FinStoch 0.5.0

A Python library for simulating stochastic processes in finance.

FinStoch

A Python library for simulating stochastic processes in finance.

Installation

pip install FinStoch

Development

pip install -e ".[dev]"

# Run tests
python -m unittest discover -s tests -p "*_test.py"

# Format / lint / type check
ruff format
flake8 --max-line-length 127
mypy . --exclude venv --ignore-missing-imports

Processes

Geometric Brownian Motion

• SDE

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

• A stochastic process where the logarithm of the variable follows a Brownian motion with drift, representing continuous growth with random fluctuations.

• Euler-Maruyama Discretization

$$ S_{t+\Delta t} = S_t.e^{\left(\mu-\frac{\sigma^2}{2}\right)\Delta t+\sigma\sqrt{\Delta t}\epsilon_t} $$

Where $\epsilon_t \sim \mathcal{N}(0, 1)$.

import numpy as np
from FinStoch.processes import GeometricBrownianMotion

S0 = 100                    # Initial value 
mu = 0.05                   # Annualized Drift coefficient (expected return rate)
sigma = 0.2                 # Annualized Volatility (standard deviation of returns)
num_paths = 10              # Number of paths to simulate
start_date = '2023-09-01'   # Start date for the simulation
end_date = '2024-09-01'     # End date for the simulation
granularity = 'D'           # Granularity in daily intervals
    
# Create the GBM model
gbm = GeometricBrownianMotion(S0, mu, sigma, num_paths, start_date, end_date, granularity)

# Simulate the GBM process
simulated_paths = gbm.simulate()

# Plot the simulated paths
gbm.plot(paths=simulated_paths, 
         title='Stock Price under Geometric Brownian Motion Assumption', 
         ylabel='Stock Price',
         fig_size=(15,5)
        )

Merton's Jump Diffusion Model

• SDE

$$ dS_t = \left(\mu - \lambda_j\left( e^{\mu_j+\frac{\sigma_j^2}{2}} - 1\right) \right) S_t dt + \sigma S_t dW_t + S_t \left( \prod_{i=1}^{dN_t} Y_i - 1\right) $$

• An extension of the geometric Brownian motion that incorporates sudden, discrete jumps $ J_t $ in addition to continuous diffusion, capturing both regular volatility and occasional large shocks.

• Euler-Maruyama Discretization

$$ S_{t+\Delta t} = S_t . e^{\left( \mu - \frac{1}{2} \sigma^2 -\lambda_j\left( e^{\mu_j+\frac{\sigma_j^2}{2}} - 1\right)\right) \Delta t + \sigma \sqrt{\Delta t} \epsilon_t + J_t } $$

Where $\epsilon_t \sim \mathcal{N}(0, 1)$ and $J_t$ is the jump component at time $t$.

import numpy as np
from FinStoch.processes import MertonJumpDiffusion

# Parameters
S0 = 100                    # Initial process value
mu = 0.05                   # Drift coefficient
sigma = 0.2                 # Volatility
lambda_j = 1                # Jump intensity
mu_j = 0.0                  # Mean of jump size
sigma_j = 0.15              # Standard deviation of jump size
num_paths = 10              # Number of simulated paths
start_date = '2023-09-01'   # Start date for the simulation
end_date = '2024-09-01'     # End date for the simulation
granularity = 'D'           # Granularity in daily intervals


# Create Merton model instance and plot
merton = MertonJumpDiffusion(S0, mu, sigma, lambda_j, mu_j, sigma_j, num_paths, start_date, end_date, granularity)

# Simulate the Merton process
simulated_paths = merton.simulate()

# Plot the simulated paths
merton.plot(paths=simulated_paths, 
         title='Stock Price under the Merton Model Assumption', 
         ylabel='Stock Price',
         fig_size=(15,5)
        )

Ornstein-Uhlenbeck model

• SDE

$$ dS_t = \theta (\mu - S_t) dt + \sigma dW_t $$

• A mean-reverting stochastic process where the variable fluctuates around a long-term mean with a tendency to revert back, driven by continuous noise.

• Euler-Maruyama Discretization

$$ S_{t+\Delta t} = S_t + \theta (\mu - S_t) \Delta t + \sigma \sqrt{\Delta t} \epsilon_t $$

Where $\epsilon_t \sim \mathcal{N}(0, 1)$.

import numpy as np
from FinStoch.processes import OrnsteinUhlenbeck 

# Parameters
S0 = 100                    # Initial value
mu = 100                    # Annualized drift coefficient
sigma = 0.2                 # Anualized volatility
theta = 0.5                 # Annualized mean reversion rate
num_paths = 10              # Number of paths to simulate
start_date = '2023-09-01'   # Start date for the simulation
end_date = '2024-09-01'     # End date for the simulation
granularity = 'D'           # Granularity in daily intervals

# Create Ornstein-Uhlenbeck model instance and plot
ou = OrnsteinUhlenbeck(S0, mu, sigma, theta, num_paths, start_date, end_date, granularity)

# Simulate the OU process
simulated_paths = ou.simulate()

# Plot the simulated paths
ou.plot(paths=simulated_paths, 
         title='Stock Price under the Ornstein Uhlenbeck model Assumption', 
         ylabel='Stock Price',
         fig_size=(15,5)
        )

Cox-Ingersoll-Ross Model

• SDE

$$ dS_t = \kappa (\theta - S_t) dt + \sigma \sqrt{S_t} dW_t $$

• A mean-reverting process with volatility that depends on the current level of the variable, ensuring the values are always non-negative.

• Euler-Maruyama Discretization

$$ S_{t+\Delta t} = S_t + \kappa (\theta - S_t) \Delta t + \sigma \sqrt{S_t} \sqrt{\Delta t} \epsilon_t $$

Where $\epsilon_t \sim \mathcal{N}(0, 1)$.

import numpy as np
from FinStoch.processes import CoxIngersollRoss 

# Parameters 
S0 = 0.03                   # Initial value
mu = 0.03                   # Long-term mean
sigma = 0.1                 # Volatility
theta = 0.03                # Speed of reversion
num_paths = 10              # Number of simulation paths
start_date = '2023-09-01'   # Start date for the simulation
end_date = '2024-09-01'     # End date for the simulation
granularity = 'D'           # Granularity in daily intervals

# Create an instance of the CoxIngersollRoss class
cir = CoxIngersollRoss(S0, mu, sigma, theta, num_paths, start_date, end_date, granularity)

# Simulate the CIR process
simulated_paths = cir.simulate()

# Plot the simulated paths
cir.plot(paths=simulated_paths, 
         title='Rate under the CIR model Assumption', 
         ylabel='Rate',
         fig_size=(15,5)
        )

Constant Elasticity of Variance Model

• SDE

$$ dS_t = \mu S_t dt + \sigma {S_t}^\gamma dW_t $$

• A stochastic process that extends the Geometric Brownian Motion process.

• Euler-Maruyama Discretization

$$ S_{t+\Delta t} = S_t + \mu S_t \Delta t + \sigma {S_t}^\gamma \sqrt{\Delta t} \epsilon_t $$

Where $\epsilon_t \sim \mathcal{N}(0, 1)$.

import numpy as np 
from FinStoch.processes import ConstantElasticityOfVariance

S0 = 100                    # Initial value
mu = 0.05                   # Annualized Drift coefficient (expected return rate)
sigma = 0.2                 # Annualized Volatility (standard deviation of returns)
gamma = 1.2                 # Elasticity coefficient
num_paths = 10              # Number of paths to simulate
start_date = '2023-09-01'   # Start date for the simulation
end_date = '2024-09-01'     # End date for the simulation
granularity = 'D'           # Granularity in daily intervals
    
# Create the CEV model
cev = ConstantElasticityOfVariance(S0, mu, sigma, gamma, num_paths, start_date, end_date, granularity)

# Simulate the CEV process
simulated_paths = cev.simulate()

# Plot the simulated paths
cev.plot(paths=simulated_paths, 
         title='Stock Price under the Constant Elasricity of Variance Model Assumption', 
         ylabel='Stock Price',
         fig_size=(15,5)
        )

Heston Stochastic Volatility Model

• SDEs

$$ dS_t = \mu S_t dt + \sqrt{v_t} S_t dW_{S,t} $$

$$ dv_t = \kappa (\theta - v_t) dt + \sigma_v \sqrt{v_t} dW_{v,t} $$

$$ dW_{S,t}\times dW_{v,t}=\rho dt $$

• A stochastic volatility model where the volatility of a variable follows its own mean-reverting process, allowing for time-varying volatility that evolves over time.

• Euler-Maruyama Discretization

$$ S_{t+\Delta t} = S_t.e^{\left(\mu-\frac{v_t}{2}\right)\Delta t+\sqrt{v_t}\sqrt{\Delta t}\epsilon_{S,t}} $$

$$ v_{t+\Delta t} = v_t + \kappa (\theta - v_t) \Delta t + \sigma_v \sqrt{v_t} \sqrt{\Delta t} \epsilon_{v,t} $$

$$ Corr(\epsilon_{S,t},\epsilon_{v,t})=\rho $$

Where $\epsilon_S$ and $\epsilon_v$ are correlated standard normal variables.

import numpy as np
from FinStoch.processes import HestonModel

# Parameters
S0 = 100                    # Initial value
v0 = 0.02                   # Initial volatility
mu = 0.05                   # Long-term mean of the value
theta = 0.04                # Long-term mean of the volatility
sigma = 0.3                 # Volatility of the volatility
kappa = 1.5                 # Speed of reversion
rho = -0.7                  # Correlation between shocks
num_paths = 10              # Number of simulation paths
start_date = '2023-09-01'   # Start date for the simulation
end_date = '2024-09-01'     # End date for the simulation
granularity = 'D'           # Granularity in daily intervals

# Initialize Heston model
heston = HestonModel(S0, v0, mu, sigma, theta, kappa, rho, num_paths, start_date, end_date, granularity)

# Simulate the Heston model
simulated_paths = heston.simulate()

# Plot the simulated paths
heston.plot(paths=simulated_paths,
            title='Asset Price under the Heston model Assumption', 
            ylabel='Asset Price',
            fig_size=(15,5)
        )

heston.plot(paths=simulated_paths,
            title='Asset Volatility under the Heston model Assumption', 
            ylabel='Asset Volatility',
            variance=True,
            fig_size=(15,5)
            )

License

This project is licensed under the MIT license found in the LICENSE file.