Hedge Calculation Method (Daily Prediction Markets)

Overview

Objective: Treat each market as a time series of daily probabilities and find markets that move oppositely. These inverse relationships become hedge candidates with measurable risk reduction.

1. Convert daily probabilities into returns

Prediction prices are probabilities $p \in (0,1)$. We transform them to a log-odds scale and take daily changes.

Clamp

Clamp: $p_t = \min(\max(p_t, \varepsilon), 1-\varepsilon)$ with $\varepsilon = 10^{-6}$

• This step assures that logit is always finite by setting a minimum and maximum value

Log-odds

$\text{logit}(p_t) = \ln\left(\frac{p_t}{1-p_t}\right)$

Return series

$r_t = \text{logit}(p_t) - \text{logit}(p_{t-1})$

Why it matters: This $\Delta$logit return treats moves near 0 or 1 symmetrically and is better behaved than raw price differences.

2. Compute relationship strength

For two markets $A$ and $B$, align overlapping days and compute Pearson correlation on the return series.

$\rho = \text{corr}(r_A, r_B)$

• $\rho > 0$: markets move together.
• $\rho < 0$: markets move opposite (hedge potential).

Stability check: We also compute rolling correlations (30/90/365 days) to test how stable the relationship is over time.

3. Minimum-variance hedge ratio (beta)

Definition: If we hedge market $Y$ using market $X$, the optimal hedge ratio is:

$\beta_{Y\leftarrow X} = \frac{\text{Cov}(Y,X)}{\text{Var}(X)} = \rho \cdot \frac{\sigma_Y}{\sigma_X}$

Interpretation: If $\beta = -0.7$, hedge 1 unit of $Y$ with 0.7 units of $X$ in the opposite direction. More negative $\beta$ means stronger hedge scaling.

4. Linear hedge quality (R²)

Signal strength: In a one-factor regression with intercept, the explanatory power is:

$R^2 = \rho^2$

5. Expected variance reduction

Residual variance: The theoretical residual variance after hedging is:

$\text{Var}(Y - \beta X) = \text{Var}(Y) \cdot (1 - \rho^2)$

We store: hedge_variance_ratio = 1 - \rho^2. Lower is better.

6. Statistical confidence (p-value)

t-test: We compute a correlation p-value using a t-test:

$t = \rho \cdot \sqrt{\frac{n-2}{1-\rho^2}}$ $p = 2P(T \ge |t|), \; \text{df} = n-2$

Edges with high p_value are filtered out.

7. Hedge score (ranking)

Each hedge candidate gets a score combining strength, stability, and liquidity:

$\text{hedge\_score} = 0.4|\rho_{90d}| + 0.4|\rho_{30d}| - 0.1\sigma_{\rho,30d} + 0.1\,\text{liquidity\_score}$

Where: $\sigma_{\rho,30d}$ is the standard deviation of rolling 30-day correlations.

Weighting:

• 0.4 on |corr_90d|
• 0.4 on |corr_30d|
• -0.1 on corr volatility
• 0.1 on liquidity score

Higher score = better hedge candidate.

Output per hedge edge

For each recommended hedge edge we store:

• corr (relationship strength)
• beta (hedge ratio)
• vol_market, vol_hedge
• r2
• p_value
• hedge_variance_ratio
• rolling correlations (30/90/365d)
• hedge_score
• rank (top 3)