Delta of a Futures Contract

Futures and spot prices move in lockstep, but the moves are not identical.  This is because the delta of a futures contract is not equal to 1.  If it were, the futures contract would be an exact replacement for the spot security, but it is not so.

To understand this, consider the following scenario for a hypothetical security that is currently selling in the spot market at $100. Interest for 3 months is 3%, and expected dividend is $1, so futures sellers are offering a futures contract with a notional of $100 for $102 today.  Just to keep the math simple, assume that at the end of the three months, the spot price stays at $100, ie the security earned no returns other than its dividend.

Now if I desire exposure to this security, I have two alternatives.  I could buy the spot security and keep it, or alternatively I could buy a futures contract.  Let us look at what happens to my cash flows under each of these scenarios:

  • Scenario 1: I buy the spot security: If I buy the spot security today for $100, at the end of 3 months I end up with $100 in the spot security plus $1 in cash from dividend. (Assume the security’s price return during this period is zero, and cash from dividends returns $1).
  • Scenario 2: I buy the futures contract: If I buy the futures contract for $102, at the end of 3 months I end up with $100 in the spot security, have paid $102 to the seller of the futures contract, and earned $3 in interest. In other words, I am in the same position as having brought the spot security.

So far so good.  What this really means is that to get $100 worth of spot security exposure, I need to buy a futures contract today with a notional of $102. The delta ratio therefore is just 100/102, or the spot price divided by the futures price.

Since the futures price = spot*(l+r)-dividend, the ‘delta ratio’ is:


Or 1/[(l+r)-(dividend yield)];

Or simply 1/(1 + r – d)

That’s about it.  It is important to know intuitively why the delta of a futures position is not the same as the delta of the cash instrument.  It could be more, or less, depending upon the carrying costs.  Another implication of the above is that as we get closer to the expiry of the futures contract, delta gets closer and closer to 1.