greeks
Theta (Θ)
Time decay — the silent killer of long options. Measures how much option value erodes each day, all else being equal.
Formula
Θ_call = [-(S · φ(d₁) · σ) / (2√T) - r · K · e⁻ʳᵀ · N(d₂)] / 365 Θ_put = [-(S · φ(d₁) · σ) / (2√T) + r · K · e⁻ʳᵀ · N(-d₂)] / 365
Variables
- Θ
- Daily time decay ($/day)
- φ(d₁)
- Standard normal PDF at d₁
- S
- Current stock price
- σ
- Implied volatility
- T
- Time to expiration (years)
Worked Example
Inputs
- S
- $100
- K
- $100
- σ
- 20%
- T
- 30 DTE
Calculation Steps
- 1
T = 30/365 = 0.0822 years - 2
First term: -(100 × 0.3989 × 0.20) / (2 × √0.0822) = -13.92 - 3
Second term: -0.05 × 100 × e⁻⁰·⁰⁰⁴¹ × 0.51 = -2.55 - 4
Annual Θ = -16.47 → Daily = -16.47 / 365
Result: Θ ≈ -$0.045/day — option loses ~4.5 cents daily
Intuition
Theta accelerates as expiration approaches — an option loses more value in its last week than in its first month. This is why income sellers target 30-45 DTE: maximum theta decay with manageable gamma risk.