PDESolvers ૮₍ ˶•⤙•˶ ₎ა

A Python package for solving partial differential equations (PDEs), including the one-dimensional heat equation and the Black-Scholes equation, using numerical methods such as explicit and Crank-Nicolson finite difference schemes. Features include built-in plotting, benchmarking and support for financial applications such as option pricing.

📦 Installation

The pdesolvers package can be installed using pip. To install the package, run the following command:

pip install pdesolvers

Updating the package to its latest version can be done with:

pip install --upgrade pdesolvers

🧩 Supported Features

Solvers

• ✅ Explicit method
• ✅ Crank-Nicolson method

Equations

• ✅ 1D Heat Equation
• ✅ Black-Scholes Equation for vanilla European options

Pricing Methods

• ✅ Monte Carlo Pricing
• ✅ Analytical Black-Scholes formula

📁 Project Structure

PDESolvers/
├── pdesolvers/                # Main Python package
│   ├── enums/                 # Enum definitions (e.g., option types, greeks)
│   ├── optionspricing/        # Modules for Monte Carlo and Black-Scholes pricing
│   ├── pdes/                  # PDE definitions (HeatEquation, BlackScholesEquation, etc.)
│   ├── solution/              # Solution classes (Solution1D, SolutionBlackScholes, etc.)
│   ├── solvers/               # Explicit, Crank-Nicolson, etc.
│   ├── tests/                 # Unit tests for Python components
│   ├── utils/                 # Helper functions
│   ├── __init__.py            # Makes it a package
├── GPUSolver/                 # GPU-accelerated module
│   ├── cpu/                   # CPU-side implementations (C++)
│   ├── gpu/                   # CUDA kernels and GPU logic
│   ├── tests/                 # Tests for GPU and C++ logic

📝 Note: The Python and C++/CUDA libraries are currently developed as separate components and are not integrated. The Python library can be used independently via PyPI, while the GPU-accelerated solvers are available as a standalone C++/CUDA project.

📊 Export Options

Use export=True flags in plotting functions or benchmarking methods to export:

• PDF: Save plots as PDF files.
• CSV: Save benchmark results as CSV files.

🚀 Usage

To use the package, you can import the desired modules and classes and create an instance of the solvers.

Example usage of the 1D heat equation solver

from pdesolvers import HeatEquation, Heat1DExplicitSolver, Heat1DCNSolver
import numpy as np

equation = (HeatEquation(1, 100,30,10000, 0.01)
            .set_initial_temp(lambda x: np.sin(np.pi * x) + 5)
            .set_left_boundary_temp(lambda t: 20 * np.sin(np.pi * t) + 5)
            .set_right_boundary_temp(lambda t: t + 5))

solution1 = Heat1DCNSolver(equation).solve()
solution2 = Heat1DExplicitSolver(equation).solve()

result = solution1.get_result()
solution1.plot()

Example usage of the Black-Scholes equation solver

from pdesolvers import BlackScholesEquation, BlackScholesExplicitSolver, BlackScholesCNSolver, OptionType, Greeks

equation = BlackScholesEquation(OptionType.EUROPEAN_CALL, 300, 295, 0.05, 0.2, 1, 100, 10000)

solution1 = BlackScholesExplicitSolver(equation).solve()
solution2 = BlackScholesCNSolver(equation).solve()

solution1.plot()
solution1.plot_greek(Greeks.GAMMA)
solution2.get_execution_time()

Example usage of Monte Carlo Pricing

from pdesolvers import MonteCarloPricing, OptionType

pricing = MonteCarloPricing(OptionType.EUROPEAN_CALL, 300, 290, 0.05, 0.2, 1, 365, 1000, 78)
option_price = pricing.get_monte_carlo_option_price()

pricing.plot_price_paths()
pricing.plot_distribution_of_payoff()
pricing.plot_distribution_of_final_prices()

Example usage of Analytical Black-Scholes formula

from pdesolvers import BlackScholesFormula, OptionType

pricing = BlackScholesFormula(OptionType.EUROPEAN_CALL, 300, 290, 0.05, 0.2, 1)
option_price = pricing.get_black_scholes_merton_price()

Using Real Historical Data

from pdesolvers import HistoricalStockData, MonteCarloPricing, OptionType

ticker = 'NVDA'

historical_data = HistoricalStockData(ticker)
historical_data.fetch_stock_data( "2024-02-28","2025-02-28")

sigma, mu = historical_data.estimate_metrics()
initial_price = historical_data.get_initial_stock_price()
closing_prices = historical_data.get_closing_prices()

pricing = MonteCarloPricing(OptionType.EUROPEAN_CALL, initial_price, 160, mu, sigma, 1, len(closing_prices), 1000, 78)

pricing.plot_price_paths(export=True)

📝 Note: You don't necessarily need to use the HistoricalStockData class — you're free to use raw yfinance data directly. Use the built-in tools only if you want to estimate metrics like volatility or mean return.

📊 Comparing Interpolated Grid Solutions

from pdesolvers import BlackScholesEquation, BlackScholesExplicitSolver, BlackScholesCNSolver, OptionType

equation1 = BlackScholesEquation(OptionType.EUROPEAN_CALL, S_max=300, K=100, r=0.05, sigma=0.2, expiry=1, s_nodes=100, t_nodes=1000)
equation2 = BlackScholesEquation(OptionType.EUROPEAN_CALL, S_max=300, K=100, r=0.05, sigma=0.2, expiry=1)

solution1 = BlackScholesExplicitSolver(equation1).solve()
solution2 = BlackScholesCNSolver(equation1).solve()

error = solution1 - solution2

📊 Additional Benchmarks

from pdesolvers import MonteCarloPricing, BlackScholesFormula, OptionType

num_simulations_list = [ 20, 50, 100, 250, 500, 1000, 2500]

pricing_1 = BlackScholesFormula(OptionType.EUROPEAN_CALL, 300, 290, 0.05, 0.2, 1)
pricing_2 = MonteCarloPricing(OptionType.EUROPEAN_CALL, 300, 290, 0.05, 0.2, 1, 365, 1000000, 78)

bs_price = pricing_1.get_black_scholes_merton_price()
monte_carlo_price = pricing_2.get_monte_carlo_option_price()

pricing_2.get_benchmark_errors(bs_price, num_simulations_list=num_simulations_list)
pricing_2.plot_convergence_analysis(bs_price, num_simulations_list=num_simulations_list, export=True)

📝 Note: The export flag used in the example above will save the plot as a PDF file in the current working directory.

🧠 Limitations

• The package currently supports only one-dimensional PDEs.
• Currently limited to vanilla European options.

🔒 License

This project is licensed under the Apache License 2.0. See the LICENSE file for details.