greeks
Charm (Delta Decay)
Second-order Greek measuring how Delta changes as time passes. Critical for understanding why your delta-hedged position drifts overnight.
Formula
Charm = -φ(d₁) · [2(r-q)T - d₂·σ·√T] / (2T·σ·√T)
Variables
- Charm
- Change in Delta per day (dΔ/dT)
- φ(d₁)
- Standard normal PDF at d₁
- d₂
- Standard Black-Scholes d₂
- σ
- Implied volatility
- T
- Time to expiration (years)
- q
- Dividend yield (continuous; use 0 if no dividends)
- r
- Risk-free rate
Worked Example
Inputs
- Option
- ATM call, 30 DTE
- Δ today
- 0.52
- Charm
- -0.015
Calculation Steps
- 1
Δ tomorrow ≈ 0.52 + (-0.015) = 0.505 - 2
After 5 days: Δ ≈ 0.52 + 5 × (-0.015) = 0.445 - 3
OTM calls lose delta faster as expiration approaches
Result: Delta drifts by ~-0.015/day toward 0 (OTM) or 1 (ITM)
Intuition
Charm explains why OTM options become worthless faster over weekends — they lose delta (and therefore theta accelerates). Market makers re-hedge Monday morning because charm shifted their deltas over the weekend.