Games of Chance (I): How Probability Revolutionalized Finance

The restless jiggles of stock price movements have been a source of puzzle for mathematicians and economists for centuries

Modern Finance has the trappings of a science like Physics (though with far less predictability than the latter). At the heart of the science of Finance is Probability, the branch of mathematics that studies quantities that are subject to random movements, what we ordinarily call “chance”. In Physics, such quantities include the path of sub-atomic particles like electrons, photons, bosons and so on. Finance, too abound with randomness – just look at the way stock prices jiggle from second to second on any trading day!

A simple mathematical process known as the Random Walk lies at the heart of modern Finance as the fundamental model for describing asset prices, including stocks, bonds and various commodities. As its name suggests, assets with the random walk property are very hard to predict purely from knowledge of their past price movements. This in turn means that probability rather than certainty lies at the heart of modern Finance.

A cast of scientific luminaries laid the probabilistic foundation for Finance, including some surprising names. But I get ahead of myself. Let us first go to the very beginning – in 16th century Italy – where the story began. At that time, a mathematician by the name of Girolamo Cardano proposed the idea of a fair game to describe gambling. A fair game (later known as a martingale) is a subtle and important concept. If a bet is “fair”, then no one can predict their winnings or losses by looking at their past performance. If they could, then the game isn’t fair because you could increase your bet when the forecast is positive and decrease your bet when it is negative. This ability would give you an edge over your opponent, and over time, that edge would snowball into untold riches for you. Unfair games are therefore improbable in the hyper-competitive world of Finance.

As intuitive as the notion of fair game was, Cordano’s insight was forgotten until 300 years later when it resurfaced in the work of an obscure French mathematician named Louis Bachelier (1870 – 1946). A young Bachelier was writing his doctoral thesis at the Sorbonne in Paris under the supervision of the great mathematician, Henri Poincare. For his thesis, Bachelier chose to analyze the Parisian stock market. In particular, he wanted to model the prices of stock warrants. These are financial contracts that give its owner the right, but not the obligation, to buy a stock at a given price before a given (expiry) date.

Bachelier wanted to derive a formula that gives the fair price of a stock warrant. While he failed in this task, he nevertheless discovered something very unusual about the movements of stock prices: they move around as though they were completely random, rather like tiny grains of pollen darting around aimlessly in liquid. Bachelier then realized that the random movements of stock prices – their very unpredictability – was empirical proof that stock prices behave like martingales, the mathematical description of a fair game. Modern Finance credits Bachelier for this discovery, which is christened as the Random Walk Model of stock prices.

Like Cardano many years before him, Bachelier’s work languished unnoticed by the scientific community. His thesis was eventually published in 1914, but he was denied tenure at the University of Dijon due to a negative letter of recommendation from the famous mathematician, Paul Levy. He spent the rest of his career in a small teaching college in the east of France. Bachelier’s fate was most unfortunate because he anticipated the work of Albert Einstein on Brownian motion by five years!

Brownian motion is the continuous-time version of the random walk, and is named after the Scottish botanist Robert Brown who discovered it in 1827. Brown, who lived from 1773 to 1858, observed under a microscope that when pollen grains are suspended in water, they never stay still, but instead jiggle randomly as shown in the following video:

To continue our story, in an amazing turn of events, Bachelier was “rediscovered” fifty years later – in the United States. The year was 1954 and the place was the University of Chicago. Leonard Jimmy Savage, a noted professor of Statistics there, accidentally came across a copy of Bachelier’s thesis in the university library. He then sent letters to his colleagues, telling them of this undiscovered gem. One of those recipients was Paul A. Samuelson, an Economics wunderkind and future Nobel Prize winner.

Mathematician Louis Bachelier (1870 – 1946) at age 20 and Economist Paul A. Samuelson (1905 – 2009)

On reading Bachelier’s thesis, Samuelson immediately understood the significance of Bachelier’s work and sought to provide an economic explanation behind the martingale property of stock prices. He came up with the answer after studying the price movements of wheat traded in the Chicago futures market. As every commodities trader knows, spot prices in wheat tend to rise from the fall harvest to the following spring due to storage costs, and then drop immediately before the next harvest, when the market anticipates a future glut. Yet, in reality, wheat prices tend to fluctuate randomly as if they are unaffected by weather patterns. How could this be?

Samuelson finally resolved the “wheat paradox” using mathematics. But the intuition behind his argument is easy enough to understand. Since the current price already contains all the known information or news, which in the case of wheat includes information about the weather, past price changes are useless for predicting the commodity’s future price. If past public information were useful for predicting future prices, investors would have used this information in the first place. Therefore, asset prices must move unpredictably every moment! Appropriately, Samuelson titled his paper, “Properly Anticipated Prices Fluctuate Randomly” and got it published in 1965. Since then, Brownian motion and its discrete counterpart, the random walk, have been the workhorses of probabilistic modelling in Finance.

This then, is the great story of a beautiful idea that flowed from mathematics to the world of abstract Finance. But if the random walk was just an abstract concept, it would useless to the practical world of investments and financial planning and would remain on the shelf as a mere curiosity. This is certainly not the case, as I will show in my next post.

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