A project on implied probability density functions for asset prices

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SPX Option-Implied Probability Distributions

This was a project completed as part of the Erdős Institute quantitative finance course, although I have updated it since then (August 2026) and sharpened the data sources and analysis. See below for an overview of the project, or go straight to the Github repository for the sweet jupyter notebooks.

This project studies how probability distributions for future S&P 500 levels can be extracted from SPX option prices and how well they match realized outcomes.

From historical option chains, the project:

  • estimates forward prices and discount rates via put-call parity,
  • recovers risk-neutral densities using the Breeden–Litzenberger relationship,
  • compares smoothing methods for noisy option data,
  • evaluates forecasts using the Continuous Ranked Probability Score (CRPS), and
  • explores a simple transformation from risk-neutral to physical probabilities using a CRRA pricing kernel.

The implementation is in Python, with analysis presented in Jupyter notebooks.

⭐ See notebooks/02_project.ipynb or reports/02_project.html (in the Github repository) for the main reports. ⭐

Motivation

Under the risk-neutral measure $Q$, a European call price is

$$ C(K,\tau) = e^{-r\tau}\mathbb{E}_Q[(S_T-K)^+]. $$

Differentiating twice with respect to strike yields the Breeden–Litzenberger result:

$$ f_Q(K) = e^{r\tau}\frac{\partial^2 C(K,\tau)}{\partial K^2}, $$

which links option prices to the risk-neutral density.

In practice, this is challenging because option quotes are discrete and noisy, and second derivatives amplify irregularities. A key issue is therefore constructing smooth call-price curves that remain arbitrage-consistent.

Methodology

1. Forward and discount rate estimation

Put-call parity implies

$$ C(K)-P(K)=D(F-K), $$

so if

$$ C(K)-P(K)=a+bK, $$

then

$$ D=-b, \qquad F=\frac{a}{D}, \qquad r=-\frac{\log D}{\tau}. $$

This allows forward prices and discount rates to be inferred directly from option data.

2. Risk-neutral density estimation

Implied-volatility smoothing

Prices are converted to implied volatilities and smoothed as a function of log-forward-moneyness:

$$ k=\log\left(\frac{K}{F}\right). $$

The smoothed surface is converted back to prices and differentiated to obtain a density. This is flexible but does not guarantee arbitrage-free prices.

Shape-constrained smoothing (Fengler)

A valid call-price curve must satisfy:

$$ \frac{\partial C}{\partial K}\leq 0, \qquad \frac{\partial^2 C}{\partial K^2}\geq 0. $$

These conditions ensure that the implied density

$$ f_Q(K)\propto C’’(K) $$

is non-negative. The Fengler approach enforces these constraints directly while fitting a smooth curve, producing an arbitrage-consistent density.

Forecast evaluation

Each implied distribution is compared to the realized SPX level using the CRPS:

$$ \int_{\mathbb{R}}\left(F_D(x)-\mathbf{1}_{x\geq y}\right)^2 dx. $$

Lower values indicate better probabilistic forecasts. The analysis considers 30-, 60-, and 90-day horizons.

From risk-neutral to physical probabilities

Risk-neutral densities are pricing objects, not real-world forecasts. To explore this gap, I use a CRRA-based transformation:

$$ f_P(s)\propto s^\gamma f_Q(s), $$

where $\gamma$ controls the pricing-kernel adjustment. Different values of $\gamma$ are evaluated using CRPS.

Repository structure

project/
│
├── README.md
├── data/
│   ├── raw/        # these directories are actually ignored
│   └── processed/  # since I don't want to use github to share data
│
├── implied_stock_distributions/
│   └── ...   # core modeling code
│
├── notebooks/
│   ├── 01_processing.ipynb     # data preprocessing
│   └── 02_project.ipynb        # read this one if you like to open jupyter notebooks!
│  
└── reports/
    └── 02_project.html         # read this one if you don't like to open jupyter notebooks!

Key tools

  • Python (NumPy, pandas, SciPy, Matplotlib)
  • Black–Scholes implied volatility
  • put-call parity
  • Breeden–Litzenberger density extraction
  • constrained spline fitting
  • CRPS for probabilistic evaluation

Key ideas

  • option-implied state prices
  • risk-neutral vs physical measures
  • volatility surface smoothing
  • static arbitrage constraints
  • probabilistic forecasting

Limitations and extensions

Results depend on smoothing choices, grid resolution, and tail treatment. The CRRA transformation is also intentionally simple and not calibrated out-of-sample.

Possible extensions include:

  • out-of-sample evaluation of $\gamma$
  • richer pricing-kernel models
  • joint strike–maturity surface fitting
  • improved tail modeling
  • comparison with ML-based forecasts

References

Breeden & Litzenberger (1978) — Prices of State-Contingent Claims Implicit in Option Prices

Fengler — Arbitrage-Free Smoothing of the Implied Volatility Surface

About

This project explores how option prices encode full probability distributions of future market outcomes, and how these distributions compare to realized returns. This was originally a project that I completed as part of a quantitative finance course with The Erdős Institute, but has been significantly updated since then.