1. Some people say that the term N[d1] in the Black-Scholes equation for a call option is the market's best estimate of the probability the option will end up in the money. From what we know about option pricing theory, this assertion is wrong. What's wrong with it?
2. You are long 100,000 3 month at-the-money put options on XYZ stock, and you have set up a delta neutral hedge by trading the stock. Suppose that tomorrow the implied volatility in option prices goes up, but the actual volatility of XYZ's price movements in the market does not change. How will that affect your position? On the other hand, what if actual volatility goes up but implied volatility is unchanged. How would your position be affected in that case?