The valuation framework of Black-76 provides a zeroth-order model in which forwards are martingales, volatility is constant, and returns are normal. The martingale attribute is robust, being a direct consequence of the assumptions underpinning all risk-neutral valuation. The assumptions of constant volatility and normal returns, however, are definitely at variance with the empirical behavior of forward prices. When does this matter?

In Part III we will explore three central features of commodities price dynamics and related structures: backwardation of volatility, skew, and correlation. Each of these features require enhancements of some form to Black-76 to even begin to sensibly price commonly occurring structures. We will not posit a holistic pricing framework that solves all problems and is consistent with all empirical results. Our goal here is to adapt Black-76 as required to (arguably) handle these three core risks while pointing out limitations and deficiencies as appropriate.

The problems that we will discuss here are those in which risk-neutral valuation is generally believed to be a defensible approach. In reality, no market is complete; certainly no one who has ever managed structured options portfolios would think otherwise. The applicability of risk-neutral pricing methods depends on just how incomplete the market is. In many practical settings, a model of price dynamics involves parameters that cannot be inferred from the market prices of instruments that actually trade liquidly.