The Thing with Black Scholes

Black Scholes (http://en.wikipedia.org/wiki/Black-Scholes) is generally not perfect. This imperfection is greater when either absolute of d1 or d2 is large. As a rule of thumb I check if the value is greater than 3, i.e. 3 SD. When d1 and d2 becomes large, i.e., when calculating the probability of the tails of the cumulative started normal distribution, the equation becomes more and more sensitive to these parameters and slight variation can produce erratic results. Also I presume the numeric error of calculating the cumulative standard normal would be large in the tails.

 

Be very of the Greeks (http://en.wikipedia.org/wiki/The_Greeks) when d1 and d2 are large. They would be imperfect and meaningless.

 

This error would be predominate in options deep in our out of the money and close to expiration. This is when d1 and d2 becomes absolutely large.

 

I guess the skew in the implied volatility surface is an attempt to correct this imperfection by the market participants. Though it has been generally termed as crash phobia after Black Monday, I guess this might be a attempt to correct and work around the imperfections in the Black Scholes equation itself through experience and learning its limitation which most probably felt during the market crash.
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Thanking you.
Best regards,
Suminda Sirinath Salpitikorala Dharmasena B.Sc. (Hon.) Comp. & I.S., Lon.

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