Spot-future parity
Encyclopedia
Spot-future parity is a parity condition that should theoretically hold, or opportunities for arbitrage
exist. Spot-future parity is an application of the law of one price
. In plain English, if I can purchase a good today for price S and conclude a contract to sell it one month from today for price F, the difference in price should be no greater than the cost of using money minus any expenses (or earnings) from holding the asset; if the difference is greater, I would have an opportunity to buy and sell the "spots" and "futures" for a risk-free profit.
The parity condition is that if an asset
can be purchased today and held until the exercise of a futures contract
, the value of the future should equal the current spot price
adjusted for the cost of money, dividend
s, "convenience yield
" and any carrying cost
s (such as storage).
The spot-future parity condition does not say that prices must be equal (once adjusted), but rather that when the condition is not met, it should be possible to sell one and purchase the other for a risk-free profit, that is, to undertake arbitrage. In highly liquid and developed markets, actual prices on the spot and futures markets may effectively fulfill the condition. When the condition is consistently not met for a given asset, the implication is that some condition of the market prevents effective arbitration; possible reasons include high transaction costs, regulations and legal restrictions, low liquidity, or poor enforceability of legal contracts.
Spot-future parity can be used for virtually any asset where a future may be purchased, but is particularly common in currency markets, commodities, stock futures markets, and bond market
s. It is also essential to price determination in swap
markets.
In the complete form:
Where:
This may be simplified depending on the nature of the asset applicable; it is often seen in the form below, which applies for an asset with no dividends, storage or convenience costs. Alternatively, r can be seen as the net total cost of carrying (that is, the sum of interest, dividends, convenience and storage). Note that the formulation assumes that transaction costs are insignificant.
Simplified form:
is the current spot price and 
is the (expected) value of the existing contract today:
which upon application of the spot-futures parity equation becomes:
Where
is the forward price today.
Arbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
exist. Spot-future parity is an application of the law of one price
Law of one price
The law of one price is an economic law stated as: "In an efficient market, all identical goods must have only one price."-Intuition:The intuition for this law is that all sellers will flock to the highest prevailing price, and all buyers to the lowest current market price. In an efficient market...
. In plain English, if I can purchase a good today for price S and conclude a contract to sell it one month from today for price F, the difference in price should be no greater than the cost of using money minus any expenses (or earnings) from holding the asset; if the difference is greater, I would have an opportunity to buy and sell the "spots" and "futures" for a risk-free profit.
The parity condition is that if an asset
Asset
In financial accounting, assets are economic resources. Anything tangible or intangible that is capable of being owned or controlled to produce value and that is held to have positive economic value is considered an asset...
can be purchased today and held until the exercise of a futures contract
Futures contract
In finance, a futures contract is a standardized contract between two parties to exchange a specified asset of standardized quantity and quality for a price agreed today with delivery occurring at a specified future date, the delivery date. The contracts are traded on a futures exchange...
, the value of the future should equal the current spot price
Spot price
The spot price or spot rate of a commodity, a security or a currency is the price that is quoted for immediate settlement . Spot settlement is normally one or two business days from trade date...
adjusted for the cost of money, dividend
Dividend
Dividends are payments made by a corporation to its shareholder members. It is the portion of corporate profits paid out to stockholders. When a corporation earns a profit or surplus, that money can be put to two uses: it can either be re-invested in the business , or it can be distributed to...
s, "convenience yield
Convenience yield
A convenience yield is an adjustment to the cost of carry in the non-arbitrage pricing formula for forward prices in markets with trading constraints....
" and any carrying cost
Cost of carry
The cost of carry is the cost of "carrying" or holding a position. If long, the cost of carry is the cost of interest paid on a margin account. Conversely, if short, the cost of carry is the cost of paying dividends, or rather the opportunity cost; the cost of purchasing a particular security...
s (such as storage).
The spot-future parity condition does not say that prices must be equal (once adjusted), but rather that when the condition is not met, it should be possible to sell one and purchase the other for a risk-free profit, that is, to undertake arbitrage. In highly liquid and developed markets, actual prices on the spot and futures markets may effectively fulfill the condition. When the condition is consistently not met for a given asset, the implication is that some condition of the market prevents effective arbitration; possible reasons include high transaction costs, regulations and legal restrictions, low liquidity, or poor enforceability of legal contracts.
Spot-future parity can be used for virtually any asset where a future may be purchased, but is particularly common in currency markets, commodities, stock futures markets, and bond market
Bond market
The bond market is a financial market where participants can issue new debt, known as the primary market, or buy and sell debt securities, known as the Secondary market, usually in the form of bonds. The primary goal of the bond market is to provide a mechanism for long term funding of public and...
s. It is also essential to price determination in swap
Swap (finance)
In finance, a swap is a derivative in which counterparties exchange certain benefits of one party's financial instrument for those of the other party's financial instrument. The benefits in question depend on the type of financial instruments involved...
markets.
In the complete form:
Where:
- F, S represent the cost of the good on the futures market and the spot market, respectively.
- e is the mathematical constant for the base of the natural logarithmE (mathematical constant)The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
. - r is the applicable interest rate (for arbitrage, the cost of borrowing), stated at the continuous compounding rate.
- y is the storage cost over the life of the contract.
- q are any dividends accruing to the asset over the period between the spot contract (i.e. today) and the delivery date for the futures contract.
- u is the convenience yieldConvenience yieldA convenience yield is an adjustment to the cost of carry in the non-arbitrage pricing formula for forward prices in markets with trading constraints....
, which includes any costs incurred (or lost benefits) due to not having physical possession of the asset during the contract period. - T is the time period applicable (fraction of a year) to delivery of the forward contractForward contractIn finance, a forward contract or simply a forward is a non-standardized contract between two parties to buy or sell an asset at a specified future time at a price agreed today. This is in contrast to a spot contract, which is an agreement to buy or sell an asset today. It costs nothing to enter a...
.
This may be simplified depending on the nature of the asset applicable; it is often seen in the form below, which applies for an asset with no dividends, storage or convenience costs. Alternatively, r can be seen as the net total cost of carrying (that is, the sum of interest, dividends, convenience and storage). Note that the formulation assumes that transaction costs are insignificant.
Simplified form:
Pricing of existing futures contracts
Existing futures contracts can be priced using elements of the spot-futures parity equation, where K is the settlement price of the existing contract, is the current spot price and is the (expected) value of the existing contract today:
which upon application of the spot-futures parity equation becomes:
Where
is the forward price today.