The first objective of our project is to use MatLab to provide simple examples of how the Monte Carlo method can be used. To do this I will create a simple program that shows the distribution of numbers of a six-sided die land on when randomly thrown for a specific number of iterations. To do this I will use the basic method of the Monte Carlo method provided by our textbook Computational Physics by Nicholas J. Girodano and Hisao Nakanishi. The second program I will create will demonstrate how the Monte Carlo method can be used to find the moment of inertia for an object with a range of different measurements for different parameters. Finally the third simple program I will create will be using the Monte Carlo method to find how much money someone needs to be invested into a stock or portfolio of stocks to retire at certain age. All these simple programs use the concept of normal or lognormal distribution and take advantage of MatLab’s built in random number generator. With these simulations we will discuss how central limit theory provides a basis for the Monte Carlo method being an acceptable

The second goal of our project, Elias Kim and I will both attempt to use the Monte Carlo algorithm for the Ising model on an L x L square lattice, separately. The Ising model is an integral part of physics because it helps describe how neighboring atoms and their electron magnetic spins interact with one another and have the ability to change the neighboring electrons spin. In addition the Ising model is used to explain ferromagnetic interaction but its influence has extended into the world of thermodynamics as well. Elias and I will then join together to compare our answers to make sure we got similar solutions.

The final objective of our project, Elias and I will team up and use our knowledge of the Monte Carlo method and Ising model to tackle financial problems. The Ising model can be used to exam the volatility of a stock or portfolio and the risk associated with it. In the world of finance, volatility is defined as a statistical measure of the dispersion of returns for a given security or market index, this can be simply calculated with the standard deviation of returns from that same security or market index. Many factors that previously were thought as random actually alter the volatility of an asset, which is what Elias Kim, and I intend to investigate.

Timeline:

Week of April 6^th : Acquire basic knowledge of Monte Carlo simulation and relevant equations and concepts. Continue researching about the Ising Model and various financial principles.

Week of April 13^th : Create the three basic Monte Carlo simulations described in the General Plan, and begin to work on the simple Ising Model described in Computational Physics.

Week of April 20^th :Finish the simple Ising Model, and compare results to Elias’ findings . Continue researching the Ising Model and its applications to the financial world.

Week of April 27^th: Begin working on the applying the Ising Model to a financial problem. Continue researching on the Ising Model and its applications to finance.

Week of May 4^th : Finish the collaboration with Elias and get ready for project presentations.

References:

1. Monte Carlo Techniques G. Cowan (RHUL).

2. Probability Review and Overview of Monte Carlo Martin Haugh

3. http://www.excelfunctions.net/Excel-Norminv-Function.html ExcelNet

4. Ising Model in Finance Pavel Dvoˇr´ak

5. Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based models

D. Sornette (ETH Zurich)

6. Phase transition in an Ising economy Miloš Borovšak

7. Statistical physics of social dynamics Claudio Castellano, Santo Fortunato, Vittorio Loreto

8. Computational Physics Nicholas J. Girodano, Hisao Nakanishi