9.43 There Is No Black-Scholes for Prediction Markets (Yet)
A belief-vol kernel needs prices that sum to $1 and carry no predictable drift. Polymarket prices sum to $0.60, $39.6M walked out, and a $2 shove still moves a market 60 days later.
Shaw Dalen and the Daedalus Research Team want to give prediction markets what Black and Scholes gave options: one shared kernel, one quotable risk factor, and a derivative layer on top. The construction is clean. Map the traded probability to log-odds so standard tools apply, force the price to be a martingale under the risk-neutral measure, and whatever is left over (belief volatility, jump intensity, cross-event dependence) becomes the thing you quote and hedge. The old article "A Black-Scholes for Beliefs: Logit Jump-Diffusion and Tradable Belief-Vol" walked through the machinery in full.
The machinery rests on two claims about the market underneath it. First, the traded price is the risk-neutral probability, so a dollar of payoff costs a coherent amount no matter which leg you buy. Second, that price carries no predictable drift beyond the convexity correction the model itself supplies. Three other papers, all reading real Polymarket and Manifold transaction data, say both claims are false right now and by wide margins. Prices do not sum to a dollar, and roughly $39.6 million walked out of the gap. Price and mean belief separate by construction under log utility, not by friction. And a two-dollar shove still moves a market price sixty days later.
The kernel, and the assumption holding it up
Write p for the traded price of a contract paying $1 if the event happens. The logit map sends p in the open interval from 0 to 1 onto the whole real line, so the modeller can use ordinary jump-diffusion tools without the price wandering outside its bounds.
$$ x_t = \operatorname{logit}(p_t) = \log\frac{p_t}{1-p_t}, \qquad dx_t = \mu(t,x_t)\,dt + \sigma_b(t,x_t)\,dW_t + \int_{\mathbb{R}} z\,\tilde N(dt,dz) $$