Buyers and sellers of options have intended rights and obligations. Is the devil in the details? Assume an underlier with ATM call delta of say 0.55. No carry. The option is a regular BS-priced option. If the seller sells the vanilla option to implement the intent to sell the stock if it is above the strike price at expiration, then is the seller not cheated? Should the seller not receive as premium 55% of the stock value to assume the intended obligation, instead of the much smaller premium of the regular call option? Is the intent described above your typical intent when you sell a call option or do you have another intent when you sell it? What is your intended obligation in words? Is the above right or wrong? How would you implement what the seller has in mind, and not something else added inside that reduce the premium of the option? I have my answers and my understanding, but I wanted to get things going by posing the above questions. Could it be that it is the buyers that are "cheated" when the option price is compared to their intent?

No The delta is the expected rate of change not a % measure of premium. The purpose of the BS model is to craft an indifference price or premium that would provide no specific edge to buyer or seller for a given volatility expectation.

Keep in mind that options prices are set by the market - meaning the price reflects what the market will bear. Assuming a given option is fairly liquid (sufficient open interest), the market maker adjusts the price to compensate for risk they are taking. For an ATM option with a .55 delta, there is an approximate (emphasis on approximate) probability of an ITM finish of 55%. Depending on time until expiration, the extrinsic part of the option is inflated or deflated according to expected volatility. The Black-Scholes model can be used to arrive at a given expected option premium. You can also take a given option premium and solve for volatility (implied volatility). Consider that the market maker has to take the opposite side of your position. If you sell, they have to buy. If you buy, they have to sell. It stands to reason that the price will be reasonably fair, which can be seen in the width of the bid/ask spread. It's kind of a convoluted answer I know - trying to keep it short. I just posted a longer web page on the related topic of option volatility and its affect on option price at: http://www.success-with-options.com/option-volatility.html I hope this helps. Mark http://www.success-with-options.com

My question is probably not clear. Let me try to clarify. Suppose, someone offers a bet. If stock finishes above strike price, he is paid one share of stock. Otherwise, he is paid nothing. What is the price of the bet? Is it not the stock price times delta? If you have that one share and you were to take the other side of the guy proposing the bet, would you accept to be paid the premium of a vanilla option?

You seem to be seeing things. What if I am doing the confusing on purpose? The vanilla call is a spread/hedge of two binary options that pay in stock and in cash. Both the seller and the buyer of a vanilla call option are sellers and buyers of two types of binary options. Are they aware of the hedge? Do they really need to hedge, and if yes why do they need to, and why should the hedge be in the ratio implicit in the vanilla option? Also, look at the change in premium in a binary that pays in stock and the vanilla option as a function of stock price. What are the pros and cons of each for the seller and buyer?

If I understand your question correctly --- yes, premiums are often "too low" as there is a discount based on the ability to hedge away price movement risk. The amount of this discount will vary depending on market's perception on how well it will be able to manage the risk. Or put another way - most options sellers would be asking a much higher price if they weren't allowed to (or weren't able to) hedge.

You have some deep insight about options as a business, and I am almost certain that you are a winner in the options business. I am wondering whether you are reading my notes( which I did not give to anyone!) from a distance I will read more of your posts as I think you have a deep grasp of the options as a business. I read one post in which you asked a question on why a short straddle you sold was not delta neutral on its visit to the same strike from an excursion it had to the north. Do you still have that question?

It's silly and confused, regardless of whether it's done on purpose. I don't know and, frankly, I don't really see the point. What sort of insight do you intend to glean from this? Hedge what? I have absolutely no idea what you're talking about. Why look at this? What does it have to do with anything? Maybe I misunderstand the point, but here's what I am getting from your posts. Please do correct me if I am wrong. You take the delta of a vanilla call (55% in your example) and choose to treat it as the probability of the underlying (call it S) ending up above the strike. You then calculate the expected value of the call payoff as 55% * S (45% * 0 + 55% * S) and suggest that this should be the price of the vanilla call. This is incorrect, for obvious reasons.

Let me explain this, then you think about the comments you made and revisit if and as you see fit: 1. Bet1: if stock finishes higher than current price one gets one share of stock. Bet1 costs 55% of stock price. 2. Bet2: If stock finishes higher than current price, one gets one dollar. Cost of bet2 is 45cents. What if one takes one bet1 (buys it), and sells S number of bet2. The outlay would be: (0.55*S - 0.45*S)=0.10*S. If stock price is in the money (in sense of vanilla) one wins bet1 and loses S bet2 bets. Let us denote by X the stock price at the end (so X>S). Bet1 is now worth X, and one has to pay S dollars for losing the S bet2 bets (each bet2 had to pay one dollar). So the net is: X-S in case if stock finishes higher than current price, and zero otherwise. The same as the vanilla option. The cost was .10S, which the same as the vanilla option.