Questioning the meaning of Delta

Even if risk neutral drift is zero, there is "vol drift" a.k.a. Ito correction. If you want to get technical, a stochastic differential equation

Code:

dS_t = rS*dt + sigma * S * dW_t

has an analytical solution

Code:

S_t = S * exp(t*r - t *0.5*sigma^2)  * exp(sigma * W_t)

The sigma square terms is there because exponential function W_t (a stochastic variable) has higher expectation than exponential function of E(W_t), i.e. Jensen's inequality.

 

Is there a better formula for that?

 

Hey this is ET, not Wilmott!

 

I think I've found a new simple formula. It gives similar Call prices for K=S, but diverging the bigger t is.
Now trying to build-in also the case K != S.
Here's an early result with t upto 5 years (without r and q yet):

Code:

S=100.000 K=100.000 s=0.300 t=0.100 --> pp=0.618 pm=0.382 C=3.729  BSM_C=3.783  diff%=-1.44
S=100.000 K=100.000 s=0.300 t=0.200 --> pp=0.618 pm=0.382 C=5.273  BSM_C=5.348  diff%=-1.41
S=100.000 K=100.000 s=0.300 t=0.300 --> pp=0.618 pm=0.382 C=6.458  BSM_C=6.548  diff%=-1.37
S=100.000 K=100.000 s=0.300 t=0.400 --> pp=0.618 pm=0.382 C=7.457  BSM_C=7.558  diff%=-1.33
S=100.000 K=100.000 s=0.300 t=0.500 --> pp=0.618 pm=0.382 C=8.338  BSM_C=8.447  diff%=-1.30
S=100.000 K=100.000 s=0.300 t=0.600 --> pp=0.618 pm=0.382 C=9.133  BSM_C=9.250  diff%=-1.26
S=100.000 K=100.000 s=0.300 t=0.700 --> pp=0.618 pm=0.382 C=9.865  BSM_C=9.987  diff%=-1.22
S=100.000 K=100.000 s=0.300 t=0.800 --> pp=0.618 pm=0.382 C=10.546  BSM_C=10.673  diff%=-1.18
S=100.000 K=100.000 s=0.300 t=0.900 --> pp=0.618 pm=0.382 C=11.186  BSM_C=11.316  diff%=-1.15
S=100.000 K=100.000 s=0.300 t=1.000 --> pp=0.618 pm=0.382 C=11.791  BSM_C=11.924  diff%=-1.11
S=100.000 K=100.000 s=0.300 t=1.100 --> pp=0.618 pm=0.382 C=12.367  BSM_C=12.501  diff%=-1.07
S=100.000 K=100.000 s=0.300 t=1.200 --> pp=0.618 pm=0.382 C=12.917  BSM_C=13.052  diff%=-1.04
S=100.000 K=100.000 s=0.300 t=1.300 --> pp=0.618 pm=0.382 C=13.444  BSM_C=13.580  diff%=-1.00
S=100.000 K=100.000 s=0.300 t=1.400 --> pp=0.618 pm=0.382 C=13.951  BSM_C=14.087  diff%=-0.96
S=100.000 K=100.000 s=0.300 t=1.500 --> pp=0.618 pm=0.382 C=14.441  BSM_C=14.576  diff%=-0.93
S=100.000 K=100.000 s=0.300 t=1.600 --> pp=0.618 pm=0.382 C=14.915  BSM_C=15.048  diff%=-0.89
S=100.000 K=100.000 s=0.300 t=1.700 --> pp=0.618 pm=0.382 C=15.374  BSM_C=15.506  diff%=-0.85
S=100.000 K=100.000 s=0.300 t=1.800 --> pp=0.618 pm=0.382 C=15.819  BSM_C=15.949  diff%=-0.81
S=100.000 K=100.000 s=0.300 t=1.900 --> pp=0.618 pm=0.382 C=16.253  BSM_C=16.380  diff%=-0.78
S=100.000 K=100.000 s=0.300 t=2.000 --> pp=0.618 pm=0.382 C=16.675  BSM_C=16.800  diff%=-0.74
S=100.000 K=100.000 s=0.300 t=2.100 --> pp=0.618 pm=0.382 C=17.087  BSM_C=17.208  diff%=-0.70
S=100.000 K=100.000 s=0.300 t=2.200 --> pp=0.618 pm=0.382 C=17.489  BSM_C=17.606  diff%=-0.67
S=100.000 K=100.000 s=0.300 t=2.300 --> pp=0.618 pm=0.382 C=17.882  BSM_C=17.995  diff%=-0.63
S=100.000 K=100.000 s=0.300 t=2.400 --> pp=0.618 pm=0.382 C=18.267  BSM_C=18.376  diff%=-0.59
S=100.000 K=100.000 s=0.300 t=2.500 --> pp=0.618 pm=0.382 C=18.643  BSM_C=18.748  diff%=-0.56
S=100.000 K=100.000 s=0.300 t=2.600 --> pp=0.618 pm=0.382 C=19.013  BSM_C=19.112  diff%=-0.52
S=100.000 K=100.000 s=0.300 t=2.700 --> pp=0.618 pm=0.382 C=19.375  BSM_C=19.469  diff%=-0.48
S=100.000 K=100.000 s=0.300 t=2.800 --> pp=0.618 pm=0.382 C=19.730  BSM_C=19.818  diff%=-0.44
S=100.000 K=100.000 s=0.300 t=2.900 --> pp=0.618 pm=0.382 C=20.080  BSM_C=20.162  diff%=-0.41
S=100.000 K=100.000 s=0.300 t=3.000 --> pp=0.618 pm=0.382 C=20.423  BSM_C=20.499  diff%=-0.37
S=100.000 K=100.000 s=0.300 t=3.100 --> pp=0.618 pm=0.382 C=20.760  BSM_C=20.830  diff%=-0.33
S=100.000 K=100.000 s=0.300 t=3.200 --> pp=0.618 pm=0.382 C=21.093  BSM_C=21.155  diff%=-0.30
S=100.000 K=100.000 s=0.300 t=3.300 --> pp=0.618 pm=0.382 C=21.420  BSM_C=21.475  diff%=-0.26
S=100.000 K=100.000 s=0.300 t=3.400 --> pp=0.618 pm=0.382 C=21.742  BSM_C=21.790  diff%=-0.22
S=100.000 K=100.000 s=0.300 t=3.500 --> pp=0.618 pm=0.382 C=22.059  BSM_C=22.100  diff%=-0.19
S=100.000 K=100.000 s=0.300 t=3.600 --> pp=0.618 pm=0.382 C=22.372  BSM_C=22.405  diff%=-0.15
S=100.000 K=100.000 s=0.300 t=3.700 --> pp=0.618 pm=0.382 C=22.681  BSM_C=22.706  diff%=-0.11
S=100.000 K=100.000 s=0.300 t=3.800 --> pp=0.618 pm=0.382 C=22.985  BSM_C=23.002  diff%=-0.07
S=100.000 K=100.000 s=0.300 t=3.900 --> pp=0.618 pm=0.382 C=23.286  BSM_C=23.294  diff%=-0.04
S=100.000 K=100.000 s=0.300 t=4.000 --> pp=0.618 pm=0.382 C=23.582  BSM_C=23.582  diff%=0.00
S=100.000 K=100.000 s=0.300 t=4.100 --> pp=0.618 pm=0.382 C=23.875  BSM_C=23.866  diff%=0.04
S=100.000 K=100.000 s=0.300 t=4.200 --> pp=0.618 pm=0.382 C=24.165  BSM_C=24.147  diff%=0.07
S=100.000 K=100.000 s=0.300 t=4.300 --> pp=0.618 pm=0.382 C=24.451  BSM_C=24.423  diff%=0.11
S=100.000 K=100.000 s=0.300 t=4.400 --> pp=0.618 pm=0.382 C=24.733  BSM_C=24.697  diff%=0.15
S=100.000 K=100.000 s=0.300 t=4.500 --> pp=0.618 pm=0.382 C=25.013  BSM_C=24.967  diff%=0.19
S=100.000 K=100.000 s=0.300 t=4.600 --> pp=0.618 pm=0.382 C=25.289  BSM_C=25.233  diff%=0.22
S=100.000 K=100.000 s=0.300 t=4.700 --> pp=0.618 pm=0.382 C=25.563  BSM_C=25.496  diff%=0.26
S=100.000 K=100.000 s=0.300 t=4.800 --> pp=0.618 pm=0.382 C=25.833  BSM_C=25.757  diff%=0.30
S=100.000 K=100.000 s=0.300 t=4.900 --> pp=0.618 pm=0.382 C=26.101  BSM_C=26.014  diff%=0.33
S=100.000 K=100.000 s=0.300 t=5.000 --> pp=0.618 pm=0.382 C=26.366  BSM_C=26.268  diff%=0.37
...

 

I now have created my own realistic Delta Greek called "MyDelta":

Code:

S=100.00 K=100.00 s=3.0 t=1.0 r=0.0 q=0.0 :
C: Value=86.638560 Delta=0.933193  MyDelta=0.500000  Gamma=0.000432 Vega=0.129518 Theta=-0.053226 Rho=0.066807  ...
P: Value=86.638560 Delta=-0.066807 MyDelta=-0.500000 Gamma=0.000432 Vega=0.129518 Theta=-0.053226 Rho=-0.933193 ...

DBG:
S @ +1SD=2008.55
S @  0SD=100.00
S @ -1SD=4.98
p_S_reaching_K(ie. ITM)=0.50 p_rest=0.50, expected_St(z=0 = 0SD = p=0.5)=100.00

 

Well, what is your interpretation of delta? The reason why delta exists is to let a trader know how much stock to hold as a hedge- but please, do continue as this is exceedingly entertaining

 

Hey, @thecoder , wanna really get your brain farting?

Delta can rise above 100% (but I think this applies only to fowards, not options) lol

 

Come on, for a single option this never can be true.
Otherwise give the params where this can happen.
Maybe you mean the delta in delta hedging where the single deltas are cumulated.

 

This might have been the intention, but in fact it's IMO simply wrong as the few examples with high volatility have clearly demonstrated.
If I were in a situation to use the delta, for example as probability for expiring ITM or in delta hedging, I would use my own delta which behaves mathematically correct, IMO.
Cf. previous posting with an example for "MyDelta".

 

The problem is your assumptions about delta are completely wrong. Delta is NOT just a measure of the probability the option expiring in the money. The delta value is also determined by how much the option will finish ITM if and when it does (conditional probability).

So in your extreme example of 300% IV, the much higher call delta (than the put) reflects the extreme asymmetric values the call could be worth vs the put if it were to expire ITM. The put's potential value is limited by the zero bound...while the call's expected value, if it expires ITM, increases with higher vol and/or more DTE. The delta of the call reflects this higher expected value if it expires ITM.