## 24 Haziran 2011

### Futures pricing

THE FORMATION OF THE FUTURES PRICES

If we suppose that there are not any restrictions about the short sales and any trading commissions , there is a cost of carry relationship between the futures markets and the spot market. This relationship can be showed by this equation:

Fo,t = So(1+c)
Fo,t= the futures price at t= 0 for the delivery time t
So= the spot price of the commodity at t= 0
c= cost of the carry of the commodity until the expiration date(% of the spot price)

If this equilibrium is unbalanced the investors have the arbitrage possibility.

If So < Fo,t the investor buy the commodity from the spot the by borrowing and sell the futures contracts. At the expiration date they delivery the commodity in return of the futures contracts and they have the gain.This situation is showed in table 1:

Table 1

The arbitrage gain when the spot price is less than the futures price when the conditions are ideal

 cash flow t=0 the investor borrows in return for an interest c +So He buys the commodity from the spot market and he stores. -So He sells the futures contracts underlying the commodity + Fo,t He deliveries the commodity at time t ,he closes out his futures position. He repays his debt and its interest at t=0 - So(1+c)

Result: Fo,t - So(1+c)
as So < Fo,t ,
Fo,t- So(1+c) > 0 ; and the investor has the arbitrage gain.

If So >Fo,t , the investors buy the futures contract of the commodity whose they pay the price at date t , lend the funds obtained by the short sale of the commodity to get interest gain up to the expiration date of the futures contracts and close out their futures position by paying the value of the futures contracts.

This situation is showed in table 2

Table 2
The arbitrage gain when the spot price is more than the futures price when the conditions are ideal

 cash flow t=0 He buys the futures contract whose the price will be paid at the expiration date t He sells short the underlying commodity. + So He lends this income until the expiration date of the contracts in return for the interest rate c. - So time t : The investor receives the commodity by paying the value of thefutures contracts which are bought - Fo,t So he acquires the commodity which is sold short He takes back the loan that he gave and its interest. + So(1+c)

Result-Fo,t + So(1+c)
as So >Fo,t ,
-Fo,t+ So(1+c) > 0 ; and the investor has the arbitrage gain.

Due to the demand variation occurred by the arbitrage operations seen in the Table 1 and 2 the futures price become equal to the equation Fo,t = So(1+c).

As we see in Table 1 and 2 the arbitrage doesn’t exist if Fo,t = So(1+c) or So(1+c) < F < So(1+c). But there are some restrictions in the real markets such as the trading commissions, the restrictions for the short sales, the cash flow from the financial assets owned (stock dividends , interest gains) and the difficulties in the storing.

The trading cost are formed by the brokerage commissions and the operating cost .

Even if the short selling is possible, some portion of the funds obtained is used to pay as a guarantee so that all funds can not be used.

Furthermore the cash flow from the financial assets owned (stock dividends , interest gains) influence the cost of carry.

k: the usable part of the fund obtained from the short sale.