The History of Financial Mathematics

First version 29 October 2026, last changed 31 October 2026 · Andreas Stockter

Financial mathematics is the mathematics of money across time and under uncertainty. It asks: What is a payment worth that only arrives in ten years? What is it worth if it is not certain that it arrives? And what does the right cost to receive it without having to take it? From these three questions there arose, over four thousand years, the methods of calculation with which merchants work out interest, insurers set their prices, savers spread their money, banks measure their possible losses and, most recently, computer programs buy and sell on their own. This text tells the story along the tools: from a clay tablet that works out when a borrowed sum doubles, through a list of deaths in Breslau and the doctoral thesis of a Paris mathematician in 1900, to a formula of 1973, a number with which banks have measured their risk since 1994, and the programs for which the Bank of England, which issues Britain’s money, is considering a kill switch in 2026. It is also the story of a recurring error: every tool that made a risk calculable tempted people to take on more of it, until the market showed what the calculation did not contain.

Like the history of AI, the history of the mathematics behind AI, the history of software development, the history of the Internet, the history of open source, the history of venture capital, the history of vibe coding, the history of e-commerce and the history of project management, this text follows a chain: each epoch solves a bottleneck the previous one left open and in doing so creates the next. First a calculation for the price of time was missing, then one for payments that depend on chance, then a description of the chance with which prices move in the market, then an answer to the question of which risk earns which reward, then a price for the right to buy later at a fixed price, then a measure of what a bank can lose, then the ability to learn from more data than a person can read, and finally the answer to the question of what happens when the methods of calculation move the prices they describe. Figure 11 at the end shows the chain as a picture.

How to read this text. Only the four basic operations of arithmetic are assumed: adding, subtracting, multiplying and dividing. Everything else, from the per cent to probability, is explained at the point where it is first needed. Every technical term is set in bold where it is explained and is used only after that; a glossary lists all the terms alphabetically. To make the terms tangible, short sections “Today” at the end of many subsections show with everyday examples where they occur today, in saving, a loan or an insurance policy; readers who only want the story can skip them. The numbers of these examples are invented and chosen to be easy to recalculate. Green boxes “Step by step” work through an example from the text one step at a time. Blue boxes “Working it out” at the end of the sections sum up the section’s rules of calculation as formulas, short rules of calculation in which letters stand for numbers; the first of them explains how to read them. They also contain derivations, worked examples and exercises whose solutions can be unfolded. Below the blue boxes are violet calculators: unfold one, change the numbers and see the result and the steps at once; they mostly start with the numbers of a worked example, not those of the exercises. Some figures can be operated with buttons and sliders, and paragraphs “Try it yourself” show how to reproduce a result with a spreadsheet or real data. Readers who only want the story can skip the blue and violet boxes. The formula sheet at the end collects all formulas with consistent symbols.

Contents: 1 Interest and time · 2 Probability and annuity · 3 Chance on the exchange · 4 Portfolio and price of risk · 5 The formula · 6 Risk, credit, crisis · 7 Factors, machines, data · 8 Language models and agents · Chain · Mixed practice · Timeline · Formula sheet · Further reading · Sources

1. Interest and time: clay tablets, usury and the book of calculation (to 1600)

The first interest calculations: Babylon

The problem. Whoever borrows seed in autumn and returns it in summer returns more than he received, because the lender has gone without his grain for so long and because the harvest can fail. Such borrowing for a time is a loan or credit; whoever borrows is the debtor, whoever lends the creditor. Interest is the price of the time that passes between borrowing and returning, and it is as old as writing. The oldest calculations we know are debt records from Mesopotamia, and the law of Hammurabi from the eighteenth century BC knew fixed rates: one fifth a year for silver, one third for barley, in line with the rates customary in the contracts of the time.¹ One fifth interest means: whoever borrows 5 units of silver returns 6 after a year, the 5 and one fifth of them besides, namely 1.

The per cent. One fifth is 1 divided by 5, as a decimal 0.2; the places after the decimal point count tenths, hundredths and thousandths, so 0.2 is two tenths and 0.05 five hundredths. Today interest is mostly given in per cent, literally “of a hundred”: 1 per cent, written 1 %, is one hundredth, that is 0.01, and one fifth is 20 per cent, because 20 hundredths are 0.2. A percentage of an amount is worked out by multiplying: 20 per cent of 5 is 0.2 · 5 = 1, 5 per cent of 300 is 0.05 · 300 = 15. More general than interest is the return of an investment, that is, of money put to work so that it becomes more: the gain as a percentage of the stake. Whoever stakes 100 and gets back 105 after a year has gained 5 of 100, a return of 5 per cent, and interest is the return of a loan.

The tablet. If a debt stands for more than a year, there are two ways of reckoning the interest. With simple interest one pays each year only on the amount borrowed: 20 per cent on 5 is 1 each year, and after three years one owes 5 + 3 = 8. With compound interest the interest is added to the debt at the end of each stretch of time, each period, and earns interest itself in the next period: 5 becomes 6 after one year, 6 plus one fifth of 6, that is 7.2, after two years, and 8.64 after three. Each year the debt is thus multiplied by the same number, here by 1 + 0.2 = 1.2; this number is called the interest factor. That interest accrues on interest was familiar to the scribes of Babylon, and they could calculate with it: an Old Babylonian tablet in the Louvre, AO 6770, asks how long a sum takes to double at one fifth interest a year. After three years it has been multiplied by 1.2 three times: 1.2 · 1.2 · 1.2 = 1.728. Such repeated multiplication by the same number is a power, written 1.2³ and read “1.2 to the power of 3”; 1.2 is called the base, 3 the exponent. The scribe knows the powers of 1.2 and sees that the answer lies between three and four years: after three years 1 has become only 1.728, after four already 2.0736. For the time in between he uses an approximation. He knows the sum at only two points, after three years and after four, and acts as if it grew evenly in between from 1.728 to 2.0736, by the same share of the increase in every month. After half a year, half of the increase would then have been added, after a quarter of a year a quarter. People estimate like this in everyday life too: someone who is at kilometre 100 at 3 p.m. and at kilometre 200 at 4 p.m. was probably at about kilometre 150 at half past three. If one plots the two known values as points in a picture, time to the right and the sum upwards, this assumption joins the points with a straight line, and one reads off the line when it reaches the height 2. Estimating a value between two known values along a straight line in this way is now called linear interpolation. His answer is four years less a good two and a half months, about 3.79 years; the box below this paragraph shows the calculation.² Compound interest is thus the oldest exponential growth people have calculated with, long before anyone had the word for it: a quantity grows exponentially if in equal times it is always multiplied by the same number.

The inverse. The tablet’s question turns compound interest around. The power calculates forwards what becomes of a sum: 1.2³ = 1.728. The logarithm calculates backwards and gives the exponent to which the base must be raised, that is, how often it must be multiplied by itself, to obtain a desired multiple: the logarithm of 1.728 to base 1.2 is 3, and the logarithm of 2 to base 1.2 is 3.8, for 1.2 to the power of 3.8 gives 2. That is the exact answer to the tablet’s question. An exponent with a decimal point fills the gap between the whole years: 1.2 to the power of 0.5, for instance, is the number that multiplied by itself gives 1.2, the square root of 1.2, written √1.2 = 1.0954, and 1.2 to the power of 3.8 is 1.2³ times 1.2 to the power of 0.8. The most useful property of the logarithm is that it turns multiplication into addition: the interest factors of successive years are multiplied, their logarithms added, and every doubling is for it a step of the same size, whether from 1 to 2 or from 1,000 to 2,000. If numbers are marked on a line so that every tenfold increase gets the same distance, on a logarithmic scale, 1, 10, 100 and 1,000 lie at equal distances, and a sum growing at a fixed rate climbs on it in equal steps, by the same distance each year (Figure 9 in section 7 shows such a scale). Tables that give the exponent for the powers of a number were already known to Babylonian scribes, for instance on the tablet MLC 2078; but they only help if the desired multiple is itself one of the powers, and 2 is not a power of 1.2. Tables with which one could calculate like this in general were first published by John Napier in Edinburgh in 1614, more than three thousand years after the tablet in the Louvre, to spare calculators, as he wrote in the preface, the troublesome multiplications and divisions of great numbers; his tables still served angle values for astronomy and navigation, and logarithms of the numbers themselves were printed by Henry Briggs in 1617.³

The calculator below this paragraph goes through the steps of tablet AO 6770 and, for comparison, computes with the logarithm as in the box “Step by step: the logarithm of 2 to base 1.2”. It is preset with the example of the tablet; with other rates and multiples it shows how close linear interpolation comes to the exact solution.

Today

If someone lends a friend 500 euros for a second-hand bicycle, the friend is the debtor and the lender the creditor. If 520 euros come back after a year, the 20 euros are the interest. With an instalment loan for a car the bank is the creditor.

The 20 euros of interest on 500 euros in the bicycle example are 20 / 500 = 0.04, a return of 4 per cent.

Whoever puts 10,000 euros into a five-year fixed deposit at 3 per cent and has the interest paid into a current account each year gets simple interest, 300 euros each year, 1,500 euros in all. If the interest stays in the deposit and earns interest itself, the money grows to 11,592.74 euros, 92.74 euros more; the period here is a year, with monthly crediting it would be a month.

For this fixed deposit the interest factor is 1.03: 10,000 euros become 10,300 euros after one year and 10,300 · 1.03 = 10,609 euros after two.

Debts grow exponentially too: if 2,000 euros on a credit card at 15 per cent interest stay unpaid for five years, they become 2,000 · 1.15⁵ = 4,022.71 euros, more than double.

The ban on interest and Fibonacci’s book of calculation

The prohibition. The Middle Ages knew interest and forbade it. Usury at first meant any interest, not only the excessive kind, and the Third Lateran Council, an assembly of the church’s bishops in Rome, excluded manifest usurers in 1179 from communion, the Lord’s Supper at mass, and from Christian burial; theology justified this with Aristotle, according to whom money is barren, and with the idea that time belongs to God.⁴ The merchants found ways around it. The bill of exchange, a written promise of payment that was redeemed in another city and in another currency, hid the interest in the exchange rate, the price at which one currency is swapped for the other. The silent partnership turned the creditor into a partner who received a share in the profit of a business instead of interest. And in the purchase of an annuity one did not lend money but paid a sum for the right to regular payments; a series of payments at different times is called a stream of payments, a series of equal payments at fixed intervals an annuity. In the discipline any series of equal payments is an annuity. The Fifth Lateran Council in 1515 allowed the montes pietatis, the church pawnshops, which lent money against a pledge, a fee for their costs, and on 7 November 1545 the Geneva reformer John Calvin wrote in a letter that only the “biting” loan to the needy was forbidden, not interest from the merchant who makes a profit with the money.⁵ With this distinction interest was released for the modern age, and the question was no longer whether one might calculate it but how.

The book of calculation. How, a merchant’s son from Pisa had written down three hundred years earlier. Leonardo Fibonacci had learned the Indo-Arabic numerals in North Africa, the digits 0 to 9 with which we write today and which make written calculation much easier than Roman numerals. In 1202 he published the Liber Abaci, a book of calculation for traders in which, beside the famous problem of how fast rabbits multiply, interest problems run for pages: what a loan owes after so many years, what a series of payments is worth today, which of two contracts is better. The economic historian William Goetzmann reads the Liber Abaci as evidence that Fibonacci was the first to work out what is today called present value, the value today of future payments: there he compares two streams of payments by reckoning both back to the same point in time.⁶ Reckoning back means reversing compound interest: instead of multiplying by the interest factor, one divides by it. 120 euros due in a year are worth 120 / 1.2 = 100 euros today at 20 per cent interest, because 100 euros would grow to 120 in a year; 144 euros due in two years are worth 144 / 1.2 / 1.2 = 100 euros. This discounting is the basic operation of the discipline because it makes payments comparable that fall due at different times: only when all of them are reckoned back to the same day can one add them up and weigh them against each other. Every bond, a debt certificate that can be sold on to others, with fixed interest payments and repayment at the end of its term, every company, every factory is valued to this day by discounting its future payments to today and adding them up. That discounting stands in a merchant’s book and not in a learned treatise says where the discipline comes from.

The occasion. The calculation was needed because trade in the Mediterranean was flourishing around 1200. Money sent over long distances arrived only after the journey, and bills of exchange were therefore traded at a discount that made up for the interest over the waiting time. Cities reckoned with time too: in the thirteenth century Pisa raised money by selling rights to future tax and monopoly revenues for a sum paid at once, and whether such an exchange pays is a question of present value. Private people reckoned the same way: in Pistoia, also in Tuscany, perpetual rents in wheat were sold as early as 1201 to 1210, the right to receive one staio of wheat every year for all time, the staio being an old measure of grain. Such a right cost 2.80 lire at the median, while a staio of wheat could be had for ¼ lira. The buyer thus paid 2.80 / 0.25 = 11.2 times one year’s delivery and received each year 0.25 / 2.80 = 8.9 per cent of his outlay in wheat. Goetzmann asks how buyers and sellers set the price without a method of discounting; section 2 shows with the perpetuity that its present value is the annual payment divided by the rate, here 0.25 / 0.0893 = 2.80.⁷ Goetzmann reads Fibonacci’s interest problems as a response to this commercial revolution. The most demanding of them concerns a soldier to whom the king pays 300 bezants a year for his fief, in four instalments of 75 at the end of each quarter; bezants was the name used in the West for gold coins from the East, after Byzantium. Then the king changes the contract and pays the 300 all at once at the end of the year. The soldier can invest his money so that 100 bezants earn 2 more each month. How much is the payment still worth to him after the change? Fibonacci reckons every payment back to the start of the year and so shows that two contracts for the same sum are worth different amounts solely because the money arrives at different times; the box below this paragraph shows the calculation.⁶

Today

Hidden interest exists to this day, for instance in the cash discount for prompt payment: if a tradesman may pay an invoice of 1,000 euros after 10 days with 2 per cent off or in full after 30 days, the 20 days’ delay costs him 20 euros on 980, about 45 per cent converted to a year with compound interest.

Annuities in this sense include the monthly instalment on a car loan, a savings instalment or a rent. A stream of payments is, for example, a mobile phone contract with a 99-euro down payment followed by 24 monthly instalments of 30 euros, 819 euros in all.

An employer offers a bonus of 5,000 euros today or 5,300 euros in two years. At 3 per cent interest the 5,300 euros are worth 5,300 / 1.03 / 1.03 = 4,995.76 euros today, slightly less than 5,000: it is better to take the money now.

Discounting also says how much one must put aside today: if one needs 10,000 euros for a car in five years, 10,000 / 1.03⁵ = 8,626.09 euros today are enough at 3 per cent.

An example of a bond: whoever lends a city 1,000 euros for five years receives 30 euros of interest each year and the 1,000 euros back at the end of the term; he can sell his debt certificate to someone else before then.

Annuities, murabaha and double-entry bookkeeping

The two forms of annuity. An annuity of equal instalments can be paid at the end or at the beginning of each period. If each instalment comes at the end, like the interest on a loan, which falls due only when the year is over, the annuity is paid in arrears; if it comes at the beginning, like rent transferred in advance, it is an annuity due. Two questions can be put to any annuity. What is it worth today? That is its present value, the sum of all discounted instalments. And what is it worth at the end of its term if each instalment goes on growing with compound interest until then, as in a savings plan? That is its future value. An instalment of an annuity due comes one period earlier than one in arrears and therefore earns interest for one period longer: present value and future value of an annuity due are those of the annuity in arrears multiplied by the interest factor. The box below this paragraph works through both forms.

The other prohibition. Christianity relaxed its prohibition of interest over the centuries; Islam keeps its own to this day. The Quran says in sura 2, verse 275, that God has permitted trade and forbidden riba, literally the increase, which the prevailing interpretation takes to mean any interest fixed in advance on a loan.⁸ Banks that work without interest arose only in the 20th century. The first is considered to be the savings bank that the economist Ahmad al-Najjar founded in 1963 in the Egyptian village of Mit Ghamr on the model of West German savings banks: it paid no interest but gave the savers a share of the profit, and it ceased operating in 1967. The Dubai Islamic Bank followed in 1975, by its own account the first Islamic commercial bank.⁹ One of the most widely used instruments of these banks is murabaha, a sale with a disclosed mark-up: the bank buys the goods the customer needs and sells them to him at a higher price, which he pays later or in instalments. The mark-up is often measured against the interest that ordinary banks charge; the Pakistani scholar Muhammad Taqi Usmani called this not desirable but permissible, because the contract remains a sale (box below).¹⁰ States and companies borrow larger sums through sukuk, certificates that represent not a claim but a share in an asset, such as a building whose rent carries the payments. When Usmani, chairman of the Shariah board of AAOIFI, the organisation that sets standards for Islamic finance, said in November 2007 that about 85 per cent of sukuk were not compatible with Islamic law, because the issuer promises to buy them back at the end for the original amount and so turns them into disguised bonds, AAOIFI resolved on 13 and 14 February 2008 that sukuk must represent real ownership of assets.¹¹ In 2025, Islamic banks, insurers and funds, common pools in which the money of many investors is invested together, managed about 4.4 trillion dollars according to the Islamic Financial Services Board, the association of their supervisors.¹²

The books. Calculating presupposes that one knows what one has. Luca Pacioli, Franciscan and teacher of mathematics, printed the Summa de arithmetica in Venice in 1494, a textbook of the whole art of reckoning. A section of it, “Particularis de computis et scripturis”, describes, as the first printed book, double-entry bookkeeping; the merchant Benedetto Cotrugli had explained it as early as 1458 in a manuscript that was printed only in 1573.¹³ An account is a list on which all additions to and withdrawals from one thing are written down, for instance the cash box, the goods or the debts of a customer. In double-entry bookkeeping every transaction is written down twice: if a trader sells cloth for 10 ducats in cash, the number stands once on the account of the cash box as an addition and once on the account of the goods as a withdrawal. For this every account has two columns, on the left the debit and on the right the credit. On the accounts for what one owns, such as cash and goods, an addition stands on the debit side and a withdrawal on the credit side. The sale thus appears twice with 10 ducats: on the debit side of the cash account and on the credit side of the goods account. Because every number appears twice, once as a debit and once as a credit, the totals of all debit entries and all credit entries must be equal, and many slips of the pen and errors of calculation become visible, though not all: a transaction left out entirely, for instance, changes both totals alike. Pacioli called it the method of Venice and made no claim to have invented it; the merchants had used it since the end of the thirteenth century. For financial mathematics it is the precondition: only when assets, debts and profit have a number at every moment can one ask how the number behaves over time.

Today

A salary that arrives at the end of the month is paid in arrears; car insurance paid in January for the whole year is paid in advance. If one saves 100 euros at the end of every month for ten years at 3 per cent a year, that is 0.25 per cent a month, the future value is 13,974.14 euros, against 12,000 euros paid in.

A household account book can be kept by double-entry bookkeeping too: if one buys a bicycle for 500 euros from a current account, the amount stands once as a withdrawal from the current account and once as an addition to the bicycle; wealth has not changed, only its form.

Tables and the chart

The tables. Calculating with compound interest was laborious as long as every power had to be formed by hand, and the great trading houses kept their tables secret. Jean Trenchant printed a book of calculation with interest problems in 1558, and in 1582 Simon Stevin of Bruges, bookkeeper and later tutor of Prince Maurice of Orange, published with the printer Plantin in Antwerp the Tafelen van Interest: the first printed, systematic account of calculation with simple and with compound interest, with ready-made tables in which anyone could look up what a guilder due after so many years is worth today at so many per cent.¹⁴ An interest table turns a calculation into a glance: with it a mere clerk in a merchant’s office could value what had previously needed a master of reckoning, a man who lived by calculating and by teaching calculation. Stevin, who introduced decimal fractions in the same decade, the numbers with a decimal point we calculate with today, thereby opened the discipline to everyone, and in the Netherlands further books with interest tables soon followed.

The chart. Figure 1 shows the difference between simple and compound interest as a chart. Such a picture is read like this: the years run to the right, the value of the sum upwards; each line connects the values a sum has year by year, and the more steeply it rises, the faster the sum grows. With simple interest the same amount is added every year, and the line is straight. With compound interest more is added each year than the year before, and the line bends ever more steeply upwards.

Working it out and practice

Try it yourself. The Babylonian problem can be recalculated in any spreadsheet, a calculating program such as Excel or LibreOffice Calc. It consists of cells named by column and row: A1 is the cell in column A, row 1. A formula in a cell begins with an equals sign; * means times, / divided by, ^ to the power of, and the name of a cell stands for the number in it. One writes the years 0 to 5 in cells A1 to A6 and the formula =100*1.2^A1 in B1, which one copies down to B6; then one sees that 100 has become 172.8 after three years and 207.4 after four. The doubling lies between, at the logarithm of 2 to base 1.2, which the formula =LOG(2,1.2) works out: 3.80 years. The merchants’ rule of thumb, which Pacioli already gives without justification in 1494,¹⁵ dividing the number 72 by the interest rate in per cent, gives 72 / 20 = 3.6 years and is more accurate for rates between five and ten per cent. Whoever calculates the logarithm for two per cent, =LOG(2,1.02), gets 35 years, for seven per cent ten, and has thereby understood why a pension plan that begins at 25 is, at seven per cent, worth twice as much as the same one that begins at 35; the box below this paragraph works it out.

The bottleneck

Around 1600 Europe could value payments that come for certain: a loan with a fixed day of repayment, an annuity for a fixed number of years. But the most important payments were not certain. Whether a ship returns from India, whether a debtor pays, how long a pensioner lives to whom the town has sold an annuity until the end of his life, on that nobody had a number; one guessed, one added surcharges, one lost. Towns sold such annuities to children and to old men at the same price, because there was no method of calculating the difference. The bottleneck of the next epoch was uncertainty, and the tool for it came from a direction that had little to do with money: the gaming table.

2. Probability, the life table and the annuity (1654–1800)

Gambling and expected value

The game. The mathematics of the uncertain began with dice. Gerolamo Cardano, physician, astrologer and gambler, wrote a book on games of chance around 1564 that was only printed in 1663 and for the first time divided the number of favourable cases by the number of all cases.¹⁶ That is to this day the simplest route to probability, the number between 0 and 1 that states how often an event, say “the die shows a six”, occurs in the long run: a die has six equally possible faces, the six is one of them, so its probability is 1 divided by 6. One writes this as a fraction, 1/6, which is only another way of writing the division, as a decimal 0.167, or as a percentage 16.7 per cent. A probability of 0 means impossible, one of 1 certain, one of ½ even odds. Two rules suffice for almost everything that follows. The probability that an event does not occur is 1 minus the probability that it occurs: no six has 1 − 1/6 = 5/6. And if two events do not depend on each other, like two tosses of the same coin, the probability that both occur is the product of their probabilities: heads twice has ½ · ½ = ¼, heads three times ⅛.

The division. The impetus for the theory came in 1654 from a correspondence: the Chevalier de Méré had asked Blaise Pascal how an interrupted game should be divided fairly, and Pascal and Pierre de Fermat solved the problem of points in letters of the summer and autumn of 1654 by asking not who was ahead but how probable each outcome of the remaining game was.¹⁷ Pascal worked it through on 29 July 1654 with an example that the box below this paragraph shows step by step: two players each stake 32 pistoles, gold coins of the time, and whoever first wins three games receives all 64; the game breaks off at 2:1, and the fair division is 48 to 16.¹⁷

The expected value. Christiaan Huygens summarised the method in 1657 in the first printed treatise of the discipline, De ratiociniis in ludo aleae, and introduced in it the concept that connects the game with money: the expected value of an uncertain gain is the sum of all possible gains, each first multiplied by its probability.¹⁸ In Pascal’s example that is ½ · 64 + ½ · 32 = 48: whoever continued the game from this position very often would receive 48 on average. What an uncertain claim is worth thereby had a number for the first time.

Today

If a mobile phone breaks within a year with probability 0.1, in the long run one in ten such phones is hit each year.

Every insurer calculates with the expected value: if repairing a mobile phone costs 300 euros and it breaks with probability 0.1 a year, the expected loss is 0.1 · 300 + 0.9 · 0 = 30 euros a year. If a phone insurance asks 60 euros, the customer pays on average 30 euros so that a loss does not catch them unprepared.

Life annuities and government bonds

The state and the annuity. The first application concerned the debts of the state. The Netherlands financed their wars with annuities for life: the buyer paid the state a sum and in return received an amount every year until his death; such a life annuity is an annuity whose length depends on the life of a person. Johan de Witt, as Grand Pensionary the leading statesman of Holland and at the same time a mathematician, demonstrated in 1671 in the Waerdye van Lyf-renten that the Republic sold the annuities too cheaply: from assumed probabilities of dying at a given age and from the interest rate he obtained, for a young and healthy life, a value of a little over 16 years’ payments, while they were sold for 14. His calculation suggested a price depending on age; such prices were introduced only in 1672 in Amsterdam under the burgomaster Jan Hudde. The treatise was hardly read, and de Witt was killed by a mob in 1672.¹⁹ There had already been an estimate: in 1662 the London draper John Graunt had made a survival table from the city’s weekly bills of mortality, according to which of 100 born 64 were alive at 6, 40 at 16, 16 at 36 and one at 76; because the bills gave no age at death, these were estimates, but the Royal Society, London’s society of natural philosophers, admitted the tradesman.²⁰ The numbers he lacked were supplied in 1693 by Edmond Halley, the astronomer, from an improbable source: the pastor Caspar Neumann had recorded births and deaths by age in Breslau for five years and sent the lists to the scholar Leibniz, the Royal Society requested them, and Halley calculated from them the first life table based on observed ages at death, a table that states for every age how many of a starting number are still alive: of a thousand one-year-olds, 855 live to the second birthday, 798 to the third, and so on into old age; Figure 2 shows the whole table as a curve. Dividing by a thousand gives probabilities: a one-year-old lives to his second birthday with probability 855 / 1,000 = 0.855. In the same paper Halley showed how to calculate with it the fair price of a life annuity for every age: one discounts each future payment, multiplies it by the probability of living to see it, and adds everything up.²¹ Present value and expected value were united in one calculation, and it stands in this form in every textbook of actuarial mathematics to this day, the mathematics with which insurers calculate their prices and reserves.

The calculation. Halley calculated at six per cent interest. How his calculation works is shown by the box below this paragraph for an annuity over just two years. Continuing the sum to the end of the table gives the life annuity: by Halley’s own table it is worth 13.44 years’ payments for a ten-year-old, 12.78 for a twenty-year-old and 7.60 for a sixty-year-old; recalculated from his life table it is 7.46, so at high ages his own table deviates a little (the “life annuity” calculator at the end of the section works through every age). The fund lately granted to the English king and queen, a pool through which the state raised money, sold life annuities at 14 per cent a year: whoever paid 100 received 14 every year until his death, and since 100 / 14 is about 7, a life cost seven years’ payments. Halley noted that young lives at the usual rate of interest were worth above thirteen: whoever nominated a young life bought for seven what was worth over thirteen.²¹

The government bond. Life annuities were not the only way in which states borrowed. A government bond is a bond issued by a state. Venice had borrowed from its citizens, often under compulsion and in proportion to each one’s wealth, and in 1262 combined these debts in one fund, the Monte. It assigned to them particular taxes from which 5 per cent interest a year was paid, and allowed the claims to be sold on to others; repayments were still made, but the creditors no longer counted on getting their money back.²² In England the Bank of England was founded in 1694, and its shareholders lent the state, as their first act, 1.2 million pounds at 8 per cent.²³ In 1749 the prime minister Henry Pelham lowered the interest on the state’s debts, and an act of 1751 combined several 3 per cent bonds into a single bond, the consols, short for consolidated annuities, issued from 1752. It paid 3 per cent and had no repayment date: the state was allowed to buy it back but did not have to. Such a bond is a perpetuity, an annuity without end, and its present value is simple. Whoever invests 100 pounds at 3 per cent receives 3 pounds every year and keeps the 100 for ever; 3 pounds a year for ever are therefore worth 100 pounds at 3 per cent interest, the instalment divided by the rate. The government repaid the last British bonds without a repayment date on 5 July 2015.²⁴ A bond with a term has a face value, the amount repaid at the end and to which the interest refers, and a coupon, the interest per year as a percentage of the face value; a sheet of such coupons, interest slips, was once attached to every bond, and one detached them and redeemed them for the interest.²⁵ The price of a bond on the market is the present value of its payments, calculated at the rate one would get elsewhere today for the same term. If this market rate is above the coupon, the bond is worth less than its face value; if below, more. Asked the other way round: which rate makes the present value of all payments equal to the price one pays? This rate is called the yield to maturity, and it is what is meant when a government bond is said to yield three per cent.

Today

If someone aged 65 pays 100,000 euros for a lifelong annuity of 5,000 euros a year, without interest he has got back what he paid only after 20 years, at 3 per cent interest only after 31; whoever dies earlier has lost, whoever lives longer has won.

Whoever wants 500 euros a month from interest alone, that is 6,000 euros a year, without touching the savings needs 6,000 / 0.03 = 200,000 euros at 3 per cent, at 4 per cent only 150,000 euros.

A bond with a face value of 1,000 euros and a 3 per cent coupon pays 30 euros every year, whatever price it is currently traded at, and 1,000 euros back at the end.

If one buys a bond of 1,000 euros with a 2 per cent coupon and two years left to run for 950 euros, one receives 20 euros of interest twice and at the end 50 euros more than one paid; the yield to maturity is 4.68 per cent, more than the coupon.

The number e, large numbers and dispersion

The number e. Between game and annuity there arose on the side one of the numbers of mathematics so important that they have been given a letter of their own. Jacob Bernoulli asked in 1683 what becomes of a debt if the interest is credited, that is added to the debt so that it earns interest itself, not once a year but ever more often. Worked through, as is usual today, with a guilder at a hundred per cent a year and interest credited once, twice, twelve times or daily, it becomes 2 guilders, 2.25, 2.61, 2.71, and Bernoulli showed that these numbers come ever closer to a fixed bound without reaching it. Such a bound is called a limit; here it lies between 2.5 and 3. It is the number Euler later called e, 2.71828…, and it came not from geometry but from compound interest.²⁶ The box below this paragraph shows the calculation behind it. The more often interest is credited, the more comes out, but the gain shrinks: going from one credit to two brings a quarter of a guilder, going from twelve to 365 only a tenth, and even whoever credits interest every second does not get beyond 2.72 guilders; why the limit lies between 2.5 and 3 is explained in the blue box at the end of the section. As early as 1737 Euler had shown that e is irrational: its digits after the decimal point never end and never repeat; in 1748 he calculated the value to 23 places.²⁷ The limit is the yield of continuous compounding, in which interest is credited at every instant, infinitely often and in infinitely small parts. In this form the discipline calculates to this day: a capital K₀ grows at an interest rate r in t years to K₀ · eʳᵗ, that is to K₀ times e to the power of r times t; for 100 euros, 5 per cent and 10 years that is 100 · e to the power of 0.5 = 164.87 euros, with yearly crediting 162.89; the second box below this paragraph works out both. The logarithm to base e is called the natural logarithm, in spreadsheets LN; it gives the time a sum needs under continuous compounding at a hundred per cent to grow to a multiple, for doubling ln 2 = 0.69 years.

Large numbers. Bernoulli’s main work, the Ars Conjectandi, appeared in 1713 after his death and proved the law of large numbers: the more often one repeats a random experiment, a trial whose outcome depends on chance, such as a throw of a die, the closer the share with which an event is observed comes to its probability.²⁸ Whoever throws a die ten times may see three sixes or none; whoever throws it six thousand times almost certainly sees about a thousand. That is the theorem insurers live by: the death of the individual is incalculable, the mortality of ten thousand is an almost certain number.

The dispersion. How certain can be calculated. A random variable is a number that depends on chance, such as the number a die shows or the number of deaths in a year. Its expected value says what comes out on average; how far the individual values scatter around it is measured by two further numbers. For every possible value one takes its distance from the expected value and squares it, that is, multiplies it by itself; in this way large distances count far more than small ones. The expected value of these squares is called the variance, and its square root the standard deviation; it has the same unit as the quantity itself and says how far a value typically lies from the expected value. For a die the expected value is 3.5 and the standard deviation 1.71, as the box below this paragraph works out. With ten thousand policyholders, each of whom dies in a year with a probability of one per cent, 0.01 · 10,000 = 100 deaths are to be expected, with a standard deviation of just under 10, which the second box below this paragraph works out. Adding up the probabilities of all possible numbers of deaths, the number lies between 80 and 120 in 96 years out of 100, no further than two standard deviations from the expected value. With a hundred policyholders one death is to be expected, with a standard deviation of likewise just under 1, and whether there are none or three is decided by chance: in relation to the expected number, the small group scatters ten times as much. From section 3 on, the standard deviation becomes the measure of risk.

The bell curve. How the number of successes in many trials scatters around its expected value was found by the Huguenot Abraham de Moivre, who will come up again with insurance, in a small Latin pamphlet dated 12 November 1733: with many coin tosses the probabilities of the possible numbers of successes follow a smooth curve.²⁹ The normal distribution has the shape of a bell, the bell curve, and it arises when many small, independent chance effects add up: a good two thirds of its values lie less than one standard deviation from the expected value, about 95 in 100 less than two, and only one in a hundred lies more than 2.33 standard deviations below. Beyond the period of this section, Carl Friedrich Gauss derived it in 1809 as the law of errors of measurement while calculating the orbits of celestial bodies from inexact observations, and Pierre-Simon Laplace proved in 1810 why it appears wherever many small chance effects add up; the name normal distribution was established only by Francis Galton and Karl Pearson towards the end of the 19th century.³⁰ It lies behind the calculation with the 10,000 insured people as behind almost everything still to come in this text, from Bachelier to the copula.

Today

For savers, more frequent crediting brings little: in one year 10,000 euros at 3 per cent become 10,300.00 euros with yearly crediting, 10,304.16 with monthly, 10,304.53 with daily and 10,304.55 euros with continuous compounding.

The law of large numbers holds for bicycles too: if one in 20 bicycles is stolen each year, the individual does not know whether theirs will be; an insurer with 10,000 customers, by contrast, can count fairly safely on about 500 thefts.

Two investments that both yield 5 per cent on average can differ completely in their spread: a fixed deposit always yields 5 per cent, its standard deviation is zero; a second investment yields 10 or 0 per cent with equal chance, so lies 5 percentage points from the mean every year, and its standard deviation is 5 percentage points.

Utility and risk aversion

Utility. Daniel Bernoulli, the nephew, answered, in a paper he had written in St Petersburg and that appeared there in 1738, a question that his cousin Nicolaus had posed in 1713 in a letter to the French mathematician Pierre Rémond de Montmort and that occupies the discipline to this day. A coin is tossed until heads falls, and the game pays one ducat if that happens on the first toss, two on the second, four on the third and so on, double with every toss. Heads on the first toss has probability ½, only on the second ½ · ½ = ¼, because tails must fall first, only on the third ⅛. Every outcome therefore contributes the same to the expected value, ½ · 1 = ½, ¼ · 2 = ½, ⅛ · 4 = ½, and the sum ½ + ½ + ½ + … never stops growing: the expected value is infinite, larger than any number one can name. And yet nobody would pay more than a few ducats to take part. Bernoulli’s solution was that people do not value the amount of money but its utility, the worth it has for them, and that the same ducat is worth more to the poor man than to the rich. He proposed measuring utility by the logarithm of wealth, so that every doubling of wealth brings the same gain in utility, whether from ten ducats to twenty or from a thousand to two thousand, and derived from it that it is reasonable to shun risk and to spread one’s wealth over several investments.³¹ Two hundred years later this became the theory with which economists describe how people decide under uncertainty, and the risk aversion he introduced is the quantity that sits in every price for risk. The mathematician John von Neumann and the economist Oskar Morgenstern showed in 1947, in the second edition of their book Theory of Games and Economic Behavior, that anyone who chooses between uncertain prospects consistently according to a few simple rules acts as if calculating the expected utility, the average of the utilities of all outcomes, each weighted by its probability; one of the rules reads: whoever prefers A to B and B to C also prefers A to C.³²

The prospects. The psychologists Daniel Kahneman and Amos Tversky showed in 1979, with questions put to test subjects, that people do not behave like this. They value gains and losses relative to what they already have, not their whole wealth, and a loss hurts more than an equal gain pleases; their prospect theory describes this with a valuation that is steeper for losses than for gains. Kahneman received for it in 2002 the Swedish central bank’s prize in economic sciences, awarded with the Nobel prizes; Tversky had died in 1996, and the prize is not awarded to the dead.³³

Today

Expected utility explains why people buy insurance although the insurer earns on it on average: if a household has 20,000 euros and faces a loss of 15,000 euros with probability 0.01, the expected loss is 150 euros. Measuring utility as Bernoulli did, with the logarithm of wealth, a household contents insurance would be worth up to 275 euros a year to the household, because the rare large loss weighs more than its average.

A coin toss in which one wins 120 euros on heads and loses 100 euros on tails has the expected value ½ · 120 − ½ · 100 = 10 euros, and yet many people turn it down.

Insurance and actuaries

Insurance. Abraham de Moivre, a French Protestant who had fled to London as a Huguenot, and a friend of Newton, turned Halley’s table into a calculating procedure in 1725 with Annuities upon Lives: he assumed that from the age of twelve the number of the living declines uniformly, by the same number every year, until at 86 no one is left, and so obtained formulas a merchant could apply without a table. In Figure 2 that is the dashed straight line; between 30 and 70 it lies close to Halley’s curve.³⁴ The application to life insurance, in which one pays in regularly and the insurer pays an agreed sum to the survivors on death, came from a man an insurer had rejected: James Dodson, teacher of mathematics, had been too old for the Amicable Society, a London insurance society, and designed a society in which the premium, the price of insurance cover, depends on the age at entry and in young years more is taken in than the deaths cost, so that a reserve arises for the later years. Dodson died in 1757, but in 1762 the Society for Equitable Assurances on Lives and Survivorships was founded on his plan, and Edward Rowe Mores, who had driven the founding, gave its chief officer a title that had not existed before: actuary, the mathematician who translates the probability of life and death into prices.³⁵ Richard Price, preacher and mathematician, advised the society from 1768, published in 1771 the Observations on Reversionary Payments with life tables from London, Norwich and Northampton, of which the Northampton table remained the standard for decades, and brought his nephew William Morgan into the house, who became actuary of the Equitable in 1775 and remained so until 1830.³⁶ Life insurance was thus the first business built entirely on mathematics: its prices came from tables, not from feeling, and its reserves from a calculation that said how much must be set aside today so that tomorrow’s promises can be kept.

Insurance after 1800. In the nineteenth century, beyond the period of this section, the tables became laws. Benjamin Gompertz, who as a Jew was excluded from the English universities, was educated privately and in 1824 became actuary of the Alliance British and Foreign Life Assurance Company in London, showed in 1825 in the Philosophical Transactions of the Royal Society that mortality, the share of people of a given age who die in the course of a year, grows over much of adult life by the same percentage with every year of age, that is exponentially, like capital at compound interest. Gompertz himself wrote this for the force of mortality, the mortality at an instant, as $B \cdot c^x$; the box below calculates in simplified form with the mortality of a whole year, which for small values grows almost exactly the same way. This Gompertz law is no law of nature, but to this day a serviceable approximation for the middle and later years.³⁷ Carl Friedrich Gauss, mathematician and director of the observatory in Göttingen, examined from 1845 to 1851 the widows’ fund of the Göttingen professors, which feared being ruined by the growing number of widows. He set the contributions of the professors and the interest on the fund’s assets against the pensions of the present widows and of those to be expected, and concluded that the fund could even raise its pensions if it tightened the conditions for drawing them.³⁸ The Swedish actuary Filip Lundberg asked a new question in his doctoral thesis of 1903: not what a single contract costs, but whether an insurer as a whole survives. In his model premiums flow in evenly, and claims arrive at random times in random amounts; the insurer is ruined if its reserve ever falls below zero. In papers of 1926 and 1928 he calculated the ruin probability, the probability that this happens at some point: if premiums cover only the expected claims, ruin is certain in the long run; if they contain a loading, the ruin probability falls with every further amount of initial capital, in the simplest case each time by the same fraction, that is exponentially. Harald Cramér, mathematician in Stockholm, turned this from 1930 onwards into a rigorous theory that has since been called the Cramér-Lundberg model.³⁹ How much capital an insurer needs to survive a bad year is law in the European Union today; section 6 tells of it.

Today

At its core the premium of a life insurance is calculated as for the phone insurance in the “Today” part on gambling and expected value: expected loss plus a loading for costs and profit. If a 35-year-old woman covers her mortgage with 200,000 euros payable on death and dies with probability 0.001 in the coming year, the expected payment is 0.001 · 200,000 = 200 euros, and the premium for that year lies above it.

The Gompertz law explains why term life insurance becomes dearer with the age at which it is taken out: if mortality grows by 9 per cent a year, at 54 it is 1.09²⁴ = 7.9 times as high as at 30.

Working it out and practice

Try it yourself. Halley’s calculation can be reproduced with a current life table, for instance that of Germany’s Federal Statistical Office or that of Eurostat, the statistical office of the European Union, which gives for every age the probability of surviving the next year. For a pension of 1,000 euros a year to a 65-year-old one writes in a spreadsheet the years 1, 2, 3 and so on in column A, in column B the probability of living to see each year, which one obtains from the table by multiplying the survival probabilities year by year, and in column C the value of the payment today, =1000*B1/1.03^A1; the sum of column C is the present value of the pension. For a man it is roughly thirteen times the annual amount, for a woman, who lives longer on average, roughly fifteen times, and whoever calculates with two per cent instead of three sees it rise by about a tenth.⁴⁰ The St Petersburg game can be tried with a coin and a friend, and the question of what one would pay to take part is answered differently by everyone, which proves Bernoulli’s point.

The bottleneck

Around 1800 actuaries could value what depended on chance, as long as chance was stable: mortality changes slowly, and ten thousand policyholders even each other out. Not so the price of a share, a part of a company that gives its owner a portion of the profit and a vote, on the stock exchange, the market on which such shares and bonds are bought and sold every day: it obeyed no life table. It changed daily, for reasons nobody could enumerate, and the mathematicians left it to the speculators, who buy in order to sell dearer later. What was missing was a description of chance that is not fixed like the balls in a lottery drum but arises from the decisions of thousands. The description came, but for half a century it was not read.

3. Chance on the exchange: Regnault, Bachelier and Brownian motion (1863–1950)

The price as a random walk

The broker. The first to take the price seriously as chance was the assistant of a stockbroker in Paris, a man who buys and sells on the exchange for others. The market price is the price at which a share or bond is traded on the exchange at a given moment. Jules Regnault published in 1863 the Calcul des chances et philosophie de la Bourse, in which he derived from price series of the French government rente, a bond of the state that paid the same interest every year and was never repaid, a perpetuity like the consols, that short-term price changes are not predictable and that the range within which a price moves grows not with time but with its square root: in four months it swings twice as far as in one, because √4 = 2.⁴¹ That is the hallmark of the random walk, a movement of steps up or down in which every step is independent of the previous one and whose distance from the start grows on average with the square root of the number of steps; the box below this paragraph shows why. How strongly a price fluctuates is measured today by its volatility, the standard deviation of its percentage changes, usually stated for a year: a share with 20 per cent volatility typically stands 20 per cent above or below the value expected after a year. Regnault wanted to warn against gambling with his calculation; without knowing it, he had described a model, a simplified description in numbers and rules with which one can work out how something behaves, and the most important pricing formulas of the exchange rest on this model a hundred years later.

The thesis. On 29 March 1900 Louis Bachelier defended at the Sorbonne, the university of Paris, his doctoral thesis entitled Théorie de la spéculation, supervised by the mathematician Henri Poincaré. He described the price of the government rente as the sum of countless small, independent random steps and derived the equation that states how probable it is to stand at a particular price after a particular time. It was the same equation with which physics describes the spread of heat: just as heat spreads ever further from a hot spot, the probability of the price spreads over time across ever more possible values. From it he calculated what an option on the rente is worth, the right but not the obligation to buy or sell it by a certain day at a price fixed in advance, and compared the result with the prices of the Paris exchange.⁴² The thesis received the grade “honorable”, not the best grade there was, but according to Courtault and others the highest for a thesis lying so far outside mathematics; Bachelier moved from post to post, Dijon turned him down in 1926 after a critical report by Paul Lévy, who had misunderstood Bachelier and, after 1931, when he found Bachelier’s results again in Kolmogorov, apologised to him by letter, and only in 1927, at 57, did he become a professor in Besançon. His model had a flaw that was only seen later: it allowed negative prices, because it let the price itself and not its percentage change fluctuate at random; a long run of steps down could push it below zero, which makes no sense for a price. A hundred and twenty years later it turned out that a negative price can make sense after all. On 20 April 2020, when in the pandemic hardly anyone needed oil and the storage tanks were full, the price of a barrel of American crude for delivery in May fell on the New York Mercantile Exchange to minus 37.63 dollars, below zero for the first time since the contract had existed: whoever held a contract for oil but had no room to store it paid to be rid of it. The Chicago exchange group CME, which owns the New York Mercantile Exchange, valued its options on oil from 22 April with Bachelier’s model, precisely because it allows negative prices, and only returned to the old models on 31 August.⁴³ But Bachelier’s model contained, five years before Einstein, the mathematics of Brownian motion, the random jittering that the botanist Robert Brown had observed in 1827 in tiny particles from pollen grains in water, and fifty years before the economists the insight that the expected gain of a speculator is zero.

Today

Whoever holds ten shares that stand at 50 euros in the morning and at 51.50 euros in the evening owns 515 instead of 500 euros in the evening without having done anything; the overview of a securities account always reckons with the latest price.

If someone tosses a coin with a friend for one euro a throw, after 100 throws he is typically about 10 euros ahead or behind, after 400 throws about 20 euros, not four times as much.

If someone puts 10,000 euros into an equity fund from which he expects 7 per cent a year and which has a volatility of 20 per cent, after a year there will typically be between 8,700 and 12,700 euros in the account; a savings account, by contrast, holds exactly the deposit plus interest.

Every savings calculator on the internet is a model: it assumes, say, that a fund brings exactly 6 per cent every year and so turns 5,000 euros into 5,000 · 1.06¹⁰ = 8,954.24 euros in ten years; in reality the fund fluctuates, but without the simplification one could not calculate at all.

An option resembles a paid reservation: if someone pays a seller 500 euros for the right, but not the obligation, to buy a car for 20,000 euros by the end of the month, and the car is then worth 22,000 euros, he buys and is 1,500 euros better off after deducting the 500; if it is worth only 18,000 euros, he lets it go, and he cannot lose more than the 500 euros.

On a small scale Brownian motion is the coin toss with the friend above, with countless tiny throws.

Physics and data

Physics. Mathematics caught up with the exchange from the other side. Albert Einstein derived Brownian motion in 1905 from the impacts of molecules, the tiny particles of which water consists, and showed that the average distance of a particle from its starting point grows with the square root of time, the same root as with Regnault.⁴⁴ Norbert Wiener gave the motion a rigorous foundation in 1923 in a paper entitled Differential Space: a calculation that assigns probabilities not only to possible end values but to whole groups of paths without jumps, so that one can work out how probable it is that a price follows a path of a certain kind, for instance never falling below 80.⁴⁵ Andrei Kolmogorov put the whole discipline on axioms in 1933 in the Grundbegriffe der Wahrscheinlichkeitsrechnung, a few basic rules from which everything else can be derived, and thereby turned the recipes of the gamblers into a theory that could reckon with arbitrary random quantities.⁴⁶ A stochastic process has since been a sequence of random variables, one for every moment, such as a price, a temperature or the position of a particle, and Brownian motion is its basic building block.

The empiricists. While the mathematicians were founding chance, statisticians, who draw conclusions from data, tested, first in America, whether the exchange obeys it. Alfred Cowles, heir to a newspaper fortune, who asked himself why no adviser had warned him of the collapse of prices in 1929, examined in 1933 in the first volume of the journal Econometrica the recommendations of sixteen financial services, twenty insurance companies and 24 market letters, circulars with advice on what to buy and sell, over several years and found that on average they did worse than chance; the title was a question, “Can Stock Market Forecasters Forecast?”, and the answer, in three words beneath the title, was: “It is doubtful.”⁴⁷ Holbrook Working showed in 1934 that a series of random increments looks like the prices of sensitive commodities, and Maurice Kendall found in 1953 in British share and commodity prices no patterns a statistician could have exploited.⁴⁸ The exchange behaved like Bachelier’s model, and none of those involved knew Bachelier.

Today

The value of a securities account on every evening of a month is a stochastic process, a series of numbers each of which one knows only on that evening; a savings book with a fixed rate is not, because its balance can be worked out in advance for every day.

Value without chance: Fisher, Williams, Macaulay

Valuation. In parallel the mathematics of value without chance arose. Irving Fisher set out interest in 1930 in The Theory of Interest as the price between present and future, determined by impatience and opportunity, and formulated the rule by which an investment, money spent today in order to get more back later, for instance on a machine, pays if the present value of its future returns is greater than its cost.⁴⁹ John Burr Williams wrote in 1938 in The Theory of Investment Value that a share is worth the present value of its future dividends, the parts of the profit that a company pays out every year to the owners of its shares, no more and no less, and gave the intrinsic value a formula; Benjamin Graham and David Dodd had described the practice for it in Security Analysis in 1934, buying below this value.⁵⁰ Frederick Macaulay introduced in the same year 1938 the duration, the average waiting time for the payments of a bond, in which every payment counts as much as it contributes to the present value. To this day it states how strongly a bond price reacts to a change in interest rates: if the rate in the market rises, the fixed payments of a bond already issued are worth less, because they are discounted more heavily, and its price falls, all the more the longer one has to wait for the money.⁵¹ These three tools reckon with certain payments and an interest rate; they are Fibonacci’s line, extended into the accounts of every company.

Today

Fisher’s rule also applies to an energy-saving fridge: if it costs 600 euros and saves 80 euros of electricity a year for ten years, the savings are worth 80 · (1 − 1.03⁻¹⁰) / 0.03 = 682.42 euros today at 3 per cent interest, more than the price; the purchase pays.

Whoever holds 50 shares of a company that pays out 1.20 euros per share receives 60 euros into his account before tax. If a rented flat brings in 6,000 euros a year after all costs for good, at 4 per cent interest it is worth 6,000 / 0.04 = 150,000 euros, by the same calculation as the perpetuity in section 2; a share with a permanent dividend of 2 euros is worth 2 / 0.05 = 40 euros at 5 per cent.

If someone holds two bonds of 100 euros each with a 3 per cent coupon, trading at 100 because the market rate is 3 per cent, one with two and one with ten years to maturity, and the market rate rises to 4 per cent, the short one falls to 98.11 euros, the long one to 91.89 euros.

Itô’s calculus

Itô. The tool with which one can calculate with Brownian motion as with ordinary numbers came from Japan in 1944. Kiyosi Itô, who had worked in the statistics bureau of the cabinet until 1943 and then moved to the university of Nagoya, found a way of adding up countless small random changes; the rule that bears his name followed in 1951. A function is a rule of calculation that turns one number into another, for instance squaring or the logarithm. If one applies a function to a price that jitters at random, its change acquires an additional part that does not exist for certain quantities, because the squares of the small random steps do not vanish but add up over time, just like the variances in the square-root law.⁵² The Itô calculus, the rules of calculation for quantities that change at random at every instant, is the reason why the option formula of 1973 became a formula and not merely an idea. What the additional part means is shown by the box below this paragraph: fluctuation costs growth. In 1944 no economist was interested; the calculus reached them only two decades later, by way of MIT, as section 4 tells.

Today

An electricity tariff is a function: with a standing charge of 10 euros a month and 30 cents per kilowatt hour it turns 200 kilowatt hours into a bill of 70 euros; compound interest 1,000 · 1.03ⁿ turns the number of years n into an account balance.

Working it out and practice

Try it yourself. A random walk arises in a spreadsheet from two columns. A1 holds =IF(RAND()<0.5,-1,1): RAND() supplies a random decimal between 0 and 1, and IF gives −1 if it is less than 0.5 (the sign < means less than), otherwise 1; that is a coin toss. B1 holds =A1, B2 =B1+A2, the running total of all steps so far. Copying both formulas 250 rows down and drawing column B as a line chart, a chart like Figure 1, shows a random walk. Whoever presses F9 gets a new path each time, and whoever lays ten of them beside the price history of a real share will not be able to tell them apart with certainty; that is the test the statistician Harry Roberts put to his colleagues in 1959,⁵³ and the button beneath Figure 3 rolls the three walks there anew. The range within which the paths mostly lie after n steps reaches twice the square root of n up and down (Figure 3): after 100 steps about ±20, that is from −20 to +20, after 400 about ±40.

The bottleneck

By 1950 chance on the exchange was described, confirmed by data and mathematically rigorous, and the parts lay in three disciplines that did not talk to each other: Bachelier with the mathematicians, who cited him, but unknown to the economists; Cowles and Working with the statisticians; Fisher and Williams with the economists. What was missing was a theory that said what an investor, someone who invests money, should hold when the future is random, and what price the market pays for someone bearing a risk, the possibility that things turn out differently than expected and he loses. The answer came from Chicago, from a doctoral student to whom a broker in his professor’s waiting room had suggested the topic.

Today

10,000 euros in a savings account at 2 per cent bring a certain 200 euros in a year; in an equity fund perhaps 700 euros are to be expected, but in a bad year it can also be a loss of 2,000 euros.

4. Portfolio, market and the price of risk (1952–1972)

Spreading instead of guessing: Markowitz

The selection. Harry Markowitz, 24 years old, published in March 1952 in the Journal of Finance fifteen pages entitled “Portfolio Selection” that founded the discipline today called financial economics, the study of investments and their prices. His observation was that investors do not just buy the share with the highest expected return but spread their money, and that theory until then could not explain why. His answer: an investor values not individual securities, that is shares and bonds, but his portfolio, the totality of what he holds, and by two numbers, the expected return and the volatility.⁵⁴ The volatility of a portfolio depends not only on the volatilities of its parts but on how they move together: two shares that do not swing in step yield together less risk than each alone, because the loss of one is often offset by the gain of the other. How closely two returns move in step is measured by the correlation, a number between −1 and 1: at 1 they always move together, at 0 independently of each other, at −1 always against each other. Two shares with 20 per cent volatility each, mixed half and half, give 20 per cent again at a correlation of 1, 17.3 at 0.5, only 14.1 at 0, and at −1 the risk would vanish altogether, as the box below this paragraph works out; if both have the same expected return, the risk falls with independent prices by almost a third without any return being lost. Diversification, spreading money across investments that do not behave alike, thereby acquired a formula and a limit. The portfolios that deliver the highest expected return for a given volatility form a curve, the efficient frontier, on which every sensible investor should lie somewhere; in Figure 4 volatility runs to the right and expected return upwards, so that good portfolios lie at the top left.

Today

If someone has 6,000 euros in an equity fund from which he expects 7 per cent, 3,000 euros in a savings account at 2 per cent and 1,000 euros in a single share from which he expects 9 per cent, that is his portfolio, and its expected return is the weighted average 0.6 · 7 + 0.3 · 2 + 0.1 · 9 = 5.7 per cent.

An ice-cream seller earns 30 per cent on his capital in a sunny summer and loses 10 per cent in a rainy one, an umbrella seller exactly the other way round; their correlation is −1. Whoever holds half of each earns (30 − 10) / 2 = 10 per cent every summer, rain or shine.

If someone puts 10,000 euros into the share of a single company that goes bankrupt with probability 0.05, with that probability he loses everything. If he spreads the money at 500 euros each over 20 such companies, which can go bankrupt independently of each other, the expected loss is the same, 500 euros, but that more than three go bankrupt and he loses 2,000 euros or more happens only with probability 0.016. For the same reason it is risky to invest one’s savings in shares of one’s own employer: if the company goes bankrupt, job and money are gone at once.

Whoever holds only bonds that bring 3 per cent at 5 per cent volatility and swaps a tenth of them for shares that bring 8 per cent at 20 per cent volatility and are uncorrelated with the bonds expects 0.9 · 3 + 0.1 · 8 = 3.5 per cent at a volatility of 4.9 per cent: more return with less risk, because the bonds alone did not lie on the efficient frontier.

The price of risk

The market. What follows if all investors calculate like this? James Tobin showed in 1958 that the choice separates when there is a safe asset, an investment without fluctuation such as a short-term government bond, which brings the risk-free rate: all hold the same portfolio of risky assets and merely mix it in different proportions with the safe asset, the cautious with much, the bold with little.⁵⁵ William Sharpe, Markowitz’s student, drew from this in September 1964 the conclusion that this common portfolio must be the market itself, the market portfolio, which contains all shares in proportion to their value on the exchange, and that the market pays for the risk of a single asset only so far as it increases the risk of the market portfolio; John Lintner and Jan Mossin reached the same result independently.⁵⁶ Their model, the Capital Asset Pricing Model or CAPM, measures this sensitivity with one number, the beta: a share with beta two moves on average twice as far as the market, rising by 2 per cent when the market rises by 1 per cent. One estimates beta by plotting the returns of the share against those of the market and looking for the straight line that fits the points best; its slope, how far it goes up when one goes one step to the right, is the beta. For its risk a share with beta two must deliver twice the risk premium, the return beyond the risk-free rate that the market brings; risk that can be diversified away is not paid for. The expected return of every asset is thereby the risk-free rate plus beta times the risk premium of the market, and the price of risk was for the first time a quantity that could be estimated from prices. In Figure 4 all mixtures of the safe asset and the market portfolio lie on a straight line, the capital market line.

Arbitrage. The second argument that shaped the epoch came from the study of how companies raise money. A company can finance itself with equity, the money of its owners, which does not have to be repaid and in return shares in profit and loss, or with debt, which must bear interest and be repaid. Franco Modigliani and Merton Miller proved in June 1958 that under ideal conditions, without taxes and costs, the value of a firm does not depend on how it finances itself, and their proof had a new form: were two identical firms valued differently, an investor could buy the cheap one, sell the dear one and gain without risk, and because that is not possible for long, the prices must be equal.⁵⁷ Arbitrage is such a gain without outlay and without risk, and the assumption that it does not exist in a functioning market became the strongest tool of the discipline: it determines prices without one having to know what investors want.

Today

For a saver the safe asset is something like a fixed deposit, in which 5,000 euros at 2.5 per cent become a certain 5,125 euros in a year; whoever puts 4,000 euros into a fixed deposit and 1,000 euros into a broadly diversified equity fund mixes exactly as Tobin describes.

If a market consisted of only three companies worth 600, 300 and 100 million euros on the exchange, whoever put 600 of 1,000 euros into the first share, 300 into the second and 100 into the third would hold the market portfolio; a fund that buys all the shares of a market in proportion to their market value comes close to it.

If the market falls by 10 per cent, a travel share with beta 1.5 falls on average by 15 per cent, a grocer with beta 0.5 by only 5 per cent, because people eat in bad times too.

A taxi with a basic fare of 4 euros and 2 euros per kilometre has the slope 2: every additional kilometre costs 2 euros more.

If the savings account brings 2 per cent and one expects 7 per cent from the equity fund, the risk premium is 5 percentage points, on 10,000 euros 500 euros a year as the reward for enduring the fluctuations.

If someone buys a house for 400,000 euros with 80,000 euros of his own money and a loan of 320,000 euros, his equity is 80,000 euros; if the value of the house rises by 10 per cent to 440,000 euros, it grows to 120,000 euros, by 50 per cent, and if it falls by 10 per cent, it shrinks to 40,000 euros, because the debt stays the same.

If one coin dealer sells a gold coin for 1,000 euros and another buys the same coin at the same time for 1,020 euros, one can buy from the first, sell to the second and earn 20 euros without risk; because many do so, the price rises at the first and falls at the second until the gap disappears.

Chance becomes theory

The evidence. Meanwhile the economists had found Bachelier. In the 1950s the statistician Jimmie Savage sent postcards to colleagues asking whether anyone knew the Frenchman; Paul Samuelson, an economist at MIT, the institute of technology near Boston, obtained the thesis and had it translated.⁵⁸ The physicist M. F. M. Osborne of the U.S. Naval Research Laboratory described share prices in 1959 as Brownian motion, but of the logarithms, so that the percentage changes are random and not the changes in euros; a price that only ever falls by a fraction of itself cannot fall below zero, and Bachelier’s negative prices disappeared.⁵⁹

Fat tails. Benoit Mandelbrot found in 1963 in cotton prices a deviation for which one needs a further concept. A distribution states how probable the possible values of a random variable are; drawing how often each value occurs gives it a shape. The normal distribution from section 2 has the shape of a bell, the bell curve, and it arises when many small, independent chance effects add up: a good two thirds of its values lie less than one standard deviation from the expected value, about 95 in 100 less than two, and only one in a hundred lies more than 2.33 standard deviations below it. Mandelbrot showed that the changes in cotton prices are random but not normally distributed: large jumps occurred far more often than the bell curve allows. Such distributions have fat tails, more probability for extreme values than the bell curve, and Mandelbrot’s warning that the variance of price changes might be infinitely large, because the outliers can become arbitrarily large, remained for forty years the most uncomfortable footnote of the discipline.⁶⁰

The efficient market. Samuelson proved in 1965 that in a market that processes all information correctly, prices must fluctuate randomly, because every predictable movement would be traded away at once: whoever knew that a price will rise tomorrow would buy today, and the price would rise already today. The price is then a martingale, a stochastic process whose best forecast for tomorrow is today’s value. Samuelson proved it for futures prices; for a share it holds apart from the expected return by which its price grows on average.⁶¹ Eugene Fama summarised the evidence in 1970 under the concept of market efficiency: a market is efficient if prices contain all available information, and Fama distinguished which information is meant, past prices, public news, or also what only insiders know, people inside a company who learn more than the public.⁶² What Cowles had counted in 1933 was thereby a theory, and the random walk was no longer a lack of knowledge but the consequence of everyone knowing everything.

Continuous time. How Itô’s calculus reached the economists is a story of neighbours. In 1965 Samuelson let prices fluctuate at random not by fixed amounts, as Bachelier had, but by fixed proportions, so that they never fall below zero, and the mathematician Henry McKean, who was working with Itô at MIT, wrote an appendix for him.⁶³ Robert Merton, who had studied applied mathematics, came to MIT in 1967 and worked out in 1969 and 1971 how an investor invests and consumes his wealth if he can change his mixture at every instant; he saw, probably first, what the Itô integral means: it adds up many small random price changes, each multiplied by the quantity one holds at that instant, and so describes nothing other than the gain of a trading strategy.⁶⁴ With that the tools were ready with which the option formula was derived in 1973.

Today

If one throws two dice and adds up the spots, 7 occurs in 6 of 36 possible cases, 2 and 12 in only one each; plotting how often each sum occurs gives a triangle with its peak at 7.

A share with 20 per cent volatility a year fluctuates by about 1.26 per cent in a day, because 20 / √250 = 1.26; according to the bell curve a day of minus 5 per cent should occur only about once in a hundred years. Fat tails mean that such days come far more often than the bell curve says; whoever reckons with it underestimates the risk of large losses.

If someone with 50 euros in his purse plays a fair coin bet for one euro a throw, the best forecast for his purse after the next throw is 50 euros, and likewise after a hundred throws. Even a rule like “double the stake after every loss” changes nothing: it brings many small gains and occasionally a large loss, and on average the result stays at zero.

If a company reports before the market opens a profit twice as high as expected, its share stands perhaps at 46 instead of 40 euros right after the opening; whoever reads the news over breakfast and then buys already pays the 46 euros.

On a small scale the Itô integral is the statement of a savings plan: if someone holds 10 fund units in the first month, which rise by 2 euros, 15 in the second, which fall by 1 euro, and 20 in the third, which rise by 3 euros, his gain is 10 · 2 − 15 · 1 + 20 · 3 = 65 euros. The Itô integral is the same sum with countless, arbitrarily small steps.

Kelly and Thorp

The gamblers. Beside the professors the gamblers calculated, and they were faster. John Kelly, an engineer at Bell Labs, the research laboratory of the American telephone company, derived in 1956 from Claude Shannon’s information theory, the mathematics of transmitting messages, what share of his wealth a gambler with an edge should stake: so much that the wealth grows exponentially fastest on average, and no more, because every higher stake slows growth again; whoever stakes about twice as much makes no progress in the long run.⁶⁵ Whoever wins a bet that doubles or loses the stake with 55 per cent and stakes everything each time is broke at the first loss; whoever stakes nothing never wins; in between lies a share at which wealth grows fastest, here 10 per cent, as the box below this paragraph shows by trial; the derivation is in the blue box at the end of the section. The Kelly criterion is this rule for the size of the stake. Edward Thorp, professor of mathematics, applied it in 1962 in Beat the Dealer to the card game blackjack, which he beat by counting cards: he remembered which cards had already been played and so recognised when the rest of the deck favoured him. In 1967, in Beat the Market, with Sheen Kassouf, he described a method for warrants, options that a company issues itself and that are traded on the exchange: he bought the share and at the same time sold so many warrants on it that gains and losses cancelled out. If the share fell, he lost on it, but the warrants he had sold also became cheaper, and he could buy them back for less; that was years before the formula that states the right ratio exactly.⁶⁶ Thorp later said he had already possessed the formula; he traded with it instead of publishing it.

Today

For a fair coin bet that doubles the stake the Kelly criterion says: stake nothing at all, because without an edge there is nothing to gain, and this holds all the more for games of chance in which the house has the edge; only whoever wins with 60 per cent should stake 20 per cent of his wealth.

Working it out and practice

Try it yourself. Markowitz can be recalculated with two shares. One downloads five years of monthly prices of two shares, for instance from the website of an exchange, and forms the monthly returns: price at the end of the month divided by the price at the end of the previous month, minus 1, as a formula for instance =B3/B2-1. The function STDEV calculates from a column of returns the standard deviation, that is the volatility of each share, AVERAGE the average and CORREL the correlation of two columns. Since these are monthly returns, STDEV gives the volatility of one month; for a year one multiplies it by the square root of 12, so that 5.8 per cent a month become 20 per cent a year, because, as Regnault saw, the fluctuation grows with the square root of time, and the mean monthly return times 12 is the expected return for a year. With the two annual volatilities in B1 and C1, the correlation in D1, the expected returns in B2 and C2 and in column A from row 4 the proportions of the first share from 0 to 1 in steps of 0.1, =A4*$B$2+(1-A4)*$C$2 gives the return of the mixture and =SQRT(A4^2*$B$1^2+(1-A4)^2*$C$1^2+2*A4*(1-A4)*$D$1*$B$1*$C$1) its volatility by the formula in the box above; the dollar signs hold cells B1 to D1 fixed when the formula is copied down, and SQRT takes the square root. Plotting return against volatility as a scatter chart yields the curved line of Figure 4; if the correlation is low enough, below the ratio of the smaller to the larger volatility, its left end lies to the left of both shares, and that is the whole of diversification in one picture. The beta of a share is given by SLOPE applied to its returns and those of an index, a number that measures the value of a fixed basket of many shares, such as the DAX for the largest German companies on the exchange. An index works like the price of a basket of goods in the supermarket: if the same basket costs 103 euros today instead of 100 a year ago, a price index with the starting value 100 stands at 103.

The bottleneck

By 1972 the discipline knew how risk is spread and how it is paid for, and it had chance as a martingale. One question remained open, and it was the most practical of all: what is an option worth, the right to buy a share later at a price fixed today? Samuelson and others had formulas, but all contained a quantity nobody knew, the expected return of the share or the risk aversion of the buyer; Bachelier had simply set the expected return to zero. In Chicago an exchange was preparing to trade exactly this right, and three economists who had met at MIT near Boston had a derivation in which the unknown quantity disappeared.

5. The formula: Black, Scholes, Merton and the derivatives (1973–1987)

The option formula

The year 1973. On 26 April 1973 the Chicago Board Options Exchange, a new exchange for options alone, opened in the smoking lounge of the Chicago Board of Trade, Chicago’s grain exchange, the first regulated trading in standardised stock options, options with uniform terms that anyone can buy and sell under the rules of the exchange: 911 contracts on sixteen stocks on the first day, all of them rights to buy.⁶⁷ An option is, as in section 3, the right, not the obligation, to buy or sell a thing by a date at a fixed price. The right to buy is called a call option, the right to sell a put option; the fixed price is called the strike price, using the right exercise, and the last day on which this is possible expiry. A European option can only be exercised on the day of expiry, an American one on any day up to and including that day. The option is the simplest derivative, a contract whose value is derived from another price. In the same spring there appeared in the Journal of Political Economy, which had at first promptly rejected the paper,⁶⁸ “The Pricing of Options and Corporate Liabilities” by Fischer Black and Myron Scholes, and in the Bell Journal of Economics Robert Merton’s “Theory of Rational Option Pricing”, which generalised the derivation and gave it the name Black-Scholes model.⁶⁹ The Black-Scholes formula gives the value of a European call option from five quantities: share price, strike price, time to expiry, risk-free interest rate and volatility. The expected return of the share does not appear, and that was the discovery.

The forerunners. What was new in 1973 was regulated trading and the formula, not the contracts. Aristotle tells in his Politics of the philosopher Thales of Miletus, who thanks to his knowledge of the stars foresaw a rich olive harvest, acquired already in winter, for a small deposit, the right to all the olive presses in Miletus and Chios and at harvest, when everyone needed them at once, let them out dearly: a right, not an obligation, for a small price paid in advance, that is an option.⁷⁰ Joseph de la Vega described in Amsterdam in 1688, in Spanish and in conversations between a philosopher, a merchant and a shareholder, in Confusión de confusiones the trading on the Amsterdam exchange, with call and put options and with futures, contracts to buy or sell a thing on a later day at a price agreed today; it is the first work about the Amsterdam exchange.⁷¹ In Osaka merchants had traded rice in the Dojima quarter since 1697, and in 1730 the shogunate, the government of Japan’s military ruler, recognised the market: the Dojima rice exchange had members, clearing houses that guaranteed every trade to both sides, and rice futures in which, against a deposit, only the difference between the agreed and the later price was paid; it counts as the world’s first organised futures exchange.⁷² What all these markets lacked was a calculation for the price. Bachelier had attempted it in 1900; Black, Scholes and Merton found it, with a trick.

The trick. Why the expected return does not appear is the heart of the discipline. Whoever has sold a call loses if the share rises, because he must then hand it over more cheaply than it is worth. He can protect himself by holding a certain quantity of the share: if the price rises, the share gains what the option costs him; if it falls, the share loses, but the option he has sold becomes cheaper, and that offsets the loss. This hedging must be adjusted continually, because the right quantity changes with the price, but if one does it, the combination of option and share is without risk, and what is without risk may, by the arbitrage argument, earn only the risk-free rate. From this condition there follows, with the Itô calculus, an equation that fixes not the option value itself but how it changes at every instant with time and price, a differential equation related to the equation of the spread of heat that Bachelier had found; its solution is the formula.⁶⁹ Investors may be as risk-averse as they like; their taste drops out of the calculation. One can therefore value the option as if all investors were risk-neutral, paying attention only to the expected value and indifferent to risk, and as if the share were expected to earn only the risk-free rate: that is risk-neutral valuation, the computational trick that has lain beneath every derivative since.

One step. How the expected return disappears is shown by an example with a single time step and an interest rate of zero, which the box below this paragraph works through step by step: a share stands at 100 and in a year at either 120 or 80, and a call option with strike 100 costs 10 today, however probable the rise is. Dividing the year into many such steps gives a tree of ups and downs, the binomial model of 1979, and in the limit of infinitely many steps the formula.

Today

If a household with oil heating pays its dealer 180 euros in summer for the right to buy 3,000 litres of heating oil in winter at no more than 1.00 euro a litre, it holds a call option with a strike price of 1.00 euro. If the oil costs 1.20 euros in winter, it saves 3,000 · 0.20 = 600 euros, 420 euros after deducting the 180; if it costs 0.90 euros, it simply buys at the market price.

If a dealer takes back a mobile phone that cost 800 euros for 300 euros after two years, that is a put option: if the phone fetches only 200 euros privately, one gives it to the dealer.

An electricity tariff with a price guarantee is a derivative: if it guarantees 30 cents per kilowatt hour for 3,000 kilowatt hours a year and the market price rises to 35 cents, the guarantee is worth 3,000 · 0.05 = 150 euros a year; if the market price falls to 25 cents, it costs the customer 150 euros.

The Black-Scholes formula also values the price cap on heating oil: if the price stands at 1.00 euro a litre today, fluctuates by 20 per cent a year and the risk-free rate is 2 per cent, the right to buy at 1.00 euro in a year is worth 8.92 cents a litre by the formula, about 268 euros for 3,000 litres; the box “Step by step: the formula with numbers” further below works it out with the same numbers, writing the price of 1.00 euro as 100 cents.

If a household agrees with its dealer in June to take 3,000 litres of heating oil in October at 0.95 euros a litre, that is a forward contract: if the oil costs 1.10 euros in October, it saves 450 euros, if it costs 0.85 euros, it pays 300 euros more than necessary, because unlike with the option it must take delivery.

A heating-oil dealer who has sold many customers a price cap hedges too: he buys part of the oil in advance and puts it in storage; if the price rises, the stored oil gains what the cap costs him.

For the heating-oil cap, risk-neutral valuation means: its price does not depend on whether the dealer thinks a rise likely, but only on today’s price, the strike price, the term, the interest rate and how strongly the oil price fluctuates.

If the price of heating oil rises or falls from 1.00 euro by 10 cents in each of two quarters, after half a year it stands at 1.20, 1.00 or 0.80 euros, at an interest rate of zero with the risk-neutral probabilities ¼, ½ and ¼; a cap of 1.00 euro pays only in the first case, and for 3,000 litres it is worth ¼ · 0.20 · 3,000 = 150 euros.

From formula to industry

The generalisation. The decade after turned the formula into a theory. John Cox, Stephen Ross and Mark Rubinstein showed in 1979 that one gets the same price if one lets the price only rise or fall in each small time step as in the box above, a tree of ups and downs that can be worked through on a pocket calculator and comes ever closer to the formula as the steps get finer; the binomial model has since been the textbook model and the tool for American options, which may be exercised early.⁷³ Michael Harrison, David Kreps and Stanley Pliska proved in 1979 and 1981 the theorem behind it: a market is free of arbitrage exactly when probabilities can be assigned to the possible prices, like the ½ in the box, under which every discounted price is a martingale, that is, stays on average where it is. If every derivative can be replicated from the share and the safe investment, there is only one such assignment, and every derivative is then the expected value of its discounted payoff under these probabilities.⁷⁴ Samuelson’s martingale of 1965 and Black’s formula of 1973 thereby turned out to be related: the same mathematics, with Samuelson under the real probabilities, with Black under the risk-neutral ones. Oldřich Vašíček described in 1977 the interest rate itself as a stochastic process fluctuating around a long-run level and founded the term-structure model, the class of models with which bonds and interest-rate derivatives, derivatives whose value depends on interest rates, are valued; David Heath, Robert Jarrow and Andrew Morton gave in 1992 the framework in which the whole yield curve is random, the interest rates for all terms from one day to thirty years.⁷⁵ Robert Merton had already shown in 1974 that the equity of an indebted firm is a call option on the firm: the owners receive what the firm is worth beyond its debts, and nothing if it is worth less than the debts, and that is the payoff of a call option with the debts as strike price. With it the debt could be valued too, and the option reached credit risk, the risk that a debtor does not pay.⁷⁶ The value of the firm and its volatility, however, cannot be read off anywhere; on the stock exchange one sees only the value of the shares and their volatility, and from these two one calculates the two unknowns back, with the option formula and Itô’s rule as two equations for two unknowns. The credit model of the firm KMV later proceeded similarly, but step by step: it estimates the volatility of the firm value from the whole price history of the shares and goes back and forth until the values no longer change.⁷⁷

The industry. The formula met an exchange that needed it and computers that could do it. Black sold option prices as tables, traders programmed the formula into pocket calculators, and within a few years the market quoted options as if it had always valued them that way; the sociologist Donald MacKenzie has shown how prices approached the formula after traders used it, not before.⁷⁸ Hayne Leland and Mark Rubinstein founded in 1980 with John O’Brien a firm that sold portfolio insurance to pension funds, pools that invest the money for company pensions. Instead of buying a put option on the whole portfolio, which did not exist, one rebuilt it, first by shifting between shares and cash, from 1982 also with futures: following the formula, one sold shares or futures on a share index as prices fell and bought them back as they rose.⁷⁹ At the same time volatility, which the formula assumes known, acquired its own mathematics: Robert Engle showed in 1982 with the example of British inflation, the rise in prices, that the range of fluctuation of a time series, a sequence of measurements over time, can be predicted from its past fluctuations, calm phases follow calm ones and wild ones wild, and Tim Bollerslev generalised the model in 1986; the models ARCH and GARCH, whose names stand for (generalised) autoregressive conditional heteroskedasticity, a variance that depends on past fluctuations, have since been the standard tools for estimating the volatility of markets.⁸⁰

The swap. The largest derivatives market was created by a deal without a formula. In August 1981 the investment bank Salomon Brothers arranged an exchange between the World Bank and IBM: the World Bank took over payments of IBM in Swiss francs, IBM in return payments of the World Bank in dollars.⁸¹ A swap is such a contract to exchange payments over years; in an interest rate swap one side pays a fixed rate and the other a floating rate, which is reset at fixed intervals to a market rate, both on the same amount, and only the interest is exchanged. The first large interest rate swap in the United States was concluded in 1982 by the student loan company Sallie Mae and ITT Financial over 100 million dollars; at the end of 1982 such contracts over 5 billion were outstanding, and in 1985 ten dealers founded an association, today’s ISDA, which standardised the contracts.⁸² At the end of 2025 interest rate derivatives referred to 669.5 trillion dollars, four fifths of the notional value of all derivatives traded outside exchanges, that is, of the amounts the contracts refer to.⁸³

Today

For a household with a mortgage both the yield curve and the term-structure model are tangible. The yield curve is the bank’s table of rates, which might quote, say, 2.0 per cent for overnight money, 3.0 per cent for five years fixed, 3.4 per cent for ten and 3.7 per cent for fifteen years. And whoever pays off 240,000 euros at 3.5 per cent with 1,100 euros a month still owes 182,627 euros after ten years; whether he then pays about 3,650 euros of interest in the first year of the follow-up financing, because the rate has fallen to 2 per cent, or about 9,130 euros, because it has risen to 5 per cent, is exactly the uncertainty that a term-structure model describes.

A mail-order retailer that offers payment by invoice bears credit risk with every order: if 20 of 1,000 customers with invoices of 80 euros each never pay, 1,600 euros are missing, and he must build 1.60 euros per order into his prices.

On a small scale portfolio insurance looks like this: a securities account of 50,000 euros is never to fall below 45,000 euros. The cushion of 5,000 euros is multiplied by four, 20,000 euros go into an equity fund, 30,000 euros safely into a fixed deposit. If the fund falls by 10 per cent to 18,000 euros, the cushion has shrunk to 3,000 euros, only 12,000 euros may remain in the fund, and 6,000 euros are sold. That works as long as one can sell while the price falls; if it jumps overnight, the sale comes too late.

If a company has a loan of 200,000 euros whose rate is adjusted to the market rate every year and concludes an interest rate swap in which it pays 3 per cent fixed and receives the market rate in return, on balance it always pays 6,000 euros a year, whether the market rate goes to 2 or to 5 per cent. Without the swap it bears the risk itself, like every borrower at a floating rate: if on a loan of 2,000 euros the rate rises from 12 to 13.5 per cent, 30 days cost 22.19 instead of 19.73 euros.

Black Monday and the smile

Black Monday. On 19 October 1987 the Dow Jones, the best-known American share index, fell by 508 points or 22.6 per cent, the largest one-day percentage loss in its history, without a new piece of news that day that would have explained it. The commission under Nicholas Brady that examined the day in January 1988 named as triggers news from the week before: an unexpectedly high trade deficit that drove up interest rates, and a proposed tax law that made the shares of takeover candidates collapse. It gave portfolio insurance a prominent role in the fall: the programs sold because prices fell, prices fell because the programs sold. Three insurers alone accounted that day for just under a tenth of share sales on the New York Stock Exchange, and such programs together for over 40 per cent of futures trading, not counting the floor traders’ own dealing.⁸⁴ The formula had assumed that one can trade at any time at prices without jumps; on this day prices jumped, and the rebuilt put option, which consisted of continual trading, was none. The market has written the lesson into prices: since 19 October 1987 put options with a strike far below the current price, that is insurance against deep falls, have been dearer than the formula provides. If one reckons back from an option price which volatility explains it by the formula, one gets the implied volatility, and if one plots it against the strike price, a skewed curve has arisen since 1987, the volatility smile (Figure 6, after the formula further below): the market pays a premium for protection against crashes that the model does not know, because the market knows Mandelbrot’s fat tails.⁸⁵

The smile. Theory answered the smile by relaxing one assumption of the formula after another. Merton had built in price jumps as early as 1976, sudden moves such as October 1987 then showed.⁸⁶ Steven Heston gave a formula in 1993 for the case in which volatility itself fluctuates randomly and is related to the price, and Bruno Dupire, Emanuel Derman with Iraj Kani, and Mark Rubinstein showed independently in 1994 how to compute from market prices a volatility that depends on price and time so that the model values all options correctly at once.⁸⁷ Patrick Hagan and colleagues objected in 2002 that this model lets the smile move with the price in the wrong direction, and proposed a model with random volatility that became known by the abbreviation SABR.⁸⁸ Since then one fits models to market prices instead of predicting prices from a model.

The prize. On 16 October 1990 Markowitz, Miller and Sharpe were awarded the Nobel Prize in economics, in 1997 Merton and Scholes; Fischer Black had died in 1995, and the prize is not awarded after death.⁸⁹ The academy’s citation named “a new method to determine the value of derivatives”, and by then the method had helped to create a market whose notional values far exceeded the stock markets: in mid-1998 the derivatives traded outside exchanges referred to 72 trillion dollars, while all listed shares in the world had been worth about 22 trillion dollars at the end of 1997.⁹⁰ The footnote of the story is that in 1997 Scholes and Merton sat on the board of a hedge fund, a fund, that is a common pool for the money of many investors, that trades with few restrictions and much borrowed money, and that this fund would show a year later what the method cannot do.

Today

If a heating-oil dealer asks not 8.92 but 12 cents a litre for the heating-oil cap in the “Today” part on the option formula, then, reckoned back through the formula, he is assuming a volatility of 27.9 instead of 20 per cent.

Working it out and practice

Try it yourself. The formula fits in any spreadsheet in five cells. With price S, strike K, time T in years, rate r and volatility σ, d1 is =(LN(S/K)+(r+σ^2/2)*T)/(σ*SQRT(T)), d2 =d1-σ*SQRT(T) and the value of the call =S*NORM.S.DIST(d1,1)-K*EXP(-r*T)*NORM.S.DIST(d2,1); instead of the letters one writes the cells in which the numbers stand. EXP(x) is e to the power of x, and EXP(-r*T), e to the power of minus r times T, is the factor by which one discounts over T years under continuous compounding. NORM.S.DIST(x,1), N(x) for short, is the probability that a normally distributed random number with expected value zero and standard deviation one is smaller than x. The box below this paragraph works the formula through for S = K = 100, T = 1, r = 2 per cent and σ = 20 per cent and arrives at 8.92 (Figure 5, where sliders change volatility and maturity and a market price can be turned back into its implied volatility). Whoever then calls up in a broker’s option list the prices of put options with different strikes and seeks for each the volatility that matches the price sees the smile himself: the number rises the further the strike lies below the price.

The bottleneck

By 1987 the discipline had the prices of individual risks and an industry that traded them. What it did not have was a measure of the sum: how much can a bank holding ten thousand positions, that is individual deals and holdings, in a hundred markets lose on a bad day, and how much equity must it hold for that? The supervisors, the authorities that watch over the banks, had been working on capital rules since the Latin American debt crisis of 1982; after the crash they asked the question more loudly, and the answer that came was a single number that reassured the discipline for a decade.

6. Risk, credit and the crisis (1988–2023)

Supervision and value at risk

Supervision. The first answer came from the banks of the states. A central bank is the bank of a state or a currency, which issues the money, sets the interest rate for the banks and often watches over them, in Germany the Bundesbank, for the euro the European Central Bank. The Basel Committee on Banking Supervision, in which the supervisors of the large industrial countries sit, adopted in July 1988 the first capital accord, named Basel I after the committee’s seat: every internationally active bank must hold equity of eight per cent of its assets, weighted by their risk. Assets are what a bank owns, above all its loans, and a risk weight states how much of a loan counts: zero for governments of the OECD, the organisation of the large industrial countries, half for a mortgage, a loan for a house secured by the house itself, and all of it for loans to firms.⁹¹ Whoever lends 100 million to a firm therefore needs 8 million of equity, whoever lends 100 million for houses 4 million; the box below this paragraph works it out for a whole bank. Basel I covered only credit risk, was coarse and easy to circumvent, but it was the first time that risk was measured worldwide in one number and backed with equity. The revision of June 2004, Basel II, allowed the large banks to estimate the risk weights with their own models, and so made supervision dependent on exactly the models it was supposed to supervise.⁹¹

The number. The second answer came from a bank. Dennis Weatherstone, chairman of J.P. Morgan from 1990, received from 1990 every day, after a meeting at 4:15 p.m., a quarter of an hour after the close of the exchange, a report that told him how much the bank could lose across all its trading departments the next day; the number in it came from a system the bank had built from older ideas in the late 1980s,⁹² the value at risk: the loss that on one day will not be exceeded with a probability of, say, 99 per cent, calculated from the volatilities and correlations of all positions.⁹³ How it is calculated for a single position is shown by the box below this paragraph: for 10 million euros in a share index that fluctuates by one per cent a day it is 233,000 euros for one day and 737,000 euros for the ten trading days that the Basel Committee prescribed in 1996.⁹⁴ In October 1994 the bank published the method under the name RiskMetrics together with free data for several hundred market factors, and within a few years the number was the standard of the industry and of the supervisors, who admitted it in 1996 for the calculation of equity.

The objection. Four mathematicians, Philippe Artzner, Freddy Delbaen, Jean-Marc Eber and David Heath, showed in 1999 what is wrong with value at risk. It says nothing about how large the loss is when the threshold is exceeded, and for two portfolios put together it can be larger than the sum of the individual values, which punishes diversification instead of rewarding it; the box below this paragraph shows it with two bonds. They set up four conditions a sensible risk measure must satisfy, among them exactly this one: the risk of two portfolios together must never be larger than the sum of their individual risks. Measures that satisfy all four they called coherent, that is consistent in themselves; the expected shortfall is the average of the losses in the worst cases, say the worst one per cent, and it is coherent, as Carlo Acerbi and Dirk Tasche proved for all distributions in 2002 (Figure 7).⁹⁵ It took seventeen years and a crisis until the supervisors adopted it.

Today

If the central bank raises its rate by 0.5 percentage points and a bank passes 0.3 points of it on to its customers, 20,000 euros in a savings account bring 60 euros more a year.

If a bank lends 20,000 euros to a tradesman’s business, under Basel I it must hold 1,600 euros of its own money for it, if it lends 20,000 euros to a household for a house, only 800 euros; because equity is expensive for the bank, this is one reason it can offer the loan for the house more cheaply.

The home loan from section 5 is a mortgage: if a household buys a house for 300,000 euros with 60,000 euros of its own money, it borrows 240,000 euros, and the house stands as security; if it does not pay, the bank can realise it. At 3.5 per cent interest and 2 per cent repayment it pays 240,000 · 0.055 / 12 = 1,100 euros a month.

For a private securities account of 50,000 euros that fluctuates by 1 per cent a day, the value at risk at 99 per cent is 2.33 · 500 = 1,165 euros for one day and 1,165 · √21 = 5,339 euros for a month of 21 trading days: on average on one day in a hundred the owner loses more than 1,165 euros.

For the account of 50,000 euros value at risk says only that on one day in a hundred more than 1,165 euros are lost; expected shortfall says how much is lost on average on these bad days, about 1,333 euros if prices are normally distributed.

The fund and leverage: LTCM

The fund. That the finest theory does not tame the market was shown in 1998 by the fund in which the theory sat. Long-Term Capital Management, founded in 1994 by John Meriwether, with Merton and Scholes on the board, bet on small price differences between almost identical securities, say a freshly issued and a six-month-old American Treasury bond, and magnified the tiny gains with leverage, the ratio of borrowed money to one’s own, which stood at around 25 to one at the start of 1998. At leverage of 25 to one a loss of just under 4 per cent on all positions was enough to eat up all of its own money, as the box below this paragraph works out. After Russia stopped paying its debts in August 1998, all positions moved in the wrong direction at once, because all investors fled at once into the safest securities: correlations that had been small in the data of the past became one. The fund lost 4.6 billion dollars in less than four months, and on 23 September 1998 the Federal Reserve Bank of New York, the New York branch of the American central bank, organised a rescue by fourteen banks and brokers with 3.625 billion dollars, for fear that the forced sale of all positions would drag down the financial system, the whole of the banks and markets.⁹⁶ The lesson Mandelbrot in 1963 and October 1987 had already taught now stood in numbers: the models were right in normal times, and the times in which one needed them were not normal.

Today

The house from section 4, bought with 80,000 euros of own money and a loan of 320,000 euros, also has leverage, four to one; if its price falls by 20 per cent, the equity is gone entirely.

Loans, tranches and the crisis of 2008

The score. Before mathematics reached the large loans, it had reached the small ones. David Durand examined in 1941 for the National Bureau of Economic Research, an American research institute, loans to consumers that are repaid in fixed instalments. He divided a sample, a selection of loans, into good ones, which had been repaid, and bad ones, where that had not happened, and looked for the characteristics that separate the two groups: steady work, a steady home, owning property and a large down payment were more common among the good loans. From this he made formulas that give each characteristic a weight, so that the weighted sum, a single figure, separates the two groups as well as possible.⁹⁷ The engineer Bill Fair and the mathematician Earl Isaac founded in 1956 a firm that sold such formulas to banks and retailers, and in 1989 it brought out the first general-purpose score, the FICO score, which turns the data of the credit bureaus, the firms that keep records of consumers’ loans, into a number between 300 and 850: the higher, the lower the risk.⁹⁸ This credit scoring, the assessment of creditworthiness with a score, has since helped decide in the United States who gets a loan and at what interest rate.

Credit. In the 1990s mathematics reached the largest market, that of debt. Merton had described in 1974 the default of a firm, its inability to pay, as an event that occurs when its value falls below its debts; Robert Jarrow and Stuart Turnbull described default in 1995 instead as a random event with a default intensity, a probability per year that can be estimated from market prices, and Darrell Duffie and Kenneth Singleton built this into a general framework in 1999.⁹⁹ With that, the credit derivative could be valued, a contract whose payment depends on the default of a debtor, above all the credit default swap, CDS for short, an insurance against the default of a bond: the buyer pays a premium every year and has his loss made good if the debtor defaults. The outstanding volume, the amount of all running contracts, stood at 62 trillion dollars at the end of 2007, about as much as the market value of all listed companies in the world, some 60 trillion dollars, although the notional amount is not the market value of the contracts.¹⁰⁰ The question on which the theory failed was that of the connection: how probable is it that two, a hundred, a thousand debtors default together? David X. Li, an actuary at the RiskMetrics Group, a firm spun off from J.P. Morgan in 1998, proposed in 2000 an answer so elegant that the industry adopted it before it tested it: a copula, a function that joins the default probabilities of individual debtors into a joint distribution, namely the Gaussian copula, named after the bell curve, which is also called the Gaussian curve, with a single adjustable number for the correlation that could be read off from the prices of credit derivatives.¹⁰¹

The tranches. Banks bundled many mortgages into securities whose payments come from the homeowners’ instalments, the mortgage securities. With this one number, packages of hundreds of bonds, loans and such mortgage securities, which the banks bundled and sold as securities, could be valued once cut into tranches, layers of different rank: the lowest bears the first defaults, the top one only when all those below are used up. The rating agencies, firms that grade the safety of debtors with marks from AAA, the best, downwards, gave the top tranche the best grade, because the model said that not all default at the same time (Figure 8).

The crisis. They defaulted at the same time. The correlation estimated from a few years of rising house prices did not hold for falling ones, and when American house prices stopped rising in 2006 and fell from 2007, tranches that had been rated as safe as government bonds lost most of their value. On 15 September 2008 the bank Lehman Brothers, with over 600 billion dollars in assets, filed the largest bankruptcy in American history, and the banks of the world found that nobody knew who owed whom how much from which contracts.¹⁰² Felix Salmon called Li’s formula on 23 February 2009 in the magazine Wired “the formula that killed Wall Street”, New York’s financial world, which was unjust to the formula and just to its use: Li himself had warned that very few users understood what they were assuming.¹⁰³ The word for this kind of failure is model risk: the loss that arises not from the market moving but from the model with which it was measured being wrong, and precisely when everyone uses the same model.

The lessons. The supervisors answered mathematics with mathematics. Basel III, published in December 2010, demanded more and better equity, a limit on borrowing that does without risk weights, and rules for liquidity, the ability to make payments at any time because one has enough cash or securities that can be sold quickly; the revision of the rules for the trading book, the positions with which a bank trades, of January 2016 provides for replacing value at risk with expected shortfall, calculated for a period of severe crisis, seventeen years after Artzner; in October 2026 this is in force neither in the EU nor in the United Kingdom nor in the United States; in the EU it is to apply from 1 January 2027, in the United Kingdom for internal models from 1 January 2028.¹⁰⁴ For insurers in the European Union Solvency II has applied since 1 January 2016, a directive, that is a European law which the states transpose into their own: every insurer must hold enough equity to survive the losses of one year with a probability of 99.5 per cent, a value at risk over one year; Lundberg’s question of ruin from section 2 has thus become an obligation.¹⁰⁵ In the United States the Federal Reserve, the central bank, and the bank supervisor OCC required in 2011 that every model be checked by staff who neither built nor use it, and the word model risk acquired guidelines, committees and posts.¹⁰⁶ What remained was the basic question that Mandelbrot, LTCM and the copula had posed together: every model estimates the future from the past, and the events against which one wants to protect oneself are those for which there is no past.

Today

An invented points table shows the principle: a steady job for more than two years 30 points, the same address for more than three years 20 points, a current account without arrears 25 points, a down payment of more than a fifth 15 points, and from 60 points the instalment loan is granted. Real methods use more characteristics, and the weights are estimated, as by Durand, from old loans.

If a debtor has a default intensity of 2 per cent a year, he keeps paying for five years with probability 0.98⁵ = 0.904; of 100 such loans about 10 therefore default within five years.

Whoever stands surety for a friend’s loan of 10,000 euros has in effect sold a CDS, usually without a premium: if the friend defaults with a probability of 3 per cent a year and nothing can then be recovered, 300 euros a year would be a fair premium for this risk.

How much the correlation in the Gaussian copula matters is shown by a surety for three friends, each of whom defaults with probability 5 per cent: if their defaults are independent, all three default together only with probability 0.05³, 1 in 8,000; if all three work for the same firm and their defaults are linked by a Gaussian copula with correlation 0.6, it is 1 in 130.

If a bank bundles 1,000 home loans of 200,000 euros each and sells the bundle as a security, the buyers receive the interest and repayments of the 1,000 households, and the bank has the money for new loans.

Every creditor with a rank knows the principle of tranches: if a bank has lent 200,000 euros with priority and a relative 60,000 euros subordinated for a house, and the sale after a default brings 230,000 euros, the bank gets everything, the relative 30,000 euros, half, and the owner nothing.

On a small scale everyone who plans for old age knows model risk: whoever saves 300 euros every month for 30 years and reckons with a return of 7 per cent expects 365,991 euros; if the money brings only 4 per cent, it is 208,215 euros, almost 158,000 euros less, and the gap shows only when it is too late.

In a household, too, liquidity is something other than wealth: whoever has 60,000 euros in a fund but an empty current account and must pay a car repair of 2,000 euros at once goes into overdraft, which at 12 per cent interest costs about 40 euros for two months; rich but not liquid, that is the reason for a reserve in a savings account.

Interest rates after the crisis

The price of the other side. The crisis also broke a habit so self-evident that nobody regarded it as an assumption. Banks discounted future payments with LIBOR, the rate at which banks in London lend each other money for three months, of which section 7 tells more, as if it were safe. Safer is an overnight indexed swap, a contract in which two banks exchange interest payments: one pays a fixed rate, the other the overnight rate, the interest on money that banks lend each other for a single night, compounded over the term. Because no money flows at the start and only the difference at the end, hardly anything can be lost. The gap between LIBOR and the fixed rate of such a swap over three months therefore measures how much banks distrust each other: before August 2007 it stood at about 0.1 percentage points, on 10 October 2008 at 3.65.¹⁰⁷ On 17 June 2010, LCH.Clearnet, a clearing house, a body that places itself between buyer and seller in every transaction and stands in for both, announced that from 29 June it would no longer discount the interest rate swaps of 218 trillion dollars in its SwapClear service with LIBOR but with the rates of overnight indexed swaps; the industry followed.¹⁰⁸ At the same time counterparty risk acquired a price, the risk that the other side of a contract defaults before it has paid. CVA, the credit valuation adjustment, is the deduction from the value of a derivative that expresses this risk, according to the Basel Committee on Banking Supervision “the price of counterparty credit risk”. By its assessment, about two thirds of the losses from counterparty risk in the crisis came not from defaults but from growing deductions, because defaults became more likely, and only about one third from actual defaults. Basel III therefore requires capital against fluctuations in CVA as well; the rules for this were already in the Basel III text of December 2010, the Committee finalised them on 1 June 2011 and revised them in 2020, applicable from 1 January 2023 under the Committee’s timetable, in the EU since 1 January 2025 (box below).¹⁰⁹

Interest rates after the crisis. After the crisis interest rates fell further, below zero. A negative interest rate means: whoever invests money gets back less than he gave. Sweden’s central bank was the first, from 8 July 2009, to charge 0.25 per cent on money that banks deposited with it, and Denmark followed in 2012.¹¹⁰ The European Central Bank lowered its deposit rate, the rate on money that banks park with it overnight, to −0.10 per cent on 11 June 2014 and to −0.50 per cent by September 2019; Switzerland went to −0.75 per cent in 2015, Japan to −0.1 per cent in 2016.¹¹¹ In between lay the euro crisis, in which investors doubted the solvency of Greece and other states of the euro area: Greece received aid of 110 billion euros in 2010 and in 2012 a haircut of 53.5 per cent of face value, and on 26 July 2012 the President of the ECB, Mario Draghi, promised in London to do “whatever it takes” to preserve the euro; in September the bank announced that it would buy bonds of such states, under conditions, without an upper limit.¹¹² The yield on ten-year German federal bonds fell below zero for the first time in June 2016, and on 13 July 2016 the federal government sold a ten-year bond with a zero per cent coupon at an average yield of −0.05 per cent: the buyers paid more than 100 and got 100 back ten years later.¹¹³ With negative rates a formula failed as well, because the usual model for interest rate options assumes positive rates; in the market Bachelier’s model, which allows negative values, prevailed for them.¹¹⁴

The turn in interest rates. Then inflation returned, the rise in prices on average, measured by a basket of goods and services: in October 2022 prices in the euro area were 10.6 per cent above the previous year, in Germany they rose by 6.9 per cent on average over 2022.¹¹⁵ Irving Fisher had shown in 1896 how interest and the value of money are connected: the nominal interest rate, the rate in money, is roughly the real interest rate, the gain in purchasing power, plus inflation, exactly (1 + nominal rate) = (1 + real rate) · (1 + inflation); a zero rate at ten per cent inflation is in real terms a loss of 9 per cent.¹¹⁶ The American central bank raised its rate from March 2022, the ECB by 0.50 percentage points on 21 July 2022, and by September 2023 its deposit rate had risen to 4.00 per cent, the yield on ten-year federal bonds in October 2023 to 2.99 per cent; in 2024 and 2025 the ECB lowered the rate to 2.00 per cent, and in 2026 it raised it again, to 2.50 per cent since 16 September 2026.¹¹⁷ Rising rates lower the price of outstanding bonds, the more so the longer their duration, and that hit investors who felt safe. British pension funds had bought long government bonds on credit; after the government’s budget of 23 September 2022 the yield on thirty-year gilts rose by 1.6 percentage points in a few days, the funds had to post more collateral, that is, lodge additional money or securities with their lenders as a pledge, and sell bonds to do so, which pushed prices down further, until the Bank of England intervened on 28 September and bought bonds worth 19.3 billion pounds by mid-October.¹¹⁸ Silicon Valley Bank in California had invested its customers’ deposits in bonds with an average duration of 5.7 years; by the end of 2022 these had lost more than 17 billion dollars in value, more than its equity of 16 billion, and when it sold bonds at a loss of 1.8 billion on 8 March 2023, customers withdrew over 40 billion the next day; on 10 March it was closed.¹¹⁹ The calculators below the box “Working it out: government bonds today” at the end of this section apply Macaulay’s duration and the other formulas to German federal bonds.

Today

If a household agreed a fixed price for heating oil in June that is worth 450 euros in October because oil has risen to 1.10 euros, the 450 euros are safe only if the dealer does not go bankrupt by then; likewise a down payment to a tradesman is lost if he becomes insolvent before doing the work.

If the heating-oil dealer defaults with probability 2 per cent and nothing can then be recovered, the fixed price worth 450 euros is worth 0.02 · 450 = 9 euros less, at 5 per cent 22.50 euros less; these 9 or 22.50 euros are the CVA.

If a bank charges 0.5 per cent a year on everything above 50,000 euros, a balance of 100,000 euros costs 250 euros a year.

A shopping basket that cost 100 euros costs 106.90 euros a year later at 6.9 per cent inflation, and 1,000 euros under the mattress buy only as much as 1,000 / 1.069 = 935.45 euros bought before.

If a savings account brings 3 per cent at 2 per cent inflation, the real rate is 1.03 / 1.02 − 1 = 0.98 per cent; if a mortgage costs 4 per cent at 3 per cent inflation, in real terms it costs only 0.97 per cent, because the debt loses value every year.

Working it out and practice

Try it yourself. Value at risk is one line in a spreadsheet. One downloads the daily prices of an index over a year, forms the 250 daily returns and asks with =PERCENTILE(B2:B251,0.01) for the quantile at one per cent, the value that only one per cent of days fell below; B2:B251 is the range of cells with the returns. That is the historical value at risk at 99 per cent, expressed as a return. The expected shortfall is the average of the two or three worst days. Then one does the same with the normal distribution, =AVERAGE(B2:B251)-2.33*STDEV(B2:B251), and compares the quantile with this value. The worst day of the year, by contrast, is compared with the mean minus 2.8 standard deviations, because that is how low the worst of 250 days lies on average under the bell curve. Whoever has 2008 or 2020 in the data sees Mandelbrot’s fat tails with the naked eye.

The bottleneck

By 2009 the discipline had learned its limits and cast them in rules. At the same time something else had grown, more quietly than the crisis: the quantity of data and the speed with which machines traded on it. The formulas of the epoch were made for people who understand them and put them into models; the next epoch belonged to programs that learn from the data without a formula and in doing so become faster than anyone who supervises them.

7. Factors, machines and data (1988–2022)

Factors and Medallion

The factors. The CAPM had failed a test. Eugene Fama and Kenneth French showed in June 1992 that beta hardly explains the differences in share returns of the years 1963 to 1990, but two other quantities do: small firms and cheap firms had persistently earned more than large and expensive ones. Cheap here means a low price relative to book value, the value of the equity in the company’s own accounts, that is its assets less its debts.¹²⁰ In 1993 they built from this the three-factor model, in which the return of a share hangs on three quantities, the market, size and value, and each of these quantities is a factor: a common influence that moves many securities at once and for bearing which the market pays a premium. Narasimhan Jegadeesh and Sheridan Titman found in the same year that shares that have risen in the last months go on rising in the next half year, a momentum that should not exist in an efficient market and that Mark Carhart added as a fourth factor in 1997.¹²¹ Fischer Black and Robert Litterman showed in 1992 at the bank Goldman Sachs how to mix Markowitz’s calculation with one’s own views about the future without its delivering portfolios that bet everything on a few securities, and made portfolio theory usable for asset managers, who invest other people’s money.¹²² When Fama received the Nobel Prize in 2013, he shared it with Lars Peter Hansen and with Robert Shiller, who had argued since 1981 that prices fluctuate too much to reflect only information; the academy honoured the thesis and its contradiction at once (Figure 9).¹²³

The machine that trades. The largest return of the epoch was earned by a mathematician who wanted no theory. James Simons, geometer and codebreaker, founded Renaissance Technologies in 1982 and the Medallion Fund in 1988, which with statisticians, physicists and specialists in speech recognition by computer sought patterns in price data without asking why they exist; by the reconstruction of the journalist Gregory Zuckerman the fund earned from 1988 to 2018 on average 66 per cent a year before and 39 per cent after fees.¹²⁴ Quantitative trading is trading by rules estimated from data and executed by programs, and Medallion was its proof that the market is not as efficient as Fama said, or at any rate not for whoever calculates faster and more precisely than everyone else. Simons died on 10 May 2024 in New York;¹²⁵ the secret of his signals, the patterns that announce a price movement, has never been published.

Today

For a saver the factor model means something like this: if he adds to a worldwide equity fund a fund that buys only cheap companies, he is betting on the value factor. If the market brings 6 per cent a year, 10,000 euros become 17,908 euros in ten years; if the factor pays a premium of 2 percentage points, it is 21,589 euros, if it lags 2 points behind for a decade, only 14,802 euros.

Whoever buys from a list of the shares that have risen most in the last six months is betting on momentum.

On a small scale a robo-adviser on the internet trades by rules like quantitative trading: if a securities account of 10,000 euros is to consist of 60 per cent shares and 40 per cent bonds and the shares rise by 25 per cent, they make up 7,500 of 11,500 euros, 65.2 per cent; the rule sells shares worth 600 euros until 60 to 40 is restored, without a person deciding.

Speed and crash

Speed. What Simons did in days, others did in microseconds, millionths of a second. High-frequency trading, in which programs place and withdraw orders to buy and sell in fractions of a second, made up a large part of the quantity traded on the American exchanges in the 2000s, and on 6 May 2010 it showed what it can do: at 2:32 p.m. an investment fund began to sell 75,000 futures contracts on the S&P 500, an index of the 500 largest American companies on the exchange, with a program that paid attention only to quantity, not to price or time; the Dow Jones was at times nearly 1,000 points down, individual shares were traded for one cent and for 100,000 dollars, and after 36 minutes most of the loss had been recovered.¹²⁶ The report of the regulators of autumn 2010 described a chain of programs reacting to one another until nobody bought any more; in April 2015 a trader was arrested in London who is said to have contributed with bogus orders he never meant to execute; he pleaded guilty in 2016 and was sentenced in 2020 to a year of home detention.¹²⁷ The flash crash began with a person who left a large sale to a program; what happened next, machines did among themselves, and it was the first public demonstration of what occupies the central banks in 2026: that programs which see the same data do the same thing.

The emergency brake. That identically built programs crash together the discipline had already experienced three years before the flash crash. In the week from 6 August 2007 quantitative equity funds, which bought shares by similar characteristics and sold others short, lost more in a few days than ever before; Amir Khandani and Andrew Lo explained this by the rapid sale of one or more large portfolios, which pushed down the prices of all similarly built funds, and on 10 August the strategies recovered strongly.¹²⁸ A short sale is the sale of a borrowed share that one hopes to buy back more cheaply later; whoever buys and sells short at the same time bets only on the gap between the shares, not on the market. The American exchanges had introduced market-wide trading halts in 1988, after October 1987, pauses during steep falls in prices; on 27 October 1997 they were triggered for the first time and in March 2020 four times within ten days.¹²⁹ After the flash crash, trading pauses for individual shares were added in the United States in June 2010, and fixed price bands from April 2013, and in the European Union every firm that trades with algorithms must since 3 January 2018 be able, as an emergency measure, to withdraw all its open orders at a stroke.¹³⁰

Today

How high-frequency trading earns is shown by a simple calculation: 0.1 cents per share is not worth mentioning for someone buying 20 shares, but on 10 million shares a day it comes to 10,000 euros.

If someone borrows 10 shares at 50 euros, sells them for 500 euros and later buys them back at 40 euros to return them to the lender, he earns 100 euros; if the share rises to 80 euros, he loses 300 euros, and there is no upper limit.

How prices arise: the order book

The order book. How prices arise from orders was a question the discipline long did not ask. Mark Garman gave the field its name with an article in 1976: market microstructure, the study of how the rules and institutions of trading shape prices.¹³¹ According to a survey from 2013, more than half of the world’s financial markets already ran on an order book, a list of all orders that want to buy or sell at a fixed price, ranked by price and, at the same price, by time; when an order meets a matching one on the other side, a trade takes place.¹³² The lowest price at which someone wants to sell is called the ask, the highest at which someone wants to buy the bid, and the gap between the two is the bid-ask spread. Harold Demsetz interpreted it in 1968 as the price of immediacy: whoever wants to trade at once instead of waiting for a counterparty pays it to the market maker, a dealer who constantly quotes a buying and a selling price at the same time.¹³³ Lawrence Glosten and Paul Milgrom showed in 1985 that the spread remains even when trading costs nothing: among the market maker’s customers are some who know more than he does, he loses to them, and the spread recovers this loss from the others (box below).¹³⁴ In the same year Albert Kyle described how a trader with knowledge splits up his orders so that the price does not give away his knowledge too quickly, and how the price moves a little further with every order, the further the larger the order.¹³⁵ Robert Almgren and Neil Chriss turned this in 2000 into a calculation for every large investor: whoever sells quickly pushes the price down; whoever sells slowly bears the risk for longer that the price moves anyway.¹³⁶ What the spread costs was shown by the switch of the American exchanges to cents. Until 2000 shares were quoted in sixteenths of a dollar, 6.25 cents; following an order of the Securities and Exchange Commission of 8 June 2000, the exchanges moved to cents step by step from 28 August 2000, and everywhere from 9 April 2001. The spreads actually paid then fell by about half on Nasdaq and by about 28 per cent on the New York Stock Exchange.¹³⁷

Today

If the same share costs 50.10 euros on one trading venue and 50.14 euros on another, that makes 4 euros on 100 shares; where an order goes is a question of microstructure.

If someone wants to sell 30 shares at once and the order book holds buyers for 20 shares at 49.90 euros and behind them for 10 shares at 49.85 euros, he receives 20 · 49.90 + 10 · 49.85 = 1,496.50 euros.

If a trading app shows “buy 50.05 euros” and “sell 49.95 euros”, buying 20 shares and selling them again at once costs 20 · 0.10 = 2 euros, as at a bureau de change that sells currency dearer than it buys it.

A used-car dealer does the same as a market maker: he buys a car for 9,000 euros and sells it for 10,000 euros, and the 1,000 euros are the price for nobody having to wait for a private buyer.

LIBOR and cryptocurrencies

The rate that was not. The basic quantity of the discipline too turned out to be man-made. LIBOR, the London interbank rate, the rate at which banks in London lend each other money, on which loans and derivatives worth hundreds of trillions of dollars hung, was formed not from transactions but from the daily submissions of a handful of banks as to what they believed they would pay; on 27 June 2012 the bank Barclays paid 290 million pounds in penalties because its traders had shifted the submissions to their advantage for years, and other banks followed.¹³⁸ A reference rate is the rate to which contracts refer, for instance a loan that bears interest every year at LIBOR plus two percentage points, and the mathematics of interest had taken it as given. The supervisors replaced it with rates from real transactions: the American committee chose on 22 June 2017 the SOFR from the market for secured overnight loans, loans for a single night against a pledge of government bonds; at the end of 2021 the LIBOR rates for sterling, euro, franc and yen ended, on 30 June 2023 the panel-based ones for the dollar; substitute synthetic rates ran until 30 September 2024.¹³⁹ The transition required rewriting millions of contracts and all interest-rate models, and it reminded the discipline that its inputs are made by people.

Money without a bank. On 31 October 2008, six weeks after the bankruptcy of Lehman Brothers, someone under the name Satoshi Nakamoto sent a paper on electronic money that does without a bank to a mailing list on cryptography, the study of encryption.¹⁴⁰ Bitcoin, the first cryptocurrency, a money whose units exist only as entries in a shared register secured with the methods of cryptography, went into operation on 3 January 2009 with the first block; it contains the headline of the London Times of the same day, that the Chancellor was on the brink of a second bailout for banks. The supply is limited to 21 million units.¹⁴¹ The register is a blockchain, a chain of blocks each of which contains the transactions of a period and a fingerprint of the previous one, so that nothing old can be changed without recalculating everything later; who may attach the next block is decided in Bitcoin by a computing race, proof of work. Ethereum, in operation since 30 July 2015, allows programs to run on the chain, and on 15 September 2022 it switched to a method without a computing race, proof of stake.¹⁴² On 2 November 2018 Uniswap went into operation on Ethereum, an automated market maker: a program that instead of an order book holds a pool of two currencies and forms the price solely from the ratio of the two quantities, so that their product stays the same, x · y = k; in the first two versions every swap cost a fee of 0.3 per cent for those who had filled the pool; since 2021 each pool chooses its fee (box below).¹⁴³ The other side showed itself in 2022. TerraUSD, a stablecoin, a cryptocurrency meant to hold a fixed value, usually one dollar, lost its peg in May; according to the American prosecution the losses came to about 40 billion dollars, and its founder Do Kwon was sentenced to 15 years in prison on 11 December 2025.¹⁴⁴ On 11 November 2022 the crypto exchange FTX filed for bankruptcy; its founder Sam Bankman-Fried was found guilty on 2 November 2023 and sentenced to 25 years in prison on 28 March 2024.¹⁴⁵ The European Union regulated the market with the MiCA regulation of 31 May 2023, whose rules for stablecoins have applied since 30 June 2024 and otherwise since 30 December 2024, and the American SEC approved on 10 January 2024 the first exchange-traded funds in the United States that hold bitcoin itself; Canada had had them since February 2021.¹⁴⁶

Today

On a loan of 100,000 euros at a reference rate of 3 per cent plus 2 percentage points, the borrower pays 5,000 euros a year; if the reference rate is reported just 0.01 points too high, that is 10 euros too much; across hundreds of trillions of dollars, billions.

If someone buys a cryptocurrency for 50 euros in an app, pays 1 per cent in fees and the price then falls by 30 per cent, 49.50 · 0.7 = 34.65 euros remain.

One can imagine a blockchain as the household account book of a shared flat, which every member has a copy of and each page of which carries a check number of the previous page: whoever forges an old page would have to change all later pages in all copies.

Whoever puts 500 euros into a coin meant to hold one euro loses 15 euros at a price of 0.97 euros and 450 euros at 0.10 euros.

Learning machines

Learning. The data grew faster than the formulas. That programs learn patterns from examples instead of from rules a person gives them, machine learning, had arrived in financial mathematics too in the 2010s, first where there was much data and little theory. A neural network is such a program made of many simple, connected steps of calculation whose weights, the numbers by which each step multiplies its inputs, it adjusts while learning until it reproduces the examples well. Shihao Gu, Bryan Kelly and Dacheng Xiu compared in 2020 a dozen methods on sixty years of American stock data with almost a thousand characteristics, numbers about the companies and their prices, and found that neural networks and tree methods, which divide the data into groups by a series of yes-or-no questions, predict returns better than the factor models, which only multiply each characteristic by a fixed number and add up. The reason is above all interactions, characteristics that only mean something together, and the most important signals stayed the same: momentum, liquidity, that is how easily a share can be traded, and volatility.¹⁴⁷ Marcos López de Prado described in a textbook in 2018 why the methods from other fields fail on the exchange if applied naively, because financial data contain few independent observations and every pattern once found is traded away by those who find it.¹⁴⁸ Hans Buehler and colleagues showed in 2019 that a neural network can learn the hedging of an option directly from simulated paths, price histories the computer generates with random numbers, and with trading costs and jumps the formula does not know: deep hedging replaces the derivation of 1973 with training, the adjusting of the network on examples, and it is the first method of the discipline that finds a hedge without being able to justify it.¹⁴⁹

The simulation. At the margin, experiments began with quantum computers, computers that use the laws of quantum physics. They are meant to estimate the expected value of an option faster than a Monte Carlo simulation, named after the casino, which works out an expected value by generating very many random cases and taking their average: for the same precision they need only the square root of the number of cases, a thousand instead of a million. For pricing options they are still not enough today; in 2025 the bank HSBC reported, with IBM, a first trial with real trading data in bonds.¹⁵⁰

Today

If a banking app stops a payment of 900 euros abroad at three in the morning, a machine-learning program may be behind it, one that has learned from thousands of examples what fraud looks like without anyone prescribing a rule for it.

A single step of calculation of a neural network for a loan application might compute, say, 0.002 · monthly income − 0.01 · monthly instalment; the network adjusts the numbers 0.002 and 0.01 and thousands of others until it matches the old cases well.

If a heating-oil dealer must pay a fee of 2 euros for every adjustment of his stock, daily rehedging on 250 days costs 500 euros a year, weekly rehedging 104 euros; a network trained for deep hedging with these costs learns when rehedging pays.

Monte Carlo simulation is met by everyone whom a savings calculator tells that a goal will be reached “with 90 per cent probability”. If someone saves 200 euros a month for 20 years in a fund that brings 6 per cent a year on average at 15 per cent volatility, and the computer generates 10,000 random paths, at the end there are about 93,000 euros in the account on average, about 83,500 euros in the middle path, and in about 9 of 100 paths less than the 48,000 euros paid in.

Working it out and practice

Try it yourself. Kenneth French provides on his page at Dartmouth College the factor returns since 1926 free of charge, month by month. Whoever lays the returns of a fund of his own beside them can test with a regression which factors the fund holds. A regression looks for the numbers with which one quantity is best composed of others, like the straight line for beta in section 4, only with several quantities at once; in a spreadsheet the function LINEST calculates it. The three numbers it delivers for market, size and value show how strongly the fund holds each factor, and the rest, the intercept or alpha, shows whether it can still do anything after deducting them. If a bank recommends a fund that charges 1.5 per cent a year in costs, and the regression shows that it holds almost nothing but the market, with an alpha near zero before costs, one does better to buy an index fund for 0.2 per cent: if the market brings 6 per cent before costs, 10,000 euros become 30,883 instead of 24,117 euros in 20 years, about 6,770 euros more. Alpha itself is calculated like this: if a fund brings 9 per cent, the risk-free rate 2 and the market 7 per cent, and the fund has a beta of 1.2, 2 + 1.2 · 5 = 8 per cent would have been expected; its alpha is 1 percentage point. For most active funds alpha after fees is negative, and that is the whole debate between active investing, in which someone picks shares, and passive investing, in which one simply buys an index, in one number.¹⁵¹ Whoever finds a characteristic that predicts returns should test it on a period he did not look at while searching; López de Prado demands this, and many discoveries do not survive it; how many is disputed among experts.¹⁵²

The bottleneck

By 2022 machines could learn from prices, trade in microseconds and find hedges no formula explains. What they could not do was read: annual reports, central bank speeches, news, half of what prices arise from lay as text; programs already counted words in it, but what a text meant was judged by people. In the autumn of 2022 machines learned to read, and the discipline acquired a tool that does not calculate but speaks, and that from the data it reads becomes a participant itself.

8. Language models, agents and the feedback loop (since 2023)

Language models in finance

The model reads. On 30 March 2023 the financial data firm Bloomberg presented BloombergGPT, a language model, a neural network that generates the most probable continuation of a text and has thereby learned what stands in annual reports, analysts’ assessments and news. It had fifty billion parameters, that many adjustable weights, and was trained on 363 billion tokens, the pieces of words into which a language model breaks up text, from forty years of the firm’s financial data and as much again of general text.¹⁵³ One week later Alejandro Lopez-Lira and Yuehua Tang of the University of Florida showed that ChatGPT, the language model that the firm OpenAI had made available to everyone at the end of 2022, when asked about headlines whether they were good or bad for a share, gave answers that were related to the returns of the next day, better than the older methods that read the mood about a share out of texts; the paper was a preprint, not yet checked by other experts, and such signals, experience shows, vanish to the extent that they become known.¹⁵⁴ In September 2023 the bank Morgan Stanley had rolled out to all its advisers an assistant based on the language model GPT-4 that finds the firm’s knowledge for them in seconds.¹⁵⁵

The supervisors. The international institutions that watch over the stability of the financial systems took up the new quantity quickly. The International Monetary Fund, the organisation of almost all the states of the world that watches over the stability of currencies and financial systems, devoted in October 2024 a chapter of its financial stability report to artificial intelligence, AI for short, that is programs that solve tasks for which human understanding was needed until now, in the markets for shares and bonds; it expected more efficient markets, but faster and more volatile ones in times of crisis.¹⁵⁶ The Financial Stability Board, a body set up by the G20, the group of twenty large economies, named on 14 November 2024 four vulnerabilities: dependence on a few providers of models and computing power, the danger of correlated markets when many use the same models, risks from attacks on computers, and risks from the models themselves, the quality of their data and their governance.¹⁵⁷ The Bank for International Settlements, the bank of the central banks in Basel, BIS for short, had in June 2024 called on the central banks to use the tools themselves in order to remain equal to them.¹⁵⁸ Besides the old danger that models are wrong, the Financial Stability Board and the BIS expressly named a new one: that many use the same models and therefore act alike.

Today

If someone asks his bank’s chatbot for the monthly instalment on a loan of 10,000 euros over 48 months at 0.5 per cent interest a month, the answer must be 234.85 euros; because a language model generates a probable text and does not calculate, it pays to check the number with the formula for the annuity payment.

Collusion and agents

Collusion. Why that is dangerous was shown by three economists in a simulation, a market rebuilt in the computer. Winston Wei Dou and Itay Goldstein of the Wharton School and Yan Ji of Hong Kong let trading programs trade against one another in 2023 that find their strategy by reinforcement learning, by trial and error in which rewarded actions become more frequent. The programs learned, without agreement, without communication and without anyone having taught them, to trade with restraint and to share the profit that competition would have eaten away; prices became less informative, and trading became harder.¹⁵⁹ Algorithmic collusion, after the algorithm, the precise rule of calculation a program carries out, is the concerted action of learning programs to the detriment of the other market participants without any person having arranged it, and it is as hard to grasp for antitrust law, which forbids agreements between competitors, as for market efficiency: there is no agreement, only the same learning rule.

The agents. In 2026 the reading model became a trading one. An AI agent is a program of language model and tools that is given a goal and chooses the steps towards it itself, researching, calculating, placing orders. Robinhood, an American online broker, opened its platform to agents on 27 May 2026: customers could open a separate account, fund it and hand over trading in it to a program of their choice through an interface, an access point through which programs talk to one another, at first only in stocks and with the possibility of disconnecting the agent at any time; within four months more than 150,000 customers had opened such an account, about 200,000 by the end of October, and at the end of September the firm announced an agent of its own that is to examine markets around the clock, build strategies and trade, on request without approval of each individual order.¹⁶⁰ A survey by the University of Cambridge of 628 financial firms, AI vendors and supervisors found in April 2026 that 52 per cent of the financial firms surveyed were piloting or deploying agents and 81 per cent expected agentic AI to be meaningfully achieved by 2030.¹⁶¹ What Thorp did by hand in 1967 and Medallion with programs in 1988, a program now did for anyone who switched it on, and it did it with the same learning rule as the neighbour’s program.

Today

Algorithmic collusion would look like this: two online shops both charge 49 euros for the same coffee machine for weeks, although either could win customers from the other at 45 euros, and neither has spoken to the other; their pricing programs have merely learned that undercutting does not pay.

A simple agent might, for instance, have the task of always keeping savings in the savings account with the best rate: it moves 10,000 euros from 1.5 to 3 per cent and so brings in 150 euros more a year, but perhaps does not check whether the higher rate applies only for three months.

Kill switch and law

The switch. The central banks saw in this an old risk at a new speed. Sarah Breeden, Deputy Governor of the Bank of England, said on 30 June 2026 at the European Central Bank’s forum in Sintra that the rulebooks were not built for agents that act on their own and that a human approving every action of an agent was not a realistic solution; the Bank was investigating with the BIS Innovation Hub and the Bundesbank in simulations how different designs of agents lead to herding, the concerted trading of many on the same news, and was asking whether market-wide safeguards were needed, on the model of trading halts, the pauses exchanges take during steep falls in prices, such as a kill switch that stops the trading of agents when faulty models threaten stability in concert.¹⁶² The securities regulators of the world had published on 25 May 2026 in their organisation IOSCO a toolkit for how examiners should supervise the use of AI from classical machine learning to agents, non-binding; the American banking supervisors had on 17 April 2026 revised their guidance on model risk and expressly excluded AI that generates text or acts on its own, in order to consult on it separately; in October 2026 the Bank of England, according to unnamed people, was considering making an AI disruption the scenario of its next stress test for the whole financial system, a calculation of how the banks would survive an assumed severe crisis.¹⁶³ Supervision was at the point where Weatherstone had stood around 1990: it wanted a number for a risk for which there was as yet no method.

The law. Europe had drawn the line earlier and moved it. The European Union’s AI Act of 2024 classifies systems that assess the creditworthiness of people or price life and health insurance as high-risk, with duties on data, documentation, human oversight and testing; trading by programs is not on the list. The duties were to apply from 2 August 2026; Regulation 2026/1744, the Digital Omnibus, published in the Official Journal of the European Union on 24 July 2026, postponed them to 2 December 2027, while the duties to mark AI-generated content as such have applied in principle since August 2026.¹⁶⁴ For credit, the oldest application of the discipline, the law has required since the General Data Protection Regulation of 2018 that, as a rule, no program alone decides about a person, with exceptions for instance for entering into a contract or with explicit consent, and in December 2023 the European Court of Justice counted the score of a credit agency as such a decision if a bank relies on it decisively; from 20 November 2026 borrowers in the EU have the right to a human being and an explanation.¹⁶⁵ For trading, the fastest application, the rules for algorithmic trading of 2018 apply, and market-wide kill switches for agents are under consideration (Figure 10).

Today

What herd behaviour looks like in the order book: if 10,000 users of an app get the same message and each sells 10 shares, 100,000 shares are to be sold while perhaps only buyers for 20,000 are waiting in the order book; the price falls until new buyers come.

The open question

The story of this text has a pattern, and the agents make it visible. Every tool that made risk calculable changed the market it described: portfolio insurance amplified in 1987 the crash it was meant to protect against; LTCM’s arbitrage models contributed in 1998 to the correlation they did not know; the copula eased until 2007 the sale of the mortgage packages whose joint default it held to be unlikely; the trading programs amplified in 2010 the crash no person wanted. Reflexivity is the circumstance that a model of the market becomes part of the market as soon as enough participants act on it, and that the data from which the next generation learns are made by the previous one. The agents of 2026 could learn and trade faster and be more alike than anything before, because they build on the same few foundation models, the large language models of a handful of firms; for now, according to Breeden, trading firms use them mostly for lower-risk tasks such as research, but that could change quickly.¹⁶² Whether that makes markets more efficient, as the theory of 1970 would expect, or more correlated, as the simulation of 2023 shows, is the question for which financial mathematics in 2026 has no formula.

Today

On a small scale reflexivity looks like this: if a widely used valuation program estimates flats in a city at 4,000 euros per square metre and forecasts a rise of 5 per cent a year, and enough buyers bid accordingly, the forecast comes true, 4,000 · 1.05³ = 4,630.50 euros after three years, and the new prices enter the next training as data.

Working it out and practice

Try it yourself. The limits of the reading model are shown by a five-minute experiment. One asks a language model to calculate the value of a call option with the numbers from section 5, price and strike 100, one year, two per cent interest, twenty per cent volatility. The answer is often right, 8.92, and sometimes confidently wrong; whoever has the spreadsheet beside it sees the difference, and whoever does not, does not. Then one asks the same model to justify the answer and gets the derivation of 1973 told without error, whether the number is right or not. That is the state of the discipline: the machine knows the story told here better than most who apply it, and it does not know when it has miscalculated.

The bottleneck

For four thousand years financial mathematics has solved bottlenecks the market set it: time, uncertainty, chance, the price of risk, the option, the measure of loss, the quantity of data. The bottleneck it faces in 2026 it has produced itself. The models are so good and so widespread that the market follows them, and a market that follows its models delivers data that confirm the models, until the day on which it does not. The mathematics that describes this does not yet exist; perhaps it is the next section of this story, and perhaps an agent will write it.

The chain in brief

1. To 1600 Babylon, Pisa and Antwerp learn to calculate with simple and compound interest, to discount payments and to keep books, against the ban on interest of the Church and of Islam (Hammurabi, Fibonacci 1202, Pacioli 1494, Stevin 1582); what does not come for certain has no price.
2. Pascal, Huygens, Graunt, de Witt, Halley, the Bernoullis, de Moivre and the Equitable (1654–1800) give the uncertain an expected value, the bell curve, a life table and a premium; the exchange remains with the speculators.
3. Regnault, Bachelier, Einstein, Wiener, Kolmogorov, Cowles and Itô (1863–1950) describe the price as a random walk and create the mathematics for it, Fisher, Williams and Macaulay the value of certain payments; nobody puts the parts together.
4. Markowitz, Tobin, Sharpe, Modigliani–Miller, Samuelson, Merton, Mandelbrot, Fama and Thorp (1952–1972) turn risk into a number with a price and chance into a theory; the option remains without a formula.
5. CBOE, Black, Scholes, Merton, Cox–Ross–Rubinstein, Harrison–Kreps, Vašíček, the first swaps and Black Monday (1973–1987) bring the formula, the theory and the industry of derivatives, and the first crash a model helped to cause.
6. Basel, value at risk, LTCM, the copula, Lehman, expected shortfall and CVA (1988–2023) measure the risk of the sum and show that the most dangerous risk is the model; negative rates and the turn in rates show that the safe rate moves too.
7. Factors, Medallion, high-frequency trading, the order book, LIBOR, cryptocurrencies, machine learning and deep hedging (1988–2022) replace formulas with data and people with programs; the programs cannot read.
8. BloombergGPT, supervisors, algorithmic collusion, agents, Breeden and the Digital Omnibus (since 2023): the machine reads and trades, and what a market does that follows its models is open.