Wall Street Quants Owe a Debt to Obscure French Studentby
The 21st century began with a major correction on Wall Street and a long period of volatility in global financial markets. The VIX, sometimes called the fear index, soared, and congressional hearings explored why major investment banks couldn’t better hedge market risk. Traders sought reprieve, and risk management became elevated from prudent practice to a tool for financial survival.
Risk is most efficiently managed through the purchase of options and other derivatives. Many investors know that the economists Fischer Black and Myron Scholes are considered the pioneers of derivatives pricing through their options-pricing theory. Their Black-Scholes formula and its variations remain the primary tools in the optimal pricing and determination of derivatives hedges.
Most don’t know that options-pricing theory actually began at the very start the 20th century, in Paris. The innovation, by an obscure graduate student named Louis Bachelier, introduced sophisticated mathematics to finance theory and eventually gave rise to quantitative analysis. It took us almost 70 years to appreciate Bachelier’s contribution -- but he should go down as the father of modern finance.
In his early 20s, following a stint in the French army, Bachelier found himself working at the Paris Stock Exchange. There he tried to understand how options on French government perpetual bonds, or rentes, could be priced.
He enrolled at the Sorbonne to study mathematical physics under one of France’s top mathematicians, Henri Poincare. His supervisor was a bit perplexed when Bachelier chose to study French bond options for his doctoral thesis, but soon agreed that his idea was brilliant.
In just a few pages, Bachelier demonstrated that the volatility of the price of an underlying security affects both the put and call price of its option. He began by postulating that the price of perpetual bonds followed what we now call a “random walk.” He then showed that the range of plausible prices of such an option diffuses, or spreads out, as the time to settlement increases, just as smoke might disperse from a chimney in a manner proportional to distance. By then, the mathematics of such “diffusion processes” were well understood, but they were about to be applied in most novel ways.
Bachelier had hit upon two of the most important concepts of modern finance -- the random walk of securities prices and the pricing of market volatility over time. Five years later in 1905, Albert Einstein used the same diffusion equation to show that the random walk of small particles colliding with even smaller particles helps explain the atomic structure. Einstein was almost universally credited with a mathematical methodology that Bachelier had developed in his thesis and applied to derivatives markets.
Bachelier’s subsequent academic career was disjointed and prone to drama. It would take 60 years for the profession to begin to appreciate his work.
But by the early 1960s, there was a general agreement in the literature that the logarithm of securities prices should follow a random walk. Theorists were also postulating that the risk-adjusted return from the options on such securities should be consistent with the return found elsewhere in the market. By 1964, the financial theorist and economist Paul Samuelson had stumbled upon Bachelier’s yet-unknown version of the random walk in his own research on a pricing formula for warrants, a close sibling of options that are issued not by market participants but by a company.
In doing so, he also discovered Bachelier’s thesis, and had it translated from French into English. With Bachelier’s insights into the random walk, Samuelson had almost assembled the entire options-pricing picture before leaving the work to his graduate assistant, Robert Merton, to complete.
Meanwhile, Black and Scholes were working on their own theory of options pricing, also in ignorance of Bachelier, Samuelson and Merton. They, too, managed to reinvent Bachelier’s options-pricing formula in a way that was elegant and easily applied, and based on a random walk not of a securities price but of its logarithm instead, a subtlety that Bachelier had missed seven decades earlier. They also incorporated the latest insights on efficient markets to complete the model in a way that Bachelier and Samuelson missed.
Black and Scholes published their results in the early 1970s, just as the Chicago Board of Trade spawned the Chicago Board of Options Exchange. For the first time, there was simultaneously a new market tool -- broadly traded options --and a new equation to price it. Black and Scholes were immortalized. Following Black’s death, Scholes and Merton went on to win the Nobel Prize for their work.
Now, the Black-Scholes equation, which was almost identical to the formula that Bachelier developed decades earlier, is programmed into most financial calculators, and is used to measure implied market risk from the pricing of the VIX. It could be called the “fear formula.” Sophisticated fund managers use it to ensure the options they use to hedge risk are properly priced.
With greater volatility in markets comes an increased probability that an option will exceed upside and downside strike prices, and hence there is a greater chance that an option will be exercised. In turn, the price of options reflect the probability of being exercised, converting a measure of volatility to a price. Alternately, the observed price of an option in relationship to its exercise price gives traders an implicit measure of the market’s anticipation of risk. If future risk can’t be observed directly, the VIX index can measure how market participants assess risk in their pricing of the VIX. Greater volatility begets greater premiums, as Bachelier first observed in the day-to-day trading of rentes on the Paris Stock Exchange.
Even with these sophisticated models at hand, we still are faced with the reality that even the best formulas to incorporate past and present conditions can’t predict the future. Our new cadre of quants on Wall Street can’t divine the future from the past, no matter how sophisticated their models may be.
We may be a little further along now than Bachelier was in 1900, but the pricing of risk in periods of major market upheaval remains as much an art as a science.
(Colin Read is the chairman of the department of finance and economics at the State University of New York, Plattsburgh. He has published a dozen books on finance theory. The opinions expressed are his own.)
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