The product that broke independent pricing
A traditional accumulator combines selections from different events. Because the events are independent, the fair price is the product of the fair prices, and the book's margin compounds: four legs each carrying a 5% overround give an overround of about 21.6% on the whole (1.05 to the fourth power), so a book that took money in proportion to the prices would keep about 18% of the stakes. That compounding makes accumulators a high-margin product for bookmakers, and they are easy to price.
The bet builder, or same-game parlay, combines selections from the same event: a team to win, a player to score, over 2.5 goals, both teams to score. These are not independent. If the home team wins 3-1, the striker scoring, the total going over and both teams scoring have all become more likely together. Multiply the individual prices and the book's price is badly wrong; in most combinations customers pick, it is too generous, because bettors instinctively combine outcomes that go together.
Same-game parlays have become one of the most important products for the largest operators. Flutter, which owns FanDuel, measures sportsbook net revenue margin as revenue as a percentage of the amount staked, its handle, and in its annual report for 2025 it attributed part of the rise in its US structural revenue margin, to 14.2%, to increased same game parlay penetration; for 2024 it credited greater adoption of higher margin same game parlay bet types in the UK and Ireland. Priced badly, the same product is a large source of pricing error. This lesson is about doing it properly.
Correlation is the whole problem
Two selections in the same event can be positively correlated (home win and over 2.5 goals when the home side is a strong attacking favourite), negatively correlated (home win and away team's striker to score), or nearly independent (a yellow card count and a first-half corner). The fair price of the combination depends on the joint distribution, not the marginals.
The naive fix is a correlation adjustment: multiply the product of marginals by a factor that reflects the estimated correlation. It is the simplest approach and it is inadequate, because the correlation between two selections depends on everything else in the parlay and on the state of the match. A fixed factor for "home win and over 2.5" is wrong in a match where the home side is 1.20 and wrong differently in one where it is 3.50.
The robust approach is to build one model of the event that generates every selection at once, and to price the combination by evaluating it under that model. For football that is the scoreline model from lesson 2, extended with player-level scoring, cards, corners and whatever else the builder offers. For basketball it is a possession-level simulation with player usage rates. The parlay's fair probability is the fraction of simulated matches in which every leg wins.
Simulation as the pricing engine
Monte Carlo is the practical tool. Simulate the match a large number of times from the model, record every quantity the builder can bet on in each simulation, and price any combination by counting. The design questions are about the simulation, not the pricing:
Fidelity. The simulation must generate the legs the builder offers with the right marginal probabilities and the right dependence. A goal in the simulation should be assigned to a scorer with probabilities that reflect who is on the pitch and the state of the match. Cards should depend on the referee and the score. Corners should be correlated with the balance of play. Every leg type added to the product is a modelling commitment.
Speed. A customer building a parlay expects a price almost instantly, and repricing happens on every leg added. Simulations are pre-computed per event and refreshed when the underlying model moves; the price of any combination is then a lookup over the stored sample. The sample size, tens of thousands of simulations per event in the examples here, sets a storage and refresh budget.
Tail precision. A parlay of six legs may win in one simulation in five thousand. The sampling error on that estimate is large: the relative error of a simulated probability grows as the expected number of successes falls, so fifty thousand simulations would contain only about ten winning scenarios, and the standard error would be roughly a third of the estimate itself. Books deal with this with variance reduction methods such as importance sampling, with analytical approximations for the tail, and, bluntly, with margin: long parlays carry wide margins partly because the price is uncertain.
Consistency with the singles. The marginal probability of each leg inside the simulation must match the price the book offers on that leg as a single, or customers will arbitrage the builder against the main market. This is a hard constraint and it is easier to satisfy by generating the singles from the same simulation than by reconciling two models.
Copulas and the middle ground
Some desks cannot afford a full event simulation for every sport and every market type. A middle ground is to model each leg's marginal separately, as now, and to model the dependence between legs with a copula: a function that joins marginals into a joint distribution with a chosen correlation structure. Gaussian copulas are a common choice because they are tractable; the correlation matrix is estimated from historical data on how legs co-occur.
Copulas are better than fixed correlation factors and worse than simulation. They capture pairwise dependence but not the full conditional structure, and the correlation matrix has to be estimated per event type and re-estimated as the product changes. Two technical limits matter for parlays. A Gaussian copula has no tail dependence short of perfect correlation: extreme outcomes become asymptotically independent, so it can understate how often extremes arrive together, which is where long parlays settle. And because bet legs are discrete outcomes, the copula joining them is not unique, so the fitted dependence is partly a modelling choice rather than something the data pins down. The honest statement is that they are a bridge for desks building toward simulation, and the simulation is where the product ends up.
Exposure across correlated markets
Correlation does not only affect pricing; it affects risk. A book's liability on a match is not the sum of its liabilities on each market, because the same scoreline settles all of them. A book that is short the home win, short over 2.5 and short the home striker to score has one bet, not three, and a 3-0 home win pays all of them at once.
Liability management therefore has to happen at the event level, in scoreline space: for every possible outcome of the event, what is the book's net position across every market and every parlay? The simulation provides exactly this, since every stored sample is a scenario and the book's payout in that scenario is computable. Desks that manage liability market-by-market are blind to their real exposure, and bet builders have made that blindness expensive.
The operational consequence is that stake limits on parlays should be set against the marginal change in event-level tail exposure, not against the parlay's own potential payout. A modest bet that lands on the book's worst scenario is riskier than a large one that hedges it.
Governance for a product that reprices itself
Bet builders take pricing decisions out of traders' hands and into a system that evaluates millions of combinations. That needs specific controls:
- Leg-type approval. Every new leg type is a modelling change and goes through the champion-challenger process before it can be combined.
- Combination limits. Combinations the model prices badly (very long, or involving legs with thin data) are simply not offered.
- Margin by leg count and correlation. Margin should rise with the number of legs and with the uncertainty of the correlation estimate, and the schedule should be explicit.
- Post-settlement audit. Every settled parlay is compared against the model's probability, and the calibration of the parlay prices is tracked exactly as single prices are. A product that pays out more than its prices implied is mispriced somewhere, and the audit finds where.
The final lesson brings the model and the market together in the decision that actually makes or loses money: how much to take, from whom, and when to stop.