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(ST)?
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(ST)?
Yes,
there can be. E0(ST)
is based on expectations and F(0,T) is based on mathematical
calculation formula of : S0 x erT
What will happen when there is a
difference between F(0,T) and E0(ST)?
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(ST)?
Yes,
let us discuss these two examples to understand two arbitrages due to the
difference between F(0,T) and E0(ST).
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
|
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