OptionMC: Monte Carlo Option Pricing

A Python package for pricing European options using Monte Carlo simulation, featuring variance reduction techniques and publication-quality visualizations. Designed for educational purposes and financial engineering applications.

Features

• European call and put option pricing with Monte Carlo simulation
• Variance reduction using antithetic variates
• Comparison with Black-Scholes analytical solutions
• Publication-quality visualizations:
  • Price convergence analysis
  • Stock price and payoff distributions
  • Parameter sensitivity analysis
• Command-line interface for quick pricing and visualization
• Comprehensive examples for educational purposes

Installation

pip install optionmc

Quick Start

from optionmc.models import OptionPricing

# Create option pricing model
model = OptionPricing(
    S0=100,     # Initial stock price
    E=100,      # Strike price
    T=1.0,      # Time to maturity (1 year)
    rf=0.05,    # Risk-free rate (5%)
    sigma=0.2,  # Volatility (20%)
    iterations=100000  # Number of simulations
)

# Calculate option prices
call_price = model.call_option_simulation()
put_price = model.put_option_simulation()

# Get analytical solutions for comparison
bs_call, bs_put = model.bs_analytical_price()

# Print results
print(f"Call Option Price: ${call_price:.4f} (Black-Scholes: ${bs_call:.4f})")
print(f"Put Option Price: ${put_price:.4f} (Black-Scholes: ${bs_put:.4f})")

Command Line Usage

# Basic option pricing
optionmc price --s0 100 --strike 95 --volatility 0.25 --time 0.5

# Using antithetic variates for variance reduction
optionmc price --method antithetic --iterations 500000

# Custom output directory
optionmc price --output-dir my_results

Mathematical Background

Black-Scholes Option Pricing: Derivation and Implementation

The Black-Scholes model provides analytical solutions for European options based on a partial differential equation approach. This package implements the closed-form solution as a benchmark for Monte Carlo simulations.

Derivation Overview

The derivation starts with several key assumptions:

• Stock prices follow geometric Brownian motion
• Markets are frictionless with no arbitrage opportunities
• Volatility and risk-free rate remain constant
• Trading occurs continuously
• No dividends are paid during the option's life

Under these assumptions, the stock price follows the stochastic differential equation:

$dS_t = \mu S_t dt + \sigma S_t dW_t$

Where $W_t$ is a Wiener process (Brownian motion), $\mu$ is the drift, and $\sigma$ is the volatility.

The Black-Scholes partial differential equation is:

$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$

Solving this equation with appropriate boundary conditions yields the closed-form solutions:

For a European call option:

$C = S_0 \Phi(d_1) - Ke^{-rT} \Phi(d_2)$

For a European put option:

$P = Ke^{-rT} \Phi(-d_2) - S_0 \Phi(-d_1)$

Where:

• $S_0$ is the initial stock price
• $K$ is the strike price
• $r$ is the risk-free rate
• $T$ is the time to maturity (in years)
• $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution
• $d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$
• $d_2 = d_1 - \sigma\sqrt{T}$

Why Use the Analytical Solution?

This package implements both Monte Carlo simulation and the analytical Black-Scholes solution for several reasons:

1. Benchmarking: The analytical solution provides a precise benchmark to validate the accuracy of Monte Carlo simulations
2. Educational Value: Comparing both methods helps users understand the trade-offs between analytical and numerical approaches
3. Efficiency: For standard European options, the analytical solution is computationally more efficient
4. Error Analysis: Having an exact solution allows for precise error measurements in the Monte Carlo implementation

Limitations of the Black-Scholes Model

Despite its elegance, the Black-Scholes model has several limitations:

1. Constant Volatility: Assumes volatility remains constant throughout the option's life, contradicting observed "volatility smiles" in markets
2. Log-normal Distribution: Assumes returns follow a normal distribution, which often understates the probability of extreme market moves
3. Continuous Trading: Assumes continuous hedging with no transaction costs, which is impractical in real markets
4. Constant Interest Rates: Assumes risk-free rates remain fixed, which may not hold for longer-dated options
5. No Early Exercise: Only applies to European options, not American options that allow early exercise
6. No Dividends: The basic model doesn't account for dividend payments (though extensions exist)

Monte Carlo Simulation

The Monte Carlo approach for option pricing follows these steps:

1. Simulate stock price paths using Geometric Brownian Motion (GBM):

   $$S_T = S_0 \exp\left((r - \frac{\sigma^2}{2})T + \sigma\sqrt{T}Z\right)$$

   where $Z \sim N(0,1)$ is a standard normal random variable.

2. Calculate payoffs at maturity:

   • Call option: $\max(S_T - K, 0)$
   • Put option: $\max(K - S_T, 0)$

3. Discount and average the payoffs to get the option price:

   • Call price: $C = e^{-rT} \mathbb{E}[\max(S_T - K, 0)]$
   • Put price: $P = e^{-rT} \mathbb{E}[\max(K - S_T, 0)]$

Variance Reduction Techniques

This package implements the antithetic variates technique, which:

1. Uses pairs of negatively correlated random variables $(Z, -Z)$ in the simulation
2. Averages the results to reduce variance
3. Typically achieves the same accuracy with fewer simulations

Package Structure

• models.py: Core option pricing algorithms
• visualization.py: Plotting functions for analysis
• cli.py: Command-line interface
• utils.py: Utility functions for saving/loading results

Limitations and Considerations

Monte Carlo Simulation Limitations

While powerful, the Monte Carlo approach has several limitations to consider:

1. Computational Intensity: Requires many iterations for accurate results, especially for:

   • Out-of-the-money options (K/S₀ >> 1 for calls, K/S₀ << 1 for puts)
   • Options with low volatility
   • Options near expiration

2. Convergence Rate: Monte Carlo error decreases as O(1/√n), meaning:

   • To halve the error, you need 4 times more simulations
   • Variance reduction techniques help but don't eliminate this issue

3. Model Assumptions:

   • Assumes geometric Brownian motion for stock prices
   • Constant volatility and interest rates
   • No transaction costs or early exercise
   • Perfect liquidity and continuous trading

4. Implementation Considerations:

   • Random number generator quality affects results
   • Numerical precision issues with very small probabilities
   • Not suitable for real-time pricing without optimizations

Package-Specific Limitations

The current implementation of OptionMC has several limitations:

1. European Options Only: Limited to European-style options; does not support American or exotic options
2. Single-Factor Models: Only models stock price as the underlying stochastic process
3. Basic Variance Reduction: Implements antithetic variates but not other techniques like control variates or importance sampling
4. No Path-Dependent Features: Cannot handle path-dependent options like Asian options or barrier options
5. No Dividend Support: Current implementation does not account for dividend payments

When to Use Alternative Methods

For certain option pricing scenarios, consider alternatives:

1. Analytical Solutions:

   • Black-Scholes for standard European options (faster and more accurate)
   • Closed-form extensions for simple dividend models or barrier options

2. Lattice Methods:

   • Binomial/trinomial trees for American options with early exercise
   • Better for visualizing the evolution of option prices over time

3. Finite Difference Methods:

   • For solving complex PDEs associated with exotic options
   • Often more efficient than Monte Carlo for low-dimensional problems

4. Fourier Transform Methods:

   • For options with known characteristic functions
   • Often more efficient for certain exotic options

Contributing

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

1. Fork the repository
2. Create your feature branch: git checkout -b feature/amazing-feature
3. Commit your changes: git commit -m 'Add some amazing feature'
4. Push to the branch: git push origin feature/amazing-feature
5. Open a Pull Request

License

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

Authors

• Sandy H. S. Herho [email protected]
• Siti N. Kaban
• Cahya Nugraha

Citation

If you use this package in your research, please cite it as:

@misc{herho2025optionmc,
  author = {Herho, S and Kaban, S and Nugraha, C},
  title = {OptionMC: A Python package for Monte Carlo pricing of European options},
  howpublished = {Available at SSRN: \url{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5224853}},
  year = {2025},
  note = {Accessed: January 10, 2025}
}