What you need (inputs)
The model uses six inputs. For WTI options we use convenience yield (q) instead of a dividend — it’s the benefit of holding the physical commodity.
S — Spot price ($/bbl)
K — Strike price ($/bbl)
T — Time to expiry (years)
r — Risk-free rate (annual)
q — Convenience yield (annual)
σ — Implied volatility (annual)
Step 1: d₁ and d₂
Two intermediate numbers drive the option value. They combine spot, strike, time, rates, and volatility.
Formulas
d₁ = [ln(S/K) + (r − q + σ²/2)×T] / (σ×√T)
d₂ = d₁ − σ×√T
Step 2: Call and put price
N(x) is the cumulative normal distribution. The formulas use N(d₁), N(d₂) and their complements.
Black–Scholes (with yield q)
Call = S×e^(−qT)×N(d₁) − K×e^(−rT)×N(d₂)
Put = K×e^(−rT)×N(−d₂) − S×e^(−qT)×N(−d₁)
Put–call parity: Call − Put = S×e^(−qT) − K×e^(−rT). The calculator checks this (should be ≈ 0).
What moves the price?
Higher volatility → higher option premium (more uncertainty = more value to the option). Longer time → usually higher premium (more time to be in the money). Spot vs strike: call is worth more when spot is above strike; put when spot is below. The graph on the right shows option value as spot moves.