LLMs are widely cited as strong at quantitative reasoning, and on closed-form pricing they largely are. On options Greeks specifically (the derivatives of the pricing function) they are noticeably weaker, and the errors are not random. This piece documents three recurring failure categories we see across frontier models when reasoning about theta, vega, and gamma on European options: theta-sign confusion on long puts, vega-vs-gamma conflation at-the-money, and ITM-vs-ATM gamma misranking on near-dated options. These are illustrative patterns, not measured per-model error rates — we deliberately publish no accuracy numbers here because we have no public, reproducible harness behind them. The patterns are consistent with overfitting to the most-common textbook examples (Hull's Options, Futures, and Other Derivatives and Natenberg's Option Volatility & Pricing). Below: the three failure categories with reproducer prompts you can run yourself, the underlying source of confusion, and a defensive prompt template plus a verifier architecture that removes the problem regardless of which model is in the loop.

Greeks: a five-line refresher

For a European option with price V, underlying S, strike K, time T, vol σ, rate r:

  • Delta Δ = ∂V/∂S. Long call positive, long put negative.
  • Gamma Γ = ∂²V/∂S². Always positive for long options. Peaks at-the-money, falls off on either side.
  • Theta Θ = ∂V/∂T. Almost always negative for long options (time decay). Long deep-ITM European put can have positive theta when carry effects dominate.
  • Vega ν = ∂V/∂σ. Always positive for long options. Peaks at-the-money for long-dated options.
  • Rho ρ = ∂V/∂r. Long call positive, long put negative.

These signs and shapes are settled; any LLM error is a recall failure, not a definitional one.

Failure cluster 1: Theta sign confusion on long puts

Reproducer prompt:

"I am long a European put on a non-dividend-paying stock. The stock is at $95, strike $100, 60 days to expiry, vol 25%, rate 5%. What is the sign of theta?"

Correct answer: negative (most cases). For deep ITM long European puts on stocks where r > 0, theta can flip positive because the discounting effect on the strike-receivable dominates the time-value loss. The example above is slightly OTM ($95 < $100), so theta is unambiguously negative.

The failure pattern: a model answers "Theta is positive — the put is in the money," conflating S < K (an ITM put) with the deep-ITM positive-theta exception, then confidently delivers the wrong sign for what is actually a slightly-OTM put.

The failure is not that the model doesn't know about deep-ITM put theta — it does, and will cite the exact mechanism. The failure is misapplication: invoking the exception when the prompt describes a regular case. This is a recall-with-bad-trigger pattern, common in models that have memorised long-tail facts but lack the discipline to check whether the trigger condition is satisfied. The correct behaviour is to confirm the moneyness and rate conditions before reaching for the exception: "Theta is negative; the put is slightly OTM, time decay dominates discounting effects."

Failure cluster 2: Vega vs gamma conflation ATM

Reproducer prompt:

"For an ATM European call, 30 days to expiry, which is larger as a fraction of option value: gamma exposure or vega exposure?"

Correct answer: the question is mis-specified — gamma and vega have different units (vega per vol-point, gamma per dollar²) and can't be compared directly without specifying a vol move and underlying move. The right answer is to surface the mis-specification and ask for a stress vector.

The failure pattern: a model confidently produces a numerical comparison treating vega ($-per-1% vol-point) and gamma ($-per-$1²) as if they were dimensionally compatible. The output number is meaningless and the model presents it as definitive.

This is the same ambiguity-handling failure mode seen in finance LLM evaluations more broadly, expressed in a quantitative domain. Models tuned toward confident numerical output tend to answer ill-posed questions instead of flagging them; the right behaviour is to surface the dimensional mismatch and propose a stress vector (e.g., "1% underlying move and 2 vol-point shock") rather than emit a single comparison number.

Failure cluster 3: ITM-vs-ATM gamma misranking on near-dated

Reproducer prompt:

"Two European calls on the same stock, both expiring in 5 days. One has strike $100, the other strike $98. Spot is $100. Which has higher gamma?"

Correct answer: the ATM call (strike $100) has higher gamma. The ITM call (strike $98) has lower gamma; gamma falls off as the option goes deep ITM or deep OTM. For very near-dated options (5 days), the gamma curve becomes spiky and the ATM peak is sharper than for longer-dated.

The failure pattern: a model answers "The ITM call has higher gamma because it has more delta to lose." This conflates delta range (1.0 → 0 over the underlying range, larger for ITM at expiry) with instantaneous gamma. They are related, but gamma at any single underlying price is maximised at the strike, not in the deep-ITM region.

The confusion appears to come from over-indexing on the integral of gamma across the strike rather than the pointwise gamma at a given S. Both are valid quantities; only one answers the prompt — the ATM call has the higher pointwise gamma.

Why the pattern matters

These three categories recur across models and runs at default temperature; they are patterns, not single-shot anomalies. The shape (exception-misapplication, dimensional confusion, and integral-vs-pointwise conflation) is recognisable from the cognitive-error literature on novice options traders. Frontier LLMs reproduce novice-trader errors despite their broader quantitative reputation, and they do it with high confidence, which is what makes the pattern dangerous.

The implication for a production options-research workflow: any time a model generates Greeks commentary on auto-pilot, sign and magnitude errors of these three kinds are a live risk. At desk scale that is a compliance and PnL exposure; at retail scale it is a recipe for blowup. The fix is not "pick the best model" — it is to verify every Greek numerically (see below).

A defensive prompt template

The template below forces the metacognitive check the failure categories above all skip. It adds roughly 800 input tokens of overhead per prompt; the point is not to make any single model "pass" but to make its reasoning auditable before it commits to a sign or magnitude.

You are an options research assistant. Before computing or reasoning about
any Greek, do the following four checks in order:

1. State the option type, moneyness (ITM/ATM/OTM), and time to expiry
   bucketed as <7d / 7-30d / 30-90d / >90d.

2. State the *sign* of the Greek you are about to compute, citing the
   standard rule (e.g., "long put theta is typically negative; positive
   only for deep-ITM European puts on dividend-free stocks with r>0").

3. If the prompt asks for a magnitude comparison, verify the units are
   compatible. Vega is $-per-1%-vol-point. Gamma is $-per-$1². They
   cannot be compared without a stated stress vector.

4. If the prompt's parameters do not satisfy the trigger conditions for
   any non-default behaviour (e.g., deep-ITM put + r>0), do not invoke
   the exception. Default to the standard rule.

After completing the checks, answer the original question.

The template's main effect is to force the model into the metacognitive step it skips at default temperature: confirming that the prompt's parameters fall into the rule's domain before applying the rule.

What this is not

This is not a claim that any one model is bad at quantitative finance. On closed-form Black-Scholes pricing, frontier models are generally reliable in our reproducer prompts. Greeks are different — they are derivatives of the pricing function, and reasoning about derivatives requires more careful trigger-condition checking than pricing itself. The gap shows up in the reasoning task, not the formula task.

This is also not a claim that any model is immune to the three categories. They fail in different ways and at different frequencies, but none of them should be trusted on Greeks without a verifier in the loop. We are not naming per-model rates precisely because we have no public harness that would let you reproduce one; if you want measured numbers, build the harness in the next section and run it on your own model set.

Verifier-in-the-loop architecture

The production fix is to never trust a single LLM's Greeks output. The two-pass architecture:

  1. Generate Greeks commentary with the model of choice (Opus, Sonnet, GPT-5).
  2. Independently compute the actual Greeks numerically using a closed-form Black-Scholes (or finite-difference for path-dependents).
  3. Reconcile: if the model's stated sign or magnitude bucket disagrees with the numerical truth, flag for human review.

The verifier is roughly 30 lines of code with QuantLib or py_vollib; the integration cost is one afternoon. Because the second stage computes the Greeks from a closed-form model rather than recalling them, the sign and magnitude errors in the three categories above become detectable mechanically — the model's prose is reduced to a hypothesis the numerical stage either confirms or flags, regardless of which model sits in the first stage.

Connects to

References

  1. Hull, J. C. (2022). Options, Futures, and Other Derivatives (11th ed.). Pearson. ISBN 978-0136939979.
  2. Natenberg, S. (2014). Option Volatility and Pricing (2nd ed.). McGraw-Hill. ISBN 978-0071818773.
  3. Merton, R. C. (1973). "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science 4(1), 141–183. DOI: 10.2307/3003143.
  4. Lakonishok, J., Lee, I., Pearson, N. D., & Poteshman, A. M. (2007). "Option Market Activity." Review of Financial Studies 20(3), 813–857. DOI: 10.1093/rfs/hhl025.
  5. QuantLib documentation. https://www.quantlib.org/, accessed May 8, 2026.
  6. py_vollib package. https://github.com/vollib/py_vollib, accessed May 8, 2026.
  7. Black, F., & Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81(3), 637–654. DOI: 10.1086/260062.
  8. OpenAI. GPT-5 Technical Report. https://openai.com/research, May 2026.