## Archive for the ‘**History: Math Finance**’ Category

## Development of Derivatives Pricing Models – I: “The Demon of Chance”

**The series looks like a wandering one… (Kendall, 1953)**

If there is any one school that can be said to be the birthplace of finance it has to be Yale, and any single economist that can take credit for being the “father of modern finance” it has to Irving Fisher (the guy most well-known for his ill-timed prediction that “[S]tock prices have reached what looks like a permanently high plateau” just before the great crash of ’29).

Trained under a physicist in late 19th century at Yale, Fisher was arguably the first quantitative economist. He popularized the idea of inter-temporal analysis in economics which forms the workhorse model for the way financial economics is taught today. The result that financial markets allow for separation of financing and investment decisions has its beginnings in the Fisher Separation Theorem. Fisher equation in monetary economics bears his name and is familiar even to 1st year business school students. His contributions abound.

Fisher’s role in the history of mathematical finance is only indirect though. Fisher was one of the founding members of the Econometric Society when it was formed in 1930. Lack of financial resources, however, prevented them from organizing regular society meetings as well as having their own publication. As Bernstein recounts in *Capital Ideas* (Ch. 1), around the time Alfred Cowles 3rd, another Yale graduate, and whose family owned the Chicago Tribune newspaper was recuperating from tuberculosis in Colorado Springs. For want of much to do and to bide his time he was also managing his family finances. For good or for worse, it turns out that his recovery was long. In that time he noticed that all of the investment advise he was getting from professional forecasters and editors were good for nothing. He analyzed their track record and found that he would have done better just tracking stocks in the market index.

Being a trained student of statistics he knew better than to just rely on a sequence of numbers. His friend helped him learn to use an IBM Hollerith (hardly a personal computer) so that he could run his regressions and pointed him to Fisher. It was no problem for Cowles to get in touch with Fisher, because his father and Fisher were classmates at Yale together. Learning about Econometric Society’s financial difficulty, and his own interest in learning more about economics of financial data, in 1932 he created an endowment for Econometric Society which paved the way for founding of *Econometrica, *a top rated journal in economics and econometrics today*. *

In July, 1933 issue of *Econometrica *results of Alfred Cowles’ painstaking work appears simply titled “Can Stock Market Forecasters Forecast?“. In it he presents the performance results of 16 financial services, 20 fire insurance companies, recommendations of the then editor of Wall Street Journal, Walter Hamilton and publications of 24 financial publications. The paper is worth reading even today, but his broad conclusion is best summarized by the last words of the article:

“[the performance]

…indicates that the most successful records are little, if any, better that what might be expected to result from pure chance. There is some evidence, on the other hand, to indicate that the least successful records are worse than what could be attributed to chance.”

Cowles carried on his studies till as late as 1960s and, as Bernstein reports in his book, results for larger sample periods and across sectors turned out to be no different.

The next set of studies that pointed towards the ‘randomness’ in stock prices were by statisticians – the important of those being by Working and Kendall before 1955. (Not that statisticians suddenly got interested in stock markets. Coincidentally, late ’20s and ’30s was also the time when some of the earliest works in time series analysis were done by Yule, Walker, Khinchine and Wold. Their work was crucial for the development of theory of ARMA process in the ’50s which allowed for systematic study of stock market data by Kendall and others.)

Kendall concluded :

“The best estimate of the change in the price between now and next week is that there is no change.”

(Remember martingale? )

While Kendall’s study didn’t exactly create a whirlwind in the world of traders, it did create enough ripples for economists to let their PhD students study stock market data as a reasonable dissertation topic – the only greybeard among the young Turks being a certain Paul Samuelson.

## Strands in the History of Mathematical Finance

If one looks at the historical accounts of the development of modern (mathematical) finance as a discipline (say, for example in Peter Bernstein‘s Capital Ideas) there seem to be three parallel strands:

*1. Development of Derivatives Pricing Models*: Here I have in mind the discovery of Louis Bachelier‘s thesis by Paul Samuelson (via Leonard Savage) and his and his students’ subsequent work on Warrant Pricing. There was enough empirical evidence accumulated by then, starting from Alfred Cowles (who was also instrumental in setting up of Econometric Society in early 20th century), Holbrook Working and Maurice Kendall.

*2. Development of Capital Asset Pricing Model*: Williams, Markowitz and Tobin paved the way for Treynor-Lintner–Sharpe–Mossin asset pricing model. The stage for thinking systematically about risk, however, was set long back by Daniel Bernoulli.

*3. Development of Stochastic Calculus and Mathematical Finance: *It took a long time for financial economists to learn about Bachelier’s work, but that’s not true of mathematicians. Levy, Kolmogorov and Doob were all aware of Bachelier, and while filling gaps in his work they laid the groundwork for stochastic calculus developed by Kiyoshi Ito.

While I hope to blog about each strand over the next few months [the key word being ‘hope’ :)], in the next post I start with a short history of developments that eventually led to the Black-Scholes-Merton Option Pricing Model as we know today.