7.9 KiB
This code simulates Geometric Brownian Motion (GBM) to model asset price paths over time, incorporating randomness and volatility in financial markets. It defines functions to generate a Wiener process (representing Brownian motion) and to compute GBM returns and price levels. Parameters such as initial stock price, volatility, drift, and the number of paths and time increments are used to create and visualize multiple simulated price trajectories. This is useful in practice for financial modeling, risk analysis, and forecasting asset prices.
import numpy as np import matplotlib.pyplot as plt
Define initial stock price, volatility, and drift parameters for the Brownian motion
s0 = 131.00 sigma = 0.25 mu = 0.35
Define simulation parameters including number of paths, time increment, and total simulation time
paths = 1000 delta = 1.0 / 252.0 time = 252 * 5
Generate a Wiener process (or Brownian motion) using normally distributed random values. This process simulates the random fluctuations of an asset's price over time, which is essential for modeling the stochastic component of Geometric Brownian Motion.
def wiener_process(delta, sigma, time, paths): """Returns a Wiener process Parameters ---------- delta : float The increment to downsample sigma sigma : float Percentage volatility time : int Number of samples to create paths : int Number of price simulations to create Returns ------- wiener_process : np.array Notes ----- This method returns a Wiener process. The Wiener process is also called Brownian motion. For more information about the Wiener process check out the Wikipedia page: http://en.wikipedia.org/wiki/Wiener_process """ # Generate a Wiener process using normally distributed random values return sigma * np.random.normal(loc=0, scale=np.sqrt(delta), size=(time, paths))
Generate returns from a Geometric Brownian Motion (GBM) model by first creating a Wiener process to simulate random fluctuations.
def gbm_returns(delta, sigma, time, mu, paths): """Returns returns from a Geometric brownian motion Parameters ---------- delta : float The increment to downsample sigma sigma : float Percentage volatility time : int Number of samples to create mu : float Percentage drift paths : int Number of price simulations to create Returns ------- gbm_returns : np.ndarray Notes ----- This method constructs random Geometric Brownian Motion (GBM). """ # Generate Wiener process for the simulation process = wiener_process(delta, sigma, time, paths) # Apply the geometric Brownian motion formula to generate returns return np.exp( process + (mu - sigma**2 / 2) * delta )
Calculate price paths starting from an initial stock price (s0) using Geometric Brownian Motion (GBM)
def gbm_levels(s0, delta, sigma, time, mu, paths): """Returns price paths starting at s0 Parameters ---------- s0 : float The starting stock price delta : float The increment to downsample sigma sigma : float Percentage volatility time : int Number of samples to create mu : float Percentage drift paths : int Number of price simulations to create Returns ------- gbm_levels : np.ndarray """ # Generate GBM returns for the given parameters returns = gbm_returns(delta, sigma, time, mu, paths) # Stack an array of ones with the returns and compute cumulative product to get price levels stacked = np.vstack([np.ones(paths), returns]) return s0 * stacked.cumprod(axis=0)
Generate price paths using the defined GBM model and plot them
price_paths = gbm_levels(s0, delta, sigma, time, mu, paths) plt.plot(price_paths, linewidth=0.25) plt.show()
Generate price paths with zero drift to observe behavior and plot them
price_paths = gbm_levels(s0, delta, sigma, time, 0.0, paths) plt.plot(price_paths, linewidth=0.25) plt.show()
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