VIX Pricer

Replication package for:

Fan, Z., Ryu, D., & Ye, Y. (2026). Valuation of VIX derivatives: Incorporating larger spikes in volatility-of-volatility dynamics.

A Python library for pricing VIX options using stochastic volatility models. This package implements both Fourier-based methods and Monte Carlo methods for efficient option pricing.

Features

• Affine Characteristic Function: Solves coupled Riccati-type ODEs for VIX direct models
• FFT Pricing: Carr-Madan FFT method for fast European option pricing
• Monte Carlo Simulation: Euler-Maruyama discretization for path simulation
• Model Validation: Built-in framework to compare FFT and MC pricing accuracy
• Flexible Architecture: Modular design for easy extension and customization

Installation

From PyPI

pip install VIX-Spike

Quick Start

Basic Option Pricing

from Pricer import ModelParameters, InitialConditions, FFTPricer

# Define model parameters
params = ModelParameters(
    T=0.0833,           # Time to maturity (1 month)
    kappa=5.0,          # Mean-reversion speed
    kappam=0.5,         # Long-term variance mean-reversion
    thetam=0.04,        # Long-run variance mean
    omegam=0.3,         # Variance volatility
    kappa1=3.0,         # Instantaneous variance mean-reversion
    theta1=0.04,        # Instantaneous variance long-run mean
    omega1=0.5,         # Vol-of-vol
    rho1=-0.7,          # Correlation
    bv=10.0,            # Shot noise decay rate
    lamb=5.0,           # Jump intensity
    muJV=0.02,          # Mean jump size
)

# Set initial conditions
initial = InitialConditions(
    lnS0=2.7,           # Initial log-VIX
    m0=0.04,            # Initial variance center
    v10=0.04,           # Initial instantaneous variance
    Lv0=0.01,           # Initial shot noise level
)

# Price a call option
strike = 15.0
risk_free_rate = 0.0
strikes, prices = FFTPricer.price_call_fft(
    r=risk_free_rate,
    T=params.T,
    params=params,
    initial=initial,
    K=strike
)

print(f"Call option price at K={strike}: {prices[0]:.4f}")

Monte Carlo Simulation

from Pricer import MonteCarloSimulator

# Create simulator
mc = MonteCarloSimulator()

# Generate paths
paths = mc.generate_paths(
    params=params,
    initial=initial,
    num_paths=10000,
    num_steps=100
)

# Price option using Monte Carlo
mc_price = mc.price_call_mc(
    r=risk_free_rate,
    T=params.T,
    params=params,
    initial=initial,
    K=strike,
    num_paths=100000
)

print(f"Monte Carlo price: {mc_price:.4f}")

Model Validation

from Pricer import ModelValidator

# Create validator
validator = ModelValidator()

# Compare FFT vs Monte Carlo across multiple strikes
results = validator.validate_pricing(
    params=params,
    initial=initial,
    strikes=[12, 13, 14, 15, 16, 17, 18],
    r=risk_free_rate,
    num_mc_paths=50000
)

print(results)

Model Description

The library implements a reduced VIX shot noise model with the following dynamics:

Log-VIX Process: $$d\ln S_t = \kappa(\theta_t - \ln S_t)dt + \sqrt{v_{1t}}dW_t^{(1)}$$

Stochastic central tendency (OU Process): $$dm_t = \kappa_m(\theta_m - m_t)dt + \omega_m dW_t^{(m)}$$

Instantaneous Variance (CIR-type): $$dv_{1t} = \kappa_1(\theta_1 - v_{1t})dt + \omega_1\sqrt{v_{1t}}dW_t^{(v)} + dL_{vt}$$

Variance Shot Noise: $$dL_{vt} = -b_v L_{vt}dt + dJ_t^{(v)}$$

Where:

• $\theta_t = m_t + L_{vt}$ (time-varying target)
• $dW_t^{(1)}$ and $dW_t^{(v)}$ have correlation $\rho_1$
• $J_t^{(v)}$ is a compound Poisson process with exponential jump sizes

Package Structure

vix-pricer/
├── Pricer/
│   ├── __init__.py                    # Package initialization
│   ├── model_config.py                # Model parameters and initial conditions
│   ├── characteristic_function.py     # Affine characteristic function solver
│   ├── fft_pricing.py                 # FFT-based option pricing
│   ├── monte_carlo.py                 # Monte Carlo simulation
│   ├── validator.py                   # Model validation framework
│   └── solvers.py                     # Numerical ODE solvers
├── pyproject.toml                     # Modern package configuration
├── setup.py                           # Legacy setup script
├── README.md                          # This file
└── LICENSE                            # MIT License

Dependencies

• Python ≥ 3.8
• NumPy ≥ 1.20.0
• SciPy ≥ 1.7.0
• Pandas ≥ 1.3.0
• tqdm ≥ 4.60.0

Development

Installing Development Dependencies

pip install -e ".[dev]"

Running Tests

pytest tests/ -v --cov=Pricer

Code Formatting

black Pricer/
flake8 Pricer/

Citation

If you use this library in your research, please cite:

@article{fan2026vix,
  author = {Fan, Z. and Ryu, D. and Ye, Y.},
  title = {Valuation of {VIX} derivatives: Incorporating larger spikes in volatility-of-volatility dynamics},
  year = {2026},
  journal = {Working Paper}
}

License

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

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

Support

For questions and support:

• Open an issue on GitHub
• Check the documentation

Acknowledgments

Developed by the VIX Spike VolOfVol Research Team for quantitative finance research and applications.