Can anyone explain the logic behind derivation of the minima for an European call option? I know schweser say’s derivation is not tested on the exam, but I can’t quite follow the thought process behind the proof. Minimum for the call option is zero (obviously option cannot have a negative value), or it is current underlying price minus PV of the strike price. So using the combination of instruments (Call, Sell Short, and Risk-Free Bond) creates a payoff of zero when the option expires in the money. It also generates a positive payoff when the option expires out of the money. But what if the expiration price of the underlying security closes at X+1? You’d expect a positive payoff, right? TIA…
Found this in an old thread… Re: the lower bound for a European call option new Posted by: JoeyDVivre (IP Logged) [hide posts from this user] Date: October 17, 2007 11:27AM All they are doing there (I guess since I don’t have the book) is a little mathematical proof that a call is always worth at least P - X/(1+r)^t (or 0) where P is current stock price and X is strike and r is risk-free rate. The reason for this might be in their proof, but an easier way to see it is that if IBM is selling at 60 and you have an option to buy it at 58 you could short the stock, get your $60, and put 58/(1+r)^t into risk-free deposits. This would leave you with 60-58/(1+r)^t dollars. At expiration, you exercise the call using the money in your risk-free deposit and cover the short with no risk. That means your call must be owrth at least 60-58/(1+r)^t dollars (in fact, it is almost certainly worth considerably more). http://www.analystforum.com/phorums/read.php?11,618736,620096#msg-620096 definitely clears things up a bit more
You might want to check out core-models.com, they have an awesome model on options pricing. They also have models on FRA’s, Equity Swaps and a couple other derivative instruments.
The derivation for the options in CFA material stinks. Schweser will show you what need to know for the exam. I don’t think you are viewing the theory behind the option correctly. Look at this way, without formulas for a few minutes. If you bought an 80 strike call that expires in 90’s. Then in 90 days if you exercised the call you would now need to fork up $80 to buy the stock. So the value of the option - needs to account for this. You could take the $80, and go buy the bond. So basically, you need the present value of interest to the strike. If you didn’t account for this interest component, then you could buy calls, short stock and pocket the interest difference in the bonds.