And as to in the money vs. out of the money, it is obviously irrelevant. You can use the 1360 Calls and 1455 Puts in my example and, because of put/call parity, the results are the same. The 1360 Calls closed 67.10 and the 1455 Puts closed 60.3. The out of the money Calls (and therefore the in the money Puts) will always be priced lower than the out of the money Puts (and therefore the in the money Calls) with regard to a standard equity index like the SPX, ES, NDX, NQ, etc. Is there still a question with regard to this? DMO, I can't really understand why someone would make a statement that has never been or will never be true in the real world.
dmo- Yeah, you're right, I did at first read the question incorrectly but I still contend that the answer is the same. Same principle behind the answer I believe. In response to opt789, as dmo stated, we are talking about options with the same implied volatility. If a put and a call are equally out of the money and are trading with the same implied volatility, the put will ALWAYS be worth less. That is not a stupid statement. It is a matter of fact.
Right you are, it's the strike that matters, not whether it's a put or a call. A put and a call at the same strike is essentially the same thing. But it is true that every model I know of uses a lognormal distribution. And you've got to know the rules before you can learn how to break them, right? I have a library of Excel add-in functions from Hoadley - and he includes a function that calculates options prices using a normal distribution rather than the standard lognormal one. One of these days I'll have to sit down and see how those prices compare to the actual prices of the index options - they'll be much closer for sure.
The straddle has a + delta beacuse its not the true at the money straddle. If you wanted the true at the money straddle you'd price the one where both the call and put were .50 delta. You're not accounting for the cost to carry that strike out to the date of settlement. Look further out in time and see how far from where the underlyings current price is to where the true delta neutral straddle is.
I probably donât understand most of the posts here. OTM calls should always be higher than OTM puts ( even if skew is not exists and IR=0 ) due to higher strike nominal , no ?
There are two factors that make the ATM straddle have a positive delta. First, as Walter says, is the effect of the lognormal distribution. That is true even if the cost of carry is zero, as is true of options on futures. The cost of carry effect - if there is a cost of carry - increases the delta of ATM straddles. As you say, a good way of thinking of that effect is that it raises the price of the underlying - so the ATM straddle isn't really ATM. But even without any cost of carry, a true ATM straddle has a positive delta due to the lognormal distribution.
Oh sorry, I thought we were talking about trading options. Feel free to continue your discussion of an arbitrarily chosen mathematical formula with speculation used to fill in the unknown, stochastic variables. When you find an index that has a flat vol curve, you be sure to let us know.
This is making my head spin. Nobody wants to admit that they're wrong. So I'll tell you what, you're all wrong and Mr. market is always right. Whether theoretically your arguments are correct or not, the market doesn't care. The reason why the market wants to price otm S&P puts higher than the calls is because the market gets what it's asking for - you can't argue with reality. There's heavy demand by portfolio managers who are competing for that insurance premium so they can sleep the night better if another '87 drops by again
I'd just like to add that there's no law forcing anyone to use a lognormal distribution in option pricing. It's true that virtually every public-domain model is based on a lognormal distribution, but you could use a normal distribution if you wanted. And if you did, then an ATM straddle with no cost of carry for the underlying would indeed have a delta of zero. You could certainly make an argument for using a normal distribution in index options - namely, that the prices it produces would more accurately reflect actual prices. The argument against it, of course, is that the index can only go to zero but can go up infinitely - which is why mathematicians have historically chosen a lognormal distribution on which to base their models.