A few weeks back, I was thinking about the concept of uncertainty in prediction markets. The price of a contract in a prediction market today gives us the probability that an event will happen. For example, the contract 2008.PRES.OBAMA is trading at 84.0, indicating that there is an 84% chance that Obama will win the presidential election.
Unfortunately, we have no idea about the stability and robustness of this estimate. How likely it is that the contract will fall tomorrow to 80%? How likely it is to jump to 90%? By treating the contract price as a "deterministic" number, we do not capture such information. We need to treat the price as a random variable with its own probability distribution, out of which we observe just the mean by looking at the prediction market.
However, to fully understand the stability of the price we need further information, beyond just the mean of the probability, revealed by the current contract price.
A first step is to look at the volatility of the price. One approach is to look at the past trading behavior, but this analysis will give us the past volatility, not the expected future volatility of the contract.
Predicting Future Volatility using Options
So, how can we estimate the future volatility of a prediction market contract?
There is a market approach to solve this problem. Namely, we can run prediction markets on the results of the prediction markets!
Recently, Intrade has introduced such contracts, the so-called X contracts (listed under "Politics->Options: US Election" from the sidebar). For example, the contract "X.22OCT.OBAMA.>80.0" pays 1 USD if the contract "2008.PRES.OBAMA" will be higher than 80.0 on Wed 22 Oct 2008. Traditionally, the threshold defined in the options contract is called strike price (e.g., the strike price for X.22OCT.OBAMA.>80.0 is 80.0).
A set of such contracts can reveal the distribution of the probability of the event for the underlying contract 2008.PRES.OBAMA. In other words, we can see not only what is the mean probability that Obama will be elected president but we can also see the expected downside risk or upside potential of the 2008.PRES.OBAMA contract. For example, the X.22OCT.OBAMA.>80.0 has a price of 90.0, indicating a 90% chance that the 2008.PRES.OBAMA contract will be above 80.0 on Oct 22nd.
Now, given enough contracts, with strike prices at various levels, we can estimate the probability distribution for the likely prices of the contract. For example, we can have contracts with strike price 10, 20, ..., 90 that will give us the probability that the contract will trade above 10, 20, ... and 90 points at some specific point in time, which corresponds to the expiration date of the options contract. So for each date, we need 9 contracts, if we need to have a 10 column histogram that describes the distribution.
Note that if we want to estimate the probability distribution dynamics we will need to setup 9 contracts for each date that we want to measure. Of course, this implies that we have plenty of liquidity in the markets if we want to rely purely on the market for such estimates.
Pricing Options and the Black-Scholes Formula
A natural question is: Can we price such "options on options" contracts?
This will at least give us some guidance on the likely prices of such contracts, if not for anything else, but to just start the market at the appropriate level. (For example, if we have a market scoring mechanism.)
There is significant research in Finance on pricing options for stocks. The Black-Scholes formula is one of the most well-known examples for deriving prices for options on stocks. The basic idea behind Black-Scholes is that the underlying stock price follows a Brownian motion, moving randomly up and down. Then by extracting the probability that this random stock move will reach various levels, it is possible to derive the option prices. (Terrence Tao has a very easy to read 3-page note explaining the Black-Scholes formula and a longer blog posting.)
Why not applying directly this model to price options on prediction markets? There are a few fundamental problems but the most important one is the bounded price of the underlying prediction market contract. The price of a prediction market contract cannot go below 0 or above 1, so the Brownian motion assumption is invalid. In fact, if we try to apply the Black-Scholes model on a prediction market, we get absurd results.
In the next post, I will review an adaptation of the Black-Scholes model that works well for prediction markets, and leads to some very interesting results!