Prediction Markets: Pricing, Mechanisms & Market Making

A binary prediction-market contract pays one unit if a precisely specified event occurs and zero otherwise. Its price is therefore expressed on the same 0–1 scale as a probability, but that resemblance is not an identity: the observed quote also reflects wealth, risk preferences, inventory, fees, market mechanism, and who is willing or able to trade. A useful market-making system must keep the contract semantics, the belief model, and the execution mechanism separate.

This chapter develops that separation. It begins with what a price means, compares order books with scoring-rule market makers, then builds an inventory-aware quoting loop in log-odds coordinates. The logit jump–diffusion and calibration stack proposed in Toward Black–Scholes for Prediction Markets is treated as a promising design, not as established empirical fact.

This is the advanced capstone. Start with binary contract payoffs, expected value, no-hedge economics, and adverse selection if price, probability, and realized PnL are not yet separate mental objects.

Start with the contract, not the chart #

Let $Y\in\{0,1\}$ be the settlement outcome and let the contract pay $Y$ at resolution. A trader who believes $\Pr(Y=1)=q$ values the expected payoff at $q$ before accounting for risk, capital, fees, and other exposures. If a frictionless risk-neutral market clears at price $p$, the familiar shorthand is

$$p=\mathbb{E}^{\mathbb{Q}}[Y]=\mathbb{Q}(Y=1),$$

where $\mathbb{Q}$ is the market's pricing measure. This is a risk-neutral probability, not automatically the population's average subjective belief and not a claim that the event will occur with frequentist probability exactly $p$.

This distinction matters operationally. A forecasting user asks whether $p$ is calibrated against outcomes. A market maker asks whether the next informed order will move the executable mid, whether the inventory can be unwound, and whether settlement wording creates correlated exposure elsewhere. Those are related questions, but no single number answers all of them.

Three liquidity mechanisms #

The contract definition does not determine how traders obtain liquidity. Three common mechanisms expose the operator to different risks.

LMSR as a cost-function market maker #

For mutually exclusive outcomes with outstanding share vector $\mathbf q$, the logarithmic market scoring rule can be expressed through the cost function

$$C(\mathbf q)=b\log\sum_i e^{q_i/b}, \qquad p_i=\frac{\partial C}{\partial q_i} =\frac{e^{q_i/b}}{\sum_j e^{q_j/b}}.$$

The parameter $b>0$ sets the liquidity scale. A larger $b$ makes the price respond less to a fixed trade, so traders obtain more depth, but the sponsor accepts a larger worst-case subsidy. A trade from $\mathbf q$ to $\mathbf q'$ costs $C(\mathbf q')-C(\mathbf q)$. Because successive cost differences telescope, the sponsor only needs the initial and final state to compute the net loss. Hanson derives the logarithmic form from a proper scoring rule and shows why it can offer infinitesimal spread with bounded loss for a finite outcome set.

Combinatorial markets add another constraint: the state space may grow exponentially, so exact prices can become computationally intractable even when the scoring rule itself is simple. The economic loss bound does not bound computation.

Log-odds give probability a usable geometry #

Direct Gaussian dynamics on $p\in[0,1]$ are awkward because unconstrained increments can cross the boundaries. Map price to log-odds instead:

$$x=\mathop{\mathrm{logit}}(p)=\log\frac{p}{1-p}, \qquad p=S(x)=\frac{1}{1+e^{-x}}.$$

Now $x$ lives on the real line. More importantly,

$$\frac{dp}{dx}=S'(x)=p(1-p).$$

The Jacobian is largest at $p=0.5$ and tends to zero near either boundary. Therefore the same log-odds shock produces a large displayed probability move in the swing zone and a small displayed move near 0 or 1. For a small logit displacement $\delta_x$,

$$\delta_p\approx p(1-p)\delta_x.$$

This is a coordinate effect, not evidence that near-certain contracts are safe. A rare but decisive information jump can still move $x$ by many units, liquidity can vanish, and the settlement payoff remains discontinuous.

Inventory-aware quoting #

Avellaneda and Stoikov model a dealer who posts a bid and ask around a stochastic mid-price. Market orders arrive with intensity that decays as a quote moves away from the mid. Under exponential utility, Brownian mid-price risk, and exponential fill intensity, the approximate reservation price is

$$r=s-q\gamma\sigma^2(T-t),$$

and the total optimal spread is approximately

$$\delta^a+\delta^b =\gamma\sigma^2(T-t) +\frac{2}{\gamma}\log\left(1+\frac{\gamma}{k}\right).$$

Here $s$ is the mid-price, $q$ is signed inventory, $\gamma$ is risk aversion, $\sigma$ is mid-price volatility, $T-t$ is the remaining risk horizon, and $k$ measures how quickly fill intensity decays with quote distance. The first formula skews both quotes to encourage inventory reduction; the second chooses their total width by balancing execution income against price and inventory risk.

For an event contract, the same structure can be applied in log-odds units:

$$r_x=x-q\gamma\sigma_x^2\tau, \qquad w_x\approx\gamma\sigma_x^2\tau +\frac{2}{\gamma}\log\left(1+\frac{\gamma}{k_x}\right),$$

then $x_{bid}=r_x-w_x/2$ and $x_{ask}=r_x+w_x/2$ are mapped through $S(\cdot)$. Every symbol must use consistent units: $q$ is contract inventory, $\sigma_x^2$ is logit variance per unit time, $\tau$ is the chosen risk horizon, and $k_x$ is calibrated against logit quote distance.

The original simulations illustrate the intended trade-off rather than establish profitability: inventory-aware quotes reduced the variance of final inventory and profit relative to symmetric quotes, while often sacrificing mean profit. That is exactly what the skew is designed to do.

The quote–risk feedback loop #

The loop has five distinct responsibilities:

1. Observe executable state. Keep bid, ask, depth, trades, tick size, and feed health. A last-trade price is not a sufficient state estimate.
2. Estimate latent belief. Convert a guarded mid to log-odds and filter microstructure noise. Mark probable jumps rather than forcing every large increment into diffusion variance.
3. Measure exposure. Combine inventory with diffusive variance, jump scenarios, settlement concentration, and related-event exposure.
4. Construct quotes. Compute a reservation skew and a base spread in one coordinate system, then map to legal prices and sizes.
5. Apply hard controls. Tick and fee floors, position and loss limits, stale-feed rejection, scheduled-news widening, and kill switches constrain the model output.

This decomposition prevents an estimator failure from becoming a risk-limit failure. If the latent-state model is unavailable or stale, the system should not silently substitute an unrelated price source or default volatility; the safe state is to stop quoting until the required inputs are valid.

Separate diffusion, jumps, and observation noise #

An observed mid moves for at least three reasons:

Diffusive belief change

Many small information updates accumulate over time.

News jump

A discrete announcement or revelation moves the credible odds abruptly.

Microstructure noise

Spread bounce, thin depth, stale quotes, and isolated trades move the observation without an equivalent change in latent belief.

Treating all three as one rolling variance creates predictable errors. Spread bounce inflates risk estimates in illiquid periods; absorbing news into a Gaussian variance makes ordinary quotes unnecessarily wide long afterward; filtering away a genuine jump leaves the dealer quoting stale beliefs.

A defensible estimation pipeline is:

1. validate settlement state and market-data freshness;
2. construct an executable or depth-aware mid and clamp only for finite logit arithmetic;
3. transform to $x$ and model observation variance from spread, depth, trade rate, and imbalance;
4. filter latent $x$ with a state-space model;
5. classify increments probabilistically into diffusion and jump components;
6. estimate diffusion variance, jump intensity, and jump-size distribution separately;
7. validate forecasts out of sample against realized quantities that match the trading horizon.

The preprint proposes a heteroskedastic Kalman filter followed by an expectation–maximization mixture for diffusion and jumps. That is an implementable candidate architecture. Its specific empirical table does not validate the architecture: the paper alternately describes real event series and a synthetic risk-neutral path, and the prose claims metric wins contradicted by the table. The source note records the exact discrepancies.

Cross-event risk is not one correlation number #

Related contracts can share slow information flow and jump together at a common announcement. Those are different hedge problems. If $x_i$ and $x_j$ have diffusive correlation $\rho_{ij}$, their instantaneous probability covariance contains

$$p_i(1-p_i)p_j(1-p_j)\sigma_i\sigma_j\rho_{ij}\,dt.$$

A simultaneous-news component adds a separate co-jump covariance. Estimating ordinary correlation on a sample that mixes both mechanisms produces unstable hedge ratios: one election-night jump can dominate months of continuous observations. Estimate continuous correlation on de-jumped intervals, stress explicit common-jump scenarios, and cap hedge ratios as either contract approaches a boundary where $p(1-p)$ becomes small.

Failure modes and controls #

Risk limits should be expressed in settlement loss as well as contract count. Ten thousand contracts at $p=0.01$ may look inexpensive, but a wrong-way resolution still pays against the full notional. Likewise, liquidity-adjusted exits matter: an inventory limit that assumes the current displayed depth survives a news event is not a limit.

What is established and what remains proposed #

The durable lesson is architectural: contract semantics determine payoff; the liquidity mechanism determines how trades move prices; the state model estimates belief risk; the control layer turns that risk and inventory into quotes; and hard limits constrain everything. Collapsing those layers into “price equals probability” is the fastest route to a model that appears elegant and fails exactly when information arrives.