Canonical Research Note: Originally published by the TrendSpike Institutional Systems & Quantitative Finance Desk. Full Black-Scholes-Merton second-order Greek derivations and dealer hedging matrices are available in the primary publication.
Python Implementation: Black-Scholes 0DTE Gamma, Vanna & Charm Engine
python
import math
def norm_pdf(x: float) -> float:
return math.exp(-0.5 * x * x) / math.sqrt(2.0 * math.pi)
def compute_0dte_greeks(spot: float, strike: float, iv: float, minutes_left: float, rate: float = 0.05) -> dict:
tau = max(minutes_left / (252.0 * 390.0), 1e-6)
sqrt_tau = math.sqrt(tau)
d1 = (math.log(spot / strike) + (rate + 0.5 * iv * iv) * tau) / (iv * sqrt_tau)
d2 = d1 - iv * sqrt_tau
pdf_d1 = norm_pdf(d1)
gamma = pdf_d1 / (spot * iv * sqrt_tau)
vanna = -pdf_d1 * (d2 / iv)
charm = pdf_d1 * ((2.0 * rate * tau - d2 * iv * sqrt_tau) / (2.0 * tau * iv * sqrt_tau))
return {
"gamma": round(gamma, 6),
"vanna": round(vanna, 6),
"charm_daily": round(charm / 252.0, 6),
}
