relation between ivol and gamma

hi frds. i am doing some forex opt trading and trying to understand the relation between the two. my question is why do gamma falls for atm opt when u take your iv up and vice versa but with otm opt gamma rises to some extent when u take your iv up.. would like to understand the mathmatics/ logics behind it .. thx very much for your help

 

its probably easier to think of it the oppsite way. as you bring vol down the ATM gamma goes up. since higher vol (longer time maturity) causes all options to be closer to ATM. as far as the math. gamma is the 2nd derivative of the pricing model ... the rate of change.

 

This is very easy to explain if you look at the relationship between volatility and the probability distribution - that is, how likely it is that at expiration the underlying will be ATM, and how likely is that at expiration the underlying will be OTM.

It should be intuitively obvious that the higher the volatility, the LESS likely it is that at expiration the underlying will be ATM. At the same time, the MORE likely it is that the underlying will be OTM.

You can imagine how the bell-shaped probability curve (actually it's a lognormal distribution curve) changes as volatility increases. The mean (middle, or ATM) probability goes down, while the tails (OTM) thicken, or get higher.

The gamma curve follows that probability curve. There's your exact mathematical explanation. If this is difficult for anyone to picture, I can dig out a graphic and post it.

 

Here you go - here's a slide from a presentation I once did - with a graphic from Shelly Natenberg's book. This should make it clear. Just keep in mind that the gamma curve and the probability curve are essentially the same.

 

liked the bell shaped aprroach very much and was quite easy to understand.. thx again

 

How about an OTM options?

If you increase IV, gamma should rise, and then when its delta is 0.5, it will start declining while gamma continue to rise.

ATM is easy to explain, but OTM is less obvious (assuming what I wrote is correct as it is intuition with no math, but my intuition is not in terms of pics).

 

Could you pls. check whether your tests are consitent or not with what I wrote about OTM options?

 

It's really easy to understand if you look at option chains. Yes, you can substitute time for IV and examine extremes.

Instead of lower IV look at an ATM option that is about to expire (late expiry Friday.) It has some value and a delta of around 50
but its delta will quickly move to 0 or 100 if the underlying moves a little. The rate of change is very high.

Go out to a 3 month option (our higher vol example) and the ATM option's delta moves relatively slowly, say from around 50 to 45 or 55 for the same movement of the underlying. The gamma is lower.

Look at the expiring chain again. Most or all of the OTM options are worthless. They have a delta of zero and a move by the underlying will only affect them if it brings them close to ATM. The gamma is zero for most of the OTM

Ah, but raise time to the 3 mo. and the otm options have value and enough movement of the underlying changes deltas for some of the OTM strikes. (those closer to ATM will obviously be more sensitive to movement.)

Now instead of adjusting time, fix time at some point and adjust vols and you will see similar results.

Higher IV decreases gamma for ATM and increases gamma for some OTM.

My answer is no different than the bell curve answer.

 

I'm puzzled by what seems to be a reference to OTM options with a delta of 0.5.

If the delta is 0.5, then we're talking about ATM options whose gammas will, as originally stated, decline as IV rises. But I don't see what that has to do with OTM options.

If you look at the graphic I posted, the effect of a rise in IV on OTM options should be as easy to understand as its effect on ATM options. Two sides of the same coin. If volatility rises then it becomes less likely that at expiration the underlying will be where it is today. By the same token, it becomes MORE likely that the underlying will be far away from where it is today.

 

Definition of ATM options is when strike is the same as underlyer price. If this definition is correct, ATM options (and even OTM options if my intuition is correct) can have a delta much higher that 0.5 (even if carry is zero). It all depends on the level of volatility. Check it out. It can be not intuitive at first, but reflect on it and let me know, and I can follow with more details.