12. In this problem you will compute January 12 2004 bid and ask volatilities (using the Black-Scholes implied volatility function) for 1-year IBM options expiring the following January. Note that IBM pays a dividend in March, June, September, and December.

a. Compute implied volatilities ignoring the dividend.

b. Take dividends into account using the discrete dividend correction to the Black-Scholes formula. For simplicity, discount all observed future dividends at a 2% continuously compounded rate. How much difference does this correction make in implied volatility?

c. Take dividends into account by computing a dividend yield for IBM based on its annualized dividend rate as of January 12. Use this dividend yield in the Black-Scholes model. How different are the implied volatilities from those you obtain in the previous part?

d. Do you observe a volatility smile?