The Formula That Killed Wall Street
A year ago, it was hardly unthinkable that a math wizard like David X. Li might someday earn a Nobel Prize. After all, financial economists—even Wall Street quants—have received the Nobel in economics before, and Li’s work on measuring risk has had more impact, more quickly, than previous Nobel Prize-winning contributions to the field. Today, though, as dazed bankers, politicians, regulators, and investors survey the wreckage of the biggest financial meltdown since the Great Depression, Li is probably thankful he still has a job in finance at all. Not that his achievement should be dismissed. He took a notoriously tough nut—determining correlation, or how seemingly disparate events are related—and cracked it wide open with a simple and elegant mathematical formula, one that would become ubiquitous in finance worldwide.
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Here’s the formula:
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More on Derivatives:
- …Improved “transparency” – a favorite remedy of politicians, commentators and financial regulators for averting future train wrecks – won’t cure the problems that derivatives pose. I know of no reporting mechanism that would come close to describing and measuring the risks in a huge and complex portfolio of derivatives. Auditors can’t audit these contracts, and regulators can’t regulate them. When I read the pages of “disclosure” in 10-Ks of companies that are entangled with these instruments, all I end up knowing is that I don’t know what is going on in their portfolios (and then I reach for some aspirin).
Derivatives contracts, in contrast, often go unsettled for years, or even decades, with counterparties
building up huge claims against each other. “Paper” assets and liabilities – often hard to quantify – become important parts of financial statements though these items will not be validated for many years. Additionally, a frightening web of mutual dependence develops among huge financial institutions. Receivables and payables by the billions become concentrated in the hands of a few large dealers who are apt to be highly-leveraged in other ways as well. Participants seeking to dodge troubles face the same problem as someone seeking to avoid venereal disease: It’s not just whom you sleep with, but also whom they are sleeping with.The Black-Scholes formula has approached the status of holy writ in finance, and we use it when
valuing our equity put options for financial statement purposes. Key inputs to the calculation include a contract’s maturity and strike price, as well as the analyst’s expectations for volatility, interest rates and dividends. If the formula is applied to extended time periods, however, it can produce absurd results. In fairness, Black and Scholes almost certainly understood this point well. But their devoted followers may be ignoring whatever caveats the two men attached when they first unveiled the formula.It’s often useful in testing a theory to push it to extremes. So let’s postulate that we sell a 100- year $1 billion put option on the S&P 500 at a strike price of 903 (the index’s level on 12/31/08). Using the implied volatility assumption for long-dated contracts that we do, and combining that with appropriate interest and dividend assumptions, we would find the “proper” Black-Scholes premium for this contract to be $2.5 million. To judge the rationality of that premium, we need to assess whether the S&P will be valued a century from now at less than today. Certainly the dollar will then be worth a small fraction of its present value (at only 2% inflation it will be worth roughly 14¢). So that will be a factor pushing the stated value of the index higher. Far more important, however, is that one hundred years of retained earnings will hugely increase the value of most of the companies in the index. In the 20th Century, the Dow-Jones Industrial Average increased by about 175-fold, mainly because of this retained-earnings factor.
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%. Under these assumptions, the mathematical expectation of loss on our contract would be $5 million ($1 billion X 1% X 50%). But if we had received our theoretical premium of $2.5 million up front, we would have only had to invest it at 0.7% compounded annually to cover this loss expectancy. Everything earned above that would have been profit. Would you like to borrow money for 100 years at a 0.7% rate? Let’s look at my example from a worst-case standpoint. Remember that 99% of the time we would pay nothing if my assumptions are correct. But even in the worst case among the remaining 1% of possibilities – that is, one assuming a total loss of $1 billion – our borrowing cost would come to only 6.2%. 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 probabilityweighted range of values of American business 100 years from now. (Imagine, if you will, getting a quote every day on a farm from a manic-depressive neighbor and then using the volatility calculated from these changing quotes as an important ingredient in an equation that predicts a probability-weighted range of values for the farm a century from now.)Though historical volatility is a useful – but far from foolproof – concept in valuing short-term options,
its utility diminishes rapidly as the duration of the option lengthens. In my opinion, the valuations that the Black- Scholes formula now place on our long-term put options overstate our liability, though the overstatement will diminish as the contracts approach maturity. Even so, we will continue to use Black-Scholes when we are estimating our financial-statement liability for long-term equity puts. The formula represents conventional wisdom and any substitute that I might offer would engender extreme skepticism. That would be perfectly understandable: CEOs who have concocted their own valuations for esoteric financial instruments have seldom erred on the side of conservatism. That club
of optimists is one that Charlie and I have no desire to join.- Excerpt from Letter to Shareholders by Warren E. Buffett
