How FRTB Sensitivities-Based Algorithms Turn Trading Risks into Capital: Delta, Vega, Curvature, Buckets, Risk Weights and Correlation Scenarios

Reader question: A trading book can contain bonds, swaps, options, equities, commodities and FX positions whose risks partly offset each other. How does the Basel standardised market-risk framework turn all those sensitivities into one reproducible capital number without simply adding every position’s absolute risk?

The FRTB sensitivities-based method (SBM) does it in layers. First, every instrument is mapped to prescribed risk factors. The bank calculates delta, vega and—where relevant—curvature risk positions. Those sensitivities are netted at identical risk factors, multiplied by supervisory risk weights, aggregated inside risk buckets using prescribed correlations, then aggregated across buckets. The process is repeated under three correlation scenarios, and the largest resulting capital requirement is selected.

The result is deliberately more structured than a simple VaR number. It exposes where capital comes from: interest-rate tenor risk, credit-spread names and sectors, equity buckets, commodities, FX, option volatility and nonlinear curvature.

What this page owns — and what it does not

This article owns the computational chain:

trading positions → prescribed risk factors → delta/vega/curvature sensitivities → risk weights → bucket aggregation → correlation scenarios → SBM capital.

It does not replace the broader FRTB market-risk overview, the separate default-risk charge, or the residual-risk add-on. Those are separate components of the standardised market-risk capital stack.

This is public prudential mathematics, not trading advice and not a calculation of any real bank’s regulatory capital.

The current Basel owner

The Basel Framework chapter MAR21 sets out the sensitivities-based method. As of this run, the current in-force MAR21 version is effective from 1 January 2023 and was last updated on 23 March 2026.

That date matters because implementations should version the rulebook. A calculation engine built from an older PDF can remain numerically consistent with itself while being inconsistent with the current standard.

The seven SBM risk classes

Basel groups sensitivities into seven risk classes:

• general interest-rate risk (GIRR);
• credit-spread risk, non-securitisations;
• credit-spread risk, securitisations outside the correlation trading portfolio;
• credit-spread risk, securitisations in the correlation trading portfolio;
• equity risk;
• commodity risk;
• foreign-exchange risk.

These are not arbitrary reporting labels. Each class has its own risk-factor definitions, buckets, supervisory risk weights and correlation parameters.

Step 1: map every instrument to prescribed risk factors

A swap does not enter the algorithm as one indivisible “swap risk”. Its value is sensitive to multiple interest-rate curve points. An equity option can create spot delta and volatility vega. A commodity spread can create sensitivities to more than one commodity risk factor.

The engine therefore needs a mapping:

instrument → risk class → bucket → risk factor → sensitivity type.

Mapping errors are often more dangerous than arithmetic errors because a perfectly calculated sensitivity in the wrong bucket receives the wrong weight and correlation structure.

Delta: first-order value sensitivity

Delta measures the first-order change in instrument value when a prescribed market variable changes.

For GIRR, Basel defines delta sensitivity using PV01-style interest-rate sensitivity. For other classes, the standard specifies prescribed relative or absolute bumps—for example percentage changes in equity, commodity or FX risk factors.

Conceptually:

s_k ≈ ΔV / Δx_k.

The regulatory implementation must use the sensitivity definition and shock convention prescribed for the relevant risk factor rather than whatever local desk Greek happens to have the same name.

Net identical risk factors before weighting

If two instruments have equal and opposite sensitivity to the same prescribed risk factor, Basel allows those sensitivities to offset before risk weighting.

For factor k:

s_k = Σ_i s_ik.

This is exact-factor netting. It does not mean “all interest-rate deltas can be netted together”. A two-year point and a ten-year point are distinct risk factors and are combined only through the correlation structure.

Risk weighting turns sensitivity into a stress-scaled position

For delta and vega, a useful generic notation is:

WS_k = RW_k × s_k,

where RW_k is the Basel supervisory risk weight.

The weight embeds a calibrated stress scale and liquidity horizon. A one-dollar sensitivity in one risk class therefore does not necessarily carry the same capital significance as one dollar in another.

Example: GIRR tenor weights

Under the current MAR21 table for delta GIRR, risk weights vary by tenor. Shorter points such as 0.25 and 0.5 years carry 1.7%, while several long tenors from five years onward carry 1.1% under the base table. Inflation and cross-currency-basis risk factors have their own treatment.

The numbers should be loaded from the applicable Basel or local rule version, not hard-coded forever.

Step 2: aggregate weighted sensitivities inside a bucket

Within one bucket, Basel combines risk-weighted sensitivities using prescribed correlations rather than a simple sum.

A generic form is:

K_b = √max(0, Σ WS_k² + Σ_k≠l ρ_klWS_kWS_l).

Here:

• K_b is bucket-level capital;
• ρ_kl is the prescribed within-bucket correlation.

This gives partial diversification credit where Basel assumes risk factors are related but not perfectly correlated.

Why the square-root aggregation matters

Suppose two weighted sensitivities are +100 and −100.

If their prescribed correlation is 100%, they can nearly cancel. If the correlation is 20%, the hedge is much less reliable under stress and the bucket capital remains substantial.

The algorithm therefore distinguishes notional offset from stress-consistent diversification.

Step 3: aggregate across buckets

Bucket-level capital values are then combined using prescribed across-bucket correlations, often written as γ_bc.

A generic structure is:

K = √max(0, Σ K_b² + Σ_b≠c γ_bcS_bS_c).

The quantity S_b represents the bucket’s signed aggregate sensitivity subject to Basel’s bounding rules.

The details differ by risk class, but the architecture is stable: factor → bucket → risk class → portfolio.

Each currency is a GIRR bucket

For general interest-rate delta risk, each currency forms a separate bucket. Multiple eligible interest-rate curves for the same currency sit inside that bucket.

This lets a USD swap sensitivity offset another USD sensitivity at the same prescribed factor while recognising high—but not necessarily perfect—correlation across distinct curves.

Basel currently uses a 99.90% within-bucket correlation for some same-tenor different-curve GIRR sensitivities, illustrating that “almost the same curve risk” is still not treated as mathematically identical.

FX uses a different architecture

For FX delta, risk factors are exchange rates relative to the reporting currency or approved base currency framework.

The current Basel standard assigns a 15% base delta risk weight to FX sensitivities, with specified treatment for selected currency pairs and first-order crosses.

FX therefore cannot be forced through the interest-rate tenor matrix just because both are market risks.

Vega: sensitivity to implied volatility

Options have value sensitivity to volatility as well as to the underlying market level.

Vega measures the change in option value for a prescribed change in implied volatility:

Vega ≈ ΔV / Δσ.

Basel maps vega using dimensions such as option maturity and, where relevant, underlying maturity or name. The vega correlation structure reflects both the corresponding delta-risk relationships and maturity relationships.

Why vega is not optional just because delta is hedged

A delta-neutral option book can still lose money when implied volatility moves.

For example, a long straddle can have near-zero spot delta at inception but substantial positive vega.

An SBM engine that stops after delta would report almost no risk for a portfolio that clearly has volatility exposure.

Curvature: capture nonlinear loss beyond delta

Curvature addresses instruments whose value changes nonlinearly under large market shocks.

The Basel method shocks the relevant risk factor upward and downward by prescribed amounts, revalues the instrument, and removes the linear delta contribution. The remaining adverse nonlinear loss forms the curvature risk position.

A stylised version is:

Curvature loss ≈ stressed revaluation loss − linear delta loss already capitalised.

This prevents double counting the first-order component while still capturing convexity or concavity.

Curvature is not the same as mathematical gamma

Traditional option gamma is a local second derivative.

FRTB curvature is based on finite prescribed stress revaluations. For highly nonlinear products, discontinuous payoffs or large shocks, the regulatory curvature charge can differ materially from a simple gamma-times-shock-squared approximation.

The safest production method is therefore to implement the Basel stress-revaluation definition rather than replace it with a local Greek unless the rules explicitly allow the approximation.

Options can create delta, vega and curvature at the same time

An option position may contribute to all three:

• delta from movement in the underlying;
• vega from movement in implied volatility;
• curvature from nonlinear response to a larger underlying shock.

The components are not duplicates because each measures a different dimension of risk.

Three correlation scenarios harden diversification assumptions

Basel does not rely on a single fixed correlation set.

The SBM is calculated under:

• a medium/base correlation scenario;
• a high-correlation scenario;
• a low-correlation scenario.

Correlation parameters are transformed according to the standard for each scenario. Delta, vega and curvature capital are aggregated under each scenario, and the bank takes the largest total.

This is a structural defence against a portfolio appearing safe only because one particular correlation assumption is favourable.

A hedge can fail differently in high and low correlation regimes

Consider a long position in one credit name hedged with a different but related name.

If correlations rise, some directional risks may offset more strongly. But a long-short relative-value trade can also become more dangerous under lower correlation because the hedge relationship breaks.

Taking the maximum across correlation scenarios makes the capital framework less dependent on one diversification story.

The SBM is only one component of the standardised market-risk charge

The standardised approach also contains:

• the default risk charge (DRC) for jump-to-default risk;
• the residual risk add-on (RRAO) for instruments with specified residual risks.

Therefore:

Standardised market-risk capital ≠ SBM alone.

The separate DRC article owns the default-jump mechanics.

Inputs and outputs

A robust SBM engine can require:

• trade and position inventory;
• pricing-model valuations;
• prescribed risk-factor mappings;
• delta sensitivities;
• vega sensitivities;
• full revaluations for curvature shocks;
• bucket attributes such as currency, sector, credit quality, market capitalisation and economy classification;
• Basel/local risk weights;
• within- and across-bucket correlations;
• correlation-scenario transformations;
• rule version and jurisdiction.

Outputs can include:

• delta capital by risk class;
• vega capital by risk class;
• curvature capital by risk class;
• bucket contributions;
• medium/high/low correlation totals;
• binding correlation scenario;
• SBM capital requirement;
• reconciliation to the wider standardised market-risk charge.

Evidence polarity: what supports confidence?

Evidence for a reliable implementation includes sensitivities that reconcile to independently bumped valuations, risk-factor mappings that reproduce desk risk reports, identical factors netting exactly, bucket assignments supported by reference data, risk weights tied to the correct rule version, and scenario totals that can be rebuilt from factor-level records.

Evidence against confidence includes unmapped trades, desk Greeks used without converting to regulatory definitions, options with delta but no vega or curvature, sector mappings that change between runs without source-data changes, negative values inside square roots before prescribed flooring, or a reported total that cannot be decomposed back to buckets.

Counterexample: a perfectly delta-hedged option book can still require large capital

Two options can offset spot delta while leaving substantial vega and curvature.

Therefore a near-zero first-order P&L sensitivity does not imply a near-zero SBM charge.

Counterexample: equal and opposite notionals do not imply zero capital

A long five-year corporate bond and short five-year bond from another issuer may have similar notional and duration.

But if the names sit in different credit buckets or carry imperfect correlation, the residual spread risk remains capitalised.

Counterexample: more diversification can raise the binding scenario

A portfolio built around cross-bucket hedges may look efficient under the medium correlation set. Once the low-correlation scenario weakens those hedges, the total can rise and become the binding capital scenario.

The largest-of-three rule intentionally allows this.

Counterexample: local gamma can understate curvature

A digital option can have highly unstable local gamma near its strike. A finite Basel stress can cross the payoff discontinuity and produce a much larger revaluation loss than a local second-order approximation suggests.

Regulatory curvature is therefore a stress-revaluation object, not merely a calculus identity.

Weak links in implementation

risk-factor mapping error. A sensitivity is assigned to the wrong tenor, currency, issuer or bucket.

unit error. PV01, percentage shocks and decimal sensitivities are mixed.

sign error. A hedge reinforces rather than offsets risk after weighting.

stale reference data. issuer sector, credit quality or market-cap classification is outdated.

premature netting. sensitivities to different factors are netted before the Basel correlation step.

vega omission. options are treated as delta-only.

curvature approximation error. local gamma replaces required stressed revaluation.

scenario omission. only the medium correlation scenario is calculated.

rule-version drift. 2026 MAR21 updates are not reflected in the parameter set.

Diagnostics: how to test the engine

• single-factor test: one sensitivity should produce weighted sensitivity equal to sensitivity × prescribed risk weight.
• exact-offset test: equal and opposite sensitivities to the identical factor should net before weighting.
• different-tenor test: equal and opposite two-year and ten-year GIRR sensitivities must not disappear through exact netting.
• bucket-correlation test: vary prescribed correlation and verify diversification changes in the expected direction.
• vega test: a delta-neutral long-volatility position should retain vega capital.
• curvature test: compare full stressed revaluation with local gamma approximation and retain the Basel-defined result.
• scenario test: reproduce medium, high and low correlation totals and select the maximum.
• reference-data test: change one issuer classification and verify only the appropriate bucket mapping changes.
• aggregation replay: rebuild portfolio capital from stored factor-level weighted sensitivities.
• rule-version test: rerun with a prior parameter set and document every resulting change.

What would falsify confidence?

Confidence should be withdrawn if factor-level sensitivities cannot reproduce the bucket result; if the engine produces identical capital under all three correlation scenarios for a diversified multi-factor portfolio without explanation; if option positions generate no vega or curvature; if the current MAR21 parameter set cannot be traced to source rules; or if the total SBM capital cannot be reconciled to delta, vega and curvature components.

Alternatives and limits

The SBM is a supervisory standardised model. It is not designed to be a bank’s perfect economic-risk model.

Internal risk systems can use historical simulation, expected shortfall, Monte Carlo full revaluation, stress tests, factor models or desk-specific Greeks. Those methods can be more granular but also more model-dependent.

The FRTB internal-models approach uses expected shortfall and modellability rules rather than the SBM architecture. The standardised approach remains important even for banks using internal models because it serves as a common supervisory benchmark and can enter other Basel constraints such as the output floor.

Connections to the surrounding knowledge estate

The FRTB overview explains how standardised and internal-model capital coexist. Default-risk charge handles jump-to-default. Adjoint differentiation explains one scalable way to compute large sensitivity sets, while correlation-matrix diagnostics provide a useful mathematical contrast with Basel’s prescribed correlations.

Verification and update triggers

Preserve the MAR21/local-rule version, risk-factor definitions, sensitivity units, bucket mappings, reference-data vintages, risk weights, correlations, curvature shock implementation and correlation-scenario parameters. Revalidate after Basel updates, local implementation changes, pricing-model migrations, new product launches, reference-data remapping or unexplained standardised-capital jumps.

Primary and high-quality references

• Basel Committee on Banking Supervision, MAR21 — Standardised approach: sensitivities-based method, current version updated 23 March 2026.
• Basel Committee, MAR20 — Standardised approach: general provisions, covering SBM, DRC and residual-risk components.
• Basel Committee, MAR11 — Definitions and application of market risk.
• Basel Committee, Basel Framework consolidated text.

Educational boundary: This article explains public Basel market-risk mathematics. It does not determine any real bank’s capital requirement, approve a model or recommend a trading position.