# Markowitz

Harry Markowitz was the first to propose a modern quantitative methodology for portfolio selection. This required knowledge of assets' volatilities and the correlation between assets. The idea was extremely elegant, resulting in novel ideas such as 'efficiency' and 'market portfolios.' In this Modern Portfolio Theory, Markowitz showed that combinations of assets could have better properties than any individual assets. What did 'better' mean? Markowitz quantified a portfolio's possible future performance in terms of its expected return and its standard deviation. The latter was to be interpreted as its risk. He showed how to optimize a portfolio to give the maximum expected return for a given level of risk. Such a portfolio was said to be 'efficient.' The work later won Markowitz a Nobel Prize for Economics but is problematic to use in practice because of the difficulty in measuring the parameters 'volatility,' and, especially, 'correlation,' and their instability.

# Sharpe, Lintner and Mossin

William Sharpe of Stanford, John Lintner of Harvard and Norwegian economist Jan Mossin independently developed a simple model for pricing risky assets. This Capital Asset Pricing Model (CAPM) also reduced the number of parameters needed for portfolio selection from those needed by Markowitz's Modern Portfolio Theory,

making asset allocation theory more practical. See Sharpe, Alexander and Bailey (1999), Lintner (1965) and Mossin (1966).

# Fama

Eugene Fama concluded that stock prices were unpredictable and coined the phrase 'market efficiency.' Although there are various forms of market efficiency, in a nutshell the idea is that stock market prices reflect all publicly available information, and that no person can gain an edge over another by fair means. See Fama (1966).

# s Sobol', Faure, Hammersley, Haselgrove and Halton...

Many people were associated with the definition and development of quasi random number theory or low-discrepancy sequence theory. The subject concerns the distribution of points in an arbitrary number of dimensions in order to cover the space as efficiently as possible, with as few points as possible (see Figure 1.1). The methodology is used in the evaluation of multiple integrals among other things. These ideas would find a use in finance almost three decades later. See Sobol' (1967), Faure (1969), Hammersley & Handscomb (1964), Haselgrove (1961) and Halton (1960).

# Thorp

Ed Thorp's first claim to fame was that he figured out how to win at casino Blackjack, ideas that were put into practice by Thorp himself and written about in his best-selling *Beat the Dealer,* the ''book that made Las Vegas change its rules.'' His second claim to fame is that he invented and built, with Claude Shannon, the information theorist, the world's first wearable computer. His third claim to fame is that he used the 'correct' formula for pricing options, formula that were rediscovered and originally published several years later by the next three people on our list. Thorp used these formula to make a fortune for himself and his clients in the first ever quantitative finance-based hedge fund. He proposed dynamic hedging as a way of

Figure 1.1: **They may not look like it, but these dots are distributed deterministically so as to have very useful properties.**

removing more risk than static hedging. See Thorp (2002) for the story behind the discovery of the Black-Scholes formulae.

# Black, Scholes and Merton

Fischer Black, Myron Scholes and Robert Merton derived the Black-Scholes equation for options in the early seventies, publishing it in two separate papers in 1973 (Black & Scholes, 1973, and Merton, 1973). The date corresponded almost exactly with the trading of call options on the Chicago Board Options Exchange. Scholes and Merton won the Nobel Prize for Economics in 1997. Black had died in 1995.

The Black-Scholes model is based on geometric Brownian motion for the asset price *S*

The Black-Scholes partial differential equation for the value *V* of an option is then