# Pricing a physical commodity forward contract

I have just started reading Options Volatility and Pricing 2nd edition and I'm a little confused on forward contract pricing. The book states

F = C * (1+r*t) + (s*t) + (i * t)


Where C = commodity price t = time to maturity r = interest rate s = annual storage cost per unit i = annual insurance cost per commodity unit

Most of this makes sense but I don't know why the interest rate is being factored into the forward price. Is this to factor in inflation for when the contract matures or is it from the assumption money will be borrowed to pay for it. The book doesn't make this abundantly clear.

I think you are partially correct, but here's the way we look at these and we make markets in many of these products.

There's a concept called "arbitrage-free pricing". Essentially there should no way to trade both sides of this and to make risk-free money.

So let's take the front month gold contract, the June (GCM6). The first delivery date for it is June 1st. The last delivery date is near the end of the month, probably about the 28th. The price of spot gold is 1300/oz. (I'm estimating since I'm not logged into bberg now).

If I buy the contract and sell you spot gold now I will be perfectly hedged. What does it cost me to repo that gold from now until delivery? Sometimes it might work out that everyone is long gold and they are looking to lend it so I might want to borrow to the last delivery day and get paid as much as I can. In that case the contract will trade at a premium since I'm going to get paid to borrow the collateral.

Now in the case of most contracts, the seller gets to decide when to deliver. So the optimal thing to do for the seller will actually be to deliver right away, even though I, as buyer wish that he wouldn't.

The pricing difference between the spot and the contract will reflect the cost of financing to collateral for the best scenario for the contract seller. In our case, let's say the anual rate is 5%. Let's say it's 15 days until earliest delivery. So the financing cost will be 1300 * .05 *15/360=\$2.71. That's what the holder of the collateral is paying to finance it. The correct price then would be 1302.71. In the real world no one is going to put on an arbitrage trade for a few cents, so it will trade in a band. But the main take away that you should have is that the difference between spot and the future is what it costs to hold the collateral. Look at natural gas contracts. They get very steep sometimes with an implied rate over 10% sometimes because it's really hard to store natural gas when there is too much of it. (And you bleed some out in storage also). I hope that's helpful. • In real life the volume of forward contracts is not that high (except maybe in foreign exchange and gold). So why do textbooks give such importance to them? It is precisely because, as you said, it is a good illustration of arbitrage free pricing, a concept that recurs again and again in Finance and therefore deserves careful discussion early on in any course. – Alex C May 8 '16 at 2:47 • This makes sense. So For the arbitrage you would be borrowing the cost of the the gold in this case and then holding the gold until delivery. Then the reason the correct price is$1302.71 is if I bought the contract and sold the underlying the outcome would be zero at delivery - no arbitrage, and its a perfect hedge. Am I understanding this correctly now? Thank you for all your help. – user20664 May 8 '16 at 4:55
• rec, you are correct. The price has to be such that there's no free money to be made by being either short or long the contract. – JoshK May 8 '16 at 18:27
• Alex, you need to differentiate between futures and forwards in that the futures are traded (generally) on-screen and a variety of participants tradde them. Forwards are really just traded OTC and every dealer will look at them in a very similar way. You generally cannot see at all the amount of forward contracts but you can very clearly see the amount of futures contracts. And in general, the less liquid the product the more choppy the pricing and the wider the vand of fair values will become. – JoshK May 8 '16 at 18:31
• JoshK Thank you again for all your help. This makes much more sense now. This seems like a dumb question, but if you didn't borrow the money to get the gold, and instead got the gold using money you already had - wouldn't you net the interest on delivery? Obviously this means risking your own capital, but it seems like a possible way to snag a few extra bucks on a contract. – user20664 May 8 '16 at 22:40

I'm going to try to answer my own question here and hopefully someone can come by and confirm I'm right before I accept my own answer.

The key to a forward contract is there is no immediate exchange of goods or money. Just an agreement to do so at a later date.

To incentivize the seller of the contract it's important to remember if the transaction happened immediately the seller could take that money and immediately begin getting interest by investing it. By tying up the asset in a forward he foregoes that opportunity.

That's the purpose of the interest rate in the above calculation. You're adding to the price the interest of investing that money at an interest rate specified above. This interest rate could be anything and it's up to the contract parties to agree on but using either the inflation rate or the risk free rate are two options that make sense.