[IamJasonBian]

Summary

• Bug fix: options/black_scholes._norm_cdf applied the A&S 7.1.26 erf approximation to x without the 1/sqrt(2) scaling (inherited from the allocation-engine port), so Φ was off by up to ~0.04. Every bs_call_price / bs_put_price consumer in options/ (simulation, backtest_forward, historical_pricing, vol compares, report) was skewed.
• bs_call_implied_vol: bisection IV solver.
• options/pricer_backtest: prices a real contract daily with three σ models and scores them vs actual closes.
• options/sensitivity: analytic Greeks + spot × IV × days-forward scenario grid CLI.

Same change as IamJasonBian/allocation-engine#62.

Backtest — NBIS 11/20/2026 $230C, 2026-09-29 → 2026-10-20 (n=16)

Model	MAE	Bias	MAPE
prev-day IV	$0.56	+$0.07	4.6%
frozen entry IV	$1.67	+$1.50	9.9%
20d realized vol	$4.12	−$4.12	22.2%

Run

python -m allocation_gym.options.pricer_backtest
python -m allocation_gym.options.sensitivity --spot 180

Test plan

• New tests: 21 passed (IV round-trip, Greeks vs finite differences, Φ vs math.erf)
• Re-run saved options/ reports — numbers change with the Φ fix

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