Derivatives Risk-Mapping Engine

Derivatives Risk Mapping Engine
GLD Options Analysis (February 8th 2026): Black-Scholes + Breeden-Litzenberger

Overview

Most options analysis focuses on implied volatility in isolation , a smile for one expiry, a term structure for another. That fragmented view often obscures where risk is actually being priced across the surface.

This project builds a derivatives risk-mapping engine that converts live options prices into market-implied probability distributions across strikes and maturities, allowing tail risk, skew, and convexity to be assessed in a single, unified framework. The engine is designed as a decision-support tool for trade selection and risk assessment, particularly around event risk and regime uncertainty.

Model Showcase - Alphabet (NASDAQ: GOOG)

To showcase the model in action, Alphabet (GOOG) is used as a representative example. The engine ingests live options data via a public market API as of 9 February, and transforms raw option prices into structured risk surfaces that reveal how volatility and probability are priced across strikes and maturities.

GOOG Implied Volatility Surface
GOOG Options (February 9th 2026): Black-Scholes Implied Volatility Surface

The implied volatility surface shows a pronounced downside skew at shorter expiries, with volatility rising sharply as strikes move below spot. This reflects concentrated demand for near-term downside protection, consistent with event-driven risk being actively hedged. As maturity increases, the surface flattens and volatility levels converge, indicating that uncertainty is not being extrapolated far into the future and that longer-dated risk is priced more symmetrically. In isolation, this surface highlights where dispersion is expensive, but does not reveal how probability mass is allocated.

GOOG Risk-Neutral Density Surface
GOOG Options (February 9th 2026): Breeden-Litzenberger Risk-Neutral Density

The risk-neutral density surface makes this allocation explicit. Probability mass is heavily concentrated in the left tail at shorter maturities, confirming that downside outcomes are not only volatile but meaningfully priced. The curvature of the surface indicates strong convexity demand, with tail probabilities decaying more slowly than under a symmetric distribution. At longer maturities, the density broadens and flattens, suggesting regime uncertainty rather than a specific anticipated shock.

Viewed together, the two surfaces suggest that near-term risk in GOOG is being actively hedged, while longer-dated structural concerns remain present but less acute. The concentration of downside probability at shorter maturities points to market sensitivity around discrete catalysts rather than a sustained deterioration in long-term fundamentals. At the same time, the broader and flatter long-dated density implies ongoing uncertainty around structural themes , such as regulation, AI competition, and margin durability , without the market assigning high conviction to extreme outcomes. In combination, this indicates a market that is cautious in the near term but not pricing a deep, persistent downside regime for GOOG.

Motivation

Options markets embed the collective expectations and risk preferences of market participants. While implied volatility captures dispersion, it does not explicitly show how probability mass is distributed or where asymmetries in risk pricing exist.

The core idea behind this engine is simple: option prices imply a probability distribution of future outcomes. By extracting and visualizing that distribution across time, it becomes possible to see where the market is paying for protection, speculating on upside, or underpricing convexity.

This approach moves analysis from volatility levels to risk structure.

What the Engine Does

The engine translates live options data into two complementary visual layers:

Together, these surfaces provide insight into:

Methodology & Architecture

1. Live Options Data Ingestion

Option chains are pulled in real time using publicly available market data. To ensure stability and robustness, contracts are filtered for liquidity, bid-ask quality, minimum time to expiry, and reasonable strike ranges relative to spot.

2. Black–Scholes Normalization

Market prices are inverted through the Black–Scholes model to compute implied volatilities. Black–Scholes is used here as a normalization and diagnostic framework, not as a belief about return distributions. This step ensures consistency across strikes and maturities and provides a benchmark for surface construction.

3. Surface Interpolation & Smoothing

Option prices are interpolated across strike–maturity space to form a smooth call price surface. Controlled smoothing is applied to mitigate microstructure noise, which is essential for numerical stability when computing higher-order derivatives.

4. Breeden–Litzenberger Density Extraction

Risk-neutral probability densities are recovered by applying the Breeden–Litzenberger relationship , the discounted second derivative of call prices with respect to strike. This yields a model-free view of how the market prices future outcomes at each maturity.

5. Visualization & Diagnostics

The extracted densities are stacked across maturities to form a 3D probability surface. This surface enables cross-sectional analysis of skew, convexity, and tail behavior in a single visual representation.

The Output

Strike dimension: where fear or speculation is concentrated

Maturity dimension: when risk matters

Surface shape: market psychology

The surface represents risk-neutral probabilities , how risk is priced, not a forecast of realized outcomes.

Applications

The engine is designed to support:

Rather than predicting markets, it provides a framework for understanding where the market is expressing fear, complacency, or speculation.

Limitations & Scope

The model assumes static interest rates and excludes dividends, and its accuracy depends on the availability of liquid option markets. It is intentionally framed as a risk-mapping and visualization tool, not a predictive model of future returns.

Implementation

Built in Python using: