This README is intended to guide the user in how to use the GBM simulation module.
The README is written in Markdown, and is much easier to read on GitHub:
https://github.com/DavisVaughan/uncc-math-6204/tree/master/assignments/hw-03
The purpose of this module is to calculate the Monte Carlo value of European options in multiple ways using varying methods of accuracy.
It seems like using a smaller time step seems to greatly increase the accuracy. Adding the second order term did not help the accuracy as much as I had expected. The biggest benefits seem to come from increasing the number of simulations. Using n = 50000 greatly increases the accuracy of the final results.
Both Euler and Milstein discretization of GBM were used to simulate the sample paths.
Using the value of S_T from the sample paths, the average payoffs of the option were calculated, and discounted back to time zero.
The exact solution to the GBM SDE was also used to compare accuracy results.
main.py - (DRIVER) a demo of the GBM functions using the parameters set in the HW-2 pdf.
gbm_simulator.py - The functions that generate the stock price simulations using either Euler or Miltstein methods.
option_value.py - Functions to calculate the value of the option from the simulated values.
option_value_exact.py - Previous HW code to generate the exact value of the options.
Because the main.py file includes the code:
if __name__ == "__main__":
print(main())the easiest way to run the example is from the terminal.
Within your command line / terminal, navigate to the folder containing the main.py script, and just run:
python2 main.py^ Make sure you are using python2.
A plot will pop up first with the 5 paths, and once you close the plot a numpy array will be printed to the console that looks like this:
MC_option_value algorithm call_put dt exact_option_value option_type \
0 28.189921 euler call .01 28.684884 european
1 29.255651 euler put .01 28.198446 european
2 28.164755 euler call .001 28.684884 european
3 28.579581 euler put .001 28.198446 european
4 27.931396 milstein call .01 28.684884 european
5 29.136742 milstein put .01 28.198446 european
6 28.118083 milstein call .001 28.684884 european
7 28.570122 milstein put .001 28.198446 european
absolute_error
0 0.494962
1 1.057205
2 0.520129
3 0.381135
4 0.753487
5 0.938296
6 0.566800
7 0.371676
Comments
IMPORTANT! - The seed is set at the C++ level using Numba, not at the Python level. Because of this, we will not get the same results unless you implement this with Numba. Setting a seed of 100 at the C++ level will generate a different stream of random numbers than setting a seed of 100 at the Python level.