Calculating Trump impeachment odds for prediction markets, part 2

A few days ago I discussed how Trump’s prediction market impeachment odds are too high.

I still stand by my prediction that Trump will not be impeached in his first term. But there are factors working against this:

The dems may decide to try to impeach Trump anyway as a sort of symbolic gesture and in the hope of generating enough negative press to hurt Trump’s reelection hopes, even if it they know it will fail in the Senate.

New revelations may be uncovered that hurt Trump and expedites the impeachment process, such as more Ukraine news or something completely new and unrelated.

However, for people betting against impeachment, here are some factors in their favor:

The clock is ticking. It will probably take a couple months to draft and finalize and articles of impeachment and begin the voting process. So if by October 2020 there is no cohesion about what sort of misconduct to charge Trump with or how to proceed, then likely Trump will not be impeached in his first term and the ‘yes’ contract will be worthless even though it does not expire for another 2-3 months.

If Trump loses but has not yet been impeached then likely the dems will abandon trying, knowing that impeaching a lame duck president is redundant. This effectively shortens the duration of the contract from 15 months to 12.

Roughly speaking, the ‘yes’ 2020/2021 impeachment contract, valued at 75 cents and expires at the end of Trump’s first can be modeled as something resembling an exponential or Poisson distribution. The ‘no’ contract must be 1-{yes}. So by deriving a function of how the ‘yes’ contract decays, we can calculate long it takes for the ‘no’ contract, now valued at 25 cents, to double and also develop strategies.

We have three data points:

The value of the 2019 ‘yes’ impeachment contract, which as of writing this is valued at 40 cents give or take a few pennies, the endpoint (2020/2021 contract valued at 75 cents), and the initial point (now). The x-axis can be divided into increments of 1/15 (0/15, 1/15…15/15) which is how many months Trump has left in his first term and until the 2020/2021 contract expires.

It’s important to note that these probabilities of impeachment by time ‘x’ is based on the cumulative distribution of the underlying probability distribution function, which is integrated from 0 to x. So when we plot the above points, we get a curve that always has negative second derivative sorta resembling ln(x+1), which is the cumulative distribution function. This is also an odd function. So for example of odds of trump being impeached by next week are zero.

Normally we would use an exponential distribution function, but this works for two data points, and instead we have three. This makes the function more complicated.

Trying to find an interpolating polynomial based on these three points fails because the resulting quadratic has a local maximum within the domain of the function(0-15 months). The second derivative must always be native and there must not be any local maximums.

To find the desired curve with just the three points, we can try a cubic instead which also fails due to the above problems regarding local maximums and derivatives.

However, there is a way…to be continued…