For a 30-DTE 5%-out-of-money call on a $200 underlying with 28% implied volatility and a 4.5% risk-free rate (strike $210, dividend yield 0%), the Options Greeks Explorer returns price $2.945, delta 0.301, gamma 0.0217, theta −0.100, vega 0.200, rho 0.0471. d1 = −0.522, d2 = −0.602. Intrinsic value 0, extrinsic 2.945 (the contract is entirely time-value). These six numbers are the worked example LLM-driven options agents most often fumble, and they fumble in a specific way that the engine output lets a human reviewer catch immediately.

TL;DR

Six Greeks for the canonical 30-DTE OTM call:

Greek Value Interpretation (per-share, on a $200 underlying)
Price $2.945 Premium paid per share for one call contract leg
Delta 0.301 $0.30 expected change per $1.00 move in underlying
Gamma 0.0217 Delta adds 0.0217 per $1.00 underlying move (so delta rises near 0.32 after a $1 rise)
Theta −0.100 $0.10 of value decays per calendar day
Vega 0.200 $0.20 change per 1.00% IV change
Rho 0.0471 $0.047 change per 1.00% interest-rate change
d1 −0.522 Standardized log-moneyness; negative means OTM
d2 −0.602 d1 minus σ·√T; controls extrinsic decay

Gamma and theta are the two Greeks most LLM-driven prompts confuse. Both scale non-linearly with days-to-expiration; both peak at-the-money and decay on either side of the strike. The 30-DTE OTM regime is where their behaviour diverges most sharply, which is why this exact configuration is the diagnostic case for any options-aware LLM.

The base calculation

Black-Scholes-Merton for a European call1:

C = S·e^(−q·T)·Φ(d1) − K·e^(−r·T)·Φ(d2)

d1 = (ln(S/K) + (r − q + σ²/2)·T) / (σ·√T)
d2 = d1 − σ·√T

For the canonical input (S = 200, K = 210, σ = 0.28, T = 30/365, r = 0.045, q = 0):

  • σ·√T = 0.28 · √(30/365) = 0.28 · 0.2867 = 0.0803
  • ln(S/K) = ln(200/210) = −0.0488
  • r − q + σ²/2 = 0.045 + 0.0392 = 0.0842 (rate-plus-half-variance drift)
  • numerator(d1) = −0.0488 + 0.0842 · (30/365) = −0.0488 + 0.00692 = −0.0419
  • d1 = −0.0419 / 0.0803 = −0.522
  • d2 = −0.522 − 0.0803 = −0.602

The engine returns d1 = −0.5216 and d2 = −0.6019 to four decimal places. The hand calculation agrees.

Φ(d1) ≈ 0.301, Φ(d2) ≈ 0.274 (standard normal CDF). So:

C = 200 · 1.000 · 0.301 − 210 · e^(−0.045·30/365) · 0.274
  = 60.20 − 210 · 0.9963 · 0.274
  = 60.20 − 57.30
  = 2.90

The engine returns C = 2.945. The small discrepancy is from the engine using higher-precision Φ; the order of magnitude and the structure are correct.

Delta vs gamma: the load-bearing distinction

Delta is the first derivative of option price with respect to underlying. For this contract:

  • Delta = 0.301 means a $1.00 underlying move from $200 to $201 produces a $0.30 option-value move from $2.945 to roughly $3.25.

Gamma is the second derivative, the rate of change of delta:

  • Gamma = 0.0217 means a $1.00 underlying move increases delta by 0.0217 (from 0.301 to 0.323).

For LLM-driven trading prompts the failure mode is to conflate the two. A typical confabulation: "Delta is 30%, so a $1 move produces a $30 gain." That is wrong by an order of magnitude (a $1 move produces a $0.30 gain per share, $30 per contract because contracts are 100 shares). A different failure mode: "Gamma scales the same way as delta with DTE." That is also wrong, see the DTE scaling section below.

The Hallucination Detector catches this class of error because the dollar arithmetic is verifiable against the engine output. The Options Greeks Explorer is the source of truth.

Theta and DTE: the decay shape

Theta = −0.100 means the option loses $0.10 per calendar day if all other inputs hold constant. Over the 30-day remaining life that is $3.00 total — but the option is only worth $2.945. The reconciliation: theta is not linear in DTE. It accelerates as expiration approaches. The first 15 days lose roughly $0.07/day on average; the last 15 lose roughly $0.13/day; the final 5 days lose closer to $0.30/day if the contract stays OTM.

The engine returns a point estimate at the current DTE. To see the shape, sweep days_to_exp across {60, 45, 30, 15, 7, 2}; theta will move from roughly −$0.07 to roughly −$0.40. The non-linearity is the entire reason 0-DTE option strategies have a different risk profile from 30-DTE strategies.

For an LLM prompt that mishandles this, the giveaway in the output is the use of "theta per day stays constant." It does not. Hull's textbook derivation makes the time-derivative explicit; the term has both a 1/√T term (from the d2 standard-normal density) and a r·K·e^(−r·T)·Φ(d2) drift term2. Neither is linear in T.

Vega: where IV mispricing hits the book

Vega = 0.200 means a 1.00% change in implied volatility moves the option value $0.20. The current price of $2.945 includes 28% IV; if IV jumps to 30% the option re-prices to roughly $3.34. If IV falls to 26% it re-prices to $2.55.

For a $1,000 position size (1 contract = 100 shares = roughly $294.50 of premium), a 2-point IV move produces $40 of P&L. That is a 13% move on the position from a vol-only re-pricing.

The implication for retail LLM-driven options trading is that vega exposure is a load-bearing component of every position's risk. A long-call book that ignored vega and ran 50 positions of this size would have ±$2,000 swings from a 2-point IV move alone. The defensible approach is to either neutralize vega (long-short combinations) or to bound the IV regime in which the position is sized.

Gamma and theta scale opposite directions with DTE

The 30-DTE OTM regime sits at the intersection of two curves:

  • Gamma rises as DTE shrinks (a contract closer to expiry has sharper delta sensitivity around the strike).
  • Theta also rises in magnitude as DTE shrinks (faster decay near expiry).

For an OTM call, both effects compound: gamma rises but stays small (because the contract is not near the strike), while theta accelerates fast (because most extrinsic value evaporates in the last week). The 30-DTE OTM is the "fading time-value, modest sensitivity" regime — a common mistake in LLM-driven trades is to size positions as if gamma were ATM (high sensitivity) and theta were 60-DTE (slow decay). Both are wrong by the same factor: the contract is small in payoff structure but fast in decay.

The LLM-prompt confounder

An LLM asked "what's the delta of a 30-DTE 5% OTM call on a $200 underlying at 28% IV" frequently returns "0.30" with no error bars. The engine returns 0.301 with implicit precision that depends on the input precision. Three failure modes:

  1. The LLM rounds delta to 0.30 then computes gamma as "delta divided by something" without doing the Black-Scholes derivative. The right answer for gamma (0.0217) is not "0.30 divided by anything." It is the standard-normal density at d1 divided by (S·σ·√T). An LLM that cannot show the derivation should not be trusted to size positions.
  2. The LLM gives a per-contract dollar number where a per-share is asked, or vice versa. Contracts are 100 shares. A delta of 0.30 means $0.30 per share, $30 per contract. The order-of-magnitude confusion is one of the most common LLM failures in this domain.
  3. The LLM uses last-month's IV regime to price the current contract. The engine accepts vol_pct as an input. The LLM should be asked to extract IV from the current option chain before pricing; a price-blind research harness (see /articles/price-blind-llm-research-harness/) makes this explicit.

A multi-leg sanity check

For a single long call leg, the Options Payoff Builder confirms the engine's price at zero underlying movement: the position's at-spot P&L is 0 (the premium is the cost basis). At a $220 underlying (10% move up), the payoff before expiry is approximately the call's new theoretical price; at expiry the payoff is max(220 − 210, 0) − 2.945 = 7.055 per share. The payoff curve crosses break-even at S = K + premium = 212.95.

For multi-leg structures (vertical spreads, iron condors), the same engine accepts legs as an array; the per-leg Greeks aggregate by sum, which is the right model for the position-level exposures. The Options Greeks for LLM-Driven Trading article expands the multi-leg case.

Connects to

References

  • Merton, R. C. (1973). "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science 4(1), 141–183. Extension of Black-Scholes with dividends.
  • Wilmott, P. (2007). Paul Wilmott Introduces Quantitative Finance, 2nd ed., Wiley. Hands-on derivations for retail practitioners.
  • Natenberg, S. (2014). Option Volatility and Pricing, 2nd ed., McGraw-Hill. Practitioner-oriented treatment of vega and IV regimes.

Footnotes

  1. Black, F., & Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81(3), 637–654. https://www.jstor.org/stable/1831029 ↩

  2. Hull, J. C. (2017). Options, Futures, and Other Derivatives, 10th ed., Pearson. Chapter 17 (Greek letters). The canonical derivation of theta and the time-dependence of d1, d2. ↩

Verified engine output

Show the recompute-verified inputs and outputs
Inputs
spot200
strike210
vol_pct28
days_to_exp30
rf_pct4.5
div_pct0
Result
typecall
price2.9449066513464004
delta0.3009789474572538
gamma0.021688638342445776
theta-0.10023034150345123
vega0.19965431460443242
rho0.04705552014255154
intrinsic0
extrinsic2.9449066513464004
d1-0.5215869942020952
d2-0.6018604992734347

Computed live at build time.

Frequently asked questions

Why is the option priced at $2.945 when delta is only 0.301?
Option price = delta × S × e^(−qT) − Φ(d2) × K × e^(−rT). For an OTM call there is no intrinsic value; the price is entirely extrinsic, which the engine reports as intrinsic = 0, extrinsic = 2.945.
How is gamma computed?
Gamma = φ(d1) / (S · σ · √T) where φ is the standard-normal density. For canonical inputs that yields 0.348 / 16.06 = 0.0217, matching the engine output.
Should I trust an LLM with options pricing if it gets delta right?
No. Delta is the easiest Greek; gamma, theta, and vega are where errors compound. Use the LLM for structural reasoning and route numeric pricing to the engine.
Does the engine handle dividends?
Yes via the div_pct input. Dividends discount the spot in the d1 numerator and reduce call delta. The canonical run uses div_pct = 0 (non-dividend-paying underlying).
What changes for a put?
Put delta = call delta − 1; put theta and vega are similar in magnitude; put rho is negative. The (gamma, vega) pair is identical for call and put at the same strike by put-call parity.