Pricing Forwards and Futures in the Binomial Model

In this lesson, we looked at pricing forwards and futures in the multi-period binomial model.

When pricing forwards, we want to set a forward price at time t=0, G(0), such that the contract is initially worthless.  We therefore use the following two formulae:

to find the price for a forward contract.  This holds even when the underlying security pays dividends, since they will enter into the equation through changing the risk-neutral probabilities, q and (1-q).



When pricing futures, we know that the futures price must be the same as the spot price at the time when the futures contract matures, so S(n) = F(n) (where F(t) is the "price" of the futures contract at time n).

A common misconception is that F(t) is how much an individual pays/receives to buy/sell a futures contract.  A futures contract actually costs nothing.  The price F(t) is only used to determine the cash flow associated with holding the contract.

Example: (F(t) - F(t-1)) is the payoff received at time t from holding one contract from time t-1 to t.

A futures contract can be thought of as a security that:

1. is always worth zero.
2. pays a "dividend" of (F(t) - F(t-1)) at every time t (can be positive, zero, or negative)

We can use the following equation to help us price a futures contract:

where we solve for:

We know that the price of the futures contract at time n-1 expiring in period n is equation to zero, which is why we set the equation equal to zero.  However in the case of a multi-period binomial model, we can combine all the periods into one using the following equation:

The law of iterated expectation then implies that F(t) = E [F(n)].  The futures price process is known as a Q-martingale process.

Inputting that t=0 and that F(n) = S(n), we have:

We can see that F(0) = G(0), which means that the forwards price is equal to the futures price, even though one pays a daily dividend and the other does not.