Thank you everyone for the replies. I appreciated all of them (even though I'm just quote-replying to this one). The one thing I guess I still need a little clarity on is the Q I wrote in my 2nd post immediately after the OP: the principle that Delta represents the approximate change in option price for a given change in the underlying is...untrue then, is it not? Or at least it's of limited value if what I quoted above is correct: that even though a near-expiry and far-expiry option may both have a Delta of 50%, their price will not change by an equivalent amount for a given move in the underlying SP...do I have that correct? And if so, then when sources claim that Delta does indeed = change in option price in relation to a move in the underlying, is that just a rough 'rule of thumb' that doesn't apply for far-out expiries?
Relationship of Delta : Underlying as function of Time to Expiration (for ATM strikes)
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Thank you everyone for the replies. I appreciated all of them (even though I'm just quote-replying to this one). The one thing I guess I still need a little clarity on is the Q I wrote in my 2nd post immediately after the OP: the principle that Delta represents the approximate change in option price for a given change in the underlying is...untrue then, is it not? Or at least it's of limited value if what I quoted above is correct: that even though a near-expiry and far-expiry option may both have a Delta of 50%, their price will not change by an equivalent amount for a given move in the underlying SP...do I have that correct? And if so, then when sources claim that Delta does indeed = change in option price in relation to a move in the underlying, is that just a rough 'rule of thumb' that doesn't apply for far-out expiries?
"Yes."
((And now we're back to gamma!))
So, you're absolutely correct: if there were no future(s) contract issue, and interest rates = 0.0%, yadda-yadda-yadda,
if you had two expiries of the same strike, they would not have identical deltas because as the instantaneous rate-of-change, the deltas would face different slopes in being of different TTE: this would be a delta/theta partial derivative, which would give reason for two very different gammas as a result. So EVEN IF {somehow} their initial measured deltas were *exactly* the same, in the next infinitesimal instant, the different gammas would produce different next-increment deltas.
If you think about a 3-dimensional graph, you can go up the slope using only delta (change=gamma), or you can partial delta-theta (change="charm") and then do theta. Either way, you're still navigating a hill.
These are not rules of thumb. The relationships are formulaic not hand wavy.
To put @tommcginnis point another way: the option price change for a $0.01 move in underlying will be almost identical. But since delta itself changes due to the existence of gamma, over a large move in underlying price (eg $10) the shorter dated option value changes more than the longer dated option because it has higher gamma.
This is in accordance with anything you might have read.
To put @tommcginnis point another way: the option price change for a $0.01 move in underlying will be almost identical. But since delta itself changes due to the existence of gamma, over a large move in underlying price (eg $10) the shorter dated option value changes more than the longer dated option because it has higher gamma.
This is in accordance with anything you might have read.
PLEASE don't mix cause & effect.
Delta doesn't *cause* a change in price -- delta IS the change in price.
Gamma doesn't *cause* a change in delta -- gamma IS the change in delta.
Gamma is a *witness* to the change in delta;
delta is a *witness* to the change in the underlying.
But fo'sho', the *geometry* of price against time *changes* as time→0.00, and the more so, as time approaches.
Lots of great graphs out there to illustrate this. I wonder which one is best?......
"Yes."
((And now we're back to gamma!))
So, you're absolutely correct: if there were no future(s) contract issue, and interest rates = 0.0%, yadda-yadda-yadda,
if you had two expiries of the same strike, they would not have identical deltas because as the instantaneous rate-of-change, the deltas would face different slopes in being of different TTE: this would be a delta/theta partial derivative, which would give reason for two very different gammas as a result. So EVEN IF {somehow} their initial measured deltas were *exactly* the same, in the next infinitesimal instant, the different gammas would produce different next-increment deltas.
If you think about a 3-dimensional graph, you can go up the slope using only delta (change=gamma), or you can partial delta-theta (change="charm") and then do theta. Either way, you're still navigating a hill.
Thank you, and thank you. Excellent explanations, and I get it (or most of it): what I'm understanding is that, taking two Call options that had just moment prior both had Deltas of 0.50, one with 1 day to expiry, the other 1 year, at the precise moment of a change in the underlying SP (let's use a big +25% just to make it extreme), the Delta of the two options will change, but to very different degrees (a relationship dictated by their respective gammas which will NOT have been identical, unlike their deltas.)
So the 1-day-to-expiry Call will now have a Delta of close to 1.00, whereas the 1-year-out Call will obviously be much less than that (likely in the 70% - 90% range, given how much longer to expiry, depending on many factors ofc, right?). So even though they had identical Deltas before the +25% gap up, their price changes by different amts...(?)
Do I have that more or less correct? If so, it kind of leads me back to my original Q (in my 2nd post), which is why most sources explaining the Greeks refer to Delta as an indicator of how an option price will change in relation to a change in the underlying. I've never seen an asterisk appended to that definition (and some posts in this thread suggest that there shouldn't be one at all), but it seems to be...at the very least very incomplete, right? Because the accuracy/veracity of that definition very much depends, so it seems, on the *magnitude* of the chg in underlying. For small moves it seems to hold rather well, but for large moves it falls apart(?)
If I'm 'getting there', but still misunderstanding something fundamental, I'd appreciate a nudge over the finish line
Delta is the instantaneous rate of change of the option. Once the underlying moves by a penny the gamma is different and so the delta will also be different. Maybe the confusion is coming from looking at your risk as delta only. You should be looking at it through a broad picture. If you buy short dated options, the delta is more volatile ATM than a longer dated option. Since options have convexity, there is no single risk measure that will tell you how much the option will move given the underlying over large moves. All risk measures in the BS world is instantaneous. Does that make sense?
Also the far dated options have higher delta due to drift (straight from the BS model).
Also the far dated options have higher delta due to drift (straight from the BS model).
Yes that's right. As @TheBigShort says all risk measures in options relate only to the current state: tweak the spot, vol, time-to-expiry, interest rate, strike and they change. If delta didn't change with spot then that means gamma=0, ie there is not optionality in the first place, so it doesn't make any sense that delta would be static over large moves in the underlying for options.