How the engines work

pure_greeks ships three pricing engines and a deterministic fallback chain that picks one for each call.

The three engines

1. Black-Scholes European (closed-form) — analytic formula for European exercise. Cheap, exact within the model.
2. CRR Binomial American — Cox-Ross-Rubinstein binomial tree, 200 steps. Captures early-exercise premium for American options. Greeks are extracted from the tree (delta and gamma from t=0 nodes, theta from t=1 vs t=0, vega and rho via finite difference over re-priced trees).
3. Intrinsic value — max(S - K, 0) for calls, max(K - S, 0) for puts. The terminal fallback when implied volatility is zero or negative.

Fallback chain

Selection order is fixed and deterministic:

1. If implied_volatility <= 0, use Intrinsic.
2. Else if exercise_style == :american, use CRR Binomial American.
3. Else use Black-Scholes European.

The engine that produced the result is always exposed on the option:

option.calculation_model  # => :crr_binomial_american | :black_scholes_european | :intrinsic

If an engine raises, the chain falls through to the next one (American → European → Intrinsic).

CRR Greeks extraction

The standard “free Greeks” technique used by QuantLib’s BinomialVanillaEngine:

• Delta ≈ (V[0]_step1 − V[1]_step1) / (S·u − S·d) — finite difference one step from expiry.
• Gamma ≈ (Δ_upper − Δ_lower) / (½(S·u² − S·d²)) where Δ_upper and Δ_lower come from step 2.
• Theta ≈ (V[1]_step2 − price) / (2·Δt) then divided by 365 for per-day units.
• Vega and rho are computed by re-pricing two more trees with σ + 0.01 and r + 0.01 respectively, then taking the forward difference.

This costs three full tree solves per option; queued for v0.2 optimization (see BENCHMARKS.md).

Why this exists

QuantLib is the industry-standard option pricer, but its Ruby binding is a binary dep that’s painful in production: you need a system install, version pinning is fragile, and it’s hard to deploy on serverless platforms. pure_greeks is a deliberately scoped subset — the vanilla American/European Greeks that most equity-option workloads actually need — implemented in pure Ruby so it installs anywhere gem install works.