Commodity Forward Arbitrage with Storage Costs and Short Sales

# Arbitrage arguments for a commodity forward on investment assets


# Arbitrage arguments for a commodity forward on investment assets












I am trying to understand the arbitrage arguments used for commodity forwards on investment assets. The theoretical price is given by $F_0 = (S_0 + U)e^{rT}$, where $U$ is the present value of all the storage costs, net of income, during the life of a forward contract. $F_0$ is the strike price, $S_0$ is the current spot price.

Consider a 1-year forward contract on an investment asset that provides no income. It costs \$2 per unit per annum to store the asset, with the equal payments being made semi-annually. Assume that the spot price is \$450 per unit and the risk-free rate is 5% per annum for all maturities. This corresponds to $r = 0.05$, $S_0 = 450$, $T = 1$, and

$$U = 1\times e^{[-0.05\times 0.5]}+1\times e^{[-0.05\times 1]} = \\\$1.9265$$

Now, the theoretical forward price $F_0$ , is given by

$$F_0 = (450 + 1.9265)\times e^{[0.05\times 1]} = \\\$475.0973$$

Now, if the actual forward price is greater than \$475.0973 (say \$480), a trader can borrow \$451.6265 at 5% rate and buy the commodity today at its spot price of \$450. Invest \$0.9753 for 6 months and \$0.9512 for 1 year to pay the \$1 storage costs at the end of six months and 1 year respectively. Enter into a forward contract to sell the commodity at \$480. Money required to repay the loan of \$451.6265 at the end of 1 year will be \$475.0973. Hence, we get a arbitrage profit at any price higher than \$475.0973 at the end of 1 year.

Next, if the actual forward price is lower than \$475.0973 (say \$474), I am facing difficulty forming the set of transactions to create the risk-less profit.

A trader can sell the commodity to get \$450 now and enter into a long futures contract to buy back the asset at \$474 in one year. Investing \$450 for 1 year will pay \$473.072 at the end of 1 year. This is lower than \$474 required to buy the asset at the end of 1 year and hence the trader to come up their own money. On selling the asset now, we don't have to pay the storage cost, but we did not have that cash to begin with. These arbitrage arguments are based on borrowing or lending money.

How to accomodate the storage cost amount in these calculations ? Please help me get this side of the arbitrage pricing argument.

## Answer by MrLCh (score 1)

https://quant.stackexchange.com/a/79999

This questions boils down to the exact terms on shorting the asset. In the real world dealing with physical assets can be quite messy (for a fun read on aluminum: Money Stuff)

Typically shorting an asset is done as follows: You borrow the asset from somebody (but this person keeps the financial exposure, as you give back the asset at a later date). You immediately sell the asset and try to buy it back later (to then give it back).

If you borrow a physical asset the lender does not need to store it anymore, but storage has become your problem. So typically the original owner of the asset pays you the saved storage costs (minus e.g. a fee for lending it out to you). But you do not need to pay storage either, as you sold the asset directly on an exchange, so this is how you end up with some extra cash. And then the replication works again.