Cross symbol margin

Factor structure, stressed betas and where diversification credit comes from.

Margin computed one symbol at a time treats a book long twelve correlated names as twelve independent risks. It is not, and the error runs in the dangerous direction: the account is charged too little for precisely the concentration that fails together. This page closes that gap, which earlier revisions listed as the largest in the risk model.

Why not shock everything jointly

A joint grid over N symbols is |G|^N scenarios. At 54 cells and 20 symbols that is 54²⁰, which nobody will evaluate. Real risk systems impose a factor structure instead.

r_{i} = β_{i} F + ε_{i}, σ_{i}^{2} = β_{i}^{2} σ_{F}^{2} + σ_{ε}^{2}

The systematic part is one shared shock and diversifies not at all. The residuals are independent by construction. Everything is mutually independent, so the losses combine in quadrature.

M = \sqrt{L_{sys}^{2} + \sum_{i} L_{ε, i}^{2}}

Linear in the number of symbols to evaluate rather than exponential.

The decomposition

Factor volatility is 18% for equities and 45% for crypto. Residual volatility follows from the identity above.

BTC is close to being the crypto factor itself, so almost none of its risk diversifies. DOGE carries volatility no beta can explain, so most of its risk does.
Symbol	Vol	Beta	Systematic	Residual
MSFT	22%	0.95	17%	14%
NVDA	42%	1.60	29%	31%
MSTR	92%	3.20	58%	72%
BTC	46%	1.00	45%	10%
ETH	58%	1.20	54%	21%
DOGE	112%	1.90	86%	72%

Correlations rise in a crash

A model using placid period correlations awards its largest diversification credit precisely when that credit is least deserved. Beta is therefore stressed toward one as the factor move worsens, and only on the downside, because correlations rising in a rally does not threaten a margin system.

β^{*} = β + (1 - β) \cdot λ \cdot min (1, | F | / 0.15)

A defensive name is pulled up toward the factor, not down. Correlations going to one means everything behaves more like the market, which for a low beta name is more exposure rather than less.

Measured credit

Equal notional per name, 30 day calls 5% out of the money. A single symbol receives nothing, and credit grows with genuine breadth.
Book	Standalone	Cross margin	Credit
One name	$2,415	$2,415	0.0%
Two crypto majors	$6,242	$5,799	7.1%
Three correlated crypto	$11,417	$10,414	8.8%
Four megacap tech	$5,776	$5,518	4.5%
Eight, both classes	$24,676	$23,294	5.6%
Twelve, wide	$39,159	$36,077	7.9%

Deliberately conservative

Credits of 4% to 9% are well below what a mature prime broker awards. That is the intended setting for a venue that has never run in production. The credit is also hard capped at 45%, because a model that can award unlimited benefit will eventually award it to a book that is not diversified at all.

Errors found building this

Three, all caught by the numbers being wrong in an obvious direction

The factor grid was built from average symbol volatility and then multiplied by beta, counting beta twice. The result charged a diversified book roughly double the standalone sum.

Volatility was shocked in both legs. Implied volatility is overwhelmingly a systematic risk, and shocking it in the residual leg as well double counted the largest risk in an option book, overcharging a lone symbol by about 70%.

The legs were added linearly. Factor and residual returns are independent, so their losses combine in quadrature.

What is still missing

Betas are static constants, not estimated from returns and not updated. A real venue would re-estimate continuously and widen the grid when estimates are unstable.
One factor per asset class. Sector structure within equities is not modelled, so a book concentrated in semiconductors receives the same credit as one spread across sectors.
The cross class correlation is a constant pair of placid and stressed values rather than a fitted quantity.