Smile-aware Greek conventions: sticky='strike' / sticky='delta' on VolSurface Greeks

[marwinsteiner]

Split out of #11, whose proposal listed smile-aware Greeks as an option
(sticky="strike" vs sticky="delta"); the shipped v0.9.0 layer took
the documented minimum -- sticky-strike only -- and #11 closes with the
v0.9.0 release, so the option gets its own issue here.

Context

These are smile-dynamics conventions, not calibration objectives: they
change what delta/gamma/theta MEAN, not how the surface is fitted, so
the work lands in the VolSurface Greeks layer, untouched by the
calibration controls.

Background: Derman's regimes of volatility

The taxonomy comes from Derman (1999), who observed that S&P 500
implied vols move with the index according to (at least) three distinct
rules, each appropriate to a market regime, each implying a different
hedge ratio. With a negatively sloped skew sigma(K) ~ sigma0 - b*K:

regime	rule of thumb	when	consequence
sticky strike	sigma(K) fixed as S moves	range-bound, trendless	Black-Scholes delta is the right hedge; ATM vol drifts down as S rises (the ATM strike slides along the fixed skew)
sticky delta / moneyness	sigma(K/S) fixed -- the smile floats with spot	stable trending market	ATM vol constant in S; call delta > sticky-strike delta by the smile-slope (vega) correction
sticky implied tree (local vol)	the local-vol surface fixed; per-strike vols fall as S rises	jumpy, fearful, post-crash	ATM vol falls at ~twice the skew slope; delta < sticky-strike delta

Three consequences for this issue:

1. The convention is a per-call choice, not a surface property.
   Which rule is the right hedge ratio is an empirical, regime-dependent
   question -- exactly why 'sticky=' belongs on the Greek calls rather
   than baked into the fit.
2. The taxonomy is three-valued, not two. The implied-tree /
   local-vol regime is deliberately OUT OF SCOPE here: a proper
   local-vol delta needs the Dupire surface, i.e. dw/dT everywhere
   (available since interp_method="monotone_cubic", Differentiability contract and C1 maturity interpolation #22) plus a
   local-vol layer that does not exist yet (filed as Dupire local volatility layer: surface.local_vol on C1 surfaces #36). The 'sticky=' parameter
   should be designed so 'local_vol' can be added later without
   breaking the API; the docs table should present all three regimes
   and say explicitly which two are implemented.
3. Derman's rules give free test oracles. With a negative skew the
   deltas must order as local-vol < sticky-strike < sticky-delta, and
   under sticky-delta the ATM vol must be invariant to a spot bump that
   drags the smile in moneyness. Both are cheap, regime-grounded
   assertions to add alongside the finite-difference checks.

• Sticky strike (current behaviour): differentiate holding sigma(K)
  fixed -- the vol at a given strike stays put as the forward moves.
• Sticky delta / sticky moneyness: sigma is held fixed per
  moneyness (or delta) level, so a spot move drags the smile along.
  Delta picks up the smile-slope correction
  delta_sd = delta_ss + vega * dsigma/dK * dK/dF-type terms
  (equivalently: differentiate through sigma(k) with k = ln(K/F)
  fixed); gamma and theta adjust correspondingly. The surface already
  exposes everything needed: dw_dk gives dsigma/dk analytically for
  the SVI family.

Proposal

surface.delta(K, T, cp="call", sticky="strike")   # default, unchanged
surface.delta(K, T, cp="call", sticky="delta")
surface.gamma(K, T, sticky=...)
surface.theta(K, T, cp="call", sticky=...)

• sticky="strike" remains the default -- existing behaviour and
  serialized surfaces are untouched.
• sticky="delta" computes the smile-slope-corrected Greeks from the
  model's analytic dw/dk (finite-difference fallback for SABR /
  DirectSVI, as elsewhere).
• price and vega are convention-independent and take no parameter.
• Document the two conventions side by side, with the standard guidance
  on when each is the right hedge ratio (index vs FX-style smiles).

Acceptance criteria

• Sticky-delta Greeks verified against finite differences of the price
  under the corresponding smile shift (bump F, move the smile in k)
• sticky="strike" bitwise-identical to today's output
• Sticky-delta call delta exceeds sticky-strike delta on a
  negative-skew fixture (Derman's regime ordering)
• Both conventions demonstrated in the examples and documented on the
  surface page (per the maintenance policy), with the three-regime
  table and the regime guidance on when each is the right hedge ratio

Reference

Derman, E. (1999). "Regimes of Volatility: Some Observations on the
Variation of S&P 500 Implied Volatilities." Goldman Sachs Quantitative
Strategies Research Notes (also summarized in Risk, April 1999).
Author's page: https://emanuelderman.com/regimes-of-volatility-risk-april-1999/
(the GS QSRN series, this note included, is collected at
https://github.com/s0ap/gs-quantitative-strategies-research-notes).