Math Finance Program Modern Statistical Inference and Econometrics Fall 2003 – Professor Hait
PROBLEM SET 2 For the following two problems, you will need the data contained in the Excel spreadsheet (IBM.xls) on the website. 1. The standard single-factor model of stock returns (the so-called “market model”) explains the daily return of a stock by the daily return of a suitable index: y = b0 + b1*x + e, where y is the daily return on the stock, x is the daily return on the index, and e is an error term (residual). The “b1” coefficient is generally referred to as the “beta”, and represents the riskiness of the stock’s return relative to the market. a. Fit this model to IBM daily returns, using SPX (S&P 500 index) as the index, and report the t-statistics, the Fstatistic, and the R2. b. Perform a test of the hypothesis that b1 = 1.0. Can you reject this hypothesis with α=5%? What about α=1%?
2. The “implied volatility” of an equity option is the market’s estimate of the future stock return volatility derived from the option price. It has been observed, contrary to the Black-Scholes assumptions, that the implied volatility of equity options tends to vary from day to day in a systematic way. One hypothesis is that there is a single-factor model of implied volatilities, so that the implied volatility of IBM is dependent on the implied volatility of an option on an index (such as SPX options). Another hypothesis is that the daily return of the stock is used to update the market’s estimate of future volatility, so that a large return would be evidence of a higher future volatility. Under this hypothesis, the level of volatility on a given day should be related to the return on that day. A third hypothesis is that the volatility of a stock is affected by the stock price level (the so called “leverage effect”), so that the level of implied volatility is dependent on the price of the stock. In the spreadsheet you will find prices, returns, and 30-day at-the-money implied volatility for IBM options, and returns and volatility for SPX options. Using this data, please test the above hypotheses. For this question, I would like: a. The exact specification of the model you are fitting (the regression equation) b. The exact hypothesis tests you are performing, and c. The results of your tests
3. (Green, #4, p.63). Suppose that the regression model is yi = α + βXi + εi where the disturbances εi have density function f(εi)=(1/λ)exp(-λεi), εi>= 0. This model is rather peculiar in that all the disturbances are assumed to be positive. Note that the disturbances have E[εi|xi] = l and Var[εi | xi] = λ2. Show that the least squares slope is unbiased but that the intercept is biased.
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