21 Jun Problem Set 6: Monte Carlo Simulation of a Mark
Problem Set 6: Monte Carlo Simulation of a Markov Process to Generate Options Price
Estimates
DUE: TUESDAY, May 6, 2019 BY 6:00 PM
In this problem, worth 25 points, the procedure of Monte Carlo simulation of Markov processes will be used to generate options price estimates and compare these to actual options prices for put and call options for at least 3 of the stocks that you used in Problem Set 1. Update the stock and options prices for calls and puts at strike prices at the money of the option at the most recent date using finance.yhaoo.com for the options. Use at least a 3-month time period to expiration such as the September 20, 2019 expiration date. Employ the implied volatility for each stock as given in the options table in Yahoo Finance. Use the Mathematica program provided on Blackboard for the course under the Problem Set 6 item. The data you will need are:
1. market values of the stock (s),
2. market implied volatility of each stock’s returns for calls and puts (sigmav and sigmaput respectively),
3. strike price of call and put options for each stock, x, (use the same strike price for calls and puts),
4. time to expiration (proportion of a year of 252 days,( t),
5. the risk-free rate for the time period say 3-months (rf), and
6. the number of iterations (nof 1000, 5,000 and 10,000).
Computational Requirements
Use Mathematica to perform your calculations.
First compute the Black -Scholes call and put prices using the respective volatility values for future comparison. These are given by functions bsfc for call option prices and bsfput for put options prices. Then, using the assumption that stock prices follow a Geometric Brownian Motion (GBM) with a drift factor of rf, so that with the change in the natural logarithm of stock prices, dlns = ln (st) – ln (s0) and etaa standard normal random variate : dlns = (rf – 0.5*sigmav^2)*dt + sigmav*eta*Sqrt[t]
stating in terms stock values st= s0*Exp[(rf – 0.5*sigmav^2)*t + sigmav*eta*Sqrt[t]] by taking the exponential of each side of the equation. This is the stock generating process.
The next step is to find the standard normal random variables eta by using the Mathematica functions eta= Table[RandomVariate[NormalDistribution[0, 1]], n]; after defining n=1000 or n=5000 or n=10000. Following this the stock prices can be computed using the Table function again as:
st = Table[ s0*Exp[((rf – 0.5*sigmav^2)*t) + sigmav*eta[[i]]*Sqrt[t]], {i, 1, n}]; (use sigmaput for put option calculations).
Using these computations the call prices and put prices can be estimated as shown in the program by computing the mean of the random call or put values and these estimates then compared to the Black-Scholes estimates and actual options prices.
The Analysis
Analyze your results for each stock option by:
1. How do the estimates of the Monte Carlo method compare with the Black-Scholes for each of the increasing number of iterations? With more iterations do the values get closer to Black-Scholes?
2. Compare the closeness of the Monte Carlo method with the actual options prices. Are they closer for the call or the put and are they closer than the Black-Scholes formula and as the number of iterations increases?
3. Which one of your stocks had the closest estimates to the actual options prices?
4. Use the tabular form provided in the program to present your results in your analysis.
Our website has a team of professional writers who can help you write any of your homework. They will write your papers from scratch. We also have a team of editors just to make sure all papers are of HIGH QUALITY & PLAGIARISM FREE. To make an Order you only need to click Ask A Question and we will direct you to our Order Page at WriteDemy. Then fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Fill in all the assignment paper details that are required in the order form with the standard information being the page count, deadline, academic level and type of paper. It is advisable to have this information at hand so that you can quickly fill in the necessary information needed in the form for the essay writer to be immediately assigned to your writing project. Make payment for the custom essay order to enable us to assign a suitable writer to your order. Payments are made through Paypal on a secured billing page. Finally, sit back and relax.
About Writedemy
We are a professional paper writing website. If you have searched a question and bumped into our website just know you are in the right place to get help in your coursework. We offer HIGH QUALITY & PLAGIARISM FREE Papers.
How It Works
To make an Order you only need to click on “Order Now” and we will direct you to our Order Page. Fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Are there Discounts?
All new clients are eligible for 20% off in their first Order. Our payment method is safe and secure.