Page 58 - DMGT513_DERIVATIVES_AND_RISK_MANAGEMENT

Unit 5: Pricing of Future Contracts

Under such circumstances, we may treat the dividend to be a rate (d) rather than a discrete Notes
amount (D). In other words, dividend is paid everyday from index and the rate at which dividend
is paid is the average of individual dividend amounts. Just like the stock price falls on ex-
dividend date by (approximately) dividend amount, we may consider that index will fall every
day at (approximately) the dividend yield. We can then simplify the pricing equation as follows.
F = Se (r–d)T ... (5.8)

Example: Suppose, the current price of BSE-SENSEX is 12400. The dividend yield (d) on
index is 1% pa and the yield on risk-free assets (r) is 10% pa, both on simple interest rate basis.
Their continuously compounded equivalents are, respectively, 0.995% pa and 9.53% pa. The
price of 90-day (0.246575 year) index futures will be
12400 e (0.0953-0.00995)0.246575 = 12,664.

Academic literature uses equation (5.8) to price index futures. However, if the index is not
broad-based or ex-dividend dates are bunched together, we should equation (5.6) in conjunction
with equation (5.5).
A stock index traces the changes in the value of a hypothetical portfolio of stocks. The value of
a futures contract on a stock index may be obtained by using the cost of carry model. For such
contracts, the spot price is the "spot index value", the carry cost represents the interest on the
value of the dividends receivables between the day of valuation and the delivery date.
Accordingly, indices are thought of as securities that pay dividends, and the futures contracts
valued accordingly.

Case 1: When the securities included in the index are not expected to pay any dividends during
the life of the contract: Here we have,
F = S e rt
0
where F is the value of futures contract, S is the spot value of index, r is the continuously
0
compounded risk-free rate of return, and t is the time to maturity (in years).
Case 2: When dividend is expected to be paid by one or more of the securities included in the
index during the life of the contract: In the event of dividends expected to be paid on some
securities, the dividend amount is discounted to present value terms and then the rule of pricing
securities with known income is applied. Thus,
F=(S )e rt
–I
O
where I is the discounted value of the dividend and other symbols are same as in case 1.
Case 3: When dividend on the securities included in the index is assumed to be paid continuously
during the life of the contract: If the dividends may be assumed to be paid continuously, with the
dividend yield rate being Y, then the futures price, F, would be given by
F=S e (r-y)t
O
5.2.2 Pricing Model for Commodity Futures

Stocks and stock indices are not the only assets. They are not even the only financial assets on
which futures contracts exist. Let us examine the pricing of commodity futures, which were
matter-of-fact the first futures contracts. When we buy the commodity in spot and hold it until
the maturity of futures contract, we need to store it in a secure warehouse and buy insurance
against contingent dangers like theft, fire, floods, etc. Accordingly, the storage and insurance
cost will be additional component in carry cost. There is another component, which is peculiar

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