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Option pricing · Black–Scholes · WTI

How to calculate option price

Call and put prices for an asset (e.g. WTI crude) can be computed with the Black–Scholes model. Change the inputs below to see the step-by-step calculation and how the option value changes with spot price.

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).

Model assumptions and limitations

Black–Scholes rests on several assumptions. Understanding them helps you judge when the model fits and when it falls short.

  • Continuous trading — No transaction costs; you can trade at any instant.
  • No arbitrage — Prices are consistent; no risk‑free profit opportunities.
  • Constant interest rates and volatility — r and σ do not change over the option’s life.
  • Log‑normal returns — The underlying price follows a geometric Brownian motion with log‑normal distribution.

For commodity options, the model is adjusted via convenience yield (or cost of carry), which reflects the benefit/cost of holding the physical asset versus the derivative.

Limitations: Black–Scholes does not capture volatility skew or kurtosis (implied volatility varies by strike and expiry). It also assumes European exercise (no early exercise). For American options (e.g. many exchange‑traded commodity options), early exercise can matter and requires numerical methods (binomial/trinomial trees) or analytical approximations.

Greeks — option sensitivities

Risk management and hedging rely on the Greeks: how option value changes with respect to key inputs.

Definitions
Delta (Δ) — ∂V/∂S: sensitivity to spot price Gamma (Γ) — ∂²V/∂S²: rate of change of delta Vega (ν) — ∂V/∂σ: sensitivity to volatility Theta (Θ) — ∂V/∂t: time decay (per day) Rho (ρ) — ∂V/∂r: sensitivity to interest rate
Black–Scholes formulas (with yield q)
Δ_call = e^(−qT)·N(d₁)    Δ_put = e^(−qT)·(N(d₁)−1) Γ = e^(−qT)·n(d₁) / (S·σ·√T)   (n = standard normal PDF) ν = S·e^(−qT)·n(d₁)·√T   (same for call and put) ρ_call = K·T·e^(−rT)·N(d₂)   ρ_put = −K·T·e^(−rT)·N(−d₂)

Delta is the main hedge ratio: e.g. Δ ≈ 0.5 means 0.5 units of spot hedge 1 unit of option exposure. Vega is crucial when volatility is uncertain.

Commodities vs financial assets

Commodity options differ from equity or FX options in important ways:

  • Storage costs — Physical commodities incur storage (tanks, warehouses). This enters the cost‑of‑carry model and affects basis.
  • Seasonality — Natural gas, power, and many agricultural markets show strong seasonal price patterns. Volatility and convenience yield vary by season.
  • Basis risk — The option may reference a futures contract or index that does not perfectly track the physical position. WTI vs Brent, or Henry Hub vs regional gas hubs, create basis that must be managed separately.

Examples: WTI vs Brent options trade on different underlyings; spreads between them reflect regional supply and transport. Natural gas options (e.g. Henry Hub) exhibit sharp seasonal volatility around winter demand peaks. When pricing or hedging, consider which underlying and tenor best match your exposure.

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.