Derivatives assignment
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 your colleagues. No late assignments will be accepted, no exceptions. Please send the assignment by email (PDF file) to
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[email protected]. 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, or 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:
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[email protected] 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). b. 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 for 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 February 19th 2016, 15:00:00. The interest rate is r=0.0028. Download the option data provided in the Excel file “VIXdata.xlsx”, which contain two worksheets: • Worksheet “Short” contains options that expire on Mar 18 2016 at 15:00 (i.e., “near-term” options) • Worksheet “Long” contains options that expire on Mar 25 2016 at 15:00 (i.e., “next-term” options) Follow the step-by-step calculation in the document “vixwhite.pdf” and answer 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. b. 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