Relationship of Delta : Underlying as function of Time to Expiration (for ATM strikes)

D

d0rian

One principle I'd like a little help with, particularly re: ATM options.

Delta is commonly referenced as an approximation of how much a change in the underlying SP will affect the price of the option. I understand Delta decay -- idea that as Time to Expiration (TTE) approaches 0, Delta of OTM options will approach 0 and Delta of ITM options will approach 1.

My Q is specifically about ATM strike prices (Delta = 0.50 or close to it). Take underlying PYPL (SP = $100.73). Consider these two $100 Calls and their Delta values (taken from IB's Option window):
  • Apr 18, 2019 (TTE = 31 Days) > Delta = 0.574
  • Jun 19, 2020 (TTE = 459 Days) > Delta = 0.612
Those Delta values don't make sense to me. Shouldn't the option with the further out expiry have a lower Delta? That relationship is certainly true for strike prices further away $100. For example, the $95 calls for PYPL are:
  • Apr 18, 2019 (TTE = 31 Days) > Delta = 0.824
  • Jun 19, 2020 (TTE = 459 Days) > Delta = 0.673
That aligns with my understanding of delta decay; so why does it get wonky around Delta = 0.50?
 
A related Q, which is really what I'm trying to understand: if Delta = expected change in option price for a given change in the underlying SP, why do 2 options that both have the same Delta (but have different expiration dates, one near, one far out) not exhibit the same change in price when underlying SP changes?

E.g. For two ATM options (Delta = ~0.50), the option price for a 04/18/2019 expiry Call will indeed change ~50% for a corresponding change in the underlying (so $5.00 decrease in the case of a $10.00 gap down using the PYPL example above.) But that same strike with a 6/19/2020 expiration will show a much more attenuated effect (perhaps a drop of only $3.00 or so). Which suggests that not all Deltas really mean the same thing. Otherwise options with similar Deltas would show exhibit similar price changes, regardless of their expiry, right?

I guess what I'm trying to ask is this: is the Delta:Underlying relationship optimized for a certain TTE (e.g. 30 days or something), or does the utility/strength of that relationship drop over time?
 
You are trying to isolate delta but ignore theta and vega. Also often further out in time options can be more sensitive to underlying stock movements given more time for the option to end up in the money at expiration. SO it makes sense further out in time options at same strike have higher delta.

But I am not sure why you are so fixated on delta, going out in time gives you higher delta, more vega and less theta in exchange for the higher cost.
 
El OchoCinco said:
You are trying to isolate delta but ignore theta and vega. Also often further out in time options can be more sensitive to underlying stock movements given more time for the option to end up in the money at expiration. SO it makes sense further out in time options at same strike have higher delta.

But I am not sure why you are so fixated on delta, going out in time gives you higher delta, more vega and less theta in exchange for the higher cost.
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And why so fixated on calls? What about puts?
It's too convenient to sometimes, to quote the guesstimate of call+put=1.00
If you look at 'the other side' of those cited expiries, you'll find additional wisdom to add to theta and vega.
The imperfections and asymmetries that you identify here will all have names that will be familiar to you... :thumbsup:
 
All vanilla options are contracts priced from the forward price of the underlying on the expiry date. In this case the PYPL stock does not pay dividends (if it did it would pull the forward price down) and so to get to the forward price you have to include the financing cost (interest rates). The forward price of PYPL in June 2020 is higher than the forward price in April 2019 in the same way that a bank account accrues more interest the longer you deposit your cash.

If the options were NOT priced like this then you could enter into a forward agreement to purchase PYPL at 100 (theoretically, I mean, as you'd have to have access to the interbank/wholesale market to do this OR alternatively buy the 100 call/sell the 100 put for net zero cost), sell the current stock short at 100 and put the cash gathered from the short-sale into an interest bearing account/bond/fund etc until expiry and make risk-free money.

So in reality the 100 strike is really NOT at-the-money in June 2020. Atm more like somewhere between 105 and 110 (see put-call parity for how to find the atm forward). This makes the 100 call more in-the-money the further forward into the future you look.

And delta us calculated taking the forward price into account.
 
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I had posted a wrong explanation here earlier about the delta differences but realised I was having a brain fart...

I think in reality for shorter dated options you are seeing the effect of gamma on option delta. Longer dated options have relatively more stable delta (less gamma) and so for any given move you will see the option value change by more (on the short-dated ones).

A $10 in PYPL is a very big move for a short-dated option and the delta +/- $10 from ATM is very different from ATM delta, right? But delta for the long-dated option doesn't change nearly as much for a $10 move in spot.

So linking it back to the value of the option: the average delta of the short-dated option over a $10 move in the underlying is somewhere far away from 0.5; but it isn't much different for the long-dated option, therefore its value changes by less.
 
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Quiet1 said:
I think in reality for shorter dated options you are seeing the effect of gamma on option delta.
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Well, of course -- but that doesn't diminish your {implicit} discussion of theta and vega and the partial derivatives between/betwixt them.
https://en.wikipedia.org/wiki/Greeks_(finance)
Gamma is certainly right in there, but intuitively, it's a reflection OF delta, not an action ON delta. For that, you lean on the other cross-Greeks that don't get so much play in shorter options or non-ATM options.

(If we're voting, I go with your first response *augmented* by Gamma. Meh! YMMV. Hell, you posted it. [!!])
 
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