Frequently asked conceptual queries...

What is lease rate in commodity markets?

That is the rate charged by the lender of commodity, for lending the commodity. In the above case, (discussed in my previous blog) the lease rate is 6%.

What is E₀(S_T)?

We assume that a commodity will reach certain price in future. The assumption or expectation is made today. That is what E₀ refers.  That is the price of the commodity at time T (in future)

What is F_(0,T)?

That is theoretical forward price of the commodity.

Can there be a difference between F_(0,T) and E₀(S_T)?

Yes, there can be.  E₀(S_T) is based on expectations and F_(0,T) is based on mathematical calculation formula of : S₀ x e^rT

What will happen when there is a difference between F_(0,T) and E₀(S_T)?

There might be an arbitrage opportunity, when the theoretical prices deviate from the expected prices. Traders generally generate arbitrage opportunities due to this difference.

Can we discuss two examples of simple arbitrages, due to the difference between F_(0,T) and E₀(S_T)?

Yes, let us discuss these two examples to understand two arbitrages due to the difference between F_(0,T) and E₀(S_T).

S0 (Spot price)	$5.00
Price quotation	Cents per bushel
Contractual commodity	Corn
Time to maturity of the contract	1 year
Risk-free rate	6%
Theoretical forward price formula	S0 x erT
Theoretical forward price	$5.00 x e0.06 x 1 year = $5.31
What the trader can expect?	The trader expects that the market prices will deviate from theoretical prices (due to many reasons) and the trader can expect the prices after one year be either greater than $5.31 or lower than $5.31

Arbitrage example one: When the trader expects the realistic prices do not match with the theoretical prices and realistic prices will be higher than $5.31; Let us say the trader identified another trader who is willing to buy at $5.35.

Some steps are initiated at Time T0	Some steps are initiated at Time TT (on maturity of contract)
Borrow $5.00 @ risk-free rate	Sell  the commodity at $5.35
Buy the commodity in the spot market for $5.00	Repay the loan with interest: -$5.31
Enter into a forward contract to sell at $5.35 (Short forward contract) after one year.	Risk-free profit  due to arbitrage is: $5.35 - $5.31 = $0.04 per bushel
The arbitrage ceases to exist, when either the theoretical prices increase or when the market prices come down. When all the traders start doing the same activity, the price would come down to $5.31 Or, when the risk-free interest increases, the forward price may go up and arbitrage ceases to exist. This is a cash and carry arbitrage (borrow money, buy the asset and sell forward)

Arbitrage example two: When the trader expects the realistic prices do not match with the theoretical prices and realistic prices will be less than $5.31; Let us say the trader identified another trader who is willing to sell at $5.25 after one year.

Some steps are initiated at Time T0	Some steps are initiated at Time TT (on maturity of contract)
Sell the asset today in the cash market at the price of S0	At time T0: We get $5.00 by selling the commodity in cash market.
Invest the proceeds at the risk-free rate of 6%	Invest at 6%; After one year, we will get $5.31 from the investment.
Buy the asset in future through long forward contract at a price of $5.25	Buy the asset $5.25 in the forward market. Create an arbitrage opportunity of $0.06 per bushel.
This is an inverse cash and carry arbitrage: Sell today, invest the proceeds; buy in future or long forward

  

How the arbitrage opportunity arrives at equilibrium (ceases to exist)?

Let us recollect, what we have discussed in the previous article:

a.       We assumed a risk-free rate of 6%.

b.      We assumed the price of 10 pencils at $5.00 (USD).

c.       We assumed that the forward prices will grow at risk-free rate.

d.      We got the forward price after one year to be $5.31, using the formula of S₀ x e^rT

e.      We assumed that some commodities are not worth storing.

f.        We assumed that prices of such commodities remain constant.

g.       In such scenario, there exists an arbitrage.

The arbitrage opportunity in such scenarios exists in the following form:

a.       Sell the commodity today, at time (T₀)

b.      At time (T₀), Get those proceeds and invest them for a period of “T”, at risk-free rate

c.       At time (T₀), enter into a long forward contract

d.     At time (T_T), use those proceeds to buy the commodity as per the long forward agreement

(Please go to the previous article to understand how arbitrage opportunity existed in such scenarios)

What we concluded?

a.       We concluded that arbitrage opportunity exists, only through theoretically, but not practically.

b.      We understood that the lender of pencils will not lend those pencils without any return or benefit.

Now, how much does he expect, and is there any arbitrage in realistic scenario?

a.      No, there cannot be arbitrage. Because, the prices of pencils is assumed to be constant for the next one year.

b.      The trader borrowed 10 pencils (worth $5.00) and sold them in the cash market and invested for 6% (risk-free rate) for one year.

c.    After one year, he will get $5.31 dollars on investment and buy those pencils again at $5.00 either in cash market (prices remain constant) or through long forward agreement.

d.      He creates an arbitrage profit of $0.31 on those 10 pencils contract.

The lender of the pencils expects some return while lending those pencils...How much he will expect?

a.       Can he expect 4%, when the risk-free rate in the market is 6%?

b.      Can he expect 5%, when the risk-free rate in the market is 6%?

c.       Can he expect 6%, when the risk-free rate in the market is 6%?

d.      Can he expect 7%, when the risk-free rate in the market is 6%?

Let us discuss all those cases, one by one...

	Risk-free rate 6%	S0 = $5.00 (for 10 pencils)	Value of forward agreement, using the formula of: S0 x erT
When Lender expects 4%	The lender expects 4%, the cost of borrowing those pencils will be $5.20; The trader with short selling today will get after one year $5.31, still there is an arbitrage profit of $0.11 dollars on this whole trade.
When Lender expects 5%	The lender expects 5%, the cost of borrowing those pencils will be $5.26; The trader with short selling today will get after one year $5.31, still there is an arbitrage profit of $0.05 dollars on this whole trade.
When Lender expects 6%	The lender expects 6%, the cost of borrowing those pencils will be $5.31; The trader with short selling today will get after one year $5.31; The cost of borrowing the pencils is equal to the value of investment proceeds. THE ARBITRAGE OPPORTUNITY CEASES TO EXIST
When Lender expects 7%	The borrower will not go for such a trade where the cost of borrowing is higher than the benefit of trade. For the lender, there is no problem. He will be ready to lend @7%, when the risk-free rate is 6% in the market.

In the market, the lender of the pencils will expect at least a benefit of minimum 6%, that is equal to the risk-free rate and with this the arbitrage opportunity ceases to exist. The remaining things,  I will discuss gradually in next articles.

Thanks a lot for your patient watch...

Surya