Modern risk management depends absolutely on modern mathematics. 7 In its absence, the ancients were as handicapped as mariners without a compass. The introduction of modern arithmetic allowed thought pioneers such as Cardano and later Galileo to develop the theory of combinations—essential for figuring the probability of outcomes when rolling dice or drawing cards. Here the modern concept of probability was born.

Though useful at the gaming table, probability theory was still not very useful for real world events. The theory of statistics was needed for that, and the first statistician might have been a Londoner, John Graunt, who studied the age distribution and the causes of death from bills of mortality in London parishes.

The primitive database (i.e., the church records) from which Graunt derived his study was itself an innovation and was less than 60 years old at the time of Graunt’s study, published in 1662. Databases would go on to become a powerful source of wealth and value, but that is another story. The astronomer and mathematician Edmund Halley extended Graunt’s work into an analysis of life expectancies, creating in 1693 a scientific basis for the valuation of annuities. Still, it would take nearly another century for a modern life insurance business based on actuarial data to evolve.

From a mathematical viewpoint, the remaining big step was the discovery of the bell-shape curve. Abraham De Moivre in the 1730s, using the binomial theorem,8 developed the concepts of the normal distribution and the standard deviation, the latter being a measure of the dispersion of the distribution about the mean. In Chapter 5, we relied on his formula, coded as NORMSDIST in the Excel spreadsheet program, to calculate Black-Scholes option values! And how did we calculate volatility? We used De Moivre’s formula for the standard deviation.