Cross Hedging
The Basis and Cross Hedging in Energy Futures Markets
Futures and Forwards
Futures and forwards both allow for purchase or sale of a commodity at a set price at a set time.
Forwards:

May be customized in terms of size, delivery date, and delivery location

Counterparty risk, no margin required
Futures:

Standardized forwards

Cleared on exchange, margin required
Standardization leads to greater volume of trading
Hedging with Futures
Say we will produce a barrel of oil in 1 month. The price of oil today is $70/barrel at our delivery location.

Our risk is that the price of oil will fall over the month (we are naturally long crude oil).

So we sell our barrel of oil one month forward (take a short position in a futures contract).

This guarantees today the price we will receive for our oil in one month.
The Basis
The hedge is never perfect, however. Consider that in our crude oil example we may not be able to enter into a contract for the same:

Delivery point

Delivery date
This creates a difference between the price of oil in the forward contract we are using to hedge and the spot price of the barrel of oil we are hedging.

This difference is called the basis.

The risk that the prices diverge is called basis risk.
Basis Risk: Prices by Futures Maturity
Closing Out the Hedge
If, due to differing delivery points and dates, we won’t deliver the crude to fulfill the contract, then we must:

Buy the forward contract back prior to expiration

Deliver on the spot market
The Basis: Example 1
So the spot price today (`S_0`) is $68. Say the price today of a barrel of oil one month forward (`F_{0,1}`) is $70. Then we define the basis as:
`basis\ =\ spot\ price\ of\ asset\ \ futures\ price\ of\ asset = \$2`
Now, imagine that when we close out the contract in one month, by buying the contract back, the forward price (`F_1`) is $65 and the spot price (`S_1`) is $50 (a basis of $15).
 Our effective selling price is: `S_1  \Delta F = \$50 + \$5 = \$55`
The Basis: More Examples
Say that `S_0 = \$70`, `F_{0,1} = \$68`, `S_1 = \$65`, and `F_1 = \$63`.

Our effective selling price is: `S_1  \Delta F = \$65 + \$3 = \$68`

Our basis went from $2 to $2.
Say that `S_0 = \$70`, `F_{0,1} = \$68`, `S_1 = \$68`, and `F_1 = \$60`.

Our effective selling price is: `S_1  \Delta F = \$68 + \$8 = \$76`

Our basis went from $2 to $8.
Basis Risk
From the above, we see our basis risk is caused by the spot price decreasing by more than the forward price.
 If we are purchasing futures at time 0 as a hedge, then our risk is the inverse.
Cross Hedging
Now, imagine that we will deliver 1,000 gallons of kerosene to a delivery point in Oregon.

Given a lack of liquidity in kerosene futures and forward contracts, you may be forced to hedge kerosene with another more liquid futures contract.

Likely candidates with heavily traded futures are: RBOB Gasoline (ticker: RB), Heating Oil (HO), Crude Oil (CL).
Hedging an asset with a similar (through not the same) asset is referred to as cross hedging.
Basis Risk in Cross Hedging
Now our basis risk is driven by different factors:
 Commodity types (hedging kerosene with RBOB)
 Delivery points (Gulf coast vs Oregon)
 Delivery dates
Choosing Amounts to Hedge
If we are to hedge 1,000 gallons of kerosene with, say, gasoline, then should we sell 1,000 gallons of gasoline?

What if gasoline price changes are twice as volatile as kerosene price changes? Then with a correlation of 1, every time kerosene increases by $1 then gasoline will increase by $2.

In this case it would make sense to sell 500 gallons of gasoline to hedge 1,000 gallons of kerosene.

What if the correlation is only 0.5? Then we should hedge with even less gasoline because it is not as good of a hedge.
So we need a way to find the hedge ratio (`h`) which is the ratio of the futures position size to the size of the commodity we want to hedge. In our case, this will be the size of the gasoline futures position divided by the size of our kerosene exposure.
Optimal Hedge Ratio
A common way to choose the optimal hedge ratio (`h`) is to find the `h` which minimizes the risk in the hedged position. Risk is measured as the variance.

In our case, where we are naturally long the commodity and are short futures, the change in the value of out hedged position is `\Delta S  h \Delta F` where `h` is the hedge ratio.

To minimize this, we can use unconstrained optimization from basic calculus.
Derivation
`Var(\Delta S  h \Delta F) = \sigma_S^2  2h\rho \sigma_S \sigma_F + h^2\sigma_F^2`
`\frac{dVar(\Delta S  h \Delta F)}{dh} = 2h\sigma_F^2  2\rho \sigma_S \sigma_F`
Setting this equal to zero and solving for `h` we have:
`h = \rho \frac{\sigma_S}{\sigma_F}`
Interpretation
An optimal hedge ratio of 0.8 means we sell 8 gallons of gasoline futures for every 10 gallons of kerosene we wish to hedge.
In the next slide, try to interpret the optimal hedge ratios. Be careful of the units of each contract (for example, CL is in $/barrel).
Optimal Hedge Ratios
Intuition
The next slide will calculate the optimal hedge ratio given the standard deviation of spot and futures price changes, as well as the correlation coefficient. Try to predict what effect a given change in the input will have on the optimal hedge ratio, and then make the change. For example:
 If you increase the standard deviation of the futures price changes, what will happen to the optimal hedge ratio?
 If you decrease the correlation coefficient, what will happen to the optimal hedge ratio?
Optimal Hedge Ratio Calculator
Credits and Collaboration
Click the following links to see the code, linebyline contributions to this presentation, and all the collaborators who have contributed to 5Minute Finance via GitHub.
Learn more about how to contribute here.