The cross hedge

Sometimes you feel you have to reduce price and market risk with a product and you find there is no future market available. In this case you might consider the cross hedge, but how is this actually done?

Well there are actually two steps necessary:

1.  Choosing a proxy commodity

2.  Adapting the product to the proxy commodity

Let’s say we want to hedge sweet corn, no future market is available but there is one for feed corn. The feed corn is then your proxy commodity, e.g. to protect you against a price fall.

This hedge requires to sell N futures with the size of Q MT to mature at time 1 and the futures price multiplies at time 0 with Q (the size of the contract)

The issue is then mainly to determine how many futures you require of feed corn. Well let’s use a little math here.

The math

Say that S₁(x) is the cash spot price of feed corn at time  1 en F_1,1 is the futures price of feed corn at time 1 for delivery at time 1. Selling the required quantity C in MT on the cash spot price gives S₁ x C dollars. The difference between the futures price gives:

-N[F_1,1 – F_0,1] x Q dollars

At maturity the futures price converges to the  spot price which makes:

-N[S₁(feed corn) – – F_0,1] x Q dollars

So that the value of the portfolio becomes:

S₁(organic sweet corn) – N[S₁(feed corn) – F_0,1 ] Q

This you divide by C and defines the hedge ratio β = NQ/C.

S₁(organic sweet corn) – β[S₁(feed corn) – F_0,1 ]

Now the idea is as with a regular delta hedge to choose β in this way that the variance of the equation will be minimized, meaning:

Var(S₁(organic sweet corn) – 2βCov[S₁(organic sweet corn), S₁(feed corn)) + β² Var(S₁(feed corn))

and when we isolate β

Β = Cov[S₁(organic sweet corn), S₁(feed corn)]  / Var(S₁(feed corn))

and then it becomes easy

N (optimum contracts) = β (C/Q)

This cross hedge can be very useful if you want to manage your risk and you are dealing with products less common in futures markets.