Article 7 introduced the Sensitivities-Based Method as the first of the Standardised Approach’s three components. This article begins actually building it. Fair warning before we start: MAR21.1 through MAR21.7 turned out to be considerably denser than a first read suggests — paragraphs 21.4 and 21.5 alone each contain several nested formulas, more concentrated technical content than almost any other stretch of the source document. Rather than force all seven paragraphs into one article, we are splitting this into two parts: this article covers the vocabulary, which instruments are in scope for each risk measure, and the full delta/vega formula (MAR21.1–21.4). Part B covers the curvature formula and how the three components combine into one SBM capital number (MAR21.5–21.7).
The Vocabulary of the Sensitivities-Based Method
The SBM works by taking the sensitivities of financial instruments to a prescribed list of risk factors, using those sensitivities to calculate delta, vega and curvature risk capital requirements. Those sensitivities are risk-weighted, then aggregated — first within risk buckets (risk factors sharing common characteristics), and then across buckets within the same risk class. Five terms anchor everything that follows.
| Term | Definition |
| Risk class | One of seven fixed categories — the same list from Article 4A’s glossary: GIRR, CSR non-securitisations, CSR securitisations (non-CTP), CSR securitisations (CTP), equity, commodity, and FX. |
| Risk factor | A variable — such as an equity price or a tenor of an interest rate curve — that affects the value of an instrument. |
| Bucket | A set of risk factors grouped together by common characteristics, such as all tenors of interest rate curves for the same currency. |
| Risk position | The portion of an instrument’s risk that relates to a specific risk factor. For delta and vega risk, the risk position is a sensitivity to that risk factor. For curvature risk, the risk position is based on losses from two stress scenarios instead. |
| Risk capital requirement | The capital a bank should hold because of the risks it takes, computed by aggregating risk positions first at the bucket level, then across buckets within a risk class. |
Sourced from MAR21.1, including sub-points (1) through (5).
https://www.bis.org/basel_framework/chapter/MAR/21.htm?inforce=20230101&published=20260323
Which Instruments Face Which Risk Charges?
Not every instrument faces all three risk measures. The starting rule is broad: all instruments held in trading desks, as defined in MAR12, and subject to the sensitivities-based method — meaning excluding instruments whose value at any point is purely driven by an exotic underlying, as set out in MAR23.3 — are subject to delta risk capital requirements. Vega and curvature are narrower, applying only to the instruments described in four specific categories.
1. Any Instrument With Optionality
The standard gives a non-exhaustive list of examples in a footnote: calls, puts, caps, floors, swaptions, barrier options, and exotic options. Anything that is an option, or that includes an embedded option — such as a convertibility feature or a rate-dependent prepayment feature — falls under this category, as long as it is subject to market risk capital requirements in the first place.
2. Any Instrument With an Embedded Prepayment Option
A prepayment option lets a debtor repay part or all of the principal before contractual maturity without compensating the lender for the interest that would otherwise have been earned. This is explicitly treated as a form of optionality under category 1 above, and the embedded option is specifically subject to vega and curvature risk with respect to the interest rate risk and credit spread risk (both non-securitisation and securitisation) risk classes.
Two further wrinkles apply here. Where the prepayment option is a behavioural option — meaning prepayment decisions are driven by borrower behaviour rather than a purely rational exercise decision — the instrument may also be subject to the Residual Risk Add-On under MAR23, and the bank’s pricing model must reflect that behavioural pattern where relevant. For securitisation tranches specifically, the underlying securitised loans may themselves carry embedded prepayment options, in which case the securitisation tranche built on top of them may also be subject to the RRAO.
3. Instruments Whose Cash Flows Are Not a Linear Function of Notional
This is really the underlying logic behind category 1. The cash flows generated by a plain-vanilla option cannot be written as a linear function of the underlying, because they equal the maximum of the spot price and the strike price — a kink, not a straight line. That is precisely why all options are subject to vega and curvature risk. By contrast, an instrument whose cash flows can be written as a linear function of the underlying notional — a coupon-bearing bond, for example — is an instrument without optionality, and is not subject to vega or curvature risk capital requirements at all.
4. Curvature Risk Can Optionally Extend to Every Delta Instrument
This category works differently from the first three — it is a permission, not an automatic requirement. Curvature risk may be calculated for all instruments subject to delta risk, not just the ones described in categories 1 through 3. The standard gives the underlying rationale directly: where a bank manages the non-linear risk of instruments with optionality and other instruments holistically, it may choose to fold instruments without optionality into its curvature risk calculation too.
This treatment is only allowed subject to two restrictions:
- The approach must be applied consistently through time — a bank cannot switch it on and off opportunistically.
- Once adopted, curvature risk must be calculated for every single instrument subject to the sensitivities-based method, not selectively for some instruments and not others.
Sourced from MAR21.2, including sub-points (1) through (4) and both footnotes in full.
https://www.bis.org/basel_framework/chapter/MAR/21.htm?inforce=20230101&published=20260323
| Risk Measure | Which Instruments Are In Scope |
| Delta | All SBM-eligible trading book instruments (excluding those purely driven by an exotic underlying, per MAR23.3). |
| Vega | Instruments with optionality, including embedded prepayment options. |
| Curvature (default) | The same instruments as vega. |
| Curvature (extended, optional) | Can be extended to all delta-eligible instruments, if applied consistently and comprehensively. |
The Four-Step Roadmap
MAR21.3 previews exactly how the rest of MAR21 is organised, and it is worth internalising this map before diving into formulas, because every article from here through Article 18 slots into one of these four steps.
- Step 1 — Risk factor definitions: the risk factors for delta, vega and curvature risk, for each risk class, are defined in MAR21.8 through MAR21.14. These are risk-class-specific, so we cover them inside each risk class’s own article (Articles 9 through 15).
- Step 2 — Delta and vega risk positions: the methods to risk-weight sensitivities and aggregate them into delta and vega risk positions, per risk class, are set out in MAR21.4 (covered below) together with MAR21.15 through MAR21.95 — the sensitivity definitions (Part C of this mini-series), and the bucket/risk-weight/correlation tables specific to each risk class (Articles 9 through 16).
- Step 3 — Curvature risk: the curvature methodology is set out in MAR21.5 (covered in Part B) together with MAR21.96 through MAR21.101 (Article 17).
- Step 4 — Portfolio-level aggregation: the risk-class-level capital requirements calculated above must be aggregated into a single capital requirement at the entire portfolio level, as set out in MAR21.6 and MAR21.7 — covered in Part B of this mini-series, and revisited in full depth in Article 18.
Sourced from MAR21.3, including sub-points (1) through (4).
Calculating the Delta and Vega Risk Capital Requirement
This is the formula engine every risk-class article from Article 9 onward will plug numbers into. For each risk class, a bank determines its instruments’ sensitivity to a set of prescribed risk factors, risk-weights those sensitivities, and aggregates the risk-weighted sensitivities — separately for delta and separately for vega — using the following four-step approach.
Step 1: Determine the Sensitivity
For each risk factor, a sensitivity is determined using the methods set out in MAR21.15 through MAR21.38 — the subject of Part C of this mini-series.
Step 2: Net Sensitivities to the Same Risk Factor
Sensitivities to the same risk factor, from every instrument in the portfolio, must be netted into a single net sensitivity for that risk factor. Sensitivities of opposite direction offset each other completely, regardless of which instrument they came from.
| A Worked Example From the Source Document Itself The standard gives its own example here: if a bank’s portfolio consists of two interest rate swaps on three-month Euribor, with the same fixed rate and the same notional but opposite direction, the GIRR sensitivity on that portfolio is zero. The two swaps cancel each other out completely at the netting stage, before risk weights or correlations even enter the calculation. |
Step 3: Risk-Weight the Net Sensitivity
The weighted sensitivity is simply the net sensitivity multiplied by the risk weight prescribed for that risk factor:
| WS_k = RW_k × s_k where: WS_k = weighted sensitivity for risk factor k RW_k = prescribed risk weight for risk factor k s_k = net sensitivity to risk factor k (from Step 2) |
Step 4: Aggregate Within Each Bucket
The risk position for a delta (or vega) bucket is found by combining the weighted sensitivities of every risk factor inside that bucket, using a prescribed correlation between each pair of risk factors. The entire quantity inside the square root is floored at zero, so the formula never asks a bank to take the square root of a negative number.
| K_b = sqrt( max( 0, Σ_k (WS_k)² + ΣΣ_(k≠l) ρ_kl × WS_k × WS_l ) ) where: K_b = risk position for bucket b ρ_kl = prescribed correlation between risk factors k and l, both within bucket b |
Step 5: Aggregate Across Buckets
The delta (or vega) risk capital requirement for the whole risk class is found by combining the bucket-level risk positions using a prescribed correlation between each pair of buckets:
| Delta (or Vega) risk capital requirement = sqrt( Σ_b (K_b)² + ΣΣ_(c≠b) γ_bc × S_b × S_c ) where: γ_bc = prescribed correlation between bucket b and bucket c S_b = Σ_k WS_k, summed over all risk factors in bucket b S_c = Σ_k WS_k, summed over all risk factors in bucket c |
There is one important safety valve built into this last step. If the values of Sb and Sc produce a negative number for the overall sum inside the square root — Σb Kb² + Σb Σc≠b γbc Sb Sc — the bank must switch to an alternative specification instead, to avoid the formula breaking down. Under that alternative specification:
| S_b = max[ min(Σ_k WS_k, K_b), −K_b ] for all risk factors in bucket b S_c = max[ min(Σ_k WS_k, K_c), −K_c ] for all risk factors in bucket c |
| What This Safety Valve Is Actually Doing In plain terms: Sb and Sc are normally allowed to be as large (positive or negative) as the sum of weighted sensitivities in that bucket. The alternative specification caps each one at plus or minus its own bucket-level risk position, Kb or Kc. This prevents the cross-bucket correlation term from swinging the overall figure negative in a way that would make the square root undefined — a technical guardrail rather than a change in economic intent. |
Sourced from MAR21.4, including sub-points (1) through (5) and all embedded formulas, in full.
Looking Ahead
With the vocabulary, the instrument scope rules, and the complete delta/vega formula now covered, Part B turns to the curvature formula — which is, if anything, more intricate than what we have covered here — and to the mechanism that ties delta, vega and curvature together into one final SBM capital requirement: the three correlation scenarios.
Frequently Asked Questions
Which instruments are subject to vega and curvature risk under FRTB?
Primarily instruments with optionality, including embedded prepayment options — anything whose cash flows cannot be written as a linear function of the underlying notional. Curvature risk can optionally be extended to all delta-eligible instruments if applied consistently and comprehensively.
How are sensitivities to the same risk factor combined?
They are netted into a single net sensitivity per risk factor across the whole portfolio, with opposite-direction sensitivities offsetting completely regardless of which instrument they originated from.
What happens if the cross-bucket aggregation formula produces a negative number under the square root?
The bank switches to an alternative specification that caps each bucket’s aggregate sensitivity at plus or minus its own bucket-level risk position, preventing the formula from becoming undefined.
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