Forward Price

[jcjc0602]

Hi David,

How are you doing!

I have a question:
The fwd price of a stock is just S0exp(rt). For Hull's book chapter27.4, equation 27.21 F=E(S), forward price is its expected future spot price, however, in a world that is fwd risk neutral wrt P(t, T). In this world, I think these two price are different.
Thanks a lot for your help!

 

[David Harper CFA FRM]

Hi jcjc,

Thanks i hope you are doing well, too!

Yes, I absolutely agree with you, and Hull does too. Elsewhere (earlier) he explains that he generally makes the unrealistic, simplifying assumption that the underlying commodity has no systematic risk (beta). Combined with the further simplification of employing the CAPM, his "default usage" is therefore an assumption that the investor's (the long forward) discount rate is (unrealistically) equal to the risk-free rate. I perceive a subtle difference: with F = E[S(t)], Hull is not implying hedgers/investors are risk neutral, rather he assumes they are but simply that the commodity has no price risk.

It is a long way of saying that Hull's true but unused formula is 5.2: F(0) = E[S(t)]*[exp(r - k)T];
i.e., F(0) <> E[S(t)]; for any discount rate (k) different than the riskfree rate (r)

Although I don't think Hull goes further on this, this is consistent with the assumption that, ordinarily, the long futures position bears risk, so charges a risk premium, and is called the "theory of normal backwardation" better explored by Kolb: that F < E(S) because the long futures investor is risk averse and demands future profit = (E(S) - F) as compensation for bearing risk (the uncertainty that the future spot will not be as expected; or simply that the E(S) has variance/dispersion).

So this risk aversion view implies that normal backwardation is the natural state. But, I personally like your statement better: that uncertain E(S) implies merely that F <> E(S). Because the short position, it seems to me, can be risk averse also and as futures are unfunded, it's not obvious to me why the long pays the premium...

Thanks, David