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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4$\begingroup$ Over-estimates or misprices? Can you provide the source where you saw that statement? $\endgroup$– SRKXDec 17, 2013 at 14:39
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$\begingroup$ hss.caltech.edu/~bcornell/PUBLICATIONS/… this is the emprical research. $\endgroup$– PashaDec 19, 2013 at 4:37
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2 Answers
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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.
---EDIT--- Here is a salient portion of Buffet's strawman argument:
Considering everything, I believe the probability of a decline in the index over a one-hundred-year period to be far less than 1%. But let’s use that figure and also assume that the most likely decline – should one occur – is 50%.
Here, I believe, is the assumption that makes Buffet think that B-S pricing would produce "absurd results." Buffet is attacking "geeks bearing formulas" (which means you, dear reader of this thread). On what basis is he taking issue with Black-Scholes? Cornell explains.
This means that Buffett has two possible beefs [with the lognormal diffusion model from B-S]. First, the equity premium, and therefore, the drift should be larger. Second, something is wrong with the volatility.
Cornell dispenses of the first horn of the dilemma: "The culprit is unlikely to be the drift." He points his finger at volatility:
The final candidate, other than the arbitrage argument on which the model is based, is the volatility. If Mr. Buffett is criticizing the use of the lognormal diffusion assumption when pricing long-term options, he is not alone. Recall that the lognormal assumption implies that volatility increases linearly with respect to the horizon over which it is measured as shown in equation (1) [lognormal distribution of B-S]. There is empirical evidence which indicates that the linearity assumption fails to hold at long horizons. For example, Siegel (2008) reports that the variance of real returns on the S&P 500 historically have failed to rise linearly with the horizon. If the long-run volatility is lower, the value of long-term put options will be less. For instance, a volatility of 15%, instead of 18%, reduces the estimated value of Mr. Buffett’s hypothetical put position to $1.5 million. It also reduces the probability that the index will be lower at expiration than at initiation.
(emphasis mine)
In short, Cornell is citing Siegel to say that the reason Buffet is getting "absurd results" is because his estimation of volatility is too high for the long run.
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1$\begingroup$ Thank you for your answer. The question is not what is the reason of mispricing, the question is what is the reason of Overestimation? $\endgroup$– PashaDec 19, 2013 at 4:39
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$\begingroup$ 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? $\endgroup$– rajah9Dec 19, 2013 at 19:39
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$\begingroup$ 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.) $\endgroup$– rajah9Dec 19, 2013 at 19:50
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$\begingroup$ Oh, and another question for the Oracle. Was that an American or European put? Any way I can exercise it before I die? $\endgroup$– rajah9Dec 19, 2013 at 19:52
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2$\begingroup$ @rajah9 I honestly think you are making some philosophical statements here, without having read the paper which OP as asking about. The question is: Why does Cornell think longdated options are overvalued under Blackscholes? I would usually think the contrary, that the BS-Options were undervalued, as they do not include fat tail risk. $\endgroup$– emcorAug 6, 2014 at 20:51
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BS is a purely mathematical construction, not based on economic fundamentals. For the SPX to fall to zero or close to it would require a lower probably than stipulate by BS (extinction event might do it, in which case both parties would fail, as well as all humans so it wouldn't matter anyway), hence BS is overestimating the probability. Setting a hypothetical reflecting barrier at, say, p=300 (the SPX is at 2050 now) and then taking the time to 100 years when pricing the option can have great ramifications. This is can also explain why selling put cash covered options tends to be a superior strategy
