Options Viewer Calculation & Methodology Guide

Overview

This page serves as the complete mathematical specification, calculation dictionary, and architectural guide for SOPHIE's Options Viewer (/option/viewer). It details every methodology, quantitative formula, indicator definition, and data convention used across the three primary sub-tools:

1. Options Chain & Volatility (Matrix, Volatility Surface, Positioning, Gamma Exposure / GEX)
2. SPX Payoff Builder (Multi-leg payoff modeling, T+0 Black-Scholes mark-to-market, Net Greeks, Risk-Neutral Probability of Profit, and Skew-Adjusted ranges)
3. Volatility Regime & VRP Research (Variance Risk Premium decomposition, 4-regime taxonomy, timing power analysis, and transition risk matrices)

All index calculations assume standard S&P 500 Index Options (^SPX) conventions: European-style exercise (no early assignment), cash settlement against the special opening/closing quotation, and a **$100 contract multiplier** ($100 per index point).

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1. Top Banner & Institutional HUD Metrics

Located in OptionsMetricsBar and OptionsViewer, these key performance indicators summarize overall market conditions and expiration-specific dynamics.

1.1 At-the-Money Implied Volatility (ATM IV)

The implied volatility of the contract closest to the spot price:

$$\text{Strike}_{\text{ATM}} = \arg\min_K |K - S|$$ $$\text{ATM IV} = \text{IV}(\text{Strike}_{\text{ATM}})$$

If both Call and Put IVs are quoted at the ATM strike, the mid-market average is reported.

1.2 Expected Move

The expected price dispersion of the underlying by expiration. Two computational methods are supported:

• Primary Formula (ATM Straddle Implied Range): Uses the standard market-maker rule-of-thumb pricing: $$\text{Expected Move}_{\$} = \left(C_{\text{ATM, Mid}} + P_{\text{ATM, Mid}}\right) \times 0.85$$ $$\text{Expected Move}_{\%} = \left(\frac{\text{Expected Move}_{\$}}{S}\right) \times 100$$
• Secondary Fallback Formula (Black-Scholes $1\sigma$ Dispersion): Used if single-leg straddle quotes are missing: $$\text{Expected Move}_{\$} = S \times \text{ATM IV} \times \sqrt{T}$$ where $T = \frac{\text{DTE}}{365}$.

1.3 Max Pain Strike

The strike price at which option buyers collectively experience the maximum loss, and option sellers (market makers) pay out the minimum cumulative capital at expiration:

$$\text{Max Pain} = \arg\min_K \left[ \sum_{i} \max(0, K - K_{c,i}) \cdot \text{OI}_{c,i} + \sum_{j} \max(0, K_{p,j} - K) \cdot \text{OI}_{p,j} \right]$$

Note: If open interest data is unavailable (such as in EOD snapshots lacking OI fields), Max Pain is suppressed to prevent arbitrary zero-tied strike outputs.

1.4 Put/Call Ratios

• Volume Put/Call Ratio: $$\text{PCR}_{\text{Volume}} = \frac{\sum \text{Put Volume}}{\sum \text{Call Volume}}$$ Measures intraday flow sentiment ($\text{PCR} > 1.0 \implies \text{Put heavy / Bearish flow}$; $\text{PCR} < 1.0 \implies \text{Call heavy / Bullish flow}$).
• Open Interest Put/Call Ratio: $$\text{PCR}_{\text{OI}} = \frac{\sum \text{Put Open Interest}}{\sum \text{Call Open Interest}}$$ Measures cumulative structural positioning.

1.5 Expiration Cycle Classification & Liquidity Ranking

SPX lists ~55 expirations (dailies, weeklies, monthlies, quarterlies, LEAPS). To keep the interface intuitive without sacrificing liquidity, the viewer implements an algorithmic cycle selector:

• Key Expirations Filter:
  1. Near-Term Dailies: All cycles with $\text{DTE} \le 2$ (0DTE and 1DTE trade heavily).
  2. Friday Weeklies: Next 4 non-monthly Friday expiries.
  3. Standard Monthlies: Next 4 standard monthlies (the 3rd Friday of the month, days 15–21, carrying the deepest open interest on the board).
  4. LEAPS Anchors: The monthly cycle nearest ~365 DTE (1-year horizon).
• Cycle Liquidity Tiering: Ranks a cycle's open interest share relative to the maximum visible cycle on screen ($\text{Share} = \text{OI}_{\text{exp}} / \text{OI}_{\max}$):
  • Deep: $\text{Share} \ge 25\%$ (solid gold pill)
  • Active: $5\% \le \text{Share} < 25\%$ (translucent gold pill)
  • Thin: $\text{Share} < 5\%$ (gray pill)

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2. Options Matrix Table & Proprietary Liquidity Scoring

Located in OptionsMatrixTable and lib/options/liquidity.ts, this double-sided chain provides standard pricing, Black-Scholes Greeks, and composite execution quality metrics.

2.1 Mid Price & Spread Percentage

$$\text{Mid Price} = \frac{\text{Bid} + \text{Ask}}{2}$$ $$\text{Spread \%} = \frac{\text{Ask} - \text{Bid}}{\text{Mid Price}}$$

2.2 Proprietary Liquidity Score Model

Unlike generic equity models, SOPHIE's liquidity scorer is calibrated specifically to SPX market microstructure, where ATM spreads trade extremely tight ($<0.3\%$) and widen rapidly OTM.

Critical Design Principle — Spread Gates the Score: A contract with large open interest but an unfillable quote (e.g., $0.00 bid /$5.00 ask) is not liquid. Therefore, Spread Score acts as a strict multiplier/gate rather than an arithmetic component that can be outvoted by open interest.

1. Spread Score ($\text{Score}_{\text{spread}}$): $$\text{Score}_{\text{spread}} = \begin{cases} 100 & \text{Spread \%} \le 0.3\% \\ 100 - 30 \times \frac{\text{Spread \%} - 0.003}{0.010 - 0.003} & 0.3\% < \text{Spread \%} \le 1.0\% \\ 70 - 40 \times \frac{\text{Spread \%} - 0.010}{0.030 - 0.010} & 1.0\% < \text{Spread \%} \le 3.0\% \\ 30 - 30 \times \frac{\text{Spread \%} - 0.030}{0.080 - 0.030} & 3.0\% < \text{Spread \%} \le 8.0\% \\ 0 & \text{Spread \%} > 8.0\% \end{cases}$$
2. Volume Score ($\text{Score}_{\text{vol}}$): Log-scaled activity indicator: $$\text{Score}_{\text{vol}} = \min\left(100, \frac{\log_{10}(\text{Volume} + 1)}{\log_{10}(1001)} \times 100\right)$$
3. Open Interest Score ($\text{Score}_{\text{OI}}$): Log-scaled resting size indicator: $$\text{Score}_{\text{OI}} = \min\left(100, \frac{\log_{10}(\text{OI} + 1)}{\log_{10}(5001)} \times 100\right)$$
4. Activity Score: $$\text{Score}_{\text{activity}} = 0.65 \times \text{Score}_{\text{vol}} + 0.35 \times \text{Score}_{\text{OI}}$$
5. Composite Liquidity Score (0–100): $$\text{Score}_{\text{composite}} = \text{round}\left( \text{Score}_{\text{spread}} \times \left(0.55 + 0.45 \times \frac{\text{Score}_{\text{activity}}}{100}\right) \right)$$

Liquidity Tiers:

• Excellent: $\text{Score} \ge 75$ (Emerald badge)
• Good: $50 \le \text{Score} < 75$ (Blue badge)
• Fair: $25 \le \text{Score} < 50$ (Amber badge)
• Poor: $\text{Score} < 25$ (Rose badge)

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3. Volatility Analytics & Surface

Located in VolatilityChartView, this module provides five analytical sub-views decoding implied volatility across strikes and maturities.

3.1 IV Smile & Skew Curve

Plots strike $K$ against implied volatility $\sigma(K)$ for the selected cycle, filtered within $\pm 22\%$ of spot to eliminate illiquid far-tail noise.

3.2 Volatility Term Structure & Forward Implied Volatility

Plots ATM implied volatility across all expirations ordered by DTE.

• Term Structure Slope:
  • Contango (Normal): $\sigma_{\text{far}} > \sigma_{\text{near}}$ (upward sloping; calm market).
  • Backwardation (Inverted): $\sigma_{\text{near}} > \sigma_{\text{far}}$ (downward sloping; near-term panic / event risk).
• Forward Implied Volatility ($\sigma_{\text{fwd}}$): Calculates the market-implied forward volatility between two expiration horizons $T_1$ and $T_2$ ($T_2 > T_1$): $$\sigma_{\text{fwd}}(T_1, T_2) = \sqrt{\frac{T_2 \sigma_2^2 - T_1 \sigma_1^2}{T_2 - T_1}}$$

3.3 Breeden-Litzenberger Risk-Neutral Density (RND)

Extracts the market's implied probability density function $f(K)$ for the underlying price at expiration directly from call option prices, based on the Breeden & Litzenberger (1978) theorem:

$$f(K) = e^{rT} \frac{\partial^2 C(K)}{\partial K^2}$$

Non-Uniform Strike Grid Numerical Second Derivative:

Real SPX chains alternate between 5-point, 10-point, and 25-point strike gaps. Using a standard uniform formula $\frac{C_{i+1} - 2C_i + C_{i-1}}{h^2}$ introduces severe error. SOPHIE implements the general non-uniform 3-point second derivative formula:

$$\frac{\partial^2 C}{\partial K^2} \approx \frac{2 \left[ \Delta K_1 \cdot C(K_{i+1}) - (\Delta K_1 + \Delta K_2) \cdot C(K_i) + \Delta K_2 \cdot C(K_{i-1}) \right]}{\Delta K_1 \cdot \Delta K_2 \cdot (\Delta K_1 + \Delta K_2)}$$

where $\Delta K_1 = K_i - K_{i-1}$ and $\Delta K_2 = K_{i+1} - K_i$.

Densities are floored at zero ($\max(0, \frac{\partial^2 C}{\partial K^2})$) and normalized so $\sum f(K_i) \Delta K_i = 100\%$.

3.4 25-Delta Skew & Kurtosis Dynamics

• 25-Delta Risk Reversal (Skew): $$\text{RR}_{25\Delta} = \text{IV}(\text{Put}_{25\Delta}) - \text{IV}(\text{Call}_{25\Delta})$$ Positive values indicate put skew (downside protection demand exceeding upside calls).
• 25-Delta Butterfly (Convexity / Kurtosis): $$\text{Fly}_{25\Delta} = \left( \frac{\text{IV}(\text{Put}_{25\Delta}) + \text{IV}(\text{Call}_{25\Delta})}{2} \right) - \text{IV}_{\text{ATM}}$$ Measures wing curvature (fat-tail probability) relative to the at-the-money base.

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4. Positioning Analysis (OI, Volume, Max Pain)

Located in PositioningChartView, this module analyzes market participant positioning.

4.1 Max Pain Cumulative Payout Curve

Simulates the total dollar liability of option sellers across candidate underlying settlement prices $K_{\text{settle}}$:

$$\text{Payout}(K_{\text{settle}}) = \frac{\sum_c \max(0, K_{\text{settle}} - K_c) \cdot \text{OI}_c \cdot 100 + \sum_p \max(0, K_p - K_{\text{settle}}) \cdot \text{OI}_p \cdot 100}{1{,}000{,}000} \quad (\text{\$ Millions})$$

4.2 Cumulative Open Interest Curve

Plots the running accumulation of Call and Put open interest across the strike spectrum:

$$\text{CumCallOI}(K) = \sum_{k \le K} \text{OI}_{\text{call}}(k), \quad \text{CumPutOI}(K) = \sum_{k \le K} \text{OI}_{\text{put}}(k)$$

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5. Gamma Exposure (GEX) & Market Maker Hedging

Located in GexChartView and detailed in the GEX Calculation Methodology spec.

5.1 Dollar Gamma Exposure Formula

$$\text{GEX}_i = \Gamma_i \times \text{OI}_i \times 100 \times S^2 \times 0.01 / 1{,}000{,}000 \quad (\text{\$M per 1\% move})$$

• $S^2$ Scaling: Converts gamma ($\Delta / \text{point}$) to shares per 1% move ($\times S \times 1\%$), then to notional dollars ($\times S$).
• Dealer Sign Convention: $$\text{Net GEX} = \sum_{\text{calls}} \text{GEX}_c - \sum_{\text{puts}} \text{GEX}_p$$

5.2 Whole-Book Scope vs. Single Expiration

By default, Net GEX, the Gamma Flip, and Walls aggregate across all listed expirations (the complete dealer book), preventing inverted regime readings from isolated front-month expiries.

5.3 Gamma Flip Level (Zero Gamma)

The hypothetical spot price $S^*$ where total dealer gamma crosses zero. Evaluated by re-pricing every contract's Black-Scholes gamma across candidate spot levels:

$$\text{Net GEX}(S') = \sum_i \pm \Gamma_i(S', K_i, T_i, \sigma_i) \cdot \text{OI}_i \cdot 100 \cdot (S')^2 \cdot 0.01$$

Solved via a 1% bracketed sweep followed by 14-step bisection root-finding.

5.4 Call Wall & Put Wall

• Directional Constraint: $\text{Call Wall} \ge S$, $\text{Put Wall} \le S$.
• Smoothing: Evaluated over a $[0.25, 0.50, 0.25]$ kernel across strikes to eliminate single-contract noise.

5.5 Higher-Order Greeks (Vanna & Charm Exposure)

• Vanna Flow: Sensitivity of dealer delta to implied volatility shifts: $$\text{Vanna}_{\$} = \pm \frac{\partial \Delta}{\partial \sigma} \times \text{OI} \times 100$$
• Charm Flow: Sensitivity of dealer delta to time decay: $$\text{Charm}_{\$} = \pm \frac{\partial \Delta}{\partial t} \times \text{OI} \times 100$$

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6. SPX Payoff Builder & Probability Range Analysis

Located in SpxPayoffBuilder, PayoffChartView, ProbabilityRangeView, and lib/options/analytics.ts.

6.1 Multi-Leg P&L at Expiration ($T=0$)

$$\text{PnL}(S_T) = \sum_{i=1}^N \text{Quantity}_i \times \text{Sign}_i \times \left[ \text{IntrinsicValue}(S_T, K_i, \text{Type}_i) - \text{Premium}_i \right] \times 100$$

where $\text{Sign}_i = +1$ for Long and $-1$ for Short.

6.2 "T+0" Mark-to-Market Curve

Plots current position value across hypothetical spot prices today ($T > 0$):

1. Implied Volatility Solving: Each leg's entry premium is solved for its exact Black-Scholes implied volatility $\sigma_i$ via bisection on mid price.
2. Current Value: $$\text{Value Today}(S') = \sum_{i=1}^N \text{Quantity}_i \times \text{Sign}_i \times \left[ C_{\text{BS}}(S', K_i, T, r, q, \sigma_i) - \text{Premium}_i \right] \times 100$$

6.3 Net Position Greeks

Aggregates Greeks across all legs, signed by position direction and scaled by quantity $\times 100$:

$$\text{Net Delta} = \sum \pm Q_i \cdot 100 \cdot \Delta_i, \quad \text{Net Gamma} = \sum \pm Q_i \cdot 100 \cdot \Gamma_i$$ $$\text{Net Theta} = \sum \pm Q_i \cdot 100 \cdot \Theta_i, \quad \text{Net Vega} = \sum \pm Q_i \cdot 100 \cdot \mathcal{V}_i$$

6.4 Risk-Neutral Probability of Profit (POP)

Evaluates the probability that the multi-leg position finishes with $\text{PnL} > 0$ at expiration under risk-neutral lognormal drift $\mu = r - q - \frac{\sigma^2}{2}$:

1. Identifies all breakeven zero-crossing strikes $B_1, B_2, \dots, B_m$.
2. Partitions the price domain into intervals $[0, B_1], [B_1, B_2], \dots, [B_m, \infty)$.
3. For each profitable interval $[L_k, U_k]$, integrates the lognormal CDF: $$P(L_k < S_T < U_k) = \Phi\left( \frac{\ln(S/L_k) + (r - q - \sigma^2/2)T}{\sigma\sqrt{T}} \right) - \Phi\left( \frac{\ln(S/U_k) + (r - q - \sigma^2/2)T}{\sigma\sqrt{T}} \right)$$
4. $\text{POP} = \sum_{k \in \text{Profitable}} P(L_k < S_T < U_k)$.

6.5 Implied Price Ranges (Confidence Intervals)

• Flat ATM Model (Symmetric): $$\text{Boundary}_{\text{lower/upper}} = S \cdot \exp\left( \left(r - q - \frac{\sigma_{\text{ATM}}^2}{2}\right)T \pm \sigma_{\text{ATM}}\sqrt{T} \cdot z_{(1+c)/2} \right)$$ For $68\%$ ($1\sigma$), $z = 1.000$; for $80\%$ ($1.28\sigma$), $z = 1.28155$.
• Skew-Adjusted (Asymmetric) Model: Reprices each boundary using the actual market implied volatility of the nearest strike to that boundary, iterated to fixed-point convergence: $$\text{Upper}^{(n+1)} = \text{Boundary}\left(S, T, r, q, \sigma(\text{Upper}^{(n)}), +z\right)$$ $$\text{Lower}^{(n+1)} = \text{Boundary}\left(S, T, r, q, \sigma(\text{Lower}^{(n)}), -z\right)$$ Produces wider downside and tighter upside boundaries consistent with real index skew.

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7. Volatility Regime & VRP Research

Located in VolRegimePanel, VrpTimingView, RegimeTransitionView, and detailed in the Volatility Regime & VRP Methodology spec.

7.1 Daily Core Signals

• 20-Day Realized Volatility ($\text{RV}_{20}$): $$\text{RV}_{20} = \text{stdev}\left(\ln \frac{S_t}{S_{t-1}}, 20\right) \times \sqrt{252} \times 100$$
• Variance Risk Premium (Vol Points): $$\text{VRP}_{\text{vol}} = \text{VIX} - \text{RV}_{20}$$
• Variance Risk Premium (Variance Points / Swap Units): $$\text{VRP}_{\text{var}} = \frac{\text{VIX}^2 - \text{RV}_{20}^2}{100}$$
• Downside Variance Share (Semivariance): $$\text{Downside Share} = \frac{\sum_{r_t < 0} r_t^2}{\sum r_t^2}$$

7.2 The 4-Regime Quadrant Taxonomy

Classifies market state across two orthogonal axes: Premium Richness ($\text{vrp\_z} > 0$) and Market Stress ($\text{vix\_rank} > 0.80$ or $\text{VIX3M} < \text{VIX}$):

7.3 Predictive Timing vs. Risk Signals

• Entry-Timing Power: Testing $\text{vrp\_z}$ quintiles against forward 21-day realized premium yields a Spearman rank correlation of $\text{IC} \approx 0.008$ (effectively zero). VRP level does not reliably time entries.
• Risk Management Power: The regime label strongly predicts transition into Crisis:
  • $P(\text{Crisis in 21d} \mid \text{Harvest}) = \mathbf{5.7\%}$
  • $P(\text{Crisis in 21d} \mid \text{Stressed Premium}) = \mathbf{26.7\%}$ (approx. $5\times$ higher risk).

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8. Summary Formula Reference Table

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Related Wiki Specs & Documentation

• GEX Calculation Methodology — Comprehensive gamma exposure, flip root-finding, and smoothing spec
• Volatility Regime & VRP Methodology — Complete Variance Risk Premium decomposition & transition risk spec
• The Volatility Surface — Morphologies of skew, smirk, and term structure
• Volume & Open Interest Analysis — Flow, positioning, and Put/Call dynamics
• Option Greeks Overview — Delta, Gamma, Theta, Vega, Rho, Vanna, and Charm