STARTING KELLY PM: OPTIMAL POSITION SIZING FOR PREDICTION MARKETS

Why This Project

Prediction markets are one of the purest forms of probability trading. You buy contracts at a price that implies a probability, and you profit if your estimate is closer to reality than the market's. The central question is: given an edge, how much of your capital should you risk?

This is exactly the problem John Kelly solved in 1956 at Bell Labs. His criterion maximizes the expected logarithmic growth rate of capital — the only strategy that is both asymptotically optimal and avoids ruin with probability 1.

I'm building Kelly PM — a position sizing engine that applies this framework to prediction markets with fractional scaling and multi-contract portfolio management.

Derivation from First Principles

The Classical Form

Consider a binary outcome with odds K (decimal). You estimate the probability of winning at p. The Kelly fraction F — the proportion of bankroll to wager — maximizes the expected value of log(wealth):

E[log(W)] = p · log(1 + F·(K-1)) + (1-p) · log(1 - F)

Taking the derivative, setting it to zero, and solving for F:

F* = (K·p - 1) / (K - 1)

This is the full Kelly optimal fraction. When F* ≤ 0, you have no edge — don't bet.

The Prediction Market Simplification

On prediction markets, you buy contracts at price P ∈ (0,1). If the outcome occurs, you receive $1. The implied odds are:

K = 1 / P

Substituting into the Kelly formula:

F = (p/P - 1) / (1/P - 1)
  = (p - P) / P · P / (1 - P)
  = (p - P) / (1 - P)

The simplified prediction market Kelly formula:

F = (p − P) / (1 − P)

where:
  F = fraction of bankroll to allocate
  p = your estimated true probability
  P = market price (implied probability)
  (p − P) = your edge
  (1 − P) = potential profit per dollar risked

The formula reduces to: position size = edge / potential profit. Elegant and intuitive.

Sensitivity Analysis

Linear Response to Edge

When market price P is fixed and your estimated probability p varies, Kelly responds linearly:

∂F/∂p = 1 / (1 − P) = constant

A 10% increase in estimated probability always produces the same increase in position size, regardless of your current estimate. Kelly scales confidence into capital allocation in a perfectly proportional way.

Nonlinear Response to Market Price

When your estimate p is fixed and market price P varies, the response is nonlinear:

∂F/∂P = (p − 1) / (1 − P)²

This is a rational function — as P decreases (cheaper contracts, higher potential payout), Kelly accelerates. This is why low-probability contracts with genuine edge produce aggressively large Kelly fractions.

The Asymmetry of Equal Ratios

Two scenarios with identical p/P ratios produce very different Kelly fractions:

Scenario	P	p	p/P	Kelly F
A	0.20	0.40	2.0×	0.250
B	0.40	0.80	2.0×	0.667

Scenario B allocates 2.67× more capital despite the same probability ratio. Why? Kelly balances win frequency and payout magnitude. In B, you win 80% of the time with a solid 2.5× payout — that combination drives Kelly much higher than winning 40% of the time with a 5× payout.

This is the core insight: it's not just about how much better your estimate is than the market's — it's the joint distribution of frequency and magnitude.

Fractional Kelly — Accounting for Estimation Error

The full Kelly formula assumes your probability estimate p is correct. In practice, it never is. The Kelly fraction is highly sensitive to overestimation of edge:

• If true edge is 5% but you estimate 10%, you bet 2× too much
• Overbetting beyond the Kelly point increases variance faster than it increases expected growth
• At 2× Kelly, your expected growth rate equals zero — the same as not betting at all

Fractional Kelly scales the optimal fraction by a constant α ∈ (0, 1):

F_fractional = α · F_kelly

The variance of log-wealth scales as α² while expected growth scales as α(2 - α). At half Kelly (α = 0.5), you retain 75% of the growth rate with only 25% of the variance. This is a remarkably favorable trade-off.

Practical guidelines for prediction markets:

• α = 0.50 — high-confidence model, backtested extensively
• α = 0.25 — standard approach, accounts for model uncertainty
• α = 0.125 — noisy or illiquid markets
• α < 0.125 — tail risk present, information shocks possible

Most serious prediction market traders operate at ¼ Kelly or below.

What I'm Building

Kelly PM will implement:

1. Edge calculator — compare user estimates against live market prices
2. Fractional Kelly engine — configurable α with regime-aware defaults
3. Multi-contract portfolio sizing — correlation-adjusted Kelly across simultaneous positions
4. Bankroll tracker — real-time P&L, drawdown monitoring, position history
5. Backtester — validate Kelly performance against historical prediction market data

The goal is a tool that turns the theory above into disciplined, systematic capital allocation — controlling greed with mathematics.