risk-neutral

Risk-Neutral Density Estimation Tools

A Python library for pricing options under various risk-neutral density assumptions, computing option-implied densities, and extracting model parameters from market data.

Features

• Option Pricers: Black–Scholes–Merton (BSM), Mixture of Log-Normals for American Options (AMLN), Edgeworth Expansion (EW), Shimko Spline Method, and Mixture of Log-Normals (MLN).
• Density Models: Compute risk-neutral probability density functions for priced assets under different distributional assumptions.
• Density Extraction: Calibrate density model parameters to market option prices via optimization (BSM, AM, EW, MLN) or direct implied-volatility inversion (Shimko).
• Utilities: Implied volatility computation (compute_implied_vol), volatility-curve fitting (fit_iv_curve).

Installation

Simply

pip install riskneutral

or

pip install git+https://github.com/Moe-Dada/risk_neutral.git

Or clone and install locally:

git clone https://github.com/Moe-Dada/risk_neutral.git
cd risk_neutral
pip install .

Quickstart

Option Pricing

from riskneutral.core_pricing import MarketParams, BSMParams, BSMPricer

# Market parameters: spot, risk-free rate, dividend yield
data = MarketParams(s0=100.0, r=0.05, y=0.02)
# BSM parameters: strike, time to expiry, volatility
params = BSMParams(k=100.0, te=1.0, sigma=0.2)
pricer = BSMPricer(market=data, params=params)
prices = pricer.price()
print(prices)  # {'d1': ..., 'd2': ..., 'call': ..., 'put': ...}

Switch pricer class for different models:

• AMPricer (Mixture of Log-Normals for American Options)
• EWPricer (Edgeworth Expansion)
• ShimkoPricer (Shimko Spline Method)
• MLNPricer (Mixture of Log-Normals)

Density Computation

import numpy as np
from riskneutral.density_computations import EwDensity, EWParams, MarketParams

mp = MarketParams(s0=100.0, r=0.05, y=0.02)
params = EWParams(k=100.0, te=1.0, sigma=0.2, skew=0.0, kurt=3.0)
model = EwDensity(market=mp, params=params)
x = np.linspace(50, 150, 200)
pdf = model.pdf(x)

Other density classes:

• ShimkoDensity (local-volatiltity skew)
• AmDensity (mixture-lognormal)
• MlnDensity (two-component lognormal mixture)

Density Extraction

import numpy as np
from riskneutral.density_extraction import DensityData, BsmDensityExtractor, BsmExtractConfig
from riskneutral.core_pricing import BSMPricer, BSMParams, MarketParams

# Simulate market option prices under BSM
r, y, te, s0, sigma = 0.03, 0.01, 0.5, 100.0, 0.25
strikes = np.array([90, 100, 110])
market_calls = np.array([
    BSMPricer(MarketParams(s0, r, y), BSMParams(K, te, sigma)).price()["call"]
    for K in strikes
])

data = DensityData(
    r=r, y=y, te=te, s0=s0,
    market_calls=market_calls,
    call_strikes=strikes
)
extractor = BsmDensityExtractor(data, BsmExtractConfig(lam=0.0))
result = extractor.extract()
print("Estimated mu, zeta:", result.params)

For other extractors, see AmDensityExtractor, EwDensityExtractor, MlnDensityExtractor, or use ShimkoDirectExtractor for implied-volatility-based extraction.

Examples

• See the examples/ folder for complete scripts demonstrating pricing, density computation, and extraction workflows.
• See examples/examples_density_computations_and_plots.py for density plots.

For example:

• Shimko's Spline Method:
• Mixture of Log-Normals:
• Edgeworth's Expansion:
• Mixture of Log-Normals for American Option

Testing

Run the full test suite with:

pytest

Contributing

Contributions and issues are welcome! Please open an issue or submit a pull request on GitHub.

License

This project is licensed under the MIT License. See the LICENSE file for details.

References

• Prof. Kam Hamdieh @ [https://profiles.rice.edu/faculty/kam-hamidieh], Hamidieh, K., 2014. Estimating the tail shape parameter from option prices. Available at SSRN 1940117.
• Dada, Moses O (2025) Three essays on volatility and tail-risk. Doctoral thesis, University of Essex. DOI [https://doi.org/10.5526/ERR-00040865]
• E. Jondeau and S. Poon and M. Rockinger (2007): Financial Modeling Under Non-Gaussian Distributions Springer-Verlag, London
• R. Jarrow and A. Rudd (1982) Approximate valuation for arbitrary stochastic processes. Journal of Finanical Economics, 10, 347-369
• C.J. Corrado and T. Su (1996) S&P 500 index option tests of Jarrow and Rudd’s approximate option valuation formula. Journal of Futures Markets, 6, 611-629