E.g.: ES futures at 800, the March 800 call and put deltas are ~ .55 and -.45. Why isn't the absolute value equal since there is no dividend and no (or very little) cost of carry for the futures? Both the BS American and Whaley models show similar deltas. I could understand if the futures were at 25 since the downside limit is 25 pts. away, but not at 800. Anyone have an explanation? Thanks.

You're on the right track - it is indeed the lognormal distribution that causes that effect. At 800, zero may seem too far away to make a difference. But with that much time remaining and volatility so high, 800 becomes more like 25 than you think. Play around with it and you'll see that's so. With volatility at 10% and that much time remaining, (and using an interest rate of zero), call and put deltas are .51 and -.49. With volatility at 45%, they become .55 and -.45. With a volatility of 85%, call/put deltas would be about .60/-.40.

I did play around with it and that's why I was so confused...I tried futures and strike at 5000 (48% vol. both at 800 and 5000), and both times the deltas were .55 and -.45. Even a .0000000001% interest rate doesn't make a difference. Futures at 800 or 5000 and vol. at 1% changed the deltas to .53 and -.47. Theoretically those deltas might be right, but in real trading, I'd prefer them to be the same. I traded bond options on the floor back in the '90s, and I would swear the call and put deltas were the same ATM using the Whaley model, but what do I know. Thanks for your reply.

Now that I know your background I can tell you exactly where the confusion is coming from. I also traded T-bond options on the floor during the 80's and 90's, and I also used the Whaley model. The main thing to keep in mind is that the volatility we were dealing with was WAY lower than the ES volatility today - probably around 8-9% for most of the time you were there. I mostly traded back months, and ATM calls always had a slightly higher delta than ATM puts (usually about .51/-.49). But if you were trading the front months, then the combination of short time remaining and low volatility could have combined to make the difference between call and put deltas seem to disappear. For example, with 30 days remaining and 8% volatility, you get a call/put delta of .504 and -.496. So if your sheets rounded deltas to two digits, then the ATM's would have indeed shown .50 and -.50. You're right that it doesn't matter if you use a futures price and ATM of 5 or 500, because it's proportional. What matters is the time and volatility. The higher the time and volatility, the more significant the lognormal distribution becomes, and the greater the difference between the absolute values of the ATM call and put deltas. If you vary the time and volatility inputs - rather than the futures price and strike inputs - you'll see the difference.

Don't ignore the effects of interest when valuing an option. Any interest over zero makes calls worth a bit more than puts. I'm not quite sure how to phrase this, but the payment of interest infers that there's a slightly bullish bias in the mode (or else no one would invest, and that gives the calls a small delta boost. Mark

Mark, it works a little differently in options on futures. I think what you're referring to is the fact that with equities, the interest rate you input is used by the model to calculate a forward price for the stock. The model then uses that forward price (and not the stock price that you input) as the underlying price. So the higher the interest rate and the more time remaining, the higher the forward price, and therefore the higher will be the call price and the lower the put price. BTW, the model subtracts the dividend from the interest rate before it calculates the forward price. So the higher the dividend, the lower the forward price, and therefore the lower the call price and the higher the put price. But in pricing options on futures, the futures price is considered to be the forward price, and so is used directly by the model as the underlying price. There is no cost to carry a futures contract, since margin can be held in T-bills, so no forward price is needed. You do input a risk-free interest rate when calculating the price of options on futures, but that interest rate is used by the model only to discount the option by its cost of carry - not to calculate a forward price for the underlying. There are many implementations of these models and each software does it a little differently, so it can be tricky to get the interest rate inputs right. I suspect Chisel is inadvertently entering a cost of carry that his model is using to (incorrectly) calculate a forward price for the ES futures - which would exaggerate the delta difference between ATM puts and calls.

Thanks for the correction. I've never traded futures options and probably should not have replied. Mark

dmo, Good point about the low vol. of the bond options. Yes, I usually traded shorter term options and the low vol. combined with delta rounding to two decimal places is my reference point. "I suspect Chisel is inadvertently entering a cost of carry that his model is using to (incorrectly) calculate a forward price for the ES futures - which would exaggerate the delta difference between ATM puts and calls." I suppose that is possible, but I'm using an Excel add-in from Montgomery Investment Technology (www.fintools.com) which I purchased about 8 yrs. ago. They have an option pricing model on their site - http://www.fintools.com/?calculators - which agrees with the numbers in my spreadsheet. Like I said, I've used a next to zero interest rate and I think it's nuts that the model will tell you to short 10 futures when buying 100 atm straddles. Is there an "improved" model that agrees with my way of thinking?

Most of your numbers make sense to me Chisel, but if you're using a 1% volatility and getting call/put deltas of .53/-.47 - and interest rates are zero or close to zero - then something's wrong somewhere. Try that one again. Unfortunately the Montgomery Investment option calculator page isn't working at the moment - I'll try it again later. I also used to use the Montgomery Investment Excel add-ins, but then they wouldn't give me a discount to upgrade - even though I was a returning customer. That pissed me off so I switched to Hoadley. Much cheaper and very good. However, no Whaley model unfortunately. That's the only thing I miss. Do I know of a model that agrees with your gut feeling about how deltas should look? I don't. Every option pricing model I'm aware of assumes a lognormal probability distribution, which means they all will give you more or less the same deltas. I think you have a point though, at least in the ES. If you buy straddles and the ES goes up, IV will drop, hurting if not killing your profits on the straddles. If you're short futures on top of that, you're going to have a tough time making money on a rally.