garch-lnmm 0.1.0

Log-normal moment-matching simulation for GARCH diffusion option pricing

garch-lnmm

Log-normal and Shifted-lognormal moment-matching simulation for European call option pricing under the GARCH diffusion model

Problem

The continuous-time GARCH diffusion model is useful for stochastic volatility option pricing, but standard Euler discretization can generate negative variance values. In practice, truncation is often applied, which may introduce discretization bias under coarse time steps.

This package implements and compares positivity-preserving moment-matching schemes, including Log-Normal Moment-Matching (LN-MM) and Shifted Log-Normal Moment-Matching (SLN-MM), against Euler conditional Monte Carlo and the Taylor closed-form analytical approximation from Barone-Adesi et al. (2003).

The conditional mean, variance, and skewness used in these schemes are computed from the exact conditional moments of the next-step GARCH diffusion variance transition.

Installation

pip install garch-lnmm

Quick Start

from garch_lnmm import GarchDiffusionMC

model = GarchDiffusionMC(
    S0=100,
    V0=0.16,
    r=0.05,
    kappa=0.75,
    theta=0.04,
    sigma=0.85,
    rho=0,
    T=5,
)

lnmm_price = model.simulate_moment_matching_cond_mc(
    N_paths=50000,
    N_steps=32,
    K=130,
    seed=42,
)

slnmm_price = model.simulate_shifted_lognormal_cond_mc(
    N_paths=50000,
    N_steps=32,
    K=130,
    seed=42,
)

print("LN-MM price:", lnmm_price)
print("SLN-MM price:", slnmm_price)

API Reference

GarchDiffusionMC(S0, V0, r, kappa, theta, sigma, rho, T)

Main simulator class for the continuous-time GARCH diffusion model.

Parameter	Type	Description
S0	float	Initial stock price
V0	float	Initial variance
r	float	Risk-free interest rate
kappa	float	Mean-reversion speed of variance
theta	float	Long-run variance level
sigma	float	Volatility of variance
rho	float	Correlation parameter
T	float	Maturity

---

simulate_euler_cond_mc(N_paths, N_steps, K, seed=None, return_stats=False)

Prices a European call using truncated Euler conditional Monte Carlo.

Parameter	Type	Description
N_paths	int	Number of Monte Carlo paths
N_steps	int	Number of time steps
K	float	Strike price
seed	int or None	Random seed
return_stats	bool	If True, return price and error statistics

Returns:

Field	Type	Description
price	float	Estimated option price
std_dev	float	Sample standard deviation
std_error	float	Monte Carlo standard error

---

simulate_moment_matching_cond_mc(N_paths, N_steps, K, seed=None, return_stats=False)

Prices a European call using the Log-Normal Moment-Matching scheme.

Parameter	Type	Description
N_paths	int	Number of Monte Carlo paths
N_steps	int	Number of time steps
K	float	Strike price
seed	int or None	Random seed
return_stats	bool	If True, return price and error statistics

Returns:

Field	Type	Description
price	float	Estimated option price
std_dev	float	Sample standard deviation
std_error	float	Monte Carlo standard error

---

simulate_shifted_lognormal_cond_mc(N_paths, N_steps, K, seed=None, return_stats=False)

Prices a European call using the Shifted Log-Normal Moment-Matching scheme.

This method computes the first three exact conditional raw moments of the next-step variance transition (V(t+h)), converts them into conditional mean, variance, and skewness, and then samples from a shifted-lognormal approximation matching these moments.

Parameter	Type	Description
N_paths	int	Number of Monte Carlo paths
N_steps	int	Number of time steps
K	float	Strike price
seed	int or None	Random seed
return_stats	bool	If True, return price and error statistics

Returns:

Field	Type	Description
price	float	Estimated option price
std_dev	float	Sample standard deviation
std_error	float	Monte Carlo standard error

---

License

MIT

Demo Notebook

References

Choi, J., Hu, L., & Kwok, Y. K. (2024).
Efficient and Accurate Simulation of the Stochastic-Alpha-Beta-Rho Model.

Motivates the shifted-lognormal moment-matching framework used in the SLN-MM extension.

Barone-Adesi, G., Rasmussen, H., & Ravanelli, C. (2003).
An Option Pricing Formula for the GARCH Diffusion Model.

Provides the conditional moments and Taylor approximation used as analytical benchmarks in this project.

Andersen, L. (2008).
Simple and Efficient Simulation of the Heston Stochastic Volatility Model.

Motivates the moment-matching methodology adopted for the variance simulation scheme.