Something funny about delta!

I'll keep this post short and sweet. 

I was plotting some graphs that showed how an option delta behaves with time to maturity and its half brother - volatility when I noticed something funny.

Some basics:

The best way to think about delta is probability of the option expiring in the money.For an option trader it would be the amount he is likely to gain or loose with changes in the value of the underlying. 

So for starters - how does it vary with the amount of time remaining for expiry. Intuitively the more time you have on your hands the more the chances that it would exceed the strike. However, it will depend on whether the option is in-the-money or out-of-the money. So if it has already exceeded the underlying, the probability is much more like 1. I have captured this in the graph below. 

So far so good.

Now let's bring in volatility. I choose vols varying from 10% to 100% and used the same range of time to expiration. The graph looked like this,

Two things are simple to see.

1. For a given maturity delta reduces as vol increases.. however this will depend on whther the option is in or out or at the money.

2. With a lower volatility it takes less time to maturity for the probability of option expiring in the money to approach 1. 

But there's more...

Volatility somehow starts overpowering the time to expiration when we look at very high vols. Contrary to what we might think, delta does not strictly decrease for an increasing volatility. The 100% vol curve lies above the 70% vol curve for T beyond 6 months. 

Possible reason.. implications?

I guess it has something to do with t being in the square-roots. But I am not sure about it.. 

However, for relatively long dated (4-6 months) options a higher vol could be much worse or better for a trader then he might think. Over long terms it makes things much much more volatile. This might push up the hedging requirements in turn.

I should get back to the book now. Too much digression can be bad for health! :)