heston-mc-variance-project 0.1.0

Monte Carlo pricing of variance derivatives under the Heston model

heston_mc

Monte Carlo pricing of variance swaps and variance options under the Heston stochastic volatility model, with control variates for improved efficiency.

Course project on efficient Monte Carlo pricing of variance derivatives under the Heston model.

For a more detailed project description, see docs/PROJECT_SCOPE.md.

What are variance derivatives?

Variance derivatives are contracts whose payoff depends on how much an asset fluctuates over time, rather than only on its final price. In this project, we focus on two examples:

• Variance swaps, whose payoff depends linearly on realized variance relative to a strike
• Variance options, whose payoff depends on the positive part of realized variance above a strike

This package is being built to simulate stock-price and variance paths under the Heston model, compute realized variance from simple returns, and use Monte Carlo methods to price these products. It also includes a control variate component to improve efficiency by reducing Monte Carlo noise.

Model

Under the risk-neutral measure, the Heston model is

$$ dS_t = (r-q)S_t dt + \sqrt{v_t} S_t dW_t^{(1)} $$

$$ dv_t = \kappa(\theta - v_t) dt + \sigma \sqrt{v_t} dW_t^{(2)} $$

$$ d\langle W^{(1)}, W^{(2)} \rangle_t = \rho dt $$

where:

• $S_t$ is the stock price
• $v_t$ is the instantaneous variance
• $r$ is the risk-free rate
• $q$ is the dividend yield
• $\kappa$ is the mean-reversion speed
• $\theta$ is the long-run variance level
• $\sigma$ is the volatility of variance
• $\rho$ is the correlation between the stock and variance shocks

Installation

pip install heston-mc-variance-project

Quick start

from heston_mc.params import HestonParams, MonteCarloConfig, default_heston_params, default_mc_config
from heston_mc.interfaces import SimulationResult, PricingResult

params = default_heston_params()
config = default_mc_config()

print(params)
print(config)

This package is currently under active development. The main pricing workflow is being assembled across the following modules:

• heston_mc.simulation
• heston_mc.realized_variance
• heston_mc.variance_swap
• heston_mc.variance_option
• heston_mc.control_variate

Demo notebook

An interactive demo will be available at notebooks/demo.ipynb:

API reference

HestonParams

Shared parameter class for the Heston stochastic volatility model.

Parameter	Type	Description
s0	float	Initial stock price
v0	float	Initial variance
r	float	Risk-free interest rate
q	float	Dividend yield
kappa	float	Mean-reversion speed of variance
theta	float	Long-run variance level
sigma	float	Volatility of variance
rho	float	Correlation between stock and variance shocks

MonteCarloConfig

Shared Monte Carlo configuration.

Parameter	Type	Description
maturity	float	Time to maturity
n_steps	int	Number of time steps
n_paths	int	Number of simulated paths
seed	int	Random seed
periods_per_year	int	Scaling used in realized variance

SimulationResult

Container for simulation output.

Field	Type	Description
stock_paths	np.ndarray	Simulated stock-price paths
variance_paths	np.ndarray	Simulated variance paths
dt	float	Time step size

PricingResult

Container for pricing output.

Field	Type	Description
price	float	Estimated derivative price
std_error	float	Monte Carlo standard error
n_paths	int	Number of simulation paths
n_steps	int	Number of time steps
runtime_seconds	float	Runtime in seconds
method_name	str	Name of pricing method

Module overview

• heston_mc.simulation — Heston path simulation
• heston_mc.realized_variance — realized variance from simple returns
• heston_mc.variance_swap — variance swap pricing
• heston_mc.variance_option — variance option pricing
• heston_mc.control_variate — control variate estimators
• heston_mc.interfaces — shared result containers
• heston_mc.params — shared parameter and Monte Carlo configuration objects

References

Broadie, M., & Jain, A. (2008). Pricing and Hedging Volatility Derivatives. The Journal of Derivatives, 15(3), 7–24. https://doi.org/10.3905/jod.2008.702503

Fouque, J.-P., & Han, C.-H. (2004). Variance reduction for Monte Carlo methods to evaluate option prices under multi-factor stochastic volatility models. Quantitative Finance, 4(5), 597–606. https://doi.org/10.1080/14697680400020317

License

MIT