Commodity Forwards and Futures

It’s all about the trade-off between holding the commodity right now or using the money to invest in other opportunities and settle a contract in the future.

A commodity at time T can be synthesized by buying zero-coupon (at price F(0,T)exp(-rT)) and a forward contract (F(0,T)) now.

Payoff now is -F(0,T)exp(-rT) and at time T is S(0,T)

But the problem is S(0,T) is not ready to be determined. There is uncertainty. Therefore, the payoff now is also

-S(0,T)exp(-alphaT) with alpha to take into account the uncertainty.

So, S(0,T)exp(-alphaT) = F(0,T)exp(-rT)

Both reflect how much you want to pay to receive the cash flow of ST in time T. And forward price discounted by risk free rate is just the NPV of future spot price.

Therefore,

F(0,T) = S(0,T) exp(-(alpha-r)T)

Of course, alpha > r due to additional uncertainty

Lease Rate:

Lender will only lend when the NPV is zero. Without lending:

NPV = S(0)(exp(-alpha+g)-1)

Alpha is the return of stock with same risk, g is the growth rate of the commodity

Therefore, borrow has to return alpha-g unit of commodity so the NPV is 0.

Lease rate = delta = alpha –g

So if alpha > g, the borrower has to compensate because lender can otherwise sell the commodity and invest in stock (NPV is zero as discounted by alpha also)

Borrow has to return exp(alpha –g) units

So, F(0,T) = S(0) exp(r-delta) = S(0) exp(-(alpha-r-g))

Just like in the financial forward where delta is the dividend yield

Contango: delta < r, upward curves

Backwardation: delta>r, downward curves

With storage cost Lamda,

F(0,T) > S0exp((r+lamda)T)

Overall:

F(0,T)=S0exp(T(r-c-delta+lambda))

C is the convenience yield

Gold: depends on total production in the future

Corn: seasonal supply constant demand

Natural gas: constant supply seasonal demand

Oil: fairly stable forward price, short term fluctuation

Basis: difference between the spot price and the forward price

Basis risk: due to change of relative value or roll over of contracts

Arbitrage can arise due to convenience yield

Commodity spread: when a commodity is the input of the other

e.g. crack spread of oil

Strip hedge – with series of forward matching the series of delivery

Stack hedge: stack and roll