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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.

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Over-estimates or misprices? Can you provide the source where you saw that statement? –  SRKX Dec 17 '13 at 14:39
    
hss.caltech.edu/~bcornell/PUBLICATIONS/… this is the emprical research. –  Pasha Dec 19 '13 at 4:37
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2 Answers

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.

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Thank you for you reply. I do read about the warren buffet's comment and yes the reason he believes is inflation. BUt he fails to provide logical financial explanation. –  Pasha Dec 19 '13 at 4:40
    
@Pasha Is that the basis of your question? If so, I think I can help with that and edit. –  quantycuenta Dec 19 '13 at 4:57
    
that is the source of my question hss.caltech.edu/~bcornell/PUBLICATIONS/… if you want to edit the question to make it more clearer. please feel free. –  Pasha Dec 19 '13 at 10:54
    
@Pasha Edited. Please, let me know if there's anything else that needs clarification. –  quantycuenta Dec 19 '13 at 18:20
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Instead of a logical proof, would you accept a little bit of hand waving?

Think about these two constants in Black-Scholes:

  • $r$, interest rate
  • $\sigma$, volatility

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.

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Thank you for your answer. The question is not what is the reason of mispricing, the question is what is the reason of Overestimation? –  Pasha Dec 19 '13 at 4:39
    
In Buffet's opinion, very long-term options (his example was 100 years) are overpriced by the B-S model. He needs to remember that B-S is predicated on log-normal prices, that is, the ratio of today's price to a future price. Index prices can't go below 0, but they can go down to a tiny fraction of a penny. If I were to engage the Oracle of Omaha on this topic, I'd ask: Who will take the other side of your $1B option with 100 years to expiration and a strike of 930? –  rajah9 Dec 19 '13 at 19:39
    
Or here's another vein that I might leave to the OP. Constructed a 100-step binomial tree, and assume the index goes up or down by sigma each year. Start at 930. What will the put be worth in year 100? In many cases it will be 0, but in the rest it will be > 0 but <= 930. (By the way, I put this into DerivaGem, S=930, K=930, r=1%, q=2%, vol=25%, T=100. For continuous dividends, the 100-year puts should cost $300.32 each.) –  rajah9 Dec 19 '13 at 19:50
    
Oh, and another question for the Oracle. Was that an American or European put? Any way I can exercise it before I die? –  rajah9 Dec 19 '13 at 19:52
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