8. Brownian Motion vs. Jump-Diffusion: Why Sports Break the Textbook

Two species of randomness

A liquid stock between news events moves in thousands of tiny increments — well-approximated by Brownian motion, the model under AS. An in-play sports contract is a different animal: the probability of "Alcaraz wins the match" barely drifts while a point is being played, then jumps the instant the point resolves. A break of serve, a wicket, a goal — each is a discontinuity. The correct model family is jump-diffusion: smooth diffusion punctuated by Poisson-arriving jumps of meaningful size.

Three structural facts about event-contract prices

• Prices are probabilities, bounded in [0,1]. The natural coordinate is the log-odds, x = ln(p/(1−p)) — where p is the contract price read as a probability (a 65¢ contract is p = 0.65) and x is that same belief re-expressed on an unbounded scale, so equal moves in x mean equally "surprising" news anywhere on the price range; recent work (Dalen 2025, "Toward Black-Scholes for Prediction Markets") models x as a jump-diffusion martingale and re-derives AS-style quotes directly in logit space — skew and spread computed on x, then mapped back through p = 1/(1+e^−x). This elegantly handles the boundary squeeze: a 2¢ move at p=0.50 and at p=0.95 are wildly different events in probability space, identical-looking in price space.
• Time-to-resolution is an expiry. Like an option, an event contract has a clock. Belief volatility collapses to zero at resolution (vol crush), and with it the AS risk terms — your (T−t) is literal here, not a stylized end-of-day.
• Inventory near resolution is unhedgeable. There is no underlying to delta-hedge with. Holding 500 YES at 90¢ in the fifth set is a position you exit through the order book or carry to settlement — nothing in between.

The first fact — that price space lies near the boundaries — deserves a picture, because it changes how you set every spread and skew in this part of the book:

The courtsider: adverse selection with a stopwatch

Sports markets host the purest predators in all of microstructure: courtsiders — people (or feeds) who learn the outcome of a point seconds before the exchange's data does. To them, your stale quote after a break point is free money. The empirical record is stark: a study of 141 Grand Slam matches on Betfair found cumulative abnormal returns of roughly 3.56% within the ~5-second window after a set ends — a yawning chasm compared to the ~1% spreads you're trying to earn. Betting exchanges impose in-play delays of a few seconds precisely to blunt this.

The model-anchor principle

In diffusive markets your fair-value anchor can be the microprice — the book itself is informative. In event markets the book is slower than the world: the best anchor is an external model fed by the event stream. For tennis: a hierarchical Markov chain (point → game → set → match) in the O'Malley/Barnett tradition, parameterized by each player's serve-win probability — closed-form, microsecond-fast, updating discretely on every point. For cricket: ball-by-ball state models (runs, wickets, balls remaining). The market maker quotes around model fair value, inventory-skewed à la AS-in-logit, and treats disagreement between model and market as either edge or alarm.