- The paper develops and verifies a stochastic-control framework that transforms latent belief diffusion into a prediction-market price process and solves the resulting inventory-sensitive HJB equation.
- Numerical results show that optimal quotes widen near settlement for uncertain outcomes, adjust for inventory, and reflect asymmetric execution risks even when inventory is zero.
- Compared with a myopic strategy, the optimal policy reduces PnL volatility by about 63%, terminal inventory by 69%, VaR by 87%, and expected shortfall by 76% while sacrificing only 0.6% of mean PnL.
This paper develops a stochastic control framework for optimal market making in prediction markets, a setting that has not previously been treated within the Avellaneda–Stoikov tradition. The key structural departure from classical market making models is that the traded asset is a binary contingent claim whose price is a conditional probability confined to (0,1), and whose value collapses to $0$ or $1$ at a fixed resolution time. The authors model the price as a nonlinear transform of a latent belief diffusion, formulate the quoting problem with both running mark-to-market inventory risk and terminal settlement risk, and prove existence and uniqueness of a classical solution to the associated Hamilton–Jacobi–Bellman (HJB) equation via a fixed-point argument in latent belief space.
Market model
The contract pays $1 if an event occurs by timeTand $0 otherwise. Settlement is modeled through an FT​-measurable random variable pT​∈(0,1) with P(Y=1∣FT​)=pT​, where Y∼{0,1} is revealed immediately after T. The price process is defined as the martingale $0$0, identified with the market-implied probability under an efficiency assumption.
Rather than modeling $0$1 directly, the authors posit a real-valued latent belief process $0$2 satisfying an SDE with drift chosen to make $0$3 a martingale: $0$4 with $0$5 and $0$6 (up to sign, the Arrow–Pratt coefficient of absolute risk aversion of $0$7). The resulting price dynamics are a driftless diffusion $0$8, where $0$9. The canonical choice is the logistic map, under which the latent variable coincides with the log-odds of the market probability—a structure connected to Hanson's logarithmic market scoring rule.
Order flow follows the standard intensity-based formulation: executions at bid and ask arrive with intensities $1$0 and $1$1, constrained to quotes in $1$2 and to a finite inventory grid $1$3. The intensity functions are assumed continuous, bounded, monotone in the quote, and to satisfy a curvature condition ensuring uniqueness of the pointwise quote optimizer.
Control problem and HJB equation
The market maker maximizes expected terminal wealth $1$4 plus a terminal penalty $1$5, less a running penalty $1$6 on instantaneous mark-to-market variance. The natural terminal penalty penalizes settlement risk via the conditional variance of $1$7, namely $1$8. Because cash enters the objective additively and does not affect any state dynamics or intensities, the four-dimensional HJB equation reduces to a three-dimensional equation for $1$9, with Hamiltonians
$1 if an event occurs by time$0
where $1 if an event occurs by time$1 and $1 if an event occurs by time$2. The argument $1 if an event occurs by time$3 is a finite difference of the value function across neighboring inventory levels, so the nonlinearity couples the equation across the inventory grid.
Optimal quotes and verification
The paper characterizes the unique maximizers of the Hamiltonians explicitly. Under the curvature condition, the maps $1 if an event occurs by time$4 and $1 if an event occurs by time$5 are strictly monotone, and the optimal quotes are obtained by inverting them against the inventory-difference state variable $1 if an event occurs by time$6 or $1 if an event occurs by time$7. The envelope theorem yields $1 if an event occurs by time$8, giving a clean recovery formula for the optimal quote from the derivative of the Hamiltonian. A verification theorem then confirms that the feedback strategy built from these maximizers is admissible and optimal whenever a bounded classical solution of the reduced HJB equation exists; the proof uses localization of the Brownian part and martingale arguments for the compensated jump integrals.
Existence and uniqueness of a classical solution
Because $1 if an event occurs by time$9 may degenerate as $and $0 or $and $1, uniform ellipticity fails on the price domain. The authors therefore transform the equation to latent space, where the operator $and $2 is uniformly parabolic under the strengthened Assumption 2 (strictly positive $and $3 of parabolic Hölder regularity, Hölder-continuous transformed intensities, and a $and $4 terminal penalty). The semilinear Cauchy problem couples inventory levels only through the finite-difference terms in $and $5. The proof proceeds in two steps: first, existence and uniqueness of a mild solution via Banach's fixed-point theorem in a weighted supremum norm, with contraction constant $and $6 for sufficiently large $and $7 and an a priori estimate controlling the Hölder norm along iterates; second, an appeal to Krylov's regularity theory showing the mild solution is classical, with uniqueness transferred back through Itô's formula. Notably, the fixed-point iteration itself constitutes a convergent numerical scheme with explicit guarantees—an alternative to the finite difference method used in the quantitative section.
Numerical analysis
The quantitative study uses the logistic transformation, a volatility specification combining a baseline level, time acceleration toward resolution, and maximal uncertainty at $and $8, and intensity functions built from exponential spread sensitivity augmented with multiplicative factors that force intensities to vanish at economically infeasible quotes ($and $9 bids, FT​0 asks). Parameters are illustrative rather than calibrated; the authors note that empirical functional forms for order intensities in prediction markets have not been systematically studied.
The reduced HJB equation is solved backward by implicit Euler with a fixed-point iteration per time step, reusing an LU decomposition of the linear operator. Three structural findings emerge:
- Spread: generally decreasing over time as liquidity parameter FT​1 rises, but widening near settlement for prices close to FT​2, where settlement risk is highest and there is little time to unwind.
- Skew without inventory: asymmetric intensities alone generate positive skew for FT​3 and negative skew for FT​4, since equal absolute spreads imply unequal execution probabilities away from the center. This effect strengthens near the boundaries.
- Inventory-driven skew: short positions push quotes upward and long positions downward, with the effect vanishing as FT​5—inventory risk becomes immaterial when the outcome is nearly determined. Running risk aversion FT​6 dominates early in the horizon, while terminal risk aversion FT​7 dominates near settlement but propagates backward in time.
A Monte Carlo study with 10,000 paths compares the optimal strategy to one maximizing instantaneous expected mark-to-market profit while ignoring inventory risk:
| Strategy |
Mean PnL |
Std(PnL) |
Avg FT​8 |
VaRFT​9 |
ESpT​∈(0,1)0 |
| Baseline |
12.47 |
28.11 |
49.37 |
32.41 |
40.50 |
| Optimal |
12.39 |
10.34 |
15.23 |
4.20 |
9.68 |
The optimal strategy sacrifices roughly 0.6% of expected PnL while cutting the PnL standard deviation by about 63%, average terminal inventory by about 69%, 5% VaR by about 87%, and 5% expected shortfall by about 76%. This is the paper's strongest quantitative claim: nearly all of the myopic strategy's expected profit can be retained while eliminating most downside exposure. One caveat is that the benchmark is simulated without the inventory bound, though the authors report it rarely exceeds it in this specification, and imposing the bound would reduce its expected PnL further.
Limitations and open questions
Several restrictions qualify the results. The model assumes a single binary contract with fixed trade size pT​∈(0,1)1, no adverse selection beyond what is embedded in the exogenous intensity specification, and order flow independent of the belief process given the current state. The martingale pricing assumption rules out predictable drift in market beliefs, which would arise if the market maker had superior information. The existence theory requires the strengthened Assumption 2—including strictly positive latent volatility and pT​∈(0,1)2 terminal penalty—which excludes some specifications covered by the global assumptions. The numerical parameters are not calibrated to any platform's data, and the paper itself flags the absence of empirically estimated intensity models for prediction markets as a gap. Whether the qualitative findings—boundary-induced skew, settlement-risk-driven spread widening near pT​∈(0,1)3—persist under calibrated intensities and multi-contract portfolios remains unaddressed.
Conclusion
The paper provides the first stochastic control treatment of market making in limit-order-book prediction markets, with prices as martingale conditional probabilities generated by a transformed latent belief diffusion. It delivers a dimension-reduced HJB equation, a verified optimal quoting strategy, and a complete classical well-posedness result obtained by transforming to latent space and applying a contraction argument. The numerical analysis identifies structural features specific to binary-settlement assets—notably intensity asymmetry generating skew even at flat inventory, and vanishing inventory risk near price boundaries—and demonstrates that the optimal strategy achieves substantial downside protection at negligible cost in expected profit relative to a myopic benchmark.
