OPTIONS DESK SPEC ADDENDUM — MERGED FINAL

Actionable Edge Mapping + Prediction Market Ladder

Base: Claude Opus 4.6 (complete, unmodified) + Cherry-picked Panel Additions

Panel: Gemini Pro | GPT-4.1 | Grok 4.1 Fast | DeepSeek V3 (unique additions tagged)

Options broker: TBD — all broker trades described generically (buy/sell the instrument, not platform steps)

Both Kalshi and Polymarket ladder structures are treated as equal primary targets throughout

This addendum supplements the original Options Desk spec. It does NOT replace the existing spec — it adds two critical layers: (1) mapping every edge to a specific trade on a specific platform, and (2) a systematic methodology for scanning prediction market ladders using options-derived probability models.

TOPIC 1: ACTIONABLE EDGE OUTPUT

1. Edge-to-Trade Mapping Table

2. Actionable Output Format

JSON output sent via Telegram and to other desks:

{
  "edge_id": "OPTIONS-20260315-SPY-BL-001",
  "desk": "options",
  "timestamp": "2026-03-15T11:30:00Z",
  "edge_description": "Breeden-Litzenberger probability extraction shows Kalshi 'S&P above 5300' is underpriced by 8c vs options-implied probability.",
  "signal_type": "breeden_litzenberger_mispricing",
  "underlying": "SPY / S&P 500",
  "confidence_level": 80,
  "vol_regime": {
    "vix": 18.5,
    "vix_percentile_52w": 35,
    "vix_term_structure": "contango (normal)",
    "vrp": 4.2,
    "realized_vol_20d": 14.3
  },
  "gex_context": {
    "net_gex": "+$2.1B",
    "dealer_position": "long_gamma",
    "gamma_wall": 5250,
    "zero_gamma_level": 5150,
    "implication": "Dealers will sell rallies above 5250 — acts as resistance"
  },
  "platforms": {
    "Kalshi": {
      "mispriced_contracts": [
        {
          "contract": "S&P above 5300 (Mar 21 weekly)",
          "current_price": 14,
          "fair_value_BL": 22,
          "edge_cents": 8,
          "edge_after_fees": 6,
          "action": "Buy YES at 14c",
          "liquidity_usd": 1200
        },
        {
          "contract": "S&P above 5350 (Mar 21 weekly)",
          "current_price": 5,
          "fair_value_BL": 11,
          "edge_cents": 6,
          "edge_after_fees": 4,
          "action": "Buy YES at 5c",
          "liquidity_usd": 600
        }
      ]
    },
    "Polymarket": {
      "mispriced_contracts": [
        {
          "contract": "S&P above 5300 (equivalent weekly)",
          "current_price": 13,
          "fair_value_BL": 22,
          "edge_cents": 9,
          "edge_after_fees": 7,
          "action": "Buy YES at 13c — better price than Kalshi this week",
          "liquidity_usd": 800
        }
      ],
      "note": "Always scan both platforms. Polymarket may price the same threshold differently. Take the better price, or split position across both if both have edge."
    },
    "Broker": {
      "action": "Buy SPY Mar 21 $530 call (if edge is directional) or sell $530/$535 call spread for credit if GEX says $535 is capped",
      "note": "GEX gamma wall at $525 may limit upside — consider the Kalshi/Polymarket trade as pure probability arbitrage rather than a directional bet"
    }
  },
  "time_sensitivity": "MEDIUM — weekly expiry in 6 days. Edge should persist 1-3 days as prediction markets adjust.",
  "risk_factors": ["OpEx Friday may pin SPY near gamma wall", "FOMC on Wednesday could override all technical levels", "GEX suggests $535 is capped — YES on 5350 is riskier"]
}

Telegram summary format:

OPTIONS EDGE: S&P Breeden-Litzenberger Mispricing

Options chain implies 22% chance S&P > 5300 by Mar 21
Kalshi: 14c | Edge = 8c (6c net)
Polymarket: 13c | Edge = 9c (7c net) — BETTER PRICE

Also: S&P > 5350 @ 5c | Fair 11c | Edge 6c (Kalshi)

Vol context: VIX 18.5 (35th pctile) | VRP 4.2 | RV 14.3
GEX: +$2.1B | Gamma wall $5250 | Dealers long gamma
Warning: GEX says upside may be capped at $5250

Broker: Buy SPY $530C | OR sell call spread for credit
Confidence: 80% | Window: 1-3 days

3. Platform Priority Ranking

TOPIC 2: PREDICTION MARKET LADDER STRUCTURE

1. Desk Data Mapping to Prediction Market Contracts

The Options Desk has the most direct mapping to prediction market contracts of any desk, because prediction market contracts ARE binary options, and the Options Desk specializes in option pricing. Both Kalshi and Polymarket run ladder structures — multiple contracts at different price thresholds for the same underlying. Scan both fully on every cycle.

2. Sharp Reference Identification

3. Probability Curve Construction Method

The Options Desk uses the Breeden-Litzenberger (BL) formula as its primary probability extraction method. This is model-free — it doesn't assume log-normal returns, doesn't require an IV smile parameterization, and incorporates all the skew and kurtosis the market is pricing.

Breeden-Litzenberger method:

The risk-neutral probability density f(K) at strike K is:

f(K) = e^(rT) * d²C/dK²

Where:

• C(K) is the call price at strike K
• r is the risk-free rate (SOFR)
• T is time to expiry

Practical implementation (finite difference):

For a discrete set of strikes K₁ < K₂ < K₃:

f(K₂) ≈ e^(rT) * [C(K₁) - 2*C(K₂) + C(K₃)] / (ΔK)²

Where ΔK = K₂ - K₁ = K₃ - K₂ (equal spacing required; interpolate if needed).

From density to cumulative probability:

P(S > K) = e^(rT) * Σ f(K_i) * ΔK for all K_i > K

Or more practically, use the call spread approximation:

P(S > K) ≈ -dC/dK ≈ [C(K - ΔK) - C(K + ΔK)] / (2 * ΔK) * e^(rT)

Inputs needed:

• Full options chain (all strikes, bid/ask mid prices) for the relevant expiry from Polygon.io
• Risk-free rate (SOFR from FRED)
• Dividend yield (for stocks with dividends — affects put-call parity)

Adjustments:

1. Bid-ask spread: Use mid prices. For illiquid strikes, apply a liquidity discount to the extracted probability (wider spread → less confident in the price).
2. 0DTE gamma effect: For same-day expirations, gamma is extreme. BL probabilities from 0DTE options may be noisy. Prefer 1-7 DTE options for extraction.
3. Earnings event adjustment: If earnings fall between now and contract expiry, the options chain has already priced in the event vol. The BL probability INCLUDES the earnings move. No additional adjustment needed.
4. Fat tails: BL naturally captures fat tails because it uses actual market prices (which incorporate the smile/skew). No additional fat-tail adjustment needed — this is a key advantage over Black-Scholes N(d2).

Recalculation frequency:

• Every 15 minutes during market hours for SPY/QQQ and the top 10 most active Kalshi/Polymarket stocks.
• Every hour for other stocks on the watch list.
• On demand: whenever a new Kalshi or Polymarket contract appears or when IV shifts >2% intraday.

[PANEL: Gemini Pro + Grok 4.1 Fast] SABR Model for Vol Smile Fitting

The BL finite-difference method requires a dense, continuous strike grid. When the raw options chain has gaps between strikes, or when you need to interpolate the vol smile smoothly across the entire surface, the SABR (Stochastic Alpha Beta Rho) model provides a parametric fit to the implied volatility smile.

When to use SABR vs raw BL:

• BL (preferred): When the chain has strikes every $1-5 for liquid names (SPY, QQQ, AAPL, NVDA). Full grid available, no need to interpolate.
• SABR (supplemental): When the chain has sparse strikes (every $5-10+), when you need to interpolate the smile to an arbitrary threshold that falls between available strikes, or when you want a smooth, arbitrage-free volatility surface across maturities.

SABR model produces implied volatility at any strike K, given four parameters:

def sabr_vol(F, K, T, alpha, beta, rho, nu):
    """
    F     = forward price of the underlying
    K     = strike price
    T     = time to expiry (years)
    alpha = vol of vol (controls overall level of vol)
    beta  = elasticity (0=normal, 1=log-normal; typically 0.5 for equities)
    rho   = correlation between spot and vol (-1 to 1; negative = skew)
    nu    = volvol (controls smile curvature)

    Returns: Black-Scholes implied vol at strike K for maturity T
    """
    if abs(F - K) < 1e-10:  # ATM approximation
        sigma_atm = alpha / (F ** (1 - beta))
        term1 = 1 + (((1 - beta) ** 2 / 24) * alpha ** 2 / (F ** (2 - 2 * beta))
                     + (rho * beta * nu * alpha / (4 * F ** (1 - beta)))
                     + ((2 - 3 * rho ** 2) * nu ** 2 / 24)) * T
        return sigma_atm * term1
    # Off-ATM formula omitted for brevity — use py-sabr or QuantLib for full implementation
    ...

Calibration process:

1. Pull the liquid strikes from Polygon.io (strikes with bid/ask spread <20% of mid).
2. Convert each option price to Black-Scholes implied vol.
3. Fit the four SABR parameters (alpha, beta, rho, nu) to minimize squared error across all strikes using scipy.optimize.minimize.
4. Use the fitted SABR surface to generate implied vol at any arbitrary strike — including Kalshi/Polymarket thresholds that fall between traded strikes.
5. Convert SABR-implied vol back to option price using Black-Scholes, then apply BL to get the probability.

Key insight: After SABR fitting, you can evaluate the probability at ANY threshold on the Kalshi or Polymarket ladder, even if no option trades at exactly that strike. This is especially valuable for thinly-traded individual stocks and for VIX contracts.

VIX-specific note: Use a Kalman filter on VIX historical mean-reversion to calibrate the VIX probability distribution. VIX is mean-reverting (not log-normal), so the SABR fit should be complemented with a mean-reversion adjustment:

# Kalman filter estimate of VIX mean-reversion speed (kappa) and long-run mean (theta)
# From VIX futures term structure: each futures price F(t) = theta + (VIX_spot - theta) * exp(-kappa * t)
# Fit kappa and theta to the futures curve daily
# Then P(VIX > X at time T) = probability from mean-reverting lognormal (CIR or Heston-like)

[PANEL: Gemini Pro] Economic Data Contracts — Normal Distribution with Historical RMSE Calibration

For CPI, NFP, GDP, and other economic data prediction market contracts, the BL method does not apply (no options chain). Use a Normal distribution calibrated to historical forecast errors:

• Mean (μ): Set to the sharp reference consensus forecast (e.g., Bloomberg consensus 3.1% CPI). Source: TradingEconomics (free).
• Standard deviation (σ): Derived from the root-mean-square error (RMSE) of consensus forecasts for that specific data release over the past 5 years. This is different from realized vol — it measures how wrong the consensus tends to be, not how volatile the series is.
• Historical database: Maintain a database of the past 5 years of major economic data releases, the corresponding consensus forecasts, and the actual outcomes. Use this to calculate the forecast error distribution.
• Formula: P(CPI > X) = 1 - NORMCDF((X - μ) / σ)
• Recalculation: When consensus estimates update (typically daily or weekly leading up to the release).

This is more precise than using historical vol of the data series, because it directly models the question "given where consensus is, how likely is a surprise above X?" — which is exactly what the Kalshi/Polymarket contract prices.

4. Ladder Scanning Methodology

Step-by-step process — applied to BOTH Kalshi and Polymarket simultaneously:

1. Enumerate all active Kalshi and Polymarket contracts: Pull stock price, S&P, Nasdaq, VIX, and economic data contracts from both platform APIs. For each, note the threshold, expiry, current price, and bid/ask spread. Map contracts from both platforms to the same underlying and threshold for direct comparison.

2. For each underlying, pull the closest-expiry options chain from Polygon.io. Build the BL probability curve. If strikes are sparse, supplement with SABR fitting (see Section 3 panel addition above).

3. For each contract threshold on each platform:

   • Look up the BL-extracted fair probability at that threshold.
   • If the threshold is between available strikes, interpolate the BL curve (or use SABR-derived probability).
   • For VIX contracts: use VIX futures + VVIX + Kalman filter to construct the VIX probability distribution.
   • For economic data contracts: use the normal distribution + historical RMSE method (see Section 3 panel addition above). The Options Desk does not have a comparative advantage over FedWatch for rate decisions — pass through those estimates.

4. Calculate raw edge: raw_edge = |BL_fair_probability - contract_price|.

5. Apply GEX overlay:

   • If a gamma wall exists between current price and the contract threshold, the probability of crossing it is LOWER than BL suggests (GEX creates a barrier). Reduce YES-side fair probability by 5-15% depending on gamma wall strength.
   • If price is above the zero-gamma level and positive GEX is pulling it toward the gamma wall, the probability of staying ABOVE the wall is HIGHER. Boost YES-side for contracts below the wall.

6. Apply VRP overlay:

   • If VRP > 6 (options expensive): BL probabilities for tail contracts are biased high (the options are overpriced, so the BL curve overstates tail probabilities). Reduce tail fair probabilities by VRP/20 (e.g., VRP=8 → reduce tails by 40%). This means tail Kalshi and Polymarket contracts are EVEN MORE overpriced than BL suggests.
   • If VRP < 2 (options cheap): BL may understate tail risk. Increase tail fair probabilities slightly.

7. Net edge after fees: Subtract platform fees for each platform separately. Kalshi and Polymarket have different fee structures — calculate net edge independently for each.

8. Filter: Minimum 3c net edge, minimum $100 liquidity, minimum 65% confidence.

9. Cross-platform comparison: For each threshold where both Kalshi and Polymarket have a contract, compare the net edges. Flag the better platform. If both have edge, flag both — split the position across platforms.

10. Rank by edge-adjusted-for-GEX: net_edge * (1 + GEX_alignment_bonus) where GEX alignment = +0.2 if GEX supports the trade direction, -0.2 if GEX opposes.

11. Output: Top 10 edges to Telegram, explicitly noting which platform (Kalshi, Polymarket, or both) for each edge. Full list to database. Share BL probability curves with Stocks Desk and Crypto Desk.

[PANEL: Grok 4.1 Fast] PDF Integration as Complement to BL Finite Difference

In addition to the BL finite-difference method, the desk can integrate the implied probability density function (PDF) directly between ladder thresholds. This is useful when a prediction market contract covers a range (e.g., "S&P closes between 5200 and 5300") rather than a simple above/below binary.

def calculate_edge(contract, implied_pdf, strikes):
    """
    contract.low  = lower threshold
    contract.high = upper threshold (use np.inf for 'above X' contracts)
    implied_pdf   = array of probability densities from BL extraction
    strikes       = corresponding strike array
    """
    # Integrate PDF over the contract's range
    mask = (strikes >= contract.low) & (strikes <= contract.high)
    fair_prob = np.trapz(implied_pdf[mask], strikes[mask])

    raw_edge = fair_prob - contract.market_price
    net_edge = raw_edge - (contract.fee_rate * 2)  # Round-trip fees

    if net_edge > 0.03 and contract.liquidity > 50:
        return net_edge
    return None

Prioritize contracts where:

• Edge > 5c
• Threshold > 1σ from current spot (tails)
• Time decay < 5% per day (avoid 0DTE noise)

5. Edge Examples with Real Numbers

Example 1: SPY Breeden-Litzenberger Tail Mispricing — Cross-Platform

• Date: March 15, 2026. SPY at $521.
• SPY Mar 21 options chain (Polygon.io): Strikes from $490 to $550, every $1.
• BL extraction at key thresholds:
  • P(SPY > $530) = 18.2% (BL)
  • P(SPY > $535) = 10.5% (BL)
  • P(SPY > $540) = 5.1% (BL)
  • P(SPY > $545) = 2.0% (BL)
• GEX context: Gamma wall at $525 (+$1.8B GEX). Dealers will sell into rallies above $525. This reduces the probability of reaching $530+ significantly.
• GEX-adjusted probabilities:
  • P(SPY > $530) = 18.2% * 0.85 = 15.5% (GEX penalizes by 15%)
  • P(SPY > $535) = 10.5% * 0.80 = 8.4%
  • P(SPY > $540) = 5.1% * 0.80 = 4.1%
• VRP: 4.2 (moderate). No significant VRP adjustment.
• Kalshi rung: "S&P above 5350" at 15c. BL says 10.5%, GEX-adjusted says 8.4%.
• Polymarket rung: "S&P above 5350" at 13c. Same BL/GEX fair value 8.4%.
• Edge direction: Both platforms are OVERPRICED at their prices vs 8.4% fair. Buy NO on both.
• Kalshi raw edge: 15% - 8.4% = 6.6c → Net 5.1c
• Polymarket raw edge: 13% - 8.4% = 4.6c → Net 3.1c (above minimum filter)
• Trade: Buy NO on "S&P above 5350" on Kalshi (primary, better price). Also buy NO on Polymarket if liquidity available. GEX gamma wall makes this threshold very unlikely to be crossed this week.

Example 2: NVDA Earnings — Options Overpricing the Move

• Setup: NVDA reports Wednesday after close. Current price $920. Options imply 9.2% move (straddle at $920 costs $85). Historical average post-earnings move: 6.8%. Options are overpricing the event by ~2.4 vol points.
• BL extraction for post-earnings probabilities:
  • P(NVDA > $960 on Thursday) = 28.5% (BL, includes elevated event IV)
  • P(NVDA > $1000 on Thursday) = 14.2% (BL)
  • P(NVDA > $1040 on Thursday) = 5.8% (BL)
• Historical-adjusted probabilities (using 6.8% avg move instead of 9.2% implied):
  • P(NVDA > $960) = 22.1% (actual historical frequency)
  • P(NVDA > $1000) = 9.0%
  • P(NVDA > $1040) = 2.8%
• Kalshi rung: "NVDA above $1000" (Thursday, post-earnings) at 18c.
• Polymarket rung: "NVDA above $1000" (Thursday, post-earnings) at 16c.
• Fair value: BL says 14.2% (biased high by expensive event IV). Historical says 9.0%. Blended: 11.5%.
• Kalshi edge: 18% - 11.5% = 6.5c → Net 5.0c
• Polymarket edge: 16% - 11.5% = 4.5c → Net 2.5c (below 3c filter — skip Polymarket, take Kalshi only)
• Trade: Buy NO on "NVDA above $1000" on Kalshi at 82c. Options are overpricing the NVDA move.
• Broker trade: Sell NVDA iron butterfly ($920 center, $960/$880 wings) to collect IV crush.

Example 3: VIX Spike — Prediction Market Overreaction

• Setup: VIX spiked from 16 to 24 today (large move). Both Kalshi and Polymarket have VIX threshold contracts.
• VIX futures curve: VIX spot 24, M1 futures 22.5, M2 futures 21.0 (backwardation — market expects VIX to decline).
• VVIX: 110 (elevated but not extreme — implies moderate uncertainty about VIX direction).
• Kalman filter: Fitted mean-reversion speed κ=4.2, long-run mean θ=17.5. At 5 days: expected VIX = 17.5 + (24 - 17.5) * exp(-4.2 * 5/252) = ~23.1. Most of the spike persists for 5 days.
• Historical: When VIX spikes >40% in 1 day, it mean-reverts to ~18-20 within 2 weeks in 75% of cases.
• Kalshi rung: "VIX above 30" (end of week, 5 days) at 25c (implied 25%).
• Polymarket rung: "VIX above 30" (end of week, 5 days) at 27c (implied 27%).
• Fair probability: Using log-normal on VIX with current spot 24, mean-reverting to 22.5, VVIX-implied vol ~25%:
  • P(VIX > 30 in 5 days) ≈ 12%
• Kalshi raw edge: 25% - 12% = 13c → Net 11.5c
• Polymarket raw edge: 27% - 12% = 15c → Net 13.5c (Polymarket actually more overpriced here)
• Trade: Buy NO on "VIX above 30" on BOTH platforms. Polymarket has the larger edge this time. The prediction markets are overreacting to the VIX spike; the futures curve says the spike will fade.

IMPORTANT NOTES

• The Options Desk's unique value is the Breeden-Litzenberger probability extraction — this is the most accurate method for pricing prediction market contracts on any underlying that has liquid options.
• Both Kalshi and Polymarket are primary targets. Neither is secondary. The same BL probability curve applies to both. Run the full scan on both platforms every cycle. Compare prices — take the better edge, or split position if both qualify.
• GEX analysis is a critical overlay. BL probabilities assume random walks, but GEX creates mechanical buying/selling at specific levels that SYSTEMATICALLY changes the probability distribution. Always apply GEX adjustments to BL probabilities.
• VRP (variance risk premium) tells you the direction of systematic bias in all options-derived probabilities. When VRP is high, BL overstates tail probabilities → prediction market tails are even more overpriced on BOTH Kalshi and Polymarket.
• The Options Desk should SHARE its BL probability curves with the Stocks Desk and Crypto Desk. They need these curves to price their own prediction market contracts accurately.
• Polygon.io ($29/mo) is non-negotiable for this desk. Without reliable options chain data at 15-minute frequency, the BL extraction is impossible.
• 0DTE options are noisy for BL extraction. Use 1-7 DTE options as the primary source. 0DTE data feeds the GEX calculation separately.
• Earnings are the highest-value event for this desk on prediction markets. The IV-historical move divergence creates systematic overpricing of prediction market contracts around earnings on both Kalshi and Polymarket.
• Broker is TBD. All broker trade descriptions in this spec are generic: describe the instrument and action (e.g., "sell 5 AAPL Jan 180 puts, delta-hedge with 200 shares"), not platform-specific UI steps. The methodology is broker-agnostic.
• SABR model is the interpolation tool when the options chain is sparse. Do not use it as a replacement for BL on liquid chains — BL is model-free and superior when strikes are dense.
• Economic data contracts use a separate probability framework (normal distribution + historical RMSE), not BL. Maintain the 5-year historical error database for CPI, NFP, GDP, and other major releases.