McGill Desautels
Daniel Andrei, Financial Derivatives FINE 448, Winter 2022
Assignment #2
Due date: Thursday, Apr. 7, 5pm
This is an open book group assignment. While you are not allowed to copy from
different groups, you are encouraged to discuss conceptual problems with all you
colleagues.
No late assignments will be accepted, no exceptions. Please send the assignment
y email (PDF file) to XXXXXXXXXX and XXXXXXXXXX.
Make sure your answers are clear and legible. Write down clearly the names of all
group members on the assignment. No need to send your codes.
Any appeal to your grade must be submitted in writing no later than one week
after the grades have been announced. A request for a re-grade will result in a full
second evaluation of all questions. The new outcome may be higher, the same, o
lower than the initial grade. For clarification questions about the assignment, please
consult with Nan Ma, the T.A. of the class. Alternatively, you can consult with me
during office hours.
No more questions will be answered 36 hours before the submission
deadline. Good luck!
1
mailto: XXXXXXXXXX
mailto: XXXXXXXXXX
1 Black-Scholes: Closed Form Solution vs. Monte-Carlo Simulation
The purpose of this exercise is to price a vanilla call option with Monte-Carlo
simulation and compare the result with the closed-form formula. The simulation will
also provide a confidence interval for the option price. We will understand how the
length of this confidence interval changes as we increase the number of simulated
paths.
The price of a stock today is S0 = 100. Consider a European call option with
maturity 3 months (T = 1/4) and strike price K = 100. We make the following
parametric assumptions: r = 0.05, σ = 0.2, and δ = 0.
a. Simulate and plot 5 paths for the stock price under the risk-neutral measure.
Use 5 minutes time-increments (there are 8× 12 = 96 increments each day and
90 days until maturity).
. Find the Black-Scholes call option price.
c. Find the option price and its 95% confidence interval by Monte-Carlo simu-
lation. You do not have to simulate the entire paths here—since the option
is European, you will have to simulate only the final price ST . Do this fo
100; 1,000; 1,000,000; and 100,000,000 simulations. Discuss how the length of
the confidence interval changes with the number of simulations. Compare the
Monte-Carlo price with the Black-Scholes price.
2 VIX Calculation
Today’s date and time is Fe
uary 19th 2016, 15:00:00. The interest rate is
= XXXXXXXXXXDownload the option data provided in the Excel file “VIXdata.xlsx”, which
contain two worksheets:
• Worksheet “Short” contains options that expire on Mar XXXXXXXXXXat 15:00 (i.e.,
“near-term” options)
• Worksheet “Long” contains options that expire on Mar XXXXXXXXXXat 15:00 (i.e.,
“next-term” options)
Follow the step-by-step calculation in the document “vixwhite.pdf” and answe
the questions below.
a. What is the time to expiration of near-term options, T1 (in years)? What is
the time to expiration of next-term options, T2 (in years)? Note: a year has
525,600 minutes.
. For both near- and next-term options, determine the forward SPX levels, F1
and F2, by identifying the strike price at which the absolute difference between
call and put prices is smallest. Next, determine K0,1 and K0,2 (the strike prices
immediately below the forward index levels, F1 and F2) for the near- and next-
term options.
2
c. Calculate the variance for both near- and next-term options, σ21 and σ
2
2.
d. Calculate the 30-day weighted average of σ21 and σ
2
2. Then take the square root
of that value and multiply by 100 to get the VIX value.
3