Question - VIX Index and Variance Risk Premium

Reference no: EM132492242

Assignment - There are three questions in this assignment. Type your answers clearly below each question. You may also use this document to provide any additional information, wherever needed. Submit this document along with an Excel sheet to complete the submission.

Question 1 - Option Value at Risk

Assume that you hold in your portfolio a European call option on the S&P500 index with maturity of 50 days and strike price $1000. The value today (i.e., at time t) of the SP&500 is $1000 and the risk-free rate (annualized) is equal to 5%. Finally, the volatility of the S&P500, in annualized calendar days, is equal to 20%. Compute the and of this portfolio according to the Delta method approximation within the Black and Scholes framework.

Hint: The book does not provide the formula for the expected shortfall, but you should be able to derive it, given the derivation of the VaR and what we have done in previous modules.

Question 2 - VIX Index and Volatility Forecasting

In this exercise, we will see how the VIX index can be used to increase our forecast of volatility.

1) Import into Excel the files VIX.xls and S&P500data.xls.

2) For each day for which the closing price on the S&P500 is available, match the value of the VIX index on the same day.

Hint: look into the VLOOKUP function in Excel.

3) Compute (in separate columns):

- The squared value of the VIX (not in percentages)

- The annualized value of the realized variance. You can do this by multiplying the daily RV by 252. Also, multiply the results by 1.4. This is done because the RV is computed only during the trading day, but we need a measure of daily (i.e., 24 hours) variance. Usually 40% of volatility happens when the market is closed, that is why we multiply by 1.4.

4) Forecast the variance for the next day according to:

(a) The HAR model

(b) The HAR model augmented by the squared VIX index (call this model HAR-VIX)

Hint: Be careful about the scaling. Everything should be in annualized decimal units.

5) Which model, the HAR or the HAR-VIX, better forecasts the next day variance?

Hint: Look at the of the regressions used to estimate the HAR and the HAR-VIX.

Question 3 - VIX Index and Variance Risk Premium

In this exercise, we will replicate the main result in the Bollerslev at al. (2011) paper.

1) Using the data from the previous exercise, compute, on each day, the average realized variance over the next 21 days.

Be careful to use annualized decimal units.

2) Using the full sample, estimate the following model:

RV_t,t+21 = a + b₁RV_t^d + b₂RV_t^w + b₃RV_t^m + b₄VIX_t² + ∈_t,

where RV_t,t+21 is the average realized variance over the next 21 days.

3) On each day in the sample, compute the expected realized variance over the next 21 days using the model estimated in the previous question:

E_t[RV_t,t+21] = a + b₁RV_t^d + b₂RV_t^w + b₃RV_t^m + b₄VIX_t²

4) Construct a monthly series of the variance risk premium by taking, on each day, the difference between the squared value of the VIX index (in decimal units) and the value of the realized variance on the same day:

VRP_t = VIX_t² - E_t[RV_t,t+21].

5) Plot the time series of VRP.

6) Construct a monthly series of future 3-month returns on the S&P500 (R_t+h) by taking, every 21 days, the difference between the log-price of the S&P500 3 months (i.e., h=63) in the future and the log-price of the S&P500 on that day (i.e., time t).

Hint: One possibility is to use the MOD function to find days that are multiples of 21.

7) Run the following regression:

R_t+63 = a + b VRP_t + ε_t

a. What is the value of the coefficient "b" in the regression?

b. What is the value of its t-statistics?

c. What is the value of the regression's?

d. Comment on the results.

Note - The submission should include the following:

1. One Excel spreadsheet that shows your work.

2. A word document that summarizes your answers and explains what you did. Think of it as a short report in which you explain your reasoning and show your results, including plots and tables if required.

Attachment:- Assignment Files - Option Value at Risk.rar

Reference no: EM132492242

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