Ito’s Dilemma

3. How do your estimates compare with the actual quoted prices? Can you explain the differences? Assuming your prices are correct, which options would you buy or sell? Our estimates are a little different with the actual quoted prices. Why? Because some assumptions are underlying the B/S formula.
1). The stock will pay no dividends until after the option expiration date.
2). Both the interest rate, r, and variance rate, δ2, of the stock are constant (or in slightly more general versions of the formula, both are known functions of time—any changes are perfectly predictable).
3). Stock prices are continuous, meaning that sudden extreme jumps such as those in the aftermath of an announcement of a takeover attempt are ruled out.

In this case, we do not take paying dividends into consideration. And we all set risk-free rate and variance rate are constant. But actually, they may have paid dividends and variance rate will change with time. And the standard variance is the annualized standard variance of returns calculated by stock prices during past 30 weeks . So the standard variance calculated may not be the proper estimates in future.

If the prices of options calculated are correct, we should sell the overpriced options with lower calculated prices compared with the actual prices and buy the underpriced options with higher calculated prices compared with the actual prices.

4. Are there any problems with the way Ito estimated the volatility numbers? Can you think of another way to estimate volatility that might yield estimates closer to the actual quotes?
The weekly sigma of returns in the case is calculated by the standard deviation during the past 30 weeks, and is used as the forecast volatility for the all six kinds of option. We calculated the implied volatility numbers of the six option and found that the implied volatility varies from each to another. So it is inappropriate to simply use one sigma of return as the forecast volatility for all options in different situation. Because the unexpected announcement and changes in policies, the representative of the historical data is limited. So the forecast volatility should be calculated according to the specific economic situation of each option and the outdate numbers should be dropped to ensure the representative of the historical data. And some statistic model can also be used to calculate the forecast volatility, such as moving average method or ARCH&GARCH method.

5. Using the Black-Scholes pricing function in Excel, calculate how sensitive IBM’s March 110 call price is to changes in stock price. How much does the call price vary for $0.50 changes in IBM share price when the option is at the money (assume stock price=$110), in the money (assume stock price=$115), and out of the money (assume stock price=$105)? What does this sensitivity analysis tell you?
After sensitivity analysis in Exhibit 2, we can conclude that out-of-the-money options has a higher change rate than in-the-money options. Out-of-the-money options are less expensive than in-the-money options but out-of-the-money options are also regarded as bearing higher risk. When it comes to returns, the out-of-the-money options often experience larger percent gains or losses than the in-the-money options, which again is due to the higher amount of risk. Since out-of-the-money options have a lower price, a small change in their price can translate into very large percent returns.