Questioning the meaning of Delta

@VolSkewTrader,
from a probabalistic point of view the distance from the mean (ie. initial S) to +1SD is equal to mean to -1SD, ie. 1SD to each of the both sides.
But seeing the same from the point of view of absolute values (ie. money values) there is a big difference.
Example for the said data set with S=K=100, s=300%, t=1 --> C=P=86.638560 :

S_t @ +1SD=2008.55
S_t @ 0SD=100.00
S_t @ -1SD=4.98
C_at_expiration_if_spot_at_plus_1SD=$1908.55
P_at_expiration_if_spot_at_minus_1SD=$95.02
CPratio=20.09

This means: a Call option expiring at S@+1SD is worth a whopping 20.09 times more than a Put option expiring at S@-1SD, eventhough the probability for each these events to occur is equal (p=15.8655%).
Btw, of course at expiration the volatility doesn't play any role anymore, just the spot S counts.

Another relation: the Call option has made a profit of
1908.55 / 86.638560 * 100 - 100 = 2102.89%
whereas the put option made only
95.02 / 86.638560 * 100 - 100 = 9.67%.

This is real, but hard to chew, IMO :)

So, this is IMO a completely new metric. The delta doesn't even reflect this relation.

And if my math above is correct, then I must conclude that the currently used option pricing models (ie. Black-Scholes-Merton) can IMO simply be not correct. What do others say? :) Come on folks, give it to me, I deserve it! :)
 
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thecoder said:
@VolSkewTrader,
from a probabalistic point of view the distance from the mean (ie. initial S) to +1SD is equal to mean to -1SD, ie. 1SD to each of the both sides.
But seeing the same from the point of view of absolute values (ie. money values) there is a big difference.
Example for the said data set with S=K=100, s=300%, t=1 --> C=P=86.638560 :

S_t @ +1SD=2008.55
S_t @ 0SD=100.00
S_t @ -1SD=4.98
C_at_expiration_if_spot_at_plus_1SD=$1908.55
P_at_expiration_if_spot_at_minus_1SD=$95.02
CPratio=20.09

This means: a Call option expiring at S@+1SD is worth a whopping 20.09 times more than a Put option expiring at S@-1SD, eventhough the probability for each these events to occur is equal (p=15.8655%).
Btw, of course at expiration the volatility doesn't play any role anymore, just the spot S counts.

Another relation: the Call option has made a profit of
1908.55 / 86.638560 * 100 - 100 = 2102.89%
whereas the put option made only
95.02 / 86.638560 * 100 - 100 = 9.67%.

This is real, but hard to chew, IMO :)

So, this is IMO a completely new metric. The delta doesn't even reflect this relation.

And if my math above is correct, then I must conclude that the currently used option pricing models (ie. Black-Scholes-Merton) can IMO simply be not correct. What do others say? :) Come on folks, give it to me, I deserve it! :)
More...

When vol is high the delta 100% reflects the disproportionate expected value of the call vs the same strike put if either one was to expire ITM...that's the whole point. The high IV call delta fully accounts for the much higher potential value of the call over its lifetime.

Good luck hedging with your "new" deltas. Everyone uses more or less the same options pricing models which spits out similar deltas. When you start trading a high vol product or longer-dated options, you are going to get your ass handed to you when you realize you've underhedged/overhedged your calls/puts, and your model's theoreticals don't match the values on the screen because your fantasy deltas are completely off.
 
thecoder said:
@VolSkewTrader,
from a probabalistic point of view the distance from the mean (ie. initial S) to +1SD is equal to mean to -1SD, ie. 1SD to each of the both sides.
But seeing the same from the point of view of absolute values (ie. money values) there is a big difference.
Example for the said data set with S=K=100, s=300%, t=1 --> C=P=86.638560 :

S_t @ +1SD=2008.55
S_t @ 0SD=100.00
S_t @ -1SD=4.98
C_at_expiration_if_spot_at_plus_1SD=$1908.55
P_at_expiration_if_spot_at_minus_1SD=$95.02
CPratio=20.09

This means: a Call option expiring at S@+1SD is worth a whopping 20.09 times more than a Put option expiring at S@-1SD, eventhough the probability for each these events to occur is equal (p=15.8655%).
Btw, of course at expiration the volatility doesn't play any role anymore, just the spot S counts.

Another relation: the Call option has made a profit of
1908.55 / 86.638560 * 100 - 100 = 2102.89%
whereas the put option made only
95.02 / 86.638560 * 100 - 100 = 9.67%.

This is real, but hard to chew, IMO :)

So, this is IMO a completely new metric. The delta doesn't even reflect this relation.

And if my math above is correct, then I must conclude that the currently used option pricing models (ie. Black-Scholes-Merton) can IMO simply be not correct. What do others say? :) Come on folks, give it to me, I deserve it! :)
More...

Why do you think markets have to care whether Black-Scholes assumed a normal distribution in the BS Formular? Stop using Delta as a gauge for ITM/OTM Probability, it's only works when the market see's it the same way. https://www.globalcapital.com/article/k6543wh6f19l/option-prices-imply-a-probability-distribution
 
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Atikon said:
Why do you think markets have to care whether Black-Scholes assumed a normal distribution in the BS Formular? Stop using Delta as a gauge for ITM/OTM Probability, it's only works when the market see it the same way.
More...
BSM and also my example use lognormal distribution. But the payout for Call and Put should still be relatively equal for the same relative distance (in units of z or SD) from mean, IMO, since the risk (or its inverse, the chance) has the same probability...
 
thecoder said:
@VolSkewTrader, unfortunately you are blindly defending the BSM, without analyzing and verifying the shown discrepancies and problems with it. :-(
More...

Not defending the BSM. I recognize its many flaws and realize its deltas and greeks are just approximations as well. But if I were to use your deltas in a high vol product or LEAP expiration, my theoretical prices would not match the market's option prices during a move in the underlying...which could be costly.
 
VolSkewTrader said:
Not defending the BSM. I recognize its many flaws and realize its deltas and greeks are just approximations as well. But if I were to use your deltas in a high vol product or LEAP expiration, my theoretical prices would not match the market's option prices during a move in the underlying...which could be costly.
More...

Ok, that's fair. My said delta was intended as "probability for expiring ITM".
I haven't tested it for delta-hedging yet. Maybe I'll do it if I find time.
 
thecoder said:
BSM and also my example use lognormal distribution. But the payout for Call and Put should still be relatively equal for the same relative distance (in units of z or SD) from mean, IMO, since the risk (or its inverse, the chance) has the same probability...
More...

The payouts are asymmetric because you are using a lognormal pricing distribution. They should NOT be relatively equal. The calls have a much wider range of potential values than the puts. The number of possible call prices x probability gives you the approximate expected value for the call....which is why the value is so much higher than the put in a 1 SD move. The # of potential call prices is entirely determined by the volatility level. The higher the vol, the more # of call prices are possible... while the potential # of put prices remains relatively fixed. That's pretty basic statistics...which you are trying to disprove.
 
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VolSkewTrader said:
The payouts are asymmetric because you are using a lognormal pricing distribution. They should NOT be relatively equal. The calls have a much wider range of potential values than the puts. The number of possible call prices x probability gives you the approximate expected value for the call....which is why the value is so much higher than the put in a 1 SD move. The # of potential call prices is entirely determined by the volatility level. The higher the vol, the more possible call prices are possible... while the potential # of put prices remains relatively fixed. That's pretty basic statistics...which you are trying to disprove.
More...

This is the behavior in the BSM model. But I've shown that this behavior is mathematically incorrect as it unfairly favors the Call over the Put.
As said, IMO the payout has to be equal for both sides for same relative distances.
This means a totally (or maybe just partially) new option pricing model would be needed.
 
thecoder said:
This is the behavior in the BSM model. But I've shown that this behavior is mathematically incorrect as it unfairly favors the Call over the Put.
As said, IMO the payout has to be equal for both sides for same relative distances.
This means a totally (or maybe just partially) new option pricing model would be needed.
More...

Payouts are NOT equal for both sides for same relative distances (1 SD move).

Extreme example:

$1 stock, 1000% Implied Vol, 30 DTE
$1 ATM Call: $.8483 | .924 delta
$1 ATM Put: $.8483 | .0759 delta

1 SD move to upside: $13.34
1 SD move to downside: -$11.34 theoretically, but $0.00 is the lower bound.

Max Call value at expiration: $12.34 (Call's max increase in value is $11.4917 for a 1 SD move)
Max Put value at expiration: $1.00 (Put's max increase in value is $.1517 for a 1+ SD move)

At 1,000% vol, call has $11.34 more potential payout than same strike put for a 1 SD move. The put's max value is k, or $1, it's strike price. That egregious possible expected payout differential is fully reflected in the dramatically different deltas.

As vol approaches infinity, the call price will move 1 to 1 (100 delta) with the underlying, while the put price will stay stagnant (zero delta) and stuck at the same price as k, the strike ($1).
 
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