Deriving forward prices

[Gareth]

This doesn't really fit into any subject category (CT8/ST6/ST5), but it's something that occurred to me... can the forward price F_0 = e^(rT)S_0 be derived using the fundamental theorem of asset pricing?

Here's what has confused me:

Value of forward contract at time 0 = V_0

V_0 = E_Q( e^(-rT) (S_T - F_0) | S_0 )

by fundamental theorem of pricing, where Q is the risk-neutral prob measure and payoff of forward is S_T - F_0.

simplifying this gives e^(-rT) x (E_Q(S_T | S_0) - F_0)

Now, under Q, S_T is martingale so:

V_0 = e^(-rT) x (S_0 - F_0) = 0 since the forward has zero initial value. This implies F_0 = S_0.

Where's the fallacy in this argument?

 

[examstudent]

gareth, ive seen this proof of forward prices by FTAP in a set of university lecture notes,.. so it can be derived this way..but as we know price of any derivative can be derived in this way

why you are not getting F0 = S o * exp (rT) is because in your argument you said that expectation under Q of teh terminal asset price ST / So is So - in fact it is So * exp (rT) - that should give the result - (excuse my poor notation and any miscomprehension of yours but still havent learned how to write equations on computer)

 

[Gareth]

oh yes the answer was staring me in the face - thanks!

Funny how quickly you can forget the details of this stuff, makes life quite tricky (im doing ST5 and now and then a bit of ST6 reappears).

 

[examstudent]

yeah but in ST5 they wont examine the proofs etc of pricing results, although one or two of the core reading chapters are direct replicas of ST6, (except there was a little proof on forward price in ST5 somewhere)

..though going back to your comment on sounfd revision, i presume you are talking about ST5 sound revision, i also have that CD and it was pretty good,
....though it would be pretty funny if the guy speaking on it tried to talk us through the forward price proof in the notes!