Given the same parameters (Price, strike, volatility, days to expiration, interest, etc.)? When you enter the same parameters in the CBOE calculator: http://www.cboe.com/LearnCenter/OptionCalculator.aspx You get different premiums for index and equity options.
Normally, the difference between models used to calculate options on indexes and options on stock is the way dividends are handled. Index options are calculated using a continuous yield, while stock options are calculated using discrete payouts. The inputs for these calculators reflects that; the one for stocks has an input for the date of the next dividend, while the one for indexes does not. However, when I put in a dividend of zero, these two calculators still come out with different prices. When I put in a dividend AND annual interest rate of zero, again they come up with different prices. I can't see how that makes sense. Hard to believe that the CBOE website would have an option calculator with a bug, but that's the only explanation I can come up with.
One is American Style the other is Euro...also dividends and tax treatment affect it (although the latter may not have anything to do with it??).
The equity options are set to American. The index options give you a choice of American or European, but whichever you set it to, they come up with different prices. I set the dividends and interest rates to zero, so that doesn't explain it. And there's no input for tax treatment...
I am not sure what CBOE does but you are supposed to use Black's model instead of Black-Scholes for options on futures. I don't have time to look it up now but I think the difference is that the no-arb relationship in BS assumes you need to invest capital in the underlying (so there is interest earned), and Black's assumes you can enter the position with 0 investment. For most traders there is a margin deposit on the futures so the "truth" is somewhere between. I am curious if anyone more familiar with this knows if Black's model works better for options on futures in the real world - are greeks noticably better, implied vols more stable, etc?
The difference between the Black model and the Black-Scholes model is the way interest rates are handled. In the BS model - which is meant for equities - you enter 1 interest rate, but internally the model uses it twice. First, the model uses it to calculate the forward price of the stock, which is what it uses internally as the price of the underlying. Then once it has generated a price for the option, the model uses the interest rate to discount the option by the cost of carry - the amount of money you would have made had you put that money in T-bills instead of buying the option. With futures, the futures price IS assumed to be the forward price. So in the Black model, the futures price that you input is used internally, unchanged, as the price of the underlying. The interest rate you input is used only to discount the price of the option by the cost of carry. In testing these two calculators that the OP mentioned, I specifically set the interest rate and dividend rate to zero, so any differences such as those mentioned above would play no role. The prices still came out different.
In this example you're pricing a call that is ten dollars in the money with 1 day remaining. It has no time value, so it should have no theta. One model correctly shows theta of zero, the other shows theta of -10, which is ridiculous, and shows that, indeed, there is a problem here.
I mostly agree with you but I still think I am right that the Black model assumes there is no cost to hedging the option in its derivation. I am not usually this anal about esoteric crap but I need to know this anyway so your post prompted me to review it. Both models assume a hedged portfolio earns the risk free interest rate. BS assumes you invest or recieve cash to construct a hedged position using the underlying so you earn/pay interest on it. The Black assumes you are hedging with zero cost forward contracts so you just discount the expected value of the option. It is hard to see this just looking at the formula but I am 99% sure it works that way, and there is a sentence on www.quantnotes.com that seems to agree. But the part that confuses me is it looks like it is just a notation transformation of the same model - if you use the forward formula F=Se^rt and solve both models for a call price they seem to be the same. But I defintely agree with your main point that setting the interest rate to zero should mechanically make the differences go away, I missed that in your first post.
I don't know if any of you guys read the Information Tab where you have to click the "I Agree" button. It's somewhat turgid but actually states explicitly what models and what simplifying assumptions it uses. Here's the pertinent excerpt: "SOME OF ITS THEORETICAL CALCULATIONS, THE OPTION CALCULATOR USES THE COX-ROSS-RUBINSTEIN BINOMIAL APPROXIMATION TO THE BLACK-SCHOLES OPTION PRICING MODEL. WHILE THIS AND OTHER APPROXIMATIONS OF THE BLACK-SCHOLES MODEL ARE THE MOST POPULAR MODELS USED FOR OPTION PRICING, OTHER MODELS EXIST THAT CONSIDER DIFFERENT FACTORS. NO MODEL CAN BE ENTIRELY ACCURATE. IN ADDITION, A NUMBER OF SIMPLIFICATIONS HAVE BEEN MADE TO THE MODEL -- FOR INSTANCE WITH REGARD TO DIVIDEND ASSUMPTIONS -- AND THE RESULTS UNDER VARIOUS CIRCUMSTANCES SUGGESTED BY THIS CALCULATOR ARE LIKELY TO BE DIFFERENT FROM THE RESULTS SUGGESTED BY OTHER CALCULATORS. MORE IMPORTANTLY, THE OPTION CALCULATOR IS AN EDUCATIONAL TOOL INTENDED TO ASSIST INDIVIDUALS IN LEARNING HOW OPTIONS WORK. IT IS NOT INTENDED TO PROVIDE INVESTMENT ADVICE, AND USERS OF THE OPTION CALCULATOR SHOULD NOT MAKE INVESTMENT DECISIONS BASED UPON VALUES GENERATED BY IT." I think what you're seeing is simply one weakness of a poor implementation, nothing more.