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.
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.
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.
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