FINANCE704 W2022 Term Project M. Milewski
Methods and Models
The purpose behind this project is to practice using various methods to price options and understand
how model selection and parameters affect the result.
General
This project will consist of two main parts:
• Part 1: Pricing a put or call option using the following methods and models:
o European Option using Black-Scholes analytical formula.
o European Option using Explicit Method with Black-Scholes PDE.
o European Option using Implicit Method with Black-Scholes PDE.
o European Option using Monte Carlo Simulation.
o American Option using Implicit Method with Black-Scholes PDE.
o American Option using Monte Carlo Simulation.
• Part 2: Markov Chain Model
o European Option using Monte Carlo Simulation.
o American Option using Monte Carlo Simulation.
Pick a Stock/ETF that is traded on a major exchange that has American Call/Put option available.
• Pick an option that has at least 60 days to expiry and has high open interest.
• You will need to record:
o Option price.
o Stock price.
o Date for option & stock price. This date will be your reference date, t=0, for your pricing
models and the expiry date will be t=T.
• Download historical price data for the stock you selected, at least one years’ worth of data will
be required.
• You will also need the current risk-free rate.
Part 1
Price the option using the various methods outlined above.
For the Monte Carlo Simulation, perform the simulation with 100, 1000, 10000, and 100000 runs.
• How does the confidence interval change with relation to the number of simulation runs?
• How many runs are sufficient?
Compare your results from the different methods to each other and to the observed option price.
Remember to justify your results using the numerical methods concepts covered in class.
Note: Do not use the built-in MATLAB functions for option pricing, you are to code the calculations.

FINANCE704 W2022 Term Project M. Milewski
Part 2 – Markov Chain Model
In this problem, you will be estimating the price of an option using a diffusion model that has the
stochastic parameter determined by a Markov chain process. The continuous time process is given by


= + ()
Where sigma is determined by the state
() = [ 2 3]

= [
0.90 0.05 0.05
0.15 0.80 0.05
0.15 0.05 0.80
]
• is the standard deviation parameter you estimated from part 1
• is the transition probability matrix
Calculate the value of a European and American option
• How did you determine the number of simulation replications?
• How does the Markov Chain model affect the price of the option compared to your previous
estimates and to the current market price?

FINANCE704 W2022 Term Project M. Milewski
Report
For this project we will be following a technical report format which is going to include details and
justification on how you designed your analysis. Basic outline:
• Abstract/Summary (1/2 Page)
o What the report is about/purpose.
o Summary of methods used in the report.
o Results/recommendations.
• Introduction/Background (1/2 - 1 Page)
o Overview what the report is about/purpose.
o Background details needed to understand the report.
o Assume the reader has a master’s level understanding of finance, so you do not have to
go into many details on financial theory but enough details that the reader understands
the model setup.
• Model Design and Analysis Results (3-6 Pages)
o Make sure this section is clearly organized.
o Describe the methods/models used for the analysis.
o Clearly state any assumptions you are making (constant risk-free rate, risk neutral
pricing, constant volatility, etc.)
o Justify your selection of parameters, initial conditions, boundary conditions, stability etc.
• Discussion (1/2 – 1 Page)
o Provide a description and interpretation of your results.
• Recommendation (1/2 Page)
o Answer the question, how well does your analysis reflect the market value of the
option? What are possible causes of differences, should any additional analysis be
performed?
Marking
• Analysis/Report
o Part 1 ~60%
o Part 2 ~30%
o Discussion/Recommendation ~10%