The Greeks

Table of Contents

1. Summary

A sensitivity is a partial derivative of value with respect to an input, almost always approximated numerically:

\begin{equation} \text{Greek} = \frac{PV(\text{input} + h) - PV(\text{input})}{h} \end{equation}

A greek is a named sensitivity, conventionally denoted by a greek letter.

Two properties follow, and both are load-bearing.

A sensitivity is defined by the input it moves, not by the instrument. Vega means sensitivity to volatility whether the trade is a vanilla, a barrier or a structure. The greek set is therefore a property of the report rather than of the product. That input is a market data requirement, so a sensitivity is a requirement paired with a greek.

A sensitivity is always local. It is measured at the current market level and is valid only near it. Every use of a greek away from that level is an extrapolation.

The first property is why a risk type decomposes. An input belongs to an asset class and a greek names which derivative was taken, so a risk type is the pair.

2. What a sensitivity is

2.1. Structural and actual

Two different questions hide behind "what is this sensitive to".

The structural sensitivity is the set of inputs an instrument could be sensitive to, independent of state or market data. It is a property of the product type and is used to slice the market data set. That is what risk classes records.

The actual sensitivity is the set of market data points the current book is measurably sensitive to. It is computed, it changes as the book changes, and it drives which data the firm needs to purchase and verify.

2.2. By order

Sensitivities are ordered by how many derivatives were taken, and the order hierarchy follows from a Taylor expansion. Delta predicts where value will be after a move; if delta is itself changing, the prediction is wrong, and gamma corrects it:

\begin{equation} \Delta P\&L \approx (D_1 - D_0) \times \Delta + \tfrac{1}{2}(D_1 - D_0)^2 \times \Gamma \end{equation}

The expansion continues in principle and stops in practice at second order, for a reason of utility rather than mathematics: there are no higher order greeks because traders do not find them useful.

2.3. By methodology

The same conceptual quantity can be obtained three ways, and they must never be silently mixed.

Method What it does Limitation
Bumped Revalue with a perturbed input and take the difference Works for any product and model; costs a revaluation
Analytic Use a closed-form formula the model supplies Faster and more precise; many models do not support it
Model The greek as produced by a specific model variant A Black-Scholes delta and a smile delta are different numbers for the same trade and the same spot

Because the method changes the number, it is part of the configuration: each greek carries a setting for unbumped, bumped, analytic, weighted or model.

3. The greeks

Greek Input moved Order
Delta Spot First
Gamma Spot Second
Vega Volatility First
Vomma Volatility Second
Rega Risk reversal, the skew First
Sega Strangle, the wings First
Rho Interest rates First
Theta The passage of time First

3.1. Delta

Sensitivity to spot, and the primary directional exposure.

In FX it carries a complication worth stating, because spot appears twice: once as the underlying of the trade, and once as the rate that converts value into the reporting currency. A book's delta is composed of both and the two cannot be separated, so a desk that cares about conversion exposure cannot isolate it. Splitting delta into a position delta and a spot risk delta would resolve it, and is not done today.

Delta also does double duty as a coordinate system. The FX volatility surface is parameterised in delta terms, with 10, 15 and 25 delta strikes, so delta is both a risk measure and an axis of the surface.

3.2. Gamma

Second-order spot sensitivity, and the rate of change of delta. Its purpose is corrective: where gamma is large, a delta-only prediction is wrong, and the error grows with the square of the move.

3.3. Vega, rega and sega

The volatility surface sensitivities.

Vega carries a convention that changes what the number means: it is the bump of the entire volatility term structure rather than a tenor-by-tenor measure. The tenor decomposition is a separate report, produced by bumping each tenor of each curve and stitching the results back into a term structure.

Rega and sega are FX-specific and follow from how the surface is quoted. An FX volatility surface is quoted as at-the-money, risk reversal and strangle per tenor rather than as a strike grid, so the natural sensitivities are to those three quoted objects. That is why FX volatility reports come as triples rather than as a single vega.

3.4. Rho

Interest rate sensitivity, computed separately for discount and projection curves, with rates bumped in absolute terms rather than relative. Its rates-desk analogue is DV01, the value change for a one basis point move.

3.5. Theta

Sensitivity to the passage of time, and the most complicated of the greeks, because sensitivity to time is not one thing. It decomposes, and the user has to choose which measure they want.

Component Meaning
Time decay The erosion of an option's time value, computed by moving time forward one day. Forwards show none
Cost of carry Interest earned on long positions less interest paid on short positions. Naturally book-level, because funding is done per book. In commodities it is storage cost

Theta also varies by how it is computed: rolling theta moves market data and deals forward a day, breakdown theta rolls one curve at a time and attributes the residual to cross effects, and backwards theta uses negative shifts.

3.6. Model greeks

Sensitivity to a model parameter rather than to a market input, obtained by bumping the parameter's term structure. A model greek is meaningful only within the model that produced it.

4. Why a risk type is an asset class crossed with a greek

Because a sensitivity is defined by the input it moves, and every input belongs to an asset class, a risk type is fully determined by the pair. Two regulatory frameworks say so in their own enumerations.

4.1. FRTB says it explicitly

The FRTB risk types are named <risk class>_<greek>:

GIRR_DELTA, GIRR_VEGA, GIRR_CURV, EQ_DELTA, EQ_VEGA, EQ_CURV, COMM_DELTA, COMM_VEGA, COMM_CURV, FX_DELTA, FX_VEGA, FX_CURV, and the credit spread family CSR_NS_, CSR_SNC_ and CSR_SC_ each with the same three suffixes.

Nothing is left to inference. The class and the greek are separate segments of the name.

Curvature is not one number in FRTB. CurvatureScenario carries an up shift and a down shift, because the framework requires both, so a curvature row needs a third coordinate that delta and vega do not.

4.2. SIMM says it implicitly

The SIMM risk types are IRCurve, IRVol, Equity, EquityVol, FX, FXVol, Commodity, CommodityVol, Inflation, InflationVol, CreditQ, CreditNonQ, CreditVol and CreditVolNonQ.

The same decomposition is there under a different spelling. The unsuffixed values are deltas and the Vol suffix is vega. Curvature is not a risk type in SIMM; it appears in MarginType alongside delta and vega, so the third greek is carried on a different axis.

4.3. What that gives

A risk type is therefore a table rather than an enumeration: 

risk_type
  asset_class  -> asset_class.code
  greek        -> greek.code

That table states at the grain of an asset class what holds at the grain of an individual input. A sensitivity pairs a market data requirement with a greek; projecting the requirement onto its asset class yields the row above. Delta and gamma share one requirement, differing in order alone, whereas vega, rega and sega do not, because each moves a different quoted object on the surface.

The framework-specific values that do not fit are not counter-examples but properties on that row. SIMM's qualifying flag on credit and FRTB's securitisation split on credit spread are attributes of the exposure, not new asset classes and not new greeks. Risk classes covers that argument.

5. Conventions that change the number

A greek is not comparable across configurations unless the configuration matches. Four things change the number without changing the name.

  • The bump. Its size, whether it is absolute or relative, and what it targets. Rates are bumped absolutely, typically one basis point; volatility defaults to a tenth of a volatility point.
  • The method. Bumped, analytic or model, as above.
  • Surface reparameterisation. A volatility surface may be restated to a different delta, strangle convention, at-the-money flag or risk reversal flag before bumping, and each choice changes what the resulting greek means.
  • The currency. Greeks are converted into an aggregation currency, and that conversion generates greeks of its own by design.

6. Open questions

  • Can FX delta be split? Position delta and spot risk delta are conceptually distinct and currently conflated. Whether the split is worth the modelling cost depends on the desk: it is small for most and material for FX.
  • Is cost of carry a deal-level quantity? It is naturally book-level, because funding is done per book. A theoretical deal-level figure is possible, and nothing settles whether it is useful.
  • How are cross effects reported? The sum of independently computed sensitivities does not equal the true total, and the residual has to go somewhere.
  • Is curvature a greek or a margin type? SIMM places it on MarginType while FRTB gives it a risk type suffix. The two frameworks disagree on which axis it belongs to.

7. Sources

The definitions, conventions and open questions are drawn from internal analysis notes on risk sensitivities, which summarise requirements material. No external standard is cited for them.

The FRTB and SIMM risk type enumerations are read from OREAnalytics/orea/simm/crifrecord.hpp, and the margin types from OREAnalytics/orea/simm/simmconfiguration.hpp.

8. See also

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