01 May Consider the dataset in Exercise 7.6. The file m tbill 3m.txt contains the yield
Consider the dataset in Exercise 7.6. The file m tbill 3m.txt contains the yield of the 3-month U.S. Treasury bill (obtained as a secondary market rate in the Federal Reserve Board H.15 publication) on the closing date of each month during the period.
(a) Using the standard deviation of the 60 most recent daily log returns of the S&P 500 futures prices, estimate the volatility s on date t in the Black-Scholes model for the period from March to December 1987.
(b) Estimate the Black-Scholes price on each date t during this period by using the estimate of s obtained in (a) and the yield of the 3-month U.S. Treasury bill on the closing date of the month prior to or at t.
(c) Plot the time series of differences between the Black-Scholes prices in (b) and the actual option prices.
(d) Compute the implied volatilities on every trading day from March to December 1987. Plot the time series of historic volatilities in (a) and of the implied volatilities. Discuss the patterns you find from your plot.
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