The Math Behind Directional Betting in Binary Markets

Originally published at chudi.dev

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Binary markets strip away ambiguity. Either the event happens or it doesn't. You either get paid $1.00 per share or you get $0. This clarity makes the math clean — and makes it immediately obvious whether you have an edge.

Most people who lose money in prediction markets are not doing the math wrong. They are skipping it entirely.

TL;DR

• Expected value = (true probability × net gain) - (loss probability × stake)
• You have edge when your true probability estimate exceeds the market's implied probability
• Kelly criterion sizes the bet to maximize long-run bankroll growth
• Fees shrink your EV and require a larger edge to break even
• The hard part is honest probability estimation — every overconfidence point is money transferred to the market

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The Mechanics of a Binary Market

A binary market works like this: you buy YES shares if you believe an event will occur, or NO shares if you believe it won't. Each share costs between $0.01 and $0.99, and pays $1.00 on resolution if correct.

The price IS the market's implied probability. A YES share at $0.65 means the market believes the event has a 65% chance of occurring.

If you believe the true probability is 75%, you have edge: the market is underpricing the YES outcome. You buy YES at $0.65 and expect to receive $1.00 with 75% probability.

Expected Value: The Only Metric That Matters

Expected value (EV) is your average outcome per dollar bet, calculated across all possible resolutions.

For a binary bet:

EV = (p_true × profit_if_win) + ((1 - p_true) × loss_if_lose)

Where:

• p_true = your estimate of the true probability
• profit_if_win = payout minus entry cost = (1 - entry_price)
• loss_if_lose = entry cost = entry_price

Example: Polymarket lists "BTC up in 5 minutes" at $0.62. You estimate the true probability is 72% based on a momentum signal.

profit_if_win = 1.00 - 0.62 = $0.38 per share
loss_if_lose = $0.62 per share

EV = (0.72 × $0.38) + (0.28 × -$0.62)
EV = $0.2736 - $0.1736
EV = $0.10 per share

You expect to earn $0.10 for every $0.62 wagered — a 16.1% expected return.

If p_true = p_market (0.62), EV = 0. The market is fairly priced for your estimate. No edge.

If p_true < p_market, EV is negative. You are betting against the odds.

The EV Formula Simplified

For binary markets, EV can be expressed more cleanly:

EV per share = p_true - p_market

When p_true = 0.72 and p_market = 0.62: EV = 0.72 - 0.62 = $0.10 per share. This holds exactly for $1 payout markets.

This is the core insight: your EV is the gap between what you believe and what the market believes. If you can't articulate why your p_true is higher than p_market, you don't have edge.

Fee Drag

Fees directly reduce EV. They do not appear in the calculation above because they need to be added separately for maker vs. taker orders.

Taker orders (fill immediately): Polymarket charges approximately 2% on winning positions. This means:

Effective EV = p_true × (1 - fee_rate) - p_market × (1 - p_true) / p_true

Simplified: if you pay 2% on wins, you lose 2% × p_true per share. At p_true = 0.72:

Fee drag = 0.02 × 0.72 × $1.00 = $0.0144 per share
Effective EV = $0.10 - $0.0144 = $0.0856 per share

The fee consumed 14.4% of your edge. At lower win probabilities, the fee drag is proportionally smaller (fewer wins to charge), but the EV itself is already thinner.

Maker orders (limit orders, filled later): Zero fee on Polymarket. The full $0.10 EV is preserved. This is why maker entry is the only viable structure for thin-edge strategies.

Break-even edge for taker entry: At p_market = 0.50 and 2% taker fee, you need p_true ≥ 0.52 just to break even. Every percentage point of edge buys you $0.01 of expected profit per share. Taker fees consume 2 of those percentage points before you start.

Kelly Criterion: How Much to Bet

Once you have confirmed positive EV, Kelly criterion calculates the optimal bet size. "Optimal" means: the fraction of bankroll that maximizes long-run geometric growth.

The Kelly formula:

f* = (p × b - q) / b

Where:

• f* = fraction of bankroll to bet
• p = probability of winning (your p_true)
• q = probability of losing (1 - p)
• b = net odds (profit per $1 wagered if you win)

For our example: p = 0.72, p_market = 0.62

b = (1 - 0.62) / 0.62 = 0.38 / 0.62 = 0.613
q = 1 - 0.72 = 0.28

f* = (0.72 × 0.613 - 0.28) / 0.613
f* = (0.4414 - 0.28) / 0.613
f* = 0.1614 / 0.613
f* = 26.3%

Full Kelly says bet 26.3% of your bankroll on this trade. With a $1,000 bankroll, that is $263.

Why You Should Use Half-Kelly

Full Kelly is theoretically optimal but practically dangerous because of estimation error in p_true. If your true probability estimate is even slightly overconfident, full Kelly over-bets.

Consider: you estimate p = 0.72 but actual win rate turns out to be 0.65. Full Kelly at 0.72 overbets significantly. The result is higher volatility, larger drawdowns, and potential ruin.

Half-Kelly:

f_half = f* / 2 = 26.3% / 2 = 13.2%

At half-Kelly, you bet $132 from a $1,000 bankroll. You give up roughly 25% of growth rate in exchange for half the variance. For most practitioners, this tradeoff is favorable.

Quarter-Kelly (6.6%) is used by very risk-averse operators or when p_true confidence is low.

The Direction Question: When Is Your Edge Real?

The math above assumes p_true is accurate. Estimating it honestly is the hardest part.

For price-movement markets (BTC up/down in 5 minutes), there are three legitimate sources of edge:

1. Information advantage: You have data the market hasn't priced yet. In a latency arb strategy, the Binance momentum move is your information — the Polymarket market hasn't repriced it yet. p_true = conditional probability of continued movement given current momentum.

2. Speed advantage: You can act on public information faster than market makers can update their quotes. This is pure latency arbitrage. Your edge is time, not information.

3. Model advantage: Your probability estimate is more accurate than the crowd's. This requires having a calibrated model that outperforms the market on a sample large enough to be statistically meaningful.

What is not a source of edge: gut feeling, recency bias, pattern-matching on a small sample, "I think BTC is going up today."

Calibration Check

A properly calibrated probability estimate means: when you say 70%, you win 70% of the time over a large sample. When you say 80%, you win 80%.

Most people's estimates are overconfident. They think 70%, they win 58%. The gap is money transferred to the market.

A simple calibration test: track every trade you make with your entry p_true estimate. After 100 trades, compare your estimated win rates to actual win rates by bucket. If your "70%" bucket wins at 58%, your estimates are 12 percentage points overconfident. Adjust.

A Worked Example: BTC 5-Minute Market

You are running a momentum detection bot. BTC has moved +0.35% in the past 60 seconds. Based on historical data, when this pattern fires, BTC continues upward and resolves YES 69% of the time.

Current market: "BTC up 5 minutes" YES at $0.58.

p_true = 0.69
p_market = 0.58
EV per share = 0.69 - 0.58 = $0.11

b = (1 - 0.58) / 0.58 = 0.72
q = 0.31

f_full = (0.69 × 0.72 - 0.31) / 0.72
f_full = (0.497 - 0.31) / 0.72
f_full = 26%

f_half = 13% of bankroll

With $2,000 bankroll: bet $260 (13%). At $0.58 per share: 448 shares.

Expected outcome:

• Win (69% of time): 448 shares × $1.00 = $448. Profit: $448 - $260 = $188.
• Lose (31% of time): 448 shares × $0.00 = $0. Loss: -$260.

EV per trade: (0.69 × $188) + (0.31 × -$260) = $129.7 - $80.6 = +$49.1

Over many trades, this compounds. That is why Kelly sizing and honest EV calculation are the foundation, not optional extras.

Common Mistakes

Mistaking win rate for edge. A 65% win rate sounds impressive. At p=0.80 entry, a 65% win rate is catastrophically negative EV. Win rate only matters relative to entry price.

Ignoring fee drag on marginal edges. An edge of 3% gross becomes 1% net after taker fees. The difference between "bet aggressively" and "don't bet" can be a single fee calculation.

Treating correlated bets as independent. Three bets on "BTC up in the next hour" are not three independent bets — they are one bet expressed three ways. Kelly sizing assumes independence. Correlated positions require reducing bet size proportionally to correlation.

Updating p_true mid-trade on noise. Your pre-trade probability estimate should be locked in at entry. Reacting to market moves mid-trade by changing your p_true estimate introduces emotion into what should be a mechanical process.

The Summary Formula

For any directional bet in a binary market:

1. Estimate p_true honestly, with a calibrated model
2. Calculate EV = p_true - p_market (simplified for $1 payout markets)
3. Subtract fee drag if using taker entry (2% on wins)
4. If EV > 0, calculate Kelly fraction: f = (p × b - q) / b
5. Bet half-Kelly to account for estimation error
6. Track and recalibrate p_true against actual outcomes after every 50+ trades

Binary markets will tell you exactly how accurate your probability estimates are. There is no narrative to hide behind — only resolution and math.