Factoring in exponential growth/decay when hedging

[Aston01]

I had a question for those more mathematically inclined then I am in regards to factoring in exponential growth and decay when it comes to hedging.

In a scenario where you started with

long call position with a delta of 100
long put position with a delta of -50​

If I understand exponential growth/decay correctly by the time the call position has lost 50% of its value the put position will have increased by 100% to result in an effective net delta of -50. From there if the underlying continued to decline the exponential growth would cause the put position to gain value at a progressively faster rate then the call position lost value.

At some point prior to the deltas inverting the larger call position is losing value faster then the growth in the smaller put position can accelerate to compensate.

Is there a way to calculate the maximum amount of exposure to unhedged loses during this inversion?

FWIW - I am not explicitly taking into account any of the other greeks in the above.

 

[i960]

If I understand exponential growth/decay correctly by the time the call position has lost 50% of its value the put position will have increased by 100% to result in an effective net delta of -50. From there if the underlying continued to decline the exponential growth would cause the put position to gain value at a progressively faster rate then the call position lost value.

Where are you getting the idea that there is exponential growth at all price points? There is exponential growth of delta (via gamma) as the underlying approaches ATM at which point gamma peaks and growth begins to taper in a logarithmic fashion as gamma decreases. A long put with a delta of -50 is already ATM and not going to gain value at a progressively faster rate - it's going to gain value at a regressive rate as it's moving away from peak gamma, and during this time gamma of the call position is starting to exponentially increase as it approaches the ATM strike of the call meaning the rate at which the call position loses value is going to accelerate (approaching ATM) while the rate at which your put position gains value is decelerating (moving away from ATM).

Is there a way to calculate the maximum amount of exposure to unhedged loses during this inversion?

What's your platform?

FWIW - I am not explicitly taking into account any of the other greeks in the above.

Gamma.

 

[Mugono]

Hi i960, I would first like to say that I very much enjoy reading your posts. I did have one query in relation to the statement:

"and during this time gamma of the call position is starting to exponentially increase as it approaches the ATM strike of the call meaning the rate at which the call position loses value is going to accelerate (approaching ATM)"


I would have thought that the rate that an ITM call option (100 delta in this example) loses value as the underlying declines would fall at a decreasing rate, maxing out at the ATM strike where gamma/theta is at the peak of the distribution/hill. Beyond this point, delta losses will be met by less and less gains from decay ('gamma bleed') ?

I look forward to any comments.

 

[donnap]

Yes, from the call delta of 1, gamma will increase on the way down. This refers to changes in delta which will decrease at an accelerating rate to .5.

Therefore the rate of the loss in value of the call will decrease, as you said. Delta decreases at a faster rate toward .5 so the rate of loss decreases at an accelerated pace. However, this ignores all other pricing factors.

BTW, common sense says that the theoretical max. risk exposure for the "inversion" would be the value of the call + the time value of the put.