# Risk Arbitrage

People have different risk profiles, and different beliefs about the future. But it seems to me like these differences should probably get washed out in markets, so that as a society we pursue investments if and only if they have good returns using some particular beliefs (call them the market’s beliefs) and with respect to some particular risk profile (call it the market’s risk profile).

As it turns out, if we idealize the world hard enough these two notions collapse, yielding a single probability distribution P which has the following property: on the margins, every individual should make an investment if and only if it has a positive expected value with respect to P. This probability distribution tends to be somewhat pessimistic: because people care about wealth more in worlds where wealth is scarce (being risk averse), events like a complete market collapse receive higher probability under P than under the “real” probability distribution over possible futures.

Let A be an arbitrary ∑-algebra (ie, a family of sets of possible worlds closed under intersections, complements, and countable unions) and consider a society which trades contracts of the form “if we are in world a, I will give you \$1” for each a in A. Suppose that all of these contracts can be resolved at some future time T. We define the price of each of contract in terms of instruments which pay out \$1 at time T. (We get into standard issues with escrow and investment for prediction markets if we don’t trust each other, but I want to set that aside for now.)

Suppose that each of these markets has a negligible bid ask spread, so that for each market there is a particular price at which I can either buy or sell a contract. Then the following facts are easy to verify at a very weak notion of “equilibrium” (ie, one in which there are no arbitrage opportunities that don’t require knowledge):

1) If P(X) is the price of a contract which pays out on the event X, then P is a probability distribution. This is proved by exhibiting straightforward arbitrage opportunities if P doesn’t form a measure.

2) At the margins, a rational individual will make an investment whose payoff is measurable with respect to A if and only if its expected return under P, is positive. This is proved by producing an explicit set of contracts which hedges this investment in such a way that it guarantees positive returns.

So to the extent we trust this idealization (that the relevant markets have low spreads) we can conclude that every rational investor is willing to make the same set of investments. This is slightly different from the way the results of prediction markets are normally talked about: the market prices do not represent rational estimates for the probability of various events, but are instead risk-adjusted estimates. Rational investors may assign different probabilities to the same event, and may have different attitudes towards risk, but market equilibrium implies that any such differences must precisely cancel out when it comes to making investment decisions.