Specifically, if \(p\) and \(\bar{p}=1-p\) are respectively the probabilities of two complementary events, and \(a\) and \(b\) are respectively the prices of contracts on them, which can be bought and sold freely, then no-arbitrage imposes that \(-a-b+10 = 0\) and statistical no-arbitrage imposes \(-\bar{p}a +p(10-a) = 0\) and \(-pb +\bar{p}(10-b) = 0\). Solving indeed gives the prices \(a=10p\) and \(b=10\bar{p}\).

However, this isn’t the end of the story.

The prediction market isn’t a closed system. Event outcomes are correlated with other payoffs outside of it. For instance, the election outcome has personal income tax consequences for certain individuals. While playing in the prediction market has no expected gain or loss, its contracts can diversify just such an external payoff to reduce its variance.

Suppose the total tax exposure of an Obama presidency is \(T_o\) and of a Romney presidency \(T_r\), and the probabilities of the two winning are respectively \(p\) and \(\bar{p}\), then the expected payoff is \(-pT_o -\bar{p}T_r\) while the variance is \(p\bar{p}(T_o-T_r)^2\).

Without loss of generality, assume \(T_o > T_r\). Then we can reduce the variance of the payoff by buying “Obama wins” contracts at normalized price \(q=a/10\). Let’s say we buy an amount worth \(N\) if expired in the money. The payoff becomes \(-T_o +(1-q)N\) with probability \(p\) and \(-T_r -qN\) with probability \(\bar{p}\). If the contracts are priced for no arbitrage as before (\(q=p\)), the expected payoff is \(-pT_o +p\bar{p}N -\bar{p}T_r -\bar{p}pN = -pT_o -\bar{p}T_r\) as before. However, the variance is \(p\bar{p}(T_o-T_r-N)^2\), which is a decrease for any \(N\in (0,2(T_o-T_r))\), with the aggregate (i.e. hedged) payoff becoming completely deterministic for \(N=T_o-T_r\). This is the point of maximal utility gain for a risk-averse hedger. One ends up “pre-paying” a portion of the potential additional tax burden in exchange for immunity from the election outcome.

The fact that hedgers exist and are biased in one direction means that the normalized price of a contract may no longer be exactly the probability of its expiring in the money. The imbalance in the market caused by risk aversion should create precisely an insurance premium to be added to the price of “Obama wins” contracts. Of course, the reality is more complicated, since not all individuals have homogeneous tax burdens under the outcomes. If the number of risk-averse hedgers is small, then the no-arbitrage assumption may still approximately hold.

“The physicist-run Prediction Company is an example of a company that has apparently extracted unusual profits from the market for over a decade. In contrast, economist-run companies like LTCM and Enron have gone belly-up. Being a physicist certainly doesn’t guarantee success … but if you are going to look for correlations in (market or any other) data then being a physicist might help.”

Joseph L. McCauley, writing in
Dynamics of markets: econophysics and finance

In any case, earthquake prediction may be a misnomer. One can never predict the precise time of an earthquake. But with more data and detection of ever smaller features, one can give more granular probabilistic predictions. So instead of saying there is a probability \(p\) of earthquake in the next 30 years, we may be able to either say (at any given moment) there is a probability \(p_1 \ll p\), or probability \(p_2 \gg p\) of one within the next year.