stays granular and bumpier.

Loss distribution over 5 years for varying pool sizes at fixed ρ and PD.

Correlation sensitivity per tranche

The "correlation smile" of structured credit: equity loves low ρ; senior hates high ρ; mezzanine is roughly correlation-neutral somewhere in between.

Expected tranche loss vs ρ (other parameters held fixed).

06 — STRESS SCENARIOS

What happens when correlation jumps?

2007–2008 lesson: the senior tranche's "AAA-ness" depends critically on the assumed ρ. A jump from 0.20 to 0.50 vaporizes that protection.

Benign — ρ = 0.10—Senior expected loss

Base — ρ = 0.20—Senior expected loss

Stressed — ρ = 0.40—Senior expected loss

Crisis — ρ = 0.70—Senior expected loss

All four scenarios use 100 names, 5% PD, 40% recovery. The senior tranche (15%–30%) is "untouchable" only under the benign assumption.

Reading the table

Loss thresholds and the probability the pool exceeds each, by correlation regime.

Loss threshold	ρ=0.10	ρ=0.20	ρ=0.40	ρ=0.70

Practitioner's note

What David Li actually proposed

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.

Where it broke

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.

Modern alternatives

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.