use the Black-Scholes option pricing formula to check whether a call option is priced correctly.

There are different websites which give details about companies options. Heres one that you can refer to: http://moneycentral.msn.com .

On that website, if you go to Markets you can find the annualized yield (rate of return) on US Government Treasury bills. The three-month one is the one that is most commonly used to determine the annual risk-free rate. (You can read up more on how to interpret US Treasury bill quotes on the New York Feds website: http://www.newyorkfed.org/aboutthefed/fedpoint/fed07.html )

On the same website, in the search box you can type in the correct symbol for the company that you have chosen (Apple), you can see information on call options, put options, and other current information on the stock.

Choose any month and date you like for the option expiration. Choose any exercise price you like for an option that is in-the-money, and another one for an option that is out-of-the-money. The column labeled Last shows the current price at which call options with different exercise prices can be purchased.

As part of your calculations you will need to calculate ², which in the Black-Scholes formula is the variance (per year) of the continuous returns on stock. This is the expected stock return volatility between the purchase date and the expiration date. Unfortunately, this represents the future, so there is no way you can know the correct volatility. What traders do instead is estimate this volatility using past data. I would like you to calculate stock return volatility using daily stock prices (closing prices) for the last 3 months (90 last trading days) of stock trading. http://finance.yahoo.com allows you to download a spreadsheet with historical prices of stock shares. You would need to select daily. To convert the resulting daily return volatility into the annual volatility, use the following formula:

²_annual = ²_daily x (total number of trading days in 2014)

For Apple, Inc (AAPL):

- What are the S, E, R, ² and T in your Black-Scholes formula for each of your two chosen options? Show calculations, where necessary, and explain in words how you got each number. Attach print-outs of website pages you used that indicate where the information is coming from; highlight the important numbers on your print-outs. (Keep in mind that the information on the websites gets constantly updated, and so if the information is not printed out at approximately the same time the option price results will not be correct.)

- What are the intrinsic values of your two chosen call options? Calculate. What do they mean?

-Are your two call options priced correctly, or are they overpriced or underpriced, according to the Black-Scholes option pricing model? By how much? Calculate and explain. Can you explain why this may be the case?

For each of your two options create a graph (like the one below) and label and put the dollar amounts of

current stock price for the stock you picked

exercise prices that you picked

current call option values that you have calculated

intrinsic values that you have calculated

time value premiums based on your calculations

- In order to replicate the payoff of your two call options at the expiration date that you selected, how many shares of stock should you buy today, and how much should you borrow at the risk-free rate? Calculate and explain.

- Are the put options for your stock overpriced or underpriced, according to the Black-Scholes option pricing model? Calculate and explain.