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Math 176 / Econ 135 Mathematics of Finance Homework 6
发布时间：2023-05-15
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Mathematics of Finance
Math 176 / Econ 135
Homework 6
Due on: Tuesday of Week 10
Volatility Project: Google (Alphabet, Inc.) and Apple, Inc.
1. Compute estimates for the mean 
GOOG and volatility 
GOOG for Google (GOOG) and mean 
AAPL and volatility 
AAPL for Apple, Inc. (AAPL) using the historical price data in the spreadsheets GOOG.csv and AAPL.csv posted on the course website. Each file contains 20 recent daily close prices from which you are to compute 19 daily returns Rn for 1 ≤ n ≤ 19. The formulas for 
and 
are in the lecture notes from Lecture 21.
2. Make two plots: the volatility smiles for puts and calls on Google (GOOG) and compare to 
GOOG .
(a) For Google (GOOG), use the CBOE options market prices found here:
https://www.nasdaq.com/symbol/goog/option-chain?dateindex=1&expir=stan
(b) For Google (GOOG), use the following 12 strike prices: the first 6 below the current stock price
and the first 6 above the current stock price.
(c) Use the options with April 6, 2023 expiration date for all computations
(d) Compute the implied volatilities for puts and calls using mid price data: mid price is the average of the bid and ask prices. Check prices during the middle of the trading day so markets are active and live bid and offer prices available
(e) Use the Black-Scholes smartphone app we presented in Lecture 21 or the following website to
compute the implied volatilities:
(f) Assume dividends are zero and the risk-free interest rate is 2% in all computations
3. Make two plots: the volatility smiles for puts and calls on Apple, Inc. (AAPL) and compare to 
AAPL .
(a) For Apple Inc. (AAPL), use the CBOE options market prices found here:
https://www.nasdaq.com/market-activity/stocks/aapl/option-chain
(b) For Apple, Inc. (AAPL), use the following 12 strike prices: the first 6 below the current stock
price and the first 6 above the current stock price.
(c) Use the options with April 6, 2023 expiration date for all computations
(d) Compute the implied volatilities for puts and calls using mid price data: mid price is the average of the bid and ask prices. Check prices during the middle of the trading day so markets are active and live bid and offer prices available
(e) Use the Black-Scholes app we discussed in class or the following website to compute the implied
volatilities:
(f) Assume dividends are zero and the risk-free interest rate is 2% in all computations
4. Plot the implied volatility term structure for Google (GOOG) and again compare to 
GOOG
(a) Plot the implied volatility on the y-axis vs. expiration date on the x-axis for Google (GOOG)
puts and calls, using expiration dates of March 2023, April 2023, June 2023, and September 2023 and a strike price as close to the current stock price as possible and put/call market mid prices shown here:
https://www.nasdaq.com/symbol/goog/option-chain?dateindex=1&expir=stan
5. Plot the implied volatility term structure for Apple, Inc. (AAPL) and again compare to 
AAPL .
(a) Plot the implied volatility on the y-axis vs. expiration date on the x-axis for Apple, Inc. (AAPL)
at-the-money puts and calls, using expiration dates of March 2023, April 2023, June 2023, and July 2023 and a strike price as close to the current stock price as possible and put/call market mid prices shown here:
https://www.nasdaq.com/market-activity/stocks/aapl/option-chain
6. Review Warren Buffett’s 2022 letter to shareholders and write a few sentences on anything that interests or inspires you. The letter may be found here:
https://berkshirehathaway.com/letters/2022ltr.pdf
