# Market Invariant for Commodity Futures

In the same sense that Meucci describes "compounded returns" as the invariant for equities and "changes in yield-to-maturity" as the invariant for fixed-income, what is the invariant for a commodity future like a corn future?

What have I tried?

Take the commodity forward pricing formula from Hull (and ignore the convexity correction for a future vs. forward, which is probably immaterial for short-term futures like corn futures):

$$F_0=S_0e^{(r+u-y)T}$$

where:

• $$F_0$$ = forward price
• $$S_0$$ = spot price
• $$r$$ = risk-free interest rate
• $$u$$ = storage costs [as a constant proportion of $$S_0$$]
• $$y$$ = convenience yield [as a constant proportion of $$S_0$$]

I observe $$S_0$$ from the market for spot corn (e.g. on USDA) and $$F_0$$ from the market for futures. I'm already estimating $$r$$ via interest rate derivatives (e.g. Fed Funds Futures, Eurodollar Futures and Options). That leaves physical cost-of-carry $$u-y$$ as the remaining unknown. I would think that measuring the change in a constant-maturity physical cost-of-carry $$u-y$$ would be the market invariant.

Random variables are market invariants if they are independent and identically distributed (i.i.d.). For example, with equities, prices are not i.i.d. (as there's clearly a relationship between today's price and yesterday's price), but returns are "generally considered" i.i.d. Likewise, there's clearly a relationship between the value of a zero-coupon bond today and yesterday, but the change in yield to maturity is "generally considered" i.i.d.

Meucci discusses it as one of the basic tenants of risk management and portfolio allocation. See here ("Prayer" 1). Basically, i.i.d. is a nice property to have in any statistical analysis.

I raise the question for futures as most work I've seen on futures performs analysis on raw future prices (or perhaps log future prices), neither of which I'd consider to be an invariant.

In his book, Meucci mentions that commodities can use returns as their invariant. This seems logical for the actual cash commodity (i.e. corn), but I'm not sure why returns make sense for a commodity future. Hence the question.

• can you please provide a source for what you mean by a "market invariant" ? invariant with respect to what ? I don't know who Meucci is so probably useful to provide additional context to understand what your question is actually. thanks! – Ezy Nov 4 '18 at 23:38