## Tuesday, July 14, 2009

### How Prices Evolve in Prediction Markets?

Last week, Nikolay presented our paper on "Modeling Volatility in Predictions Markets" at the EC'09 conference. One of the questions that we are answering in this paper is, "what is the most likely price of a prediction market contract at some point in the future?"

Let's start with the expected price. If we assume that the markets are efficient, then the current price of the contract is the best possible estimate for the future expected price. However, the current price is NOT the most likely price in the future. In fact the probability of the contract will have the same price in the future is decreasing with time. Why? Because the final price of the contract as we get closer to the expiration will get closer to 0 or 1, as the uncertainty about the outcome decreases over time. So, while the expected price will be equal to the current price, most of the future prices will be closer to 0 and 1.

Below you can see some 3d plots of the "future price density" as a function of the future price P and the time to expiration t. We assume that "now" is t=0 and the contract expires at t=1.

If the current price is 0.5, then the future price density, as a function of the future price P and the time to expiration t, is:

As you can see, the possible prices, when we are close to t=0, are clustered around the current price (in this case 0.5). Then, as we move closer to the expiration, the probability density moves closer to 0 and 1. As this contract had price 0.5, the plot is completely symmetric around the axis P=0.5.

If we have a current contract price at 0.4, then the density becomes more skewed towards 0:
And here is an even more skewed plot, with the current contract price at 0.25:

Just in case you want to create your own plots, here is the Maple code:
with(stats);normpdf:=(x,mu,sigma)->statevalf[pdf,normald[mu,sigma]](x);spdf:=x -> normpdf(x,0,1);normicdf:=(p,mu,sigma)->statevalf[icdf,normald[mu,sigma]](p);sicdf:=x->normicdf(x,0,1);f:= (pnow,pfuture,lambda) -> spdf ( sqrt(1/lambda) * sicdf(pnow) - sqrt(1/lambda-1) * sicdf(pfuture))*sqrt(1/lambda-1)/spdf(sicdf(pfuture));plot3d(eval((f(p,P,t)), {p=0.5}), P=0..1, t=0.1..0.75, axes=boxed, shading=zhue, orientation=[-120, 50]);

So, what can we do with these results? One application is to price the X contracts on Intrade: In the "X" contracts, the question is about the future price movement of a prediction market contract (e.g., "will the contract for Democrats winning the 2012 election be above 0.75 on December 31st, 2010?").

These X contracts are similar to the existing "call" and "put" options on the stock market, where people try to guess where the price of a stock will be in the future. There is a significant difference, though: When a trader prices and trades a call/put option for a share, (e.g., using the vanilla Black-Scholes formula) the trader needs to guess the future volatility of the share price. Through this process, the trade gives to the public valuable information about the future volatility of share price. For prediction markets, trading an X contract does not reveal the same information. Our work shows what the exact form of future price distributions, without the need to provide any volatility estimates. (Volatility can be largely determined by the current price and time to expiration; see the past blog post and the EC'09 paper for details.) So, pricing an X contract requires just to plug in the current price, time to expiration, and strike price (information that is already public) to find the "correct" price for the X contract.

So, am I saying that the X contracts are completely useless? No. But the information revealed by trading these contracts is significantly less compared to the information revealed by trading options on stocks.