aleatory.processes.GBM

class aleatory.processes.GBM(drift=1.0, volatility=0.5, initial=1.0, T=1.0, rng=None)

Geometric Brownian Motion

A Geometric Brownian Motion object.

Notes

A Geometric Brownian Motion \(\{X(t) : t \geq 0\}\) is characterised by the following SDE.

\[dX_t = \mu X_t dt + \sigma X_t dW_t \ \ \ \ t\in (0,T]\]

with initial condition \(X_0 = x_0\geq0\), where

• \(\mu\) is the drift

• \(\sigma>0\) is the volatility

• \(W_t\) is a standard Brownian Motion.

The solution to this equation can be written as

\[X_t = x_0\exp\left((\mu + \frac{\sigma^2}{2} )t +\sigma W_t\right)\]

and each \(X_t\) follows a log-normal distribution.

Examples

from aleatory.processes import GBM
process = GBM()
fig = process.draw(n=100, N=5, figsize=(12, 7))
fig.show()

Constructor, Methods, and Attributes

__init__(drift=1.0, volatility=0.5, initial=1.0, T=1.0, rng=None)

Parameters:

• drift (float) – the parameter \(\mu\) in the above SDE

• volatility (float) – the parameter \(\sigma>0\) in the above SDE

• initial (float) – the initial condition \(x_0\) in the above SDE

• T (float) – the right hand endpoint of the time interval \([0,T]\) for the process

• rng (numpy.random.Generator) – a custom random number generator

Methods

__init__([drift, volatility, initial, T, rng])	param float drift: the parameter \(\mu\) in the above SDE
draw(n, N[, T, marginal, envelope, title, ...])	Simulates and plots paths/trajectories from the instanced stochastic process.
estimate_covariances([times])
estimate_expectations()
estimate_quantiles(q)
estimate_stds()
estimate_variances()
get_marginal(t)
marginal_expectation([times])
marginal_stds([times])
marginal_variance([times])
plot(n, N[, T, title, suptitle])	Simulates and plots paths/trajectories from the instanced stochastic process.
plot_covariance([times, title])
plot_kernel([times, colormap, matrix_shape, ...])
plot_kernel3d([times, title])
plot_mean_variance([times, title])
plot_paths_and_kernel(n, N[, T, title, ...])	Plots the paths of the process and the covariance kernel.
process_covariance([times])
process_expectation()
process_stds()
process_variance()
sample(n)	Generate a realization.
sample_at(times)	Generate a realization using specified times.
simulate(n, N[, T])	Simulate paths/trajectories from the instanced stochastic process.

Attributes

T	End time of the process.
drift	Geometric Brownian motion drift parameter.
initial	Geometric Brownian motion initial point.
rng
volatility	Geometric Brownian motion volatility parameter.