Valuation models
Crack, spark and calendar spread options
Crack spreads such as 3:2:1, spark and clean spark spreads with an emissions leg, and calendar spreads between delivery months, exact for two legs by conditioning and Black-76, with Kirk's approximation alongside, and Monte Carlo for three or more; calendar vols and correlation from Schwartz, Schwartz-Smith or Gabillon.
Inputs
| Leg | Future | Weight (signed) | Vol % | |
|---|---|---|---|---|
Results
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Methodology
Conventions
- Each leg is a lognormal future to expiry; the payoff is the weighted sum less the strike.
- Two legs: exact, by conditioning on the short leg and Black-76 on the long one; Kirk shown alongside. More legs: Monte Carlo.
- Calendar spreads take both months' vols and their correlation from the futures model.
References
- Kirk (1995)
- Carmona and Durrleman (2003)
- Schwartz and Smith (2000)
Formulas
- Payoff
- max(Σ wᵢ Fᵢ(T) − K, 0)
- Kirk
- σ² = σ₁² − 2ρσ₁σ₂ a + σ₂² a², a = F₂/(F₂ + K)