An analytic framework for pricing energy derivatives

Despite its obvious shortcoming, Black´s formula for futures options is still widely used for pricing energy derivatives. The lognormality assumption that underlies this formula is inconsistent with the market implied distribution for many commodities and as a result, out-of-the-money options are mispriced by Black´s formula. Our objective is to develop a self-consistent term-structure pricing framework based on the general diffusions and derive simple pricing formulas similar to Black´s one with a few additional parameters that can be easily estimated from market prices of liquid options. We assume the following risk neutral dynamics for futures prices: df(t,T)=σ₁(f,t,T)dz₁+σ₂ (f,t,T)dz₂, dz₁dz₂=0. The value of the discounted European call option V(t,f) on T-maturity futures struck at K is determined as the solution to the following diffusion problem ∂V/∂t+½(σ₁²(f,t,T)+σ ₂²(f,t,T))∂²V/∂f², V(T,f)=(f-K)⁺