Every moneyline is a binary option.
This is the trade ticket.
A moneyline bet pays $1 or $0 at terminal time. That's a binary option on the score differential, with the pregame line setting the drift and game-level scoring variance setting the volatility. Under a Brownian motion model — Stern (1994), well-replicated for NBA — the Greeks have closed-form expressions. Inputs on the left. Greeks and payoff curves on the right.
dog fair: 37.0%
vig: 3.98%
implied drift μ: 4.48 pts
Cheap dogs are options.
How much option, exactly?
Pregame, an underdog at p₀ looks like a cheap lottery ticket. The model says: across 40,000 simulated game paths per price, the expected peak in-game probability divided by entry follows a tight log-linear law — E[max pdog] / p₀ ≈ 1.05 − 0.91·ln(p₀) with R² ≈ 0.9998. The 2× hit rate is essentially constant. Cheap dogs do not double up more often; they double up by larger amounts.
The takeaway,
stated plainly
The folk wisdom that "below 25¢ is the line" for live-betting underdog plays is partially right and partially wrong. Right: cheap underdogs are structurally more convex. Wrong: there's no threshold. The relationship is smooth. The biggest marginal gains in convexity come from going from 10% to 5%, not from crossing some arbitrary 25% line.
The non-obvious finding: the probability of any 2× repricing is approximately constant across pregame underdog prices. What changes is not how often you get a favorable move; it's how big that move is relative to what you paid. This is the precise content of "cheap optionality" — same hit rate, larger relative payoff.
Notes &
caveats
i. σ = 13.5 points-per-game is the consensus NBA score-differential volatility from Stern (1994) and replicated in Polson & Stern (2015). Other sports require a different model — NFL has discrete possessions that violate the Brownian assumption — so this v1 is NBA-only. Don't generalize.
ii. The model assumes constant within-game volatility. Real basketball has stochastic vol from foul rate, pace changes, and end-game intentional fouling. Greeks should be read as first-order approximations, not as exact prices a market maker would charge.
What this is
not
iii. This is not a betting system, an edge-finder, or financial advice. It is a visualization tool — a way of seeing the structural similarity between sports markets and derivatives markets.
iv. The empirical claim of Module 02: expected peak in-game repricing of an underdog follows a tight log-linear law — E[peak]/p₀ ≈ 1.05 − 0.91·ln(p₀), R² = 0.9998 from 40k Monte Carlo paths per price. Cheap underdogs are structurally convex. Folk wisdom had the shape right and the threshold wrong.