Real Time Greeks Recalculation On Market Moves

When to Use

Invoke this skill when an options book must show current Greeks on a live tape and revaluing every contract on every tick does not fit the CPU budget. The engine decides, per position and per tick, between two things: advancing the cached Greeks with a second-order expansion in spot,

$$\Delta(S_0 + h) \approx \Delta_0 + \Gamma_0 h, \qquad V(S_0 + h) \approx V_0 + \Delta_0 h + \tfrac{1}{2}\Gamma_0 h^2$$

or repricing the contract outright with Black-Scholes-Merton. The expansion is cheap and, over a small move, close enough. Over a large one it is not — which is why the Basel market-risk standard does not accept a delta measure alone for options and computes its curvature charge from repriced instruments under an up and a down shock (BCBS, MAR21.5). Small move, expand; large move, reprice.

The part that decides whether this works is not the expansion. It is the anchor: what "the move" is measured against.

When NOT to Use

As a pricing or implied-vol engine. The reval consumes the implied vol handed in for that strike and inherits every property of the surface that produced it. Build the surface first — options-implied-volatility-surface-construction.
For American-style contracts where early exercise is live. The closed form is European with a continuous dividend yield. Deep-ITM puts and calls into a dividend need a different model — american-vs-european-style-option-exercise-handling.
To net a multi-underlying book. A tick is one price for one underlying, and this engine reports only that underlying's nets. Cross-underlying aggregation, dollar-normalisation and limit auditing belong to options-greeks-real-time-portfolio-aggregation.
As a hedger or a kill switch. The report is an observation. Sizing the offsetting trade is greeks-based-portfolio-hedging-automation; halting is kill-switch-and-drawdown-circuit-breakers, which must be independent of this path.
Through expiry on at-the-money strikes. Delta is discontinuous through the pin and gamma is unbounded; no expansion of either is stable at any refresh rate. The engine degrades to always-reprice inside the pin horizon and you should still read options-pin-risk-management-at-expiry.
As a latency optimisation you have not measured. If the book is small enough to reprice wholesale inside the tick budget, do that — it is strictly more accurate and has no anchor to get wrong. Approximate only once the profile says you must.

Prerequisites

Contract terms per leg: symbol, underlying_symbol, option_type (CALL/PUT), strike, signed position_qty, and multiplier.
multiplier — deliverable units per contract, read from the contract master. Required, no default. 100 for a standard unadjusted US equity option; different for OCC-adjusted contracts, index products and crypto.
Current market inputs per leg, refreshed each tick: implied_vol (e.g. 0.20) and time_to_expiry_years ($>0$). Optionally risk_free_rate and dividend_yield (both default to 0.0, i.e. an explicit zero assumption).
One spot per tick, per underlying, passed to the call — not held per position.
A caller-supplied monotonic tick_timestamp_s. Without it the staleness trigger cannot fire.
Trigger thresholds (RecalculationTriggerConfig). The defaults are illustrative starting points, not a standard — see references/standards.md.

Workflow

Validate the tick and every leg before anything is published.
- Reject a non-positive or non-finite spot, a blank symbol, a non-positive multiplier, strike, implied_vol or time_to_expiry_years, an unknown option_type, and a duplicate symbol within one tick.
- Decision point — a NaN spot does not read as an error, it reads as a small move. abs(nan) > threshold is False, so an unvalidated NaN never trips the reval trigger and pins the entire book on the cached Greeks indefinitely. Raise on the tick.
- Decision point — reject the whole tick, not the bad leg. A partial Greeks snapshot is a risk number with an unknown fraction of the book missing from it. That is worse than no number, because it looks like one.
- Decision point — option_type must be validated, not branched on. if t == "CALL": ... else: # PUT turns "C" into a put and flips the sign of delta. A mis-signed delta does not look wrong; it looks like the other side of the book.
Restrict the tick to its own underlying.
- A tick carries one price for one name. Legs on any other underlying are excluded and counted, never repriced at this spot.
- Decision point — this is the structural fix for stale-spot Greeks. If each position carries its own spot field, a book ends up netting Greeks computed at three different prices for the same name and nothing in the output says so. One spot per tick makes that unrepresentable.
Test the current tick against the anchor — the last full revaluation, never the last tick. $$\text{drift} = \frac{|S_t - S_{\text{anchor}}|}{S_{\text{anchor}}}$$
- Decision point — this is the defect the whole skill exists to prevent. Reset the baseline to the previous tick and a monotone run of sub-threshold ticks never trips the threshold: sixty consecutive $+0.1%$ ticks against a $0.5%$ threshold move spot $+6.2%$ with zero revaluations, because each individual step is "small". The book is carried that entire distance on a frozen gamma. Anchor on the last reval and the same sequence reprices repeatedly.
- Reprice when any of these fires, and record which one: no anchor yet; inside the near-expiry horizon; spot drift past the threshold; implied vol moved past its band; the anchor is older than the staleness cap; the stepped delta left its admissible band.
- Decision point — a vol move invalidates the cache with no spot move at all. $\Gamma$ and $\nu$ are functions of $\sigma$. A book on a flat tape through a vol repricing has stale gamma and stale vega and a spot-only trigger will never notice.
- Decision point — a quiet book still drifts. Theta and charm move the Greeks with no tick at all. Without a staleness cap, an illiquid name's Greeks are as old as its last print.
- Decision point — order the tests by validity, not by cost. Expiry is checked before spot: inside the pin horizon the expansion is the wrong shape, not merely imprecise, so how small the move was is irrelevant.
Step or reprice.
- Step: $\Delta \leftarrow \Delta_0 + \Gamma_0 h$, with $\Gamma$, $\nu$, $\Theta$ carried forward frozen. Their own drift (speed, vanna, charm) is second order over a sub-threshold move — which is exactly what step 3 bounds.
- Decision point — bound the stepped delta. A call delta lives in $[0, 1]$ and a put delta in $[-1, 0]$. A linear step through a high-gamma region can produce $1.4$; publishing it is worse than the cost of the reprice it should have triggered.
- Reprice: full BSM with continuous dividend yield. Re-anchor spot, vol, expiry, timestamp, price and all four Greeks in one atomic update — a half-updated anchor is a permanent, silent bias.
Report the nets and the evidence to calibrate on.
- Nets for the ticked underlying, summed with math.fsum so ordering cannot change the answer, and scaled by the deliverable exactly as options-greeks-real-time-portfolio-aggregation scales them.
- Every reval reports spot_taylor_value_error_per_unit: the true value change at the new spot (at the anchor's vol and expiry, so the comparison isolates spot) minus the delta-gamma estimate it replaced. This is how the threshold gets calibrated — negligible error means CPU is being burned on a threshold that is too tight; material error means the published delta was wrong between revals.

Full procedure: see references/workflows.md. Standards reference: see references/standards.md. Printable pre-flight checklist: see assets/checklist.md.

Common Pitfalls

Anchoring the move test on the previous tick. The single defect that makes this architecture dangerous rather than merely approximate. A trending tape of individually-small ticks never trips the threshold, and the book runs arbitrarily far from the last real price on a frozen gamma. Anchor on the last full revaluation.
Treating a small move as a small error. The expansion freezes gamma, so the delta error grows with $\tfrac{1}{2},(\partial\Gamma/\partial S),h^2$ — and near expiry or near the strike $\partial\Gamma/\partial S$ is large. The same 0.5% move that is negligible on a one-year option is not negligible on a one-day one.
Triggering on spot only. Gamma and vega are functions of implied vol. A vol repricing on a flat tape leaves every cached Greek stale and a spot-only trigger silent.
No staleness cap. Theta and charm move the Greeks with no tick at all, so an illiquid name's "real-time" Greeks are as old as its last print.
Clamping expiry with max(1e-4, T). An expired contract has no Black-Scholes delta. Substituting a floor reports a confident number for a position that has none, and the book keeps hedging against it.
Silently defaulting option_type to PUT. An if CALL / else branch sign-flips delta on any typo, and a sign-flipped delta reads as a legitimate position on the other side.
Hard-coding the multiplier at 100. The OCC holds the premium multiplier at 100 through corporate actions and changes the deliverable instead — 5 shares after a 1-for-20 reverse split. Greeks scale with the deliverable, so 100 overstates that position exactly $20\times$. A Deribit BTC option is 1 BTC per contract.
Omitting the dividend yield. Call delta carries a factor of $e^{-qT}$; assuming $q = 0$ on a yielding name biases delta by roughly $qT$ — small per position, systematic across the book, and always in the same direction.
Applying one underlying's tick to the whole position list. An AAPL print repricing MSFT options at AAPL's spot produces Greeks that are not wrong by a little.
Letting a stale anchor survive a feed gap. After a gap the anchor's provenance is unknown, and a stale anchor is worse than none: no anchor forces a reval, a stale one suppresses it. Reset on gap and on session boundary.
Applying an out-of-order tick. A late-arriving earlier print advances the anchor backwards and leaves the book permanently mismarked. Reject it and resequence.
Publishing a delta outside $[-1, 1]$. An impossible Greek downstream is not caught by anything that consumes it; the hedger will size against it.
Reading a fast refresh as a fresh number. The output is only as current as the vol and expiry fed in. Stepping quickly off a stale surface is a stale risk number, delivered promptly.

Verification

Closed form against hand-derived values. At $S = K = 100$, $\sigma = 0.20$, $T = 1$, $r = q = 0$: $d_1 = +0.10$, $d_2 = -0.10$, $\Delta_{\text{call}} = 0.539827837$, $\Delta_{\text{put}} = -0.460172163$, $\Gamma = 0.019847627$, $\nu = 0.396952547$/vol pt, $\Theta = -0.010875412$/calendar day, price $= 7.965567455$.
Parity. $\Delta_{\text{call}} - \Delta_{\text{put}} = e^{-qT}$ (not 1) once $q > 0$; $C - P = Se^{-qT} - Ke^{-rT}$.
First sight reprices. A contract with no anchor $\implies$ FULL_BLACK_SCHOLES / NO_ANCHOR, and spot_taylor_value_error_per_unit is None — there is no truth to compare against and none is invented.
Taylor step. Anchor at $S=100$, tick to $100.20$ ($+0.20%$) $\implies$ TAYLOR_EXPANSION, $\Delta = 0.539827837 + 0.019847627 \times 0.20 = 0.543797363$, $\Gamma/\nu/\Theta$ unchanged, and the anchor still reads $S = 100$.
Anchor-drift regression (the critical one). Anchor at $100$, tick to $100.40$ ($\implies$ Taylor, drift $0.40%$), then to $100.80$ $\implies$ FULL_BLACK_SCHOLES / SPOT_MOVE_THRESHOLD, drift $0.80%$. Measured against the previous tick the second step is $0.398%$ and would wrongly step. Likewise 60 consecutive $+0.1%$ ticks must produce $\ge 5$ revaluations; a last-tick baseline produces zero while spot travels $+6.18%$.
Boundary. Drift of exactly $0.5%$ against a $0.5%$ threshold steps; $0.6%$ reprices. Down moves trigger symmetrically.
Non-spot triggers. A $+1$ vol-point move at an unchanged spot $\implies$ IV_MOVE_THRESHOLD with drift $0.0$; a $61$s-old anchor on a flat tape $\implies$ ANCHOR_AGE; $T = 0.5/365 \implies$ NEAR_EXPIRY on any move; a step that puts a call delta above 1 $\implies$ TAYLOR_DELTA_OUT_OF_BOUNDS and a repriced delta $\le 1$.
Underlying isolation. An AAPL tick against a book holding one AAPL and one MSFT leg $\implies$ one result, positions_skipped_other_underlying == 1, and no anchor created for the MSFT leg.
Negative checks. A NaN/Inf/zero/negative spot, a blank symbol, a duplicate symbol in one tick, an out-of-order timestamp, option_type="C", a string-typed number, a non-positive multiplier/strike/implied_vol/time_to_expiry_years, and a zero or negative config threshold must each raise. One bad leg must reject the whole tick and create no anchors.
Run python -m unittest discover -s skills/real-time-greeks-recalculation-on-market-moves/scripts and confirm a 100% pass rate.

Real Time Greeks Recalculation On Market Moves

When to Use

When NOT to Use

Prerequisites

Workflow

Common Pitfalls

Verification

Hands off to 9

Handed off from 6

When to Use

When NOT to Use

Prerequisites

Workflow

Common Pitfalls

Verification

Related Skills

Hands off to 9

Handed off from 6