Option pricing models

Definition

Pricing models translate assumptions about the underlying’s future distribution into option values and Greeks. The canon: Black-Scholes-Merton (1973) — continuous lognormal diffusion, closed-form European prices, the framework that defined the field; Cox-Ross-Rubinstein binomial trees (1979) — discrete steps that handle American early exercise; and the practical truth that markets quote prices FIRST and the model runs in reverse, turning prices into implied volatility (opt-implied-volatility). The model is a translation device, not an oracle.

How it works / structure

• BSM machinery: inputs S, K, T, r, dividend yield, σ → price and Greeks; its assumptions (constant vol, continuous hedging, lognormal returns, no jumps) are all false in ways the market prices — the skew (opt-volatility-skew) IS the market’s correction schedule to BSM’s flat-vol assumption.
• Binomial/CRR: lattice of up/down steps converging to BSM in the limit; evaluates early exercise at each node — the standard American-equity-option engine (opt-dividend-effects drives the exercise decisions it finds).
• Beyond the canon (named, not detailed): jump-diffusion (Merton 1976) prices gap risk; stochastic-volatility models (Heston) price vol-of-vol and generate skews internally — the platform names them where relevant and quotes IV surfaces rather than re-deriving them.
• Engine usage (executable facts): the platform prices and Greeks with CRR for American equity options and BSM for European/index, quotes all vol in implied terms, and treats model choice as a pinned replay parameter.

When it applies

Greeks generation (greek-delta through opt-second-order-greeks), scenario repricing (risk-scenario-analysis full revaluation), IV extraction and surface construction, and early-exercise boundary evaluation. Model literacy is what keeps “the model says it’s mispriced” claims honest — usually the model is missing what the market prices.

Risk profile & failure modes

• Assumption leakage: delta-hedging P&L, exercise boundaries, and scenario values all inherit model assumptions; BSM deltas on a heavily skewed surface are systematically off vs smile-adjusted deltas — a known, material effect in risk aggregation.
• Calibration circularity: models calibrated to today’s prices reproduce today’s prices; predictive content lives only in the assumptions’ realism.
• “Mispricing” hubris: a vanilla model disagreeing with the market usually reveals the model’s missing jump/vol-of- vol term, not free money (lens-quantitative humility applies).

Evidence & limits

Black-Scholes (1973) and Merton (1973) are the founding derivations; CRR (1979) the discrete workhorse. The empirical literature documenting systematic BSM violations (smiles, jumps, stochastic vol) is equally established — the model canon and its documented failures together are the field. The platform’s stance: models are coordinate systems for quoting and hedging; edges claimed FROM a model require evidence the model captures something the market misses, which is rare and must be replayed.

Falsifiable-thesis examples

Illustrations only, not signals:

• “CRR and BSM prices for X’s 60-day ATM call differ by less than the bid-ask spread (early-exercise premium negligible here)” — falsified by the computed comparison.
• “Smile-adjusted deltas reduce this book’s daily hedged P&L variance vs BSM deltas over the quarter” — falsified by the paired hedging replay.

Cross-references

• The inversion that matters: opt-implied-volatility; the correction schedule: opt-volatility-skew, opt-term-structure
• Model-free anchor: opt-put-call-parity
• Outputs consumed: greek-delta, greek-gamma, opt-second-order-greeks, mgmt-delta-hedging