Modeling Volatility in Prediction Markets, Part II

In the previous post, I described how we can estimate the volatility of prediction markets using additional prediction market contracts, aka options on prediction markets. I finished indicating that techniques that can be used to price options for stocks, are not directly applicable in the prediction market context.

Now, I will review a different modeling approach that builds on the spirit of Black-Scholes but is properly adapted for the prediction market context. This model has been developed by Nikolay, and is described in the paper "Modeling Volatility in Prediction Markets".

Modeling Prediction Markets as Competitions

Let's consider the simple case of a contract with a binary outcome. For example, who will win the presidential election? McCain or Obama?

The basic modeling idea is to assume that each competing party has an ability $S_i(t)$ that evolves over time, moving as a Brownian motion. (A simplified example of such ability would be the number of voters for a party, the number of points in a sports game, and so on.) At the expiration of the contract at time $T$, the party $i$ with the higher ability $S_i(T)$ wins.

Actually, to have a more general case, we can use a generalized form of the Brownian motion, an Ito diffusion, that allows for the abilities to have a drift $\mu_i$ over time (i.e., the average rate of growth), and different volatilities $\sigma_i$. The quantity that we need to monitor is the difference of the two ability processes $S(t)=S_1(t)-S_2(t)$. If at the expiration of the contract at time $T$ we have $S(T)>0$, then party 1 wins. If $S(T)$ is less than 0, then party 2 wins.

Interestingly, the difference $S(t)$ is also an Ito diffusion, with $\mu=\mu_1-\mu_2$, $\sigma=\sqrt{\sigma_1^2+\sigma_2^2-2\rho \sigma_1 \sigma_2}$, where $\rho$ is the correlation of the two ability processes. Under this scenario, the price of the contract $\pi(t)$ at time $t$ is:

$\pi(t) = Pr\{ S(T)>0 | S(t) \}$

which can be written as:

$\pi(t) = N\Big(\frac{S(t) + \mu \cdot (T-t)}{\sigma \cdot \sqrt{T-t} } \Big)$

where $N(x) =\frac{1}{2} \Big[ 1 + erf\Big( \frac{x}{\sqrt{2}} \Big) \Big]$ is the CDF of the normal distribution with mean 0, and standard deviation 1 and $erf(x)$ is the error function.

Notice that as time $t$ gets closer to the expiration, the denominator gets close to 0, which makes the ratio closer to $\infty$ or $-\infty$, and price $\pi(t)$ gets close to 0 or 1. However, if $S(t)$ is close to 0 (i.e., the two parties are almost equivalent), then we observe increasingly higher instability as we get close to expiration, as small changes in the difference $S(t)$ can have a significant effect in the outcome.

For example, consider two parties: party 1 with an ability that has positive drift $\mu_1=0.2$ and volatility $\sigma_1=0.3$, and party 2 with negative drift $\mu_2=-0.2$ and higher volatility $\sigma_2=0.6$. In this case, assuming no correlation, the difference is a diffusion with drift $\mu=0.4$ and volatility $\sigma=0.67$. Here is one scenario of the evolution, and below you can see the price of the contract, as time evolves.

As you may observe from the example, the red line (party 1) is for the most time above the blue line (party 2), which causes the green line (the difference) to be above 0. As the contract gets close to expiration, the contract gets closer and closer to 1 (i.e., party 1 will win). Close to the end, the blue line catches up, which causes the prediction market contract to have a big swing from almost 1 to 0.5, but then swings back up as party 1 finally finishes at the expiration above party 2.

So far, we generated a nice simulation but our results depend on knowing the parameters of the underlying "ability processes". Since we never get to observe these values, what is the use of all this exercise?

Well, the interesting thing is that by using the price function, we can now proceed to derive its volatility. Without going into the details, we can prove that the volatility of the prediction market contract is:

$V(t) = \frac{1}{\sqrt{T-t}} \cdot \varphi( N^{-1}( \pi(t) ) )$

where $N^{-1}(x)$ is the inverse CDF of the standard normal distribution and $\varphi(x)=\frac{exp( (-x^2)/2)}{\sqrt{2\pi}}$ is the density of the standard normal distribution.

In other words, volatility depends only on the current price of the contract and time to expiration! Anything else is irrelevant! Drifts do not matter: they are priced already in the current price of the contract, since we know where the drift will lead at expiration. The magnitude of the volatilities are also priced into the current contract price: higher volatilities cause the contract price to get closer to 0.5, as it is easier for $S(t)$ to move above and below 0 when it has high volatility. Furthermore, the direction of the volatilities of the underlying abilities is indifferent as they can move the difference into either direction with equal probability. (The only assumption is that the volatilities of the underlying abilities processes do not change over time.)

Volatility Surface

So, what this model implies for the volatility of the prediction markets?

First of all, the model says that volatility increases as we move closer to the expiration, as long as the price of the contract is not 0 or 1. For example, assuming that now we have $t=0$ and expiration is at $T=1$, the volatility is expected to increase as follows:

So, how volatility changes with different contract prices? As you can see, volatility is highest when the contract trades at around 0.5, and gets close to 0 when price is 0 or 1.

And just to combine the two plots and present a nice 3d plot, with the present being at $t=0$ and expiration at $T=1$:
The experimental section in the paper "Modeling Volatility in Prediction Markets", indicates that the actual volatility observed in the InTrade prediction markets fits well the current model.

Now, given this model, we can judge what is a "noise movement" and what is actually a "significant move" in prediction markets. Furthermore, we can provide an "error margin" for each day, indicating the confidence bounds for the market price.

I will post more applications of this model in the next few days. We will see how to price the X contracts on InTrade, and a way to compute correlations of the outcomes of state elections, given simply the past movements of their corresponding prediction markets.