pricing
Black-Scholes Put
Calculates the theoretical fair value of a European put option. Mirror image of the call formula using negative d values.
Formula
P = K · e⁻ʳᵀ · N(-d₂) - S · N(-d₁)
Variables
- P
- Put option price
- S
- Current stock price
- K
- Strike price
- r
- Risk-free interest rate
- T
- Time to expiration (years)
- N(x)
- Cumulative normal distribution
Worked Example
Inputs
- S
- $100
- K
- $100
- σ
- 20%
- r
- 5%
- T
- 0.25
Calculation Steps
- 1
d₁ = 0.175, d₂ = 0.075 (from same inputs as call example) - 2
N(-d₁) = N(-0.175) ≈ 1 - 0.5695 = 0.4305 - 3
N(-d₂) = N(-0.075) ≈ 1 - 0.5299 = 0.4701 - 4
P = 100 × e⁻⁰·⁰¹²⁵ × 0.4701 - 100 × 0.4305 = 46.43 - 43.05
Result: P ≈ $3.38 (verified by put-call parity: C - P = S - K·e⁻ʳᵀ)
Intuition
A put is cheaper than the equivalent call at ATM because the stock has a natural upward drift (risk-free rate). Put-call parity connects the two prices exactly.