Logit-Space Pricing

Black-Scholes framework for prediction markets with Greeks and PnL decomposition

Prediction market prices are probabilities bounded to [0, 1]. The logit transform ln(p/(1-p)) maps them to unbounded real space where normal Black-Scholes math works. This gives you proper Greeks, boundary-aware spreads, and PnL attribution for binary contracts.

Core transforms

python
import horizon as hz

x = hz.logit(0.65)       # 0.619
p = hz.sigmoid(0.619)    # 0.65

Greeks for binary contracts

python
greeks = hz.logit_greeks(
    prob=0.65,
    belief_vol=0.15,       # volatility in logit space
    time_to_resolve=30.0,  # days
)

print(f"Delta: {greeks.delta:.4f}")    # price sensitivity to belief shift
print(f"Gamma: {greeks.gamma:.4f}")    # convexity
print(f"Vega:  {greeks.vega:.4f}")     # sensitivity to belief-vol
print(f"Theta: {greeks.theta:.4f}")    # time decay

PnL decomposition

python
pnl = hz.logit_pnl_decompose(
    entry_prob=0.55,
    exit_prob=0.65,
    belief_vol=0.15,
    dt=5.0,  # days held
)

print(f"Delta PnL:  {pnl.delta_pnl:.4f}")
print(f"Gamma PnL:  {pnl.gamma_pnl:.4f}")
print(f"Theta PnL:  {pnl.theta_pnl:.4f}")
print(f"Residual:   {pnl.residual:.4f}")

Separates your P&L into directional (delta), convexity (gamma), time decay (theta), and unexplained components.

When to use

  • Pricing and hedging binary contracts (prediction markets, binary options)
  • Computing Greeks for risk management on probability-priced assets
  • Decomposing P&L to understand what’s driving returns

Next

Belief-Volatility Surface Streaming estimation of belief-vol. Characteristic Function Pricing Heston, Merton, Variance Gamma models.