Page 57 - DMGT513_DERIVATIVES_AND_RISK_MANAGEMENT

Derivatives & Risk Management

Notes Carry costs and benefits are modelled either as continuous rates or as discrete flows. Some
costs/benefits such as the cost of funds (i.e., the risk-free interest rate) are best modelled
continuously. The dividend yield on a broad-based stock portfolio and the interest income on a
foreign currency deposit also fall into this category. Other costs/benefits like quarterly cash
dividends on individual common stocks, semi-annual coupons on bonds, and warehouse rent
payments for holding an inventory of grain are best modelled as discrete cash flows. In the
interest of brevity, only continuous costs are considered here.
Thus, the futures price (F) should be equal to spot price (S) plus carry cost minus carry return. If
it is otherwise, there will be arbitrage opportunities as follows.

1. When F > (S + CC – CR): Sell the (overpriced) futures contract, buy the underlying asset in
spot market and carry it until the maturity of futures contract. This is called "cash-and-
carry" arbitrage.
2. When F < (S + CC – CR): Buy the (under priced) futures contract, short-sell the underlying
asset in spot market and invest the proceeds of short-sale until the maturity of futures
contract. This is called "reverse cash-and-carry" arbitrage. The "reverse cash-and-carry"
arbitrage assumes that the short-sellers receives the full proceeds of short-sale. In practice,
this may be only partially true or even impossible.
Thus, it makes no difference whether we buy or sell the underlying asset in spot or futures
market. If we buy it in spot market, we require cash but also receive cash distributions (e.g.,
dividend) from the asset. If we buy it in futures market, the delivery is postponed to a later day
and we can deposit the cash in an interest-bearing account but will also forego the cash
distributions (like dividend) from the underlying asset. However, the difference in spot and
futures price is just equal to the interest cost and the cash distributions.
5.2.1 Pricing Model for Index Futures

We have so far considered the futures contract on a stock. Futures on stock index is somewhat
different. Index futures pose additional problems to arbitrage play. The problems stem from the
fact that the index futures does not exist physically (which, of course, is the reason that the index
futures is compulsorily cash-settled). Nevertheless, we can replicate the index by buying or
selling all the component stocks in proportion to their value in index. This may be further
simplified by buying or selling just one stock as proxy for the index after considering its beta.
This is a technical issue and is beyond the scope of the present unit. Let us confine our discussion
now to the pricing of index futures.
Carry cost for index futures is similar – it is simply the interest cost of holding the index at
current market price. The first term in the equation will, therefore, continue to be SerT. The
second term, carry return, needs some adjustment. Because there are many component stocks
and all of them may have cash dividend during the life of futures contract, we will have to sum
the present-value of all such cash distributions. Thus,

Carry Return = D e  rt 1  D e  rt 2  ... D e  rt n ... (5.7)

1
n
2
where D , D ... are different cash dividends from component stocks to be paid on t , t ...t dates,
1 2 1 2 n
respectively. The value of e is 2.7183. We can simplify the pricing equation for index futures if
the following three conditions are met.
1. Index is broad-based
2. All component stocks pay dividend

3. Dates on which dividend is paid are uniformly distributed without being bunched

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