Spot and Forward Prices
How can we develop a relationship between the current price of an asset (spot) and its future
value? First we have to think about how these two prices are set. Clearly there is a relationship
between them, but what?
Consider if the spot price were 100 and the forward price, for delivery in a year, were 110.
Would you rather buy it now for 100 or spend a little more to lock in the price?
There are a couple things that we immediately notice we're missing. First, since we're comparing
money in two different times, we need to worry about the relative values of these dollars
the interest rate, which gives the price of next year's dollars. We also need to know something
about how/if the value of the underlying asset changes
if we're buying ripe tomatoes then
they'll go bad long before a year is up; if we're buying oil then we have to store it somewhere; if
we're buying stock shares they pay dividends.
Interest rate: assume the rate is given as "
" and that we're working in continuous time so the
present value of each dollar, paid in a year's time, is
, where T=1 so it is
In the example above, where spot is 100 and forward is 110, if the interest rate is low then we
could borrow money today to buy at spot, sell it at the forward price, make $10 per transaction
and if the $100 borrowed costs, say, $3 or $4, then that's a nice profit from the arbitrage. On the
other hand, if the interest rate were very high then the opposite transaction would be more
worthwhile. If I have $100 I could put it in the bank and get more than $110 after a year. Sell
short at the spot rate (100) and buy forward at 110 to lock in the price at which I return the
underlying asset. The difference (how much more I earn from interest over the 110 forward
price) is arbitrage profit. In either case, the arbitrage trades work to change demand and supply
to bring the prices back into line.
We might be confused because we might think that the forward price is a predictor of the price
that will be set at that future date. But it's not
the spot price is a predictor. Why? Again, we
consider what actions might be taken by a smart financial trader. Suppose that it is known that,
on Friday, the price of some asset will jump from 50 to 75. Clearly, someone who holds the asset
on Friday will get a huge return on their money. So what is likely to be the demand for that asset
on Thursday? Wednesday? Tuesday? Today? The argument gets more complicated if the asset
is difficult to store or if it changes value when held. (Below we discuss the implications of the
CAPM model to show that, under some circumstances, the futures price can be an unbiased