Will highly appreciate if anybody can provide logical financial proof why the Black-Scholes option pricing model overestimates the value for long-term options as described in this paper "Warren Buffett, Black-Scholes and the Valuation of Long-dated Options" by Bradford Cornell.
Buffett seems to suggest that inflation and retained earnings not being taken into account in Black Scholes is the source of mispricing.
This is not my specialty, so please take this with a flat of salt, but I have problems with the fact that as time approaches infinity, all call prices approximate half of the underlying price. Even more strangely, as time approaches infinity, all put prices approximate negative half of the underlying price.
I haven't seen a model yet that corrects this problem.
As for inflation, in his 100 year bond case, if inflation stayed constant, it would be reflected in $r$, and if it varied, yes that would be a problem, but inconstant variables is no revelation.
As for retained earnings, Black Scholes assumes lognormality, so $\mu$ is cancelled out no matter how its derived. This again is no revelation. If he wanted to take that into account as well as more extreme events, he could use the Variance Gamma Process, but comparing these two models precisely at the money really doesn't show too much difference. Variance Gamma helps with in or out of the money.
The reason why Buffett is complaining about BS is because he's forced to use it by the SEC, so if memory serves, he had to book an immediate ~50% loss since he sold less than the BS price.
The author of the paper you linked, seemed to agree that if inflation remained relatively constant at ~2%, BS would misprice, but this is not true to me since inflation is more or less reflected in $r$ thus would remain constant except in the final years.
To me, again, the problem with any option model is that $K$ is discounted away while $S$ is not. At 100 years, this effect would surely show its head since discounting at the 30 year treasury, ~4%, for 100 years will have eroded ~98% of $K$'s value away. What's more, a call would be BS priced at ~$200 million with those characteristics.
If Buffett had to write down 50%, again depending on my faulty memory, his counterparty may have seriously underpaid based upon BS.
Please remember that this is not my specialty, so I can only examine this in a very superficially mathematical way.
Instead of a logical proof, would you accept a little bit of hand waving?
Think about these two constants in Black-Scholes:
Also think about a long-term option, say, one whose expiration date is a year from now. Will $r$ and $\sigma$ be the same over the year? Probably not. And yet a constant interest rate and volatility are two assumptions. That's the source of the mispricing.