[IamJasonBian]

Summary

• Bug fix: _norm_cdf applied the A&S 7.1.26 erf approximation to x without the 1/sqrt(2) scaling, so Φ was off by up to ~0.04 (e.g. Φ(1) = 0.870 vs 0.841). Every bs_call_price / bs_put_price — including the existing protective-put/collar sims — was skewed.
• bs_call_implied_vol: bisection IV solver.
• option_pricer_backtest: prices a real contract daily with three σ models and scores them vs actual closes.
• option_sensitivity: analytic Greeks + spot × IV × days-forward scenario grid CLI.

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%

IV rose ~70% → ~93% as the underlying fell, so realized vol badly underprices and frozen IV drifts.

Run

python -m trading_system.backtests.option_pricer_backtest
python -m trading_system.backtests.option_sensitivity --spot 180

Test plan

• pytest tests/backtests — 48 passed (IV round-trip, Greeks vs finite differences, Φ vs math.erf)
• Re-check any saved put/collar backtest reports — their numbers change with the Φ fix

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