We now have a well-stocked cupboard of dodgy option pricing arguments to reply to at some point or another, and it’s not often a new one turns up. Yet that’s what happened the other day. One of our firm-friendly friends told us that Warren Buffett thinks long-dated options are over-priced by Black-Scholes, and that he has proved this claim both by theoretical methods and by practical means, i.e. by making a ton of money. Let’s take a look.
Theory
Let’s take the theoretical approach first. Buffett stated, in his 2008 newsletter, that if we wrote a 100-year $1 billion put option on the S&P 500 at the end of 2008, with an at-the-money strike price of 903, then the standard Black-Scholes premium for this contract would be $2.5m. But this result is irrational, he says. Many factors will be pushing the value of the S&P up, for example inflation, and particularly 100 years of retained earnings.
In the 20th Century, the Dow-Jones Industrial Average increased by about 175-fold, mainly because of this retained-earnings factor.
Moreover, we would only have to pay out in 1% of cases, by his estimate, and even in the extreme case where the FTSE went to zero and we had to pay out the whole $1bn, well that’s in 100 years’ time. Assuming we could invest the premium at 6.2%, the premium would compound up to $1bn and we would be able to pay the other party’s ‘winnings’.
Clearly, either my assumptions are crazy or the formula is inappropriate. The ridiculous premium that Black-Scholes dictates in my extreme example is caused by the inclusion of volatility in the formula and by the fact that volatility is determined by how much stocks have moved around in some past period of days, months or years. This metric is simply irrelevant in estimating the probability weighted range of values of American business 100 years from now.
Now this argument isn’t actually new, being a version of the actuarial fallacy, namely that we should price options using probabilistic and statistical methods. This line of reasoning is what nearly sunk AIG, of course. But this pearl of wisdom comes from Buffett, who is to us moderns what Aristotle was to the medieval schoolmen. ‘Buffett hath said it’, dixit ergo etc.
Well, what with central bank money printing and retained earnings, it is of course highly probable that any stock index will be perhaps 200 or more times its current level in the next century. But that is irrelevant to the pricing of an option, as Kevin and I keep saying. The question is what a rational investor will decide when choosing between an investment in the underlying asset, or selling a put option. The delta or asset-equivalent position of the option will be long, moreover for very long dates the delta will be highly stable. This stability makes long-dated options easy to hedge, if you can get the drift terms (interest rate and dividend) right, which is not quite so easy.
Being short a long-dated option is very much like being long the underlying, and there is little to decide between them with the gamma being low, except for the volatility charge you pay the trader.
As for the volatility charge, if you think the volatility quoted is too high, then you sell the put, if not, you buy the underlying asset. Usually the quote will be too high, as this is how the trader makes his money. I estimate (very roughly) that one percent vol point would be worth about £200m for the size Buffett was trading in. We don’t know the counterparty, but we can be sure they made a handsome profit by hedging the put. Buffett made a lot of money by going long the S&P for more than ten years, so both sides will have been happy. For that reason, there is nothing irrational about the B-S formula. Buffett’s assumptions are not crazy, but that does not mean the formula is not appropriate.
Indeed, it has to be precisely that way for the formula to work.
Practice
Turning to Buffett’s winnings, the chart at the top shows the annual profits 2008-2017 from his short put positions. The blue line in the chart below shows a simulation of the P/L using the Black-Scholes with inputs based on the assumptions set out in his newsletters, and you see it closely resembles Buffett’s official filings. The red line shows the P/L from a fixed position equivalent to Buffett’s estimated starting delta. Not only does the line closely resemble the B-S option P/L, proving the standard result that the exposure of a long dated option will closely resemble that of a position in the underlying, but the position is also more profitable. This outcome is probably just chance – we don’t know exactly what option positions Buffett was running – but it casts more than a shadow of doubt on Buffett’s extravagant claim in his 2010 newsletter that ‘Black-Scholes produces wildly inappropriate values when applied to long-dated options’.
To repeat, Black-Scholes values are just fine. The fact you can make a lot of money by writing long-dated put options and waiting is no more remarkable than the fact you can also make a lot of money by just going long for the same amount of time (and with considerably less risk).
One other thing. Buffett argues that even if the option falls deeply into the money, we could still break even by investing the premium at 6.2%, and waiting for compound interest to kick in. Er, actually no. He was writing in 2008 when the idea that long interest rates could fall to present levels of below 2%, and stay that way for some while, was unthinkable.
But ‘In the business world, the rearview mirror is always clearer than the windshield’. I wonder who said that?