Using Log-Odds Changes to Model Binary Prediction Prices

# Finding an invariant/stationary quantity in prediction markets?


# Finding an invariant/stationary quantity in prediction markets?












I am looking at trade data for specific outcome of an event on a prediction market (Kalshi, but this could apply to others) and trying to model market microstructure effects such as inferring mid price and spread, forecasting future trades, etc.

Let us say we have an event with a single market of "yes"/"no"for the outcomes of the event.

The "yes" and "no" tokens can be treated as binary options where, upon an outcome occurring, the corresponding token ("yes" if it happened, "no" if it did not as we are avoiding edge cases of neither for now) pays out $1 and the other pays out nothing. The trading then halts.

Due to the structure of Kalshi, there is a price for "yes"and "no"tokens that must both be positive, less than 1 and, at any time, the prices must sum to 1. Buying a "yes" is also equivalent to selling a "no" (and vice-versa)) and margin/shorting is not allowed meaning we do not need to worry about arbitrage constraints on separate order books for each outcome ("yes" or "no").

Due to these conditions, we can effectively focus on the orderbook of just one token since the best bid of the "no" token is the same as the best ask of the "yes" token and vice-versa. We choose "yes" going forward.

I am attempting to apply a traditional quant methodology of creating something approximately invariant/stationary as detailed in "The Prayer" by Meucci (here).

My problem is that I am not sure how to do this since the value of the each token must converge to either 0 or 1 upon the outcome occurring which seems to break invariance/stationarity and we of course do not know the value at expiration in advance since that is part of what we might be trying to infer/the market is trying to infer/represent expectations of.

For spot equities/cryptocurrency/etc., we can look at the log returns of prices. For vanilla options, we can look at the changes in the implied (log) volatility over the life of the options. For bonds, we can look at the change in the yield-to-maturity over the life of the bond(? I am not too familiar with bond pricing/how default plays into this, but this makes sense to me as cited here).

This market seems to be similar to bonds/vanilla options, but different enough that I am not sure how to proceed since, to my knowledge, bond prices are nonstationary, but usually converge to a pre-determined amount at maturity with almost-certainty (excluding defaults) and vanilla options converge to intrinsic value at expiry and, for this market, these don't seem to directly apply.

Thanks!

## Answer by Jien Weng (score 2, accepted)

https://quant.stackexchange.com/a/85468

This is a great question. I am not entirely sure if this fully solves your problem, but applying The Prayer (Meucci) to binary assets is tricky because of the hard boundary conditions at 0 and 1.

The most common approach to find an invariant in this space is to map the price $p$ (which represents a probability) into Log-Odds (or Logit) space:

$$ L_t = \ln \left( \frac{p_t}{1 - p_t} \right) $$

While your price $p_t$ is trapped in $(0, 1)$ (and must converge to 0 or 1), $L_t$ exists in $(-\infty, +\infty)$. It removes the compression effects near the boundaries that break stationarity assumptions.

And so, the changes in log-odds ($\Delta L_t$) are a much stronger candidate for an invariant (i.i.d) than simple price returns.

You can think of this as analogous to modeling Yield changes in bonds (as you mentioned) or Implied Volatility changes in vanilla options, rather than the price itself. The invariant you are likely looking for is the innovation in this log-odds process.