Jump Diffusion

Equipped with the background knowledge from the geometric_brownian_motion class, it

is now straightforward to implement a class for the jump diffusion model described by

Merton (1976). Recall the stochastic differential equation of the jump diffusion, as shown

in Equation 16-3 (see also Equation 10-8 in Chapter 10, in particular for the meaning of

the parameters and variables).

Equation 16-3. Stochastic differential equation for Merton jump diffusion model

dSt = (r – rJ)Stdt + StdZt + JtStdNt

An Euler discretization for simulation purposes is presented in Equation 16-4 (see also

Equation 10-9 in Chapter 10 and the more detailed explanations given there).

Equation 16-4. Euler discretization for Merton jump diffusion model

The Simulation Class

Example 16-4 presents the Python code for the jump_diffusion simulation class. This

class should by now contain no surprises. Of course, the model is different, but the design

and the methods are essentially the same.

Example 16-4. Simulation class for jump diffusion

DX Library Simulation

jump_diffusion.py

import numpy as np

from sn_random_numbers import sn_random_numbers
from simulation_class import simulation_class

class jump_diffusion(simulation_class):
”’ Class to generate simulated paths based on
the Merton (1976) jump diffusion 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   object



            Methods

            =======

            update  :

                            updates parameters

            generate_paths  :