Geometric Brownian Motion

Geometric Brownian motion is a stochastic process as described in Equation 16-1 (see

also Equation 10-2 in Chapter 10, in particular for the meaning of the parameters and

variables). The drift of the process is already set equal to the riskless, constant short rate r,

implying that we operate under the equivalent martingale measure (see Chapter 15).

Equation 16-1. Stochastic differential equation of geometric Brownian motion

dSt = rStdt + StdZt

Equation 16-2 presents an Euler discretization of the stochastic differential equation for

simulation purposes (see also Equation 10-3 in Chapter 10 for further details). We work in

a discrete time market model, such as the general market model ℳ from Chapter 15, with

a finite set of relevant dates 0 < t 1 < t 2 < ... < T.

Equation 16-2. Difference equation to simulate the geometric Brownian motion

The Simulation Class

Example 16-3 now presents the specialized class for the GBM model. We present it in its

entirety first and highlight selected aspects afterward.

Example 16-3. Simulation class for geometric Brownian motion

DX Library Simulation

geometric_brownian_motion.py

import numpy as np

from sn_random_numbers import sn_random_numbers
from simulation_class import simulation_class

class geometric_brownian_motion(simulation_class):
”’ Class to generate simulated paths based on
the Black-Scholes-Merton geometric Brownian motion model.

            Attributes

            ==========

            name    :   string

                            name    of  the object

            mar_env :   instance    of  market_environment

                            market  environment data    for simulation

            corr    :   Boolean

                            True    if  correlated  with    other   model   simulation  object



            Methods

            =======

            update  :

                            updates parameters

            generate_paths  :