In a previous article, we reviewed the idea of
, the relationship that governs the price of futures contracts. The first principle to keep in mind is that, with interest rates held constant, there is a linear relationship between the value of a futures or forward contract and the spot value of the underlying asset. If the price of a futures contract deviates from this relationship, then we tend to say that arbitrage conditions exist, such that traders can make a risk-free profit by selling the expensive asset and buying the cheap one.
However, there are some additional factors specific to different markets that affect the prices of futures and forwards. Correctly pricing and trading those markets requires that we understand these peculiar factors. To see how these factors affect futures prices, let's begin with a base formula for a futures contract price, again using the relationship to the spot price and the risk-free interest rate:
(1) F = S(t) * exp(r * (T-t))
That formula in English says that the futures price is equal to the spot price (S) right now (at time t) times the constant e to the power of the product of the interest rate (r) and the difference between the delivery time (T) and today.* More simply, it says that the futures should be priced at the spot price plus compensation for the time value of money. One conclusion that follows from this relationship so far is that futures prices are not predictions about the future - they're not expressions of market expectations beyond the information already contained in the spot price.
When the asset underlying a futures contract is a foreign currency, the price of the contract will reflect any difference between the interest rate for the domestic currency and that of the foreign currency (f). The change required to the formula is small - we just subtract the rate of interest f from the domestic rate:**
(2) F = S(t) * exp((r-f) * (T-t))
Again, if a trader priced an FX contract without making this adjustment, he would mistakenly perceive the market price an arbitrage opportunity.
As we've noted before, storage costs for most commodities are not trivial. Many agricultural products have a low price per weight, many energy products are difficult or dangerous to store, etc. Sellers demand to be compensated for the cost of storing assets until delivery, and that compensation is reflected in the futures price. At the same time, sellers sometimes derive an immediate benefit from holding the asset, the "convenience yield," and this benefit is also priced. Note that this benefit reduces the impact of storage costs.
(3) F = S(t) * exp((r+s-c) * (T-t))
The formula above includes storage costs (s) and any convenience yield (c).
Equity index futures
Here are two important differences between equity futures and the asset classes above. First, equity futures are cash-settled, which means that storage and convenience costs are not involved. The second difference is that most of the components of an equity index will pay some sort of dividend, and those dividends (q) are reflected in index futures prices:
(4) F = S(t) * exp((r-q) * (T-t))
Alternatively, if you calculate the value of dividends to expiration in terms of spot index points, you can just subtract those dividends from the base formula (1) we used above. Here is a helpful example from CME Group.
* Since interest rates are quoted in annual terms, it's important to input times annualized as well, e.g. 90/360 for a 90-day contract.
** Some texts give the formula as F = S(t) * r / f. The difference is that the formula we give above assumes continuous compounding.
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