how to prove future price is risk neutral expectation?

bquant · May 1st, 2008, 8:01 pm

according to non-arb pricing, the future price of a non-dividend asset is the risk netral expectationof the asset at a future time. How to prove this?

manolom · May 2nd, 2008, 7:45 am

The fundamental theorem of asset pricing that states that a asset divided by a numeraire is a martingale in the space of measure associated to the numeraire. In the case of risk-neutral pricing, the numeraire is the continuosly compounded bank-account that are worth 1 at time 0.

ChicagoGuy · May 2nd, 2008, 8:18 pm

You would have to look at a an of N dimensional linear spaces, define a specific probability measure, blah blah blah. Its not a 2 line proof. Look at Pliskas "introduction to mathematical finance" if you want it in the discrete case. You should note that its an if and only if statement. To existance of a risk neutral measure implies non-arbitrage and non-arbitrage implies the existance of a risk neutral measure.

bilbo1408 · May 3rd, 2008, 4:13 am

What do you mean prove it? If there was some sort of known future price of the asset, I believe it would be taken advantage of.....I have always understood the risk-neutral measure to be that the expected future price is the spot price compounded at the risk-free rate......

Paul · May 3rd, 2008, 7:05 am

By "future price" do you mean "price of the futures contract" or "price in the future"? I assume the former, in which case the no-arb argument involves entering into the futures contract, simultaneously selling the asset and banking the cash. It may not be a two-line proof, but in PWIQF2 it was only about two paras (mostly chat), and certainly doesn't require heavy machinery like numeraires and martingales. (Technically this is the argument for forward contracts rather than futures, the latter are a bit messier because of the marking to market.)P