The Money Formula. Wilmott Paul

and its current movement is a function not only of earlier fluctuations, but also of the present market position. (Chased its own tail)

      The determination of these fluctuations is subject to an infinite number of factors: it is therefore impossible to expect a mathematically exact forecast. Contradictory opinions in regard to these fluctuations are so divided that at the same instant buyers believe the market is rising and sellers that it is falling. (Sounds like we're wasting our time, people)

      Undoubtedly, the Theory of Probability will never be applicable to the movements of quoted prices and the dynamics of the Stock Exchange will never be an exact science. (Thought this was a science exam?)

      However, it is possible to study mathematically the static state of the market at a given instant, that is to say, to establish the probability law for the price fluctuations that the market admits at this instant. Indeed, while the market does not foresee fluctuations, it considers which of them are more or less probable, and this probability can be evaluated mathematically. (Too much on finance! – this was a real comment on Bachelier's thesis by France's leading probability theorist, Paul Lévy)

      Bachelier's starting assumption, which he called his “Principle of Mathematical Expectation,” was that the mathematical expectation of a speculator is zero. As in the random walks of Figure 2.1, some bets will win, and others will lose, but these cancel out in the long run. Note that we are referring here to the mathematical chances of success – a speculator's psychological expectations may be very different. He then assumed that prices move in a random walk, with price changes following a normal distribution, and referred to what he called the “Law of Radiation (or Diffusion) of Probability,” which described how the future price became more uncertain as you went further into the future. The results are very similar to Figure 2.3 (the displacements at each iteration were there set to plus or minus a fixed amount, in this case 1, rather than being normally distributed, but the effects are almost identical over large enough times). From this, he derived a method for pricing options, which grant the purchaser the right to buy or sell an asset at a fixed price at some time in the future. As discussed further later, the technique he developed is essentially a special case of the ones commonly used today.

      1

      We're translating from the Spanish. We think that “¡Vaya mierda!” is slang for “Have a great day!” but we're not sure.

      2

      This was estimated by the economist Tim Harford and Paul for the BBC Radio 4 program More or Less based on data from the website of the Bank for International Settlements. This “headline” figure, which is open to interpretation, includes both the contracts traded through an exchange and the over-the-counter market in which two parties trade directly. It is also what is called the “notional” value. If a contract specifies that it will pay you 1 % of

1

We're translating from the Spanish. We think that “¡Vaya mierda!” is slang for “Have a great day!” but we're not sure.

2

This was estimated by the economist Tim Harford and Paul for the BBC Radio 4 program More or Less based on data from the website of the Bank for International Settlements. This “headline” figure, which is open to interpretation, includes both the contracts traded through an exchange and the over-the-counter market in which two parties trade directly. It is also what is called the “notional” value. If a contract specifies that it will pay you 1 % of $1 million in a year's time then that would be recorded as a notional of $1 million, whereas it's really justworth about $10,000. So it's tricky to say what amount really is at risk in that $1.2quadrillion.

3

By accident. He set the exchange rate for silver too high, so silver coins left the country.

4

Flynn (1941).

5

Manuel (1974).

6

Keynes (1946).

7

Muir (2003).

8

Patterson (2009, p. 8).

9

Gitlin (2014).

10

Smith (1776).

11

Samuelson (1973).

12

Kennedy (2005).

13

Alexander Carlyle, quoted in Özler (2012).

14

Hamilton (1858, p. 77).

15

Özler (2012).

16

Fleischacker (2002), Kiladze (2015).

17

Greene (1961, p. 88).

18

Edgeworth (1881, p. 16).

19

Jevons (1957).

20

Quetelet (1842).

21

Quoted in Bernstein (1998, p. 160).

22

We find this a bit disturbing. But not as disturbing as Alan Greenspan's extreme fondness for Ayn Rand. As he wrote in The Age of Turbulence, “Ayn Rand became a stabilizing force in my life… I was intellectually limited until I met her” (Greenspan, 2007).

23

Bockman (2013, p. 47).

24

Haldane (2014).

25

Para (1995).

26

E.g., Cliff Asness (co-founder of AQR Capital Management) (Patterson, 2009, p. 265).

27

Since displacements to the right are positive and displacements to the left are negative, the average is always zero, so it is more convenient to use the root mean square (RMS) – defined as the square root, of the average, of the squares. Measured this way, the deviation of the hypothetical stock from its starting point grows with the square root of time.

28

The standard deviation is just the RMS again, see note above, but with all distances measured from the mean rather than from zero.

29

Galton (1889).

30

Quoted in Bernstein (1998, p. 200).

31

Bachelier (1900).