Chapter 2: Forward and Futures Prices. Copyright
c
b y Peter Ritchken 1999 18
on this index. The multiplier of the contract is 500. Therefore, the contract controls
400 × 500 = $200, 000. Of this amount (4/16)200, 000 = $50, 000 is in A, (7/16)200, 000 =
$87, 500 is in B, and the remaining (5/16)200, 000 = $62, 500 in C. The initial replicating
portfolio consists 50, 000/40 = 1, 250 shares of A, 87, 500/35 = 2, 500 shares of B,and
62, 500/25 = 2500 shares of C.
Assume the futures contract settles in 30 days, and interest rates are 10%. The calculations
for the theoretical futures price are shown below.
Stock A B C
Borrow Funds 50,000 87,500 62,500
Number of Shares Bought 1,250 2,500 2,500
Dividend Income Received and reinvested
at 10% until settlement date 628.43 1256.18 -
The amount owed after 30 days is 200, 000e
−0.10(30/365)
= 201, 650.61. The net amount
owed, after adjusting for dividends, is therefore 201, 650.61 − 1884.61 = 199, 766. The
theoretical futures price is therefore 199, 766/500 = $399.53.
The stock index arbitrage strategy just described ensures that futures prices do not
deviate to far from the implied net cost of carry. However, in the analysis we made several
simplifying assumptions. First, we assumed that transaction costs could be ignored. While
computer entry systems, such as DOT, has made the trading of portfolios more efficient,
stock index arbitrageurs estimate the total round trip costs to be at least 0.5% of the
portfolio value. Second, the assumption that index prices and replicating portfolio prices
are exactly equal is not precise. Third, the size and timing of all dividends on all stocks
in the index was assumed to be known with certainty. While this assumption may be
valid for some short term contracts, any uncertainty in dividends will impact the arbitrage
scheme. In addition, the reverse cash-and-carry strategy requires selling short stocks in the
index. Such a strategy implicitly assumed that the proceeds of the sales from selling stock
short would be immediately available for investment at the riskless rate. In fact, margin is
required for short sales, and these funds may not be available to the investor. Moreover,
short selling all stocks in the index may be difficult or may take some time. Specifically,
stocks can only be sold short after an uptick in the stock price. Finally, certain rules, such
as circuit breakers, can impede arbitrage.
As a result of the above real world considerations, the futures price on an index may
not be at full net carry. Indeed, there exist price bands such that no riskless arbitrage
strategies exist as long as the futures price trades within the band. Only when the futures
price moves outside this range will it become possible for arbitrageurs to initiate cash-and-
carry or reverse cash-and-carry arbitrage. Implementing systems that actually monitor the
stock index futures-stock portfolio relationship requires that these features be considered.