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