By ADMIN
Spot-Futures Parity
We can be a little bit more precise concerning the relationship between spot and futures prices for financial futures. To illustrate, suppose we had a futures contract on shares of common stock in a single company (actually there are no such contracts in the U.S.). This particular stock does not pay dividends.
For concreteness, suppose the contract calls for delivery of 1,000 shares of stock in one year. The current (i.e., cash or spot) price is $50 per share. Also, 12-month T-bills are yielding 6 percent. What should the futures price be? To answer, notice that you can buy 1,000 shares of stock for $50 per share, or $50,000 total. You can eliminate all of the risk associated with this purchase by selling one futures contract. The net effect of this transaction is that you have created a risk-free asset. Since the risk-free rate is 6 percent, your investment must have a future value of $50,000 × 1.06 = $53,000. In other words, the futures price should be $53 per share.
Suppose the futures price is, in fact, $52 per share. What would you do? To make a great deal of money, you would short 1,000 shares of stock at $50 per share and invest the $50,000 proceeds at 6 percent. Simultaneously, you would buy one futures contract.
At the end of the year, you would have $53,000. You would use $52,000 to buy the stock to fulfill your obligation on the futures contract and then return the stock to close out the short position. You pocket $1,000. This is just another example of cash-futures arbitrage.
More generally, if we let F be the futures price, S be the spot price, and r be the risk-free rate, then our example illustrates that
F = S(1 + r)
In other words, the futures price is simply the future value of the spot price, calculated at the risk-free rate. This is the famous spot-futures parity condition. This condition must hold in the absence of cash-futures arbitrage opportunities.
