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'The Physics of Wall Street': The Most Arrogant Book in the World? Part 17


Fat tails and options.

Editor's note: The following column is the 17th part of an ongoing series of articles by Aaron Brown examining the claims made in The Physics of Wall Street: A Brief History of Predicting the Unpredictable, a new book by James Owen Weatherall. Click here to read Part 1.

James Weatherall's The Physics of Wall Street misunderstands Nassim Taleb's book The Black Swan, which was discussed in previous weeks. Now we'll look at how he denies Taleb credit with respect to option pricing and fat tails.

The Physics of Wall Street takes a common sense view of finance that was overturned in an intellectual revolution from about 1950 to 1980. The older view fails to appreciate how Harry Markowitz's Modern Portfolio Theory rebutted the ancient idea that every trade represents a disagreement between buyer and seller over the value of the items traded (this specific contention is made several times in the book). Under MPT investors agree not only on the value of securities, but on the probability distribution of their future prices, yet they trade anyway.

The commonsense view also misses the revolutionary impact of the Black-Scholes option pricing formula, and the body of work from about 1966 to 1975 that it represents. This work has such broad importance that it cannot be summarized with any one idea, but among the most important is that the price of a derivative does not depend on the probability distribution of its payouts. It's not surprising that Weatherall fails to understand that; since he thinks all of finance is guessing future prices, he naturally thinks that pricing an option consists of guessing the distribution of future prices of the underlying. But Louis Bachelier solved that problem in 1900 and it had absolutely no impact on finance or the world. The Black-Scholes formula shares mathematical similarities, but has an entirely different conceptual basis (actually three different conceptual bases, but that's another story). It's also worth noting that the Black-Scholes formula could not have revolutionized finance by giving better prices for options, because it doesn't price any of the options that traded at the time (or that have traded much since) and options were an insignificant part of the financial system in the early 1970s anyway.

The basic option pricing insight is easy to see (once it's pointed out, that is). Say a stock is selling for $70 today and will be either $50 or $100 tomorrow. What is the value of the option to purchase 100 shares of stock tomorrow for $80 per share?

The option will be worth either $2,000 or $0 tomorrow, depending on whether the stock is worth $100 or $50. Either way, it will be worth the same as 40 shares of stock minus $2,000. Forty shares of stock is worth $2,800 today. Ignoring the time value of money for simplicity, $2,000 tomorrow is worth $2,000 today. So the option is worth $2,800 minus $2,000 or $800. The general technique we have used is called "change of numeraire." By pricing the option in terms of shares of stock rather than dollars, we have answered a question that otherwise lacked the necessary information. Change of numeraire turned out to be an incredibly powerful tool for attacking quantitative financial problems.

The important point is that the price of the option does not depend on the probability of the stock going to $100. The option has to be worth $800 whether the stock has a 99% chance of going to 100, a 1% chance, or anything else.

Of course a real stock might go up or down any amount, it does not have only two possible prices tomorrow. But I can get around that by pricing the option tick by tick. I can compute the value of the option when it has only one tick to go before expiry, that allows me to price it two ticks before expiry and so on back to any time I want. As long as I can transact costlessly at each price, I know what the option is worth any given number of ticks before expiry, regardless of the probabilities of ticking up or down. But the terminal distribution of the stock price depends on the probabilities of ticking up or down. In theory, almost any terminal distribution of the stock price is consistent with the option price.

This change in focus from the value of the underlying to its trading characteristics is part of a trend that brought us from a financial system based on value and credit (which is still what most people, even most economists, assume about the world) to one based on liquidity.

Change of numeraire did not solve the practical problem of pricing options. I cannot guarantee that I can transact at every tick, and I cannot transact costlessly at any tick. Also, real options are written for periods of calendar time, not numbers of ticks. But change of numeraire did convince people that market microstructure was the key to option pricing, and that ability to trade the underlying efficiently was the key to option businesses. At the same time, it took attention away from attempts to model probability distributions of future underlying prices.

Weatherall tells some of the story of how option trading proved to be a dangerous activity in the decade before the stock market crash of 1987. Lots of people, including some of the most sophisticated and successful traders blew up ("blow up" is a technical term in finance, it does not mean merely losing money, but losing more than you have prepared for, so much that you cannot continue trading). But he somehow got the idea that this was caused by fat-tailed distributions of stock returns. That is false in theory, because traders were not making predictions about the distributions of stock returns. It is also false in practice, because before the 1987 crash the blow-ups were not associated with larger-than-expected moves in underlyings, rather they occurred at the same time as unusual microstructure events. A large part of the answer turned out to be that option trading changed microstructure (for example, "pinning" stock prices to popular exercise prices).

One of the most sophisticated and successful option shops was O'Connor & Associates. Weatherall credits them with avoiding blow-ups by figuring out that stock price return distributions had fat tails. Now I have great respect for the people at O'Connor and they certainly knew what they were doing, including a lot of stuff that other people didn't know. But everyone knew stock price return distributions had fat tails, it's obvious from looking at history. And knowing about fat tails was no help in avoiding option blow-ups. I suspect O'Connor's secrets had more to do with trading and execution than modeling return distributions.

In any event, O'Connor's secrets remained secret while another trader got famous for linking fat tails to option pricing…and explaining it to the world. Since Weatherall assumes the two topics were linked all along, he can't see the value of this work. It was Nassim Taleb who figured out why option traders blow up, and wrote the definitive book on managing option positions (Dynamic Hedging: Managing Vanilla and Exotic Options). He built these insights into a comprehensive philosophical system in subsequent popular bestsellers. It is epistemology, not economics, that is the connection between fat tails and financial blow-ups.

Next week, we conclude with the main message of The Physics of Wall Street. Is the solution to all financial problems for everyone to think like physicists? (Spoiler: I think the answer is "no.")

Links to previous stories in this series: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13, Part 14, Part 15, Part 16.
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