Futures as an unbiased predictor

ig0r · Post by **ig0r** » Sun Jan 20, 2008 4:12 am

In hull 6e page 121 (chapter 5) of options futures and other derivatives, he argues that futures are not an unbiased predictor of expected spot at maturity. He argues that one can determine whether the future should be less than or greater than expected spot depending on its correlation to equities. If one knows whether futures are underpriced or overpriced relative to expected spot, can't one simply trade the future against the spot (which will on average realize the expected spot) and arbitrage this out? Also, how does this reconcile with the exchange rate forward parity which states that forwards/futures are an unbiased predictor?

sammyzee · Post by **sammyzee** » Sun Jan 20, 2008 5:39 pm

The fair price of a future exists because at prices above and below that pure arbitrage opportunities exist. What the fair price is, and what the bid/offer prices are is a different. Massive use of a future for speculation will probably cause either the bid or offer to move depending on which way everyones going, but they will both still be either side of the fair price. If you have the bid/offer on the same side of the fair price, you're a) giving your counterparty an aribitrage opportunity, and b) losing the ability to fully hedge your position.

functor · Post by **functor** » Sun Jan 20, 2008 6:50 pm

I'm not sure, but here's a possibility. I'll look at a forward contract on a stock (if interest rates are deterministic forward=future). Then F_0 = S_0 * e^{rT}. But if you take S_0 to follow a lognormal process with drift mu and diffusion term sigma, then under the real world measure, E(S(T)] = S_0 * e^{mu * T}.

Thus if r > mu the forward price is more than the expected future value of the stock and vice-versa if r < mu. Of course this difference goes away if you use the risk-neutral measure when talking about the expected value.

ig0r · Post by **ig0r** » Sun Jan 20, 2008 9:19 pm

functor:

right, thats basically what he says and gives bounds depending on the beta of spot. What if r>mu and F_0 > E[S(T)]? Cant we short forward and buy spot; or are you saying that spot will not realize E[S(T)] unless we assume risk-neutrality?

functor · Post by **functor** » Mon Jan 21, 2008 12:39 am

I don't have the book -- and don't know much about beta and CAPM. But what I was thinking was along the follows:

Say an asset is currently trading at 100 dollars, and the risk-free interest rate is zero. We want to price a forward contract to recieve the asset a year from now. Assume no carry costs or anything.

If the forward price is 99 dollars, then sell the stock and buy the forward. In a year we receieve the stock from the forward, and cancel out our short position, net profit = 1 dollar.

If the forward price is 101 dollars, then sell the forward contract and buy the stock for 100. In one year we sell our stock to cancel our short position on the forward contract, net profit = 1 dollar.

In both cases, the way the asset price moved in the future was completely irrelevant, in particular E[S(T)] was completely irrelevant. The only thing that is relevant is the current price, and the current price equals E[S(T)] only under the risk-neutral measure (if interest rates are non-zero then S_0 = E[S(T)] discounted).