Case Study 2: Scalar Market Trading: GDP Growth Brackets

Overview

This case study demonstrates how to analyze and trade a scalar bracket market for an economic indicator. We will take a realistic GDP growth bracket market, extract the implied distribution, compare it to an independent econometric model, identify bracket-level mispricings, and construct a trading strategy.

Scenario

A prediction market platform offers quarterly GDP growth brackets for Q2 2028. The market has been open for several weeks and has attracted moderate trading volume. Here are the current prices:

The overround is 6%.

Part 1: Extract the Implied Distribution

Step 1: Remove Overround

Using the multiplicative method (simplest for initial analysis):

Step 2: Compute Implied Moments

Using bracket midpoints (with -2.0% for the first bracket and 5.0% for the last):

$$\hat{\mu} = 0.0283 \times (-2.0) + 0.0566 \times (-0.5) + 0.0849 \times 0.25 + 0.1226 \times 0.75 + 0.1698 \times 1.25$$ $$+ 0.1981 \times 1.75 + 0.1604 \times 2.25 + 0.0943 \times 2.75 + 0.0566 \times 3.5 + 0.0283 \times 5.0$$

$$\hat{\mu} = -0.0566 - 0.0283 + 0.0212 + 0.0920 + 0.2123 + 0.3467 + 0.3609 + 0.2593 + 0.1981 + 0.1415$$

$$\hat{\mu} \approx 1.547\%$$

For the variance, we compute $\hat{\sigma}^2 = \sum q_k (m_k - \hat{\mu})^2$:

$$\hat{\sigma}^2 = 0.0283(−2.0 − 1.547)^2 + 0.0566(−0.5 − 1.547)^2 + \ldots$$

$$\hat{\sigma}^2 \approx 1.857$$

$$\hat{\sigma} \approx 1.363\%$$

Implied skewness: Computing the third moment, we find $\hat{\gamma} \approx -0.12$, indicating a slight left skew (heavier left tail). This is consistent with the general observation that GDP growth has asymmetric downside risk.

Step 3: Build the Implied CDF

The cumulative distribution at each bracket boundary:

The median (50th percentile) falls between 1.5% and 2.0%, closer to 1.6%.

Part 2: Econometric Model Comparison

Our Econometric Model

We maintain a Bayesian VAR model that incorporates: - Current employment data - Consumer spending trends - Manufacturing PMI - Yield curve information - Federal Reserve policy stance - Leading economic indicators

Our model produces a predictive distribution for Q2 2028 GDP growth:

$$\text{GDP growth} \sim N(\mu = 2.1\%, \sigma = 1.1\%)$$

Note: Our model predicts a higher mean (2.1% vs. market's implied 1.55%) and lower variance (1.1% vs. market's implied 1.36%).

Bracket-by-Bracket Comparison

Key Observations

1. The center-right brackets (1.0--2.0%) are significantly underpriced: The market assigns 36.8% to the 1.0--2.0% range, while our model assigns 49.1%. This 12 percentage point gap is the largest concentration of mispricing.

2. Both tails are overpriced: The market assigns 8.49% to outcomes below 0% and 8.49% to outcomes above 3.0%. Our model assigns only 3.05% and 2.53% respectively. The market appears to have excess uncertainty (too much variance).

3. The mean shift creates a systematic pattern: Because our model has a higher mean, all brackets below our mean are relatively overpriced and all brackets above are relatively underpriced, with the effect strongest near the center.

4. The variance difference amplifies the tail effect: Our lower variance means we assign even less probability to extremes, making the tails appear even more overpriced.

Part 3: Trading Strategy Construction

Strategy 1: The Mean-Shift Trade

Buy the brackets where GDP growth is 1.0--2.0% and sell where it is below 0% and above 3.0%.

Position: - Buy Bracket 5 (1.0--1.5%): Strong edge of 7.15 percentage points - Buy Bracket 6 (1.5--2.0%): Strong edge of 5.12 percentage points - Sell Bracket 1 (Below -1.0%): Edge of 2.59 percentage points - Sell Bracket 2 (-1.0% to 0.0%): Edge of 2.85 percentage points - Sell Bracket 9 (3.0--4.0%): Edge of 3.38 percentage points - Sell Bracket 10 (Above 4.0%): Edge of 2.58 percentage points

Sizing (Quarter-Kelly, conservative given model uncertainty):

Net cost: $396 + $357 - $2.40 - $3.60 - $3.60 - $1.50 = $741.90

Strategy 2: The Variance-Compression Trade

This strategy focuses purely on the variance disagreement, without taking a view on the mean. It buys center brackets and sells tail brackets.

Position: - Buy Brackets 4-7 (0.5--2.5%): Center of the distribution - Sell Brackets 1-2 and 9-10: Tails

This trade profits if GDP growth is moderate (between 0.5% and 2.5%) and loses if GDP growth is extreme.

Strategy 3: Combined (Recommended)

We combine elements of both strategies, emphasizing the brackets with the strongest edge:

Core positions: - BUY 250 contracts of Bracket 5 (1.0-1.5%) at $0.18 = $450 - BUY 200 contracts of Bracket 6 (1.5-2.0%) at $0.21 = $420 - BUY 100 contracts of Bracket 4 (0.5-1.0%) at $0.13 = $130 - SELL 80 contracts of Bracket 1 (Below -1%) at $0.03 = $2.40 credit - SELL 60 contracts of Bracket 10 (Above 4%) at $0.03 = $1.80 credit

Net investment: $1,000 - $4.20 = $995.80

Part 4: Payoff Analysis

Scenario-by-Scenario Payoffs

For the combined strategy on a $10,000 bankroll:

Expected Value

$$E[\text{P\&L}] = \sum_k q_k \times \text{P\&L}_k$$

$$= 0.0024 \times (-1073) + 0.0281 \times (-993) + 0.0735 \times (-992) + 0.1585 \times (-97)$$ $$+ 0.2413 \times 843 + 0.2493 \times 377 + 0.1598 \times (-992) + 0.0618 \times (-992)$$ $$+ 0.0228 \times (-992) + 0.0025 \times (-1054)$$

$$\approx -2.58 - 27.90 - 72.89 - 15.38 + 203.34 + 93.96 - 158.52 - 61.31 - 22.62 - 2.64$$

$$\approx -\$66.55$$

Wait --- this negative expected value warrants investigation. Let us recalculate more carefully.

The issue is that our position sizes are too conservative relative to the cost. Let us recalculate with proper Kelly sizing from the optimizer (see code/case-study-code.py).

Corrected Analysis with Optimizer

Running the proper multi-outcome Kelly optimizer with our model probabilities and market prices, the quarter-Kelly optimal allocation is:

Note: The optimizer does not recommend the sell positions because the short-selling risk is asymmetric in this market structure --- the potential loss from a short position that pays out is large relative to the small credit received.

With corrected positions (on $10,000 bankroll): - Buy $180 of Bracket 4 at 0.13 = 138 contracts - Buy $520 of Bracket 5 at 0.18 = 289 contracts - Buy $380 of Bracket 6 at 0.21 = 181 contracts - Total cost: $1,080

Revised payoffs:

Hmm, let us be more precise. When buying $520 at price 0.18 per contract, we buy $520/$0.18 = 2,889 "shares" at $0.18. If Bracket 5 wins, each share pays $1, so the payout is $2,889. Net P&L = $2,889 - $1,080 = +$1,809.

Let me redo this properly:

Corrected: - Buy 1,385 shares of Bracket 4 at $0.13 each = cost $180.05 - Buy 2,889 shares of Bracket 5 at $0.18 each = cost $520.02 - Buy 1,810 shares of Bracket 6 at $0.21 each = cost $380.10 - Total cost: $1,080.17

Payoffs:

Expected P&L: $$E = 0.104 \times (-1080) + 0.1585 \times 305 + 0.2413 \times 1809 + 0.2493 \times 730$$ $$+ 0.1598 \times (-1080) + 0.0871 \times (-1080)$$

$$= -112.32 + 48.34 + 436.51 + 181.99 - 172.58 - 94.07 = +\$287.87$$

Expected return: $287.87 / $10,000 = 2.88%

This is more reasonable. The expected return is positive, driven by the concentration of probability in the 1.0--2.0% range where we hold our largest positions.

Risk Metrics

• Probability of profit: 65.0% (if GDP is 0.5--2.0%)
• Maximum loss: $1,080 (10.8% of bankroll)
• Maximum gain: $1,809 (18.1% of bankroll)
• Expected return: 2.88%
• Probability of max loss: 35.0%

Part 5: Distribution Shape Analysis

Is the Market Right to Have Fat Tails?

Our normal distribution model assigns very low probability to extreme outcomes. But GDP growth can experience large shocks (e.g., pandemic, financial crisis). Should we really trust a normal distribution?

Arguments for our model (thinner tails): - The current economic environment is stable - No major systemic risks on the horizon - The Fed has tools to manage volatility - Historical quarterly GDP growth has been relatively well-behaved in the absence of crises

Arguments for the market (fatter tails): - Black swan events are always possible - The market may be incorporating geopolitical risks - Participants may have information our model does not capture - GDP measurement revisions create additional uncertainty

Robustness Check: Student's t Distribution

To check robustness, we also fit a Student's t distribution (which has fatter tails than normal) to the bracket data:

$$\text{GDP growth} \sim t(\nu = 5, \mu = 2.0\%, \sigma = 0.9\%)$$

With heavier tails, the mispricing pattern changes:

The t-distribution model still finds the center brackets underpriced, but the tail mispricings are smaller. This suggests the center-bracket trade is more robust than the tail-selling strategy.

Part 6: Monitoring and Exit Strategy

Key Data Releases to Monitor

1. Advance GDP Estimate: First official estimate, released about 4 weeks after quarter end. This will be the resolution event.
2. Monthly GDP Proxies: Industrial production, retail sales, employment reports provide monthly updates.
3. GDPNow (Atlanta Fed): A real-time model updated frequently. If GDPNow aligns with our model, it confirms our view.

Exit Rules

1. Price target reached: If Bracket 5 (1.0-1.5%) rises from 0.18 to 0.24 or higher, our edge has mostly evaporated. Consider selling.
2. Model update reversal: If incoming data causes our model to shift the mean below 1.5%, our primary thesis is weakened. Reduce positions.
3. Volatility spike: If a crisis occurs (financial stress, geopolitical shock), the distribution should widen. Our low-variance thesis breaks down. Exit quickly.
4. Time decay: As the quarter progresses and data accumulates, uncertainty resolves. If prices have not moved in our favor by mid-quarter, reevaluate.

Position Management Timeline

• Weeks 1-4: Full position. Monitor for major data surprises.
• Weeks 5-8: Reassess with mid-quarter data. Adjust sizing if model estimates have changed.
• Weeks 9-12: Begin reducing position as resolution approaches. Lock in profits on brackets that have moved favorably.
• Post-quarter: Hold through official GDP release if position is still positive EV.

Conclusion

This case study demonstrates the scalar market trading workflow:

1. Extract implied distribution from bracket prices (remove overround, compute moments).
2. Compare to independent model (our Bayesian VAR predicts higher mean, lower variance).
3. Identify bracket-level opportunities (center brackets underpriced, tails overpriced).
4. Size with Kelly criterion (quarter-Kelly given model uncertainty).
5. Analyze risk (65% chance of profit, max loss = 10.8% of bankroll).
6. Check robustness (center-bracket trade survives alternative distributional assumptions).
7. Plan monitoring and exits (data releases, price targets, model updates).

The key insight for scalar markets: the smoothness constraint imposed by any reasonable continuous distribution creates structure that discrete bracket markets can violate, generating trading opportunities.

Full code implementation is in code/case-study-code.py and code/example-03-scalar-analysis.py.