Glossary Financial Mathematics

All technical terms of the article The History of Financial Mathematics in alphabetical order. The number at the end names the section in which the term is introduced.

A

• Account – List on which all additions to and withdrawals from one thing are written down, for instance the cash box or the goods; in double-entry bookkeeping every transaction is entered on two accounts. 1
• Accrued interest – Interest accumulated on a bond since the last interest date, which the buyer pays to the seller; for German federal bonds on an actual/actual basis. 6
• Actuary – Mathematician who translates probabilities of life, death and loss into premiums and reserves; title since 1762 at the Equitable Life. 2
• AI agent – Program of language model and tools that is given a goal and chooses the steps towards it itself, researching, calculating, placing orders; since May 2026 at Robinhood for retail customers. 8
• Algorithmic collusion – Concerted action of learning trading programs to the detriment of the other market participants without agreement or communication, only through the same learning rule; demonstrated in the simulation of Dou, Goldstein and Ji (2023). 8
• Alpha – What a fund earns beyond the factors: the intercept of the regression of its returns on market, size and value; negative after fees for most active funds. 7
• Amortisation – Part of an instalment that reduces the debt after the interest on the outstanding debt has been paid; with a constant annuity payment it grows from instalment to instalment. 1
• Annuity – Series of equal payments at fixed intervals, for instance yearly; in the Middle Ages, bought as a contract, a way around the prohibition of interest. 1
• Annuity due – Annuity whose instalments are paid at the beginning of each period, like rent; present and future value are one interest factor larger than in arrears. 1
• Annuity payment – Constant instalment with which a loan is repaid with interest in n periods: R = K₀ · r / (1 − (1 + r)⁻ⁿ); 10,000 euros at 5 per cent over three years give 3,672 euros a year. 1
• Arbitrage – Gain without outlay and without risk; the assumption that it does not exist in a functioning market has determined prices since Modigliani and Miller (1958) without knowing investors’ preferences. 4
• Automated market maker – Program that instead of an order book holds a pool of two currencies and forms the price from the ratio of the quantities, so that their product stays the same (x · y = k); Uniswap since 2 November 2018. 7

B

• Beta – Measure of the Capital Asset Pricing Model (Sharpe 1964) for how strongly an asset fluctuates on average with the market; beta two means double movement and double risk premium, diversifiable risk is not paid for. 4
• Bid-ask spread – Gap between the lowest selling price (ask) and the highest buying price (bid); for Demsetz in 1968 the price of immediacy, for Glosten and Milgrom in 1985 also the compensation for losses to better-informed traders. 7
• Bill of exchange – Written promise of payment redeemed in another city and currency; in the Middle Ages a way of hiding interest in the exchange rate. 1
• Binomial model – Model of Cox, Ross and Rubinstein (1979) in which the price can only rise or fall in each time step; a tree that converges to the Black-Scholes formula as the steps get finer and can value options exercisable early. 5
• Black-Scholes formula – Formula of 1973 for the value of a European option from price, strike, maturity, interest rate and volatility; the expected return does not appear, because the option can be replicated without risk by trading the share. 5
• Blockchain – Shared register made of a chain of blocks, each containing the transactions of a period and a fingerprint of the previous one; basis of Bitcoin and Ethereum. 7
• Bond – Tradable debt certificate with fixed interest payments and repayment at the end of its term; valued like any stream of payments by discounting, its sensitivity to changes in interest rates measured since Macaulay (1938) by duration. 1
• Break-even inflation – Inflation rate at which a conventional and an inflation-linked bond of the same maturity earn the same; roughly the difference of their yields. 6
• Brownian motion – Random jittering, observed by Brown in 1827 in pollen, derived by Bachelier in 1900 for prices and by Einstein in 1905 for particles, made rigorous by Wiener in 1923; the basic building block of stochastic processes. 3

C

• Call option – Right but not obligation to buy a thing by a date at the strike price; at expiry worth what the price lies above the strike, otherwise nothing. 5
• Central bank – Bank of a state or a currency that issues the money, sets the interest rate for the banks and often watches over them, such as the Bundesbank or the European Central Bank. 6
• Compound interest – Interest in which each period is reckoned on the sum together with the accrued interest, so that capital grows exponentially; calculated on Babylonian tablets four thousand years ago, source of the number e. 1
• Continuous compounding – Compounding in which interest is credited at every instant; under it a capital K₀ grows in t years at an interest rate r to K₀ · eʳᵗ. 2
• Convexity – Curvature of the curve linking a bond price to its yield; it makes losses from rising rates somewhat smaller and gains from falling rates somewhat larger than duration says. 6
• Copula – Function that joins the distributions of individual random quantities into a joint distribution; the Gaussian copula with one correlation parameter (Li 2000) valued the mortgage packages of the financial crisis. 6
• Correlation – Number between −1 and 1 measuring how closely two returns move in step: 1 entirely together, 0 independently, −1 always against each other; determines in Markowitz’s formula how much risk diversification removes, for two shares with 20 per cent volatility each, mixed half and half, 20, 17.3 or 14.1 per cent at 1, 0.5 or 0. 4
• Counterparty risk – Risk that the other side of a contract defaults before it has paid; its price is the CVA. 6
• Coupon – Interest of a bond per year as a percentage of its face value, originally the interest slip detached from the certificate and redeemed for the interest. 2
• Covariance – Measure of how two quantities move together: their correlation times their two standard deviations. 8
• Credit derivative – Contract whose payment depends on the default of a debtor, above all the credit default swap; 62 trillion dollars notional outstanding at the end of 2007. 6
• Credit risk – Risk that a debtor does not pay or does not pay in full; since Merton in 1974 it can be valued with the option formula. 5
• Credit scoring – Assessment of creditworthiness with a score, a weighted sum of a few characteristics; first formulas by Durand in 1941, general-purpose FICO score since 1989. 6
• Cryptocurrency – Money whose units exist only as entries in a shared register secured with the methods of cryptography; the first was Bitcoin, since 3 January 2009. 7
• CVA – Credit valuation adjustment, deduction from the value of a derivative for the risk that the other side defaults; in the financial crisis about two thirds of the losses from counterparty risk, to be backed with capital since Basel III. 6

D

• Debit and credit – The two columns of an account, debit on the left, credit on the right; on accounts for possessions such as cash and goods, additions stand on the debit side and withdrawals on the credit side. In double-entry bookkeeping every transaction stands once as a debit and once as a credit, so the totals of both sides are equal. 1
• Deep hedging – Method of Buehler and colleagues (2019) in which a neural network learns the hedging of a derivative from simulated paths, with trading costs and jumps, instead of deriving it from a formula. 7
• Default intensity – Probability per unit of time with which a debtor defaults, estimated from market prices by Jarrow and Turnbull (1995); at the intensity λ he survives t years with probability exp(−λ · t). 6
• Delta – Number of shares that the hedge of an option requires; by Black-Scholes N(d1) for the call, 0.58 in the example of section 5. 5
• Derivative – Contract whose value is derived from another price, such as an option, a futures contract, a swap; valued since 1973 by replication and risk-neutral expectation. 5
• Derivative of a function – How steeply a function rises at a point: its increase over a tiny step divided by the step; at the highest point it is zero. 4
• Discounting – Reverse of compound interest: division of a future payment by (1 + rate) to the power of the years to obtain its present value. 1
• Distribution – Statement of how probable the possible values of a random variable are; drawn, it takes a shape, for the normal distribution that of a bell. 4
• Diversification – Spreading of wealth across investments that do not move alike; reduces the volatility of the portfolio below that of its parts, with a limit that Markowitz calculated. 4
• Dividend – Part of the profit that a company pays out to the owners of its shares; by Williams a share is worth the present value of its future dividends. 3
• Double-entry bookkeeping – Method of accounting in which every transaction is recorded twice, as a debit on one account and a credit on another, so that the sums must balance; printed by Pacioli in 1494 as the “method of Venice”. 1
• Duration – Average maturity of a bond’s payments weighted by their present values; introduced by Macaulay in 1938, measure of the sensitivity of a bond price to changes in interest rates. 3

E

• Efficient frontier – Set of all portfolios that deliver the highest expected return for a given volatility; on it, according to Markowitz, every sensible investor should lie. 4
• Equity – Money of the owners of a company, which does not have to be repaid and shares in profit and loss; the counterpart of debt. 4
• Exchange rate – Price at which one currency is swapped for another. 1
• Expected shortfall – Average loss in the worst cases beyond the value-at-risk threshold; according to Artzner and colleagues (1999) a coherent risk measure, since 2016 the basis of the trading-book rules. 6
• Expected utility – Average of the utilities of all possible outcomes, each weighted by its probability; according to von Neumann and Morgenstern (1947) anyone who chooses consistently by a few simple rules acts this way. 2
• Expected value – Sum of all possible outcomes of an uncertain event, each weighted with its probability; introduced by Huygens in 1657, the first number for the value of an uncertain claim. 2
• Exponential growth – Growth in which a quantity is always multiplied by the same number in equal times, so that it increases ever faster; the oldest example is compound interest. 1

F

• Face value – Amount a bond repays at the end of its term and to which its interest refers. 2
• Factor – Common influence that moves many securities at once and for bearing which the market pays a premium; market, size and value in Fama and French (1992/93), momentum in Carhart (1997). 7
• Fat tails – Property of distributions in which extreme outcomes are far more frequent than the bell curve allows; shown by Mandelbrot in 1963 in cotton prices, visible in option prices since 1987. 4
• Financial mathematics – Mathematics of money across time and under uncertainty: present value of future payments, price of uncertain payments and price of rights to payments; from the interest of Babylon through life table, portfolio theory and option formula to the risk models of the banks and the learning programs. Introduction
• Floating rate – Interest rate that is reset at fixed intervals to a market rate; in an interest rate swap the counterpart of the fixed rate. 5
• Formula – Short rule of calculation in which letters stand for numbers that are put in; compound interest, for instance: Kₙ = K₀ · (1 + r)ⁿ. Introduction
• Forward rate – Rate agreed today for a later period, computed from spot rates: (1 + s₂)² = (1 + s₁) · (1 + f). 6
• Function – Rule of calculation that turns one number into another, for instance squaring or the logarithm. 3
• Fund – Common pool in which the money of many investors is invested together; special forms include pension funds and hedge funds. 1
• Future value – Value of a payment or annuity at the end of the term if each payment goes on growing with compound interest until then; counterpart of present value. 1
• Future value of an annuity – Future value of n equal instalments R at the end of each period: R · ((1 + r)ⁿ − 1) / r; for an annuity due (1 + r) times as much. 1
• Futures – Contracts to buy or sell a thing on a later day at a price agreed today; futures on share indices served portfolio insurance until 1987. 5

G

• Gompertz law – Observation by Benjamin Gompertz (1825) that the mortality of adults grows by the same percentage with every year of age, that is exponentially; a serviceable approximation for the middle and later years. 2
• Government bond – Bond issued by a state; forerunners in the Monte of Venice (1262), in England through the Bank of England (1694) and the consols (1751). 2

H

• Hedge fund – Fund, that is a common pool for the money of many investors, that trades with few restrictions and much borrowed money; Long-Term Capital Management is the best-known example. 5
• Hedging – Holding of a counter-position that offsets price movements; continually adjusted, it makes option plus share riskless and is the core of the derivation of 1973. 5
• Herding – Concerted trading of many market participants on the same news; with agents built on the same foundation models the concern of the central banks in 2026. 8
• High-frequency trading – Trading in which programs place and withdraw orders in fractions of a second; a large part of volume since the 2000s, a contributing cause of the flash crash of 6 May 2010. 7

I

• Implied volatility – Volatility that, inserted into the Black-Scholes formula, yields the observed option price; the language in which the market has talked about risk since 1987. 5
• In arrears – Said of an annuity whose instalments are paid at the end of each period, like the interest on a loan. 1
• Index – Number that measures the value of a fixed basket of many shares, such as the DAX; it measures how the market as a whole moves. 4
• Inflation – Rise in prices on average, measured by a basket of goods and services; in the euro area 10.6 per cent year on year in October 2022. 6
• Interest – Price of the time between borrowing and returning, usually as a share of the borrowed sum per year; in Mesopotamia one fifth for silver, one third for barley. 1
• Interest factor – Number by which a sum is multiplied each period: 1 plus the interest rate, so 1.2 at 20 per cent. 1
• Interest rate swap – Swap in which one side pays a fixed and the other a floating rate on the same amount; only the interest is exchanged. Interest rate derivatives referred to 669.5 trillion dollars at the end of 2025. 5
• Interest table – Printed table from which one reads off what a sum is worth after so many years at so many per cent; kept secret by the trading houses, published by Stevin in 1582. 1
• Internal rate of return – Interest rate at which the net present value of an investment becomes zero, its return; for 1,000 euros against three times 400 euros 9.7 per cent. 3
• Intrinsic value – Value of a share as the present value of its future dividends, after Williams (1938); yardstick of buying below value in Graham and Dodd. 3
• Investment – Money spent today in order to get more back later, for instance on a machine; by Fisher’s rule it pays if the present value of the returns exceeds the cost. 3
• Itô calculus – Differential and integral calculus for random paths, founded by Itô in 1944; the rule that a function of a jittering path acquires an additional term from the squares of the steps makes the option formula possible. 3
• Itô integral – Sum of many small random price changes, each multiplied by the quantity held at that instant; according to Merton the gain of a trading strategy. 4

K

• Kelly criterion – Kelly’s rule (1956) for the stake of a gambler with an edge: so much that wealth grows exponentially fastest on average, and no more. 4
• Kill switch – Market-wide safeguard on the model of trading halts that stops the trading of AI agents when faulty models threaten stability in concert; considered by the Bank of England in June 2026. 8

L

• Language model – Program that generates the most probable continuation of a text; since BloombergGPT (2023) it reads reports, news and minutes for the financial markets. 8
• Law of large numbers – Theorem of Jacob Bernoulli (1713): the more often a random experiment is repeated, the closer the observed share comes to the true probability; the reason insurance works. 2
• Leverage – Ratio of borrowed capital to one’s own; magnifies gains and losses, at LTCM about 25 to one at the start of 1998. 6
• Life annuity – Stream of payments that runs until the death of a person; means of finance of the towns and states of the early modern period, first valued by age by de Witt in 1671 and Halley in 1693. 2
• Life insurance – Contract in which one pays premiums regularly and the insurer pays an agreed sum on death; calculated from life tables since the Equitable of 1762. 2
• Life table – Table stating for every age how many of an initial number born are still alive or how probable survival of the next year is; Halley’s table from Breslau data (1693) was the first from observations. 2
• Limit – Number that a sequence of numbers comes ever closer to without exceeding it, such as e = 2.71828… as the limit of (1 + 1/n)ⁿ with ever more frequent compounding. 2
• Linear interpolation – Estimating a value between two known values on the assumption that the quantity runs along a straight line between them; this is how the Babylonian tablet AO 6770 found the doubling time at twenty per cent interest, 3.79 instead of exactly 3.80 years. 1
• Liquidity – Ability to make payments at any time because one has enough cash or securities that can be sold quickly; required of banks since Basel III. 6
• Loan – Handing over money or goods for a time against return, usually with interest; whoever borrows is the debtor, whoever lends the creditor. 1
• Logarithm – Inverse of the power: the exponent to which a base must be raised to give a number; the logarithm of 2 to base 1.2 is 3.8, the doubling time at twenty per cent interest. Turns multiplication into addition, so that every doubling is a step of the same size; first tables by Napier 1614, to base e the natural logarithm (LN). 1
• Logarithmic scale – Division of an axis in which every tenfold increase gets the same distance, so that 1, 10, 100 and 1,000 lie equally far apart; a sum at a fixed rate climbs on it in equal steps. 1

M

• Machine learning – Methods by which a program learns patterns from examples instead of from rules; in financial mathematics since the 2010s for prediction, hedging and risk measurement. 7
• Market efficiency – Fama’s thesis (1970) that the prices of a market contain all available information, in weak, semi-strong and strong form depending on the kind of information. 4
• Market maker – Dealer who constantly quotes a buying and a selling price at the same time and earns the spread between them. 7
• Market microstructure – Study of how the rules and institutions of trading shape prices; term coined by Mark Garman in 1976. 7
• Market portfolio – Portfolio containing all shares in proportion to their value on the exchange; by Sharpe all investors hold it, mixed with the safe asset. 4
• Market price – Price at which a share or bond is traded on the exchange at a given moment. 3
• Martingale – Stochastic process whose best forecast for tomorrow is today’s value; according to Samuelson (1965) the form prices take in a market that processes all information. 4
• Model – Simplified description in numbers and rules with which one can work out how something behaves, such as Bachelier’s random walk for the price; every model leaves something out. 3
• Model risk – Loss that arises not from the movement of the market but from the model with which it was measured being wrong; greatest when everyone uses the same model. 6
• Modified duration – Duration divided by (1 + yield): the percentage by which a bond price changes roughly when the yield changes by one percentage point. 6
• Momentum – Observation of Jegadeesh and Titman (1993) that shares that have recently risen go on rising in the next half year; inexplicable in an efficient market, carried as a factor since 1997. 7
• Money market instrument – Security with a maturity of at most one year, often without a coupon, sold below face value; for instance the German Federal Treasury discount paper. 6
• Monte Carlo simulation – Estimate of an expected value as the mean of many randomly generated cases; its error falls only with the square root of the number of cases. 7
• Mortgage – Loan for a house or flat secured by the house itself; under Basel I with the risk weight 0.5. 6
• Mortgage security – Security into which a bank bundles many mortgages; its payments come from the homeowners’ instalments. 6
• Murabaha – Sale with a disclosed mark-up in Islamic finance: the bank buys goods and sells them to the customer at a higher price against later payment or instalments. 1

N

• Natural logarithm – Logarithm to base e, written ln, in spreadsheets LN; ln 2 = 0.693. 2
• Negative interest rate – Interest below zero: whoever invests money gets back less than he gave; the ECB charged interest on banks’ deposits from June 2014 to 2022, down to −0.50 per cent. 6
• Negative number – Number below zero, written with a minus in front, such as −1 for a step down; a number and its counterpart with a minus together give zero. 3
• Net present value – Present value of all payments of an investment minus its cost; by Fisher (1930) an investment pays if it is positive. 1,000 euros for three times 400 euros give +30.84 euros at 8 per cent. 3
• Neural network – Program made of many simple, connected steps of calculation whose weights are adjusted on examples while learning; the basis of deep hedging and of language models. 7
• Nominal interest rate – Interest in money, regardless of what the money buys; after Fisher (1 + nominal rate) = (1 + real rate) · (1 + inflation). 6
• Normal distribution – Distribution of the bell curve that arises when many small, independent chance effects add up; a good two thirds of the values lie less than one standard deviation from the mean, only one in a hundred more than 2.33 below it. According to Mandelbrot (1963) price changes have fatter tails. 2

O

• Option – Right, not obligation, to buy (call) or sell (put) a thing by a date at a fixed price; traded in standardised form on the CBOE since 1973. 3
• Order book – List of all orders that want to buy or sell at a fixed price, ranked by price and time; more than half of the world’s financial markets run on electronic order books. 7
• Overfitting – A model or a strategy adapts to the chance of the data on which it was found and fails on new data; of a hundred strategies without advantage, on average five pass a test at the five per cent level. 7
• Overnight indexed swap – Contract in which two banks exchange a fixed rate for the overnight rate compounded over the term; almost free of default risk, since 2010 the basis for discounting interest rate derivatives. 6

P

• Per cent – Hundredth, written %: 1 per cent is 0.01, 20 per cent is 0.2; a percentage of an amount is obtained by multiplying, 5 per cent of 300 is 0.05 · 300 = 15. 1
• Percentage point – Difference between two percentages: from 4 to 5 per cent is one percentage point more, although 5 is a quarter more than 4. 2
• Period – Stretch of time after which interest is calculated and credited, usually a year, a month or a day. 1
• Perpetuity – Annuity without end, such as the British consols; its present value is the instalment divided by the rate. 2
• Portfolio – Totality of the investments someone holds; after Markowitz (1952) the object of valuation, not the single security. 4
• Portfolio insurance – Strategy of the 1980s of replicating a put option on a portfolio synthetically by selling futures as prices fall; according to the Brady Report the most important factor in the crash of 19 October 1987. 5
• Power – Repeated multiplication by the same number, written 1.2³ and read “1.2 to the power of 3”; 1.2 is the base, 3 the exponent. With compound interest, the value after n years is the initial capital times the n-th power of the interest factor. 1
• Premium – Price of insurance cover; at the Equitable from 1762 dependent on the age at entry and higher in young years than the current deaths, so that a fund arises for old age. 2
• Present value – Value today of future payments, calculated by reckoning each payment back to today with the interest rate; used by Fibonacci in 1202 to compare contracts, basis of every valuation. 1
• Present value of an annuity – Present value of n equal instalments R at the end of each period: R · (1 − (1 + r)⁻ⁿ) / r, derived via the geometric series. 1
• Probability – Share of cases in which an outcome occurs, between zero and one; calculated since Cardano, Pascal and Fermat, since Kolmogorov in 1933 a measure on a set of events. 2
• Prospect theory – Theory of Kahneman and Tversky (1979): people value gains and losses relative to what they already have, and losses weigh more than equal gains. 2
• Put option – Right but not obligation to sell a thing by a date at the strike price; an insurance against falling prices. 5
• Put-call parity – Relation C − P = S − K · e⁻ʳᵀ between the prices of a European call and put with the same strike and term; follows from the arbitrage argument alone, without a model. 5

Q

• Quantile – Value below which a given share of cases lies: the 1 per cent quantile of daily returns is fallen below on only one day in a hundred; value at risk is a quantile. 6
• Quantitative trading – Trading by rules estimated from data and executed by programs; the Medallion Fund (since 1988) is its best-known example. 7

R

• Random experiment – Trial whose outcome depends on chance, such as a throw of a die; repeated often, the observed share of an event approaches its probability. 2
• Random variable – Number that depends on chance, such as the gain of a game or the number of deaths in a year; described by its possible values and their probabilities. 2
• Random walk – Movement of independent random steps whose distance from the start grows on average with the root of the number of steps; observed by Regnault in 1863 in prices, model of the exchange in Bachelier in 1900. 3
• Rating agency – Firm that grades the safety of debtors and securities with marks from AAA downwards. 6
• Real interest rate – Gain in purchasing power: roughly the nominal rate minus inflation; a zero rate at ten per cent inflation is in real terms a loss of 9 per cent. 6
• Reference rate – Interest rate to which contracts refer; LIBOR was formed from banks’ submissions and manipulated, replaced from 2017 to 2023 by rates from real transactions such as SOFR. 7
• Reflexivity – Circumstance that a model of the market becomes part of the market as soon as enough participants act on it, and the data of the next generation are made by the previous one; the pattern of 1987, 1998, 2007 and 2010. 8
• Regression – Method that looks for the numbers with which one quantity is best composed of others; with one quantity it is the best straight line through the points, with several the factor model. 7
• Return – Gain of an investment as a percentage of the stake; whoever stakes 100 and gets back 105 after a year has a return of five per cent. Interest is the return of a loan. 1
• Riba – Arabic for “increase”: the surcharge forbidden in the Quran (sura 2, verse 275), in the prevailing interpretation any interest fixed in advance on a loan. 1
• Risk – Possibility that things turn out differently than expected and one loses; in financial mathematics measured for instance by volatility or value at risk. 3
• Risk premium – Return that a risky investment is expected to bring beyond the risk-free rate, as a reward for its risk. 4
• Risk weight – Factor of the Basel capital rules (since 1988) stating how much of a loan counts in calculating required capital; since Basel II (2004) partly from the banks’ own models. 6
• Risk-free rate – Interest rate of a safe asset, the yardstick against which the return of risky investments is measured. 4
• Risk-neutral valuation – Computational trick of valuing a derivative as if all investors were risk-neutral and every asset were expected to earn only the risk-free rate; possible because risk aversion cancels out in replication. 5
• Ruin probability – Probability that an insurer’s reserve at some point falls below zero; according to Lundberg and Cramér it falls with initial capital, in the simplest case exponentially. 2

S

• Safe asset – Investment without fluctuation, such as a short-term government bond; it brings the risk-free rate. 4
• Share – Part of a company that gives its owner a portion of the profit and a vote and is traded on the stock exchange. 2
• Sharpe ratio – Additional expected return above the risk-free rate per unit of volatility, (μ − r) / σ; slope of the straight line on which the mixtures of market portfolio and safe investment lie. 4
• Short sale – Sale of a borrowed share that one hopes to buy back more cheaply later; combined with a purchase, a bet on the gap between two shares. 7
• Simple interest – Interest reckoned each year only on the original amount; 20 per cent on 5 gives 5 + 3 = 8 after three years, with compound interest 8.64. 1
• Slope – How steeply a straight line rises: how far it goes up when one goes one step to the right; beta is the slope of a share’s returns against those of the market. 4
• Spot rate – Rate for money from today to a date with no payments in between; the spot rates of all maturities form the yield curve. 6
• Square root – The number that multiplied by itself gives a given number, written √: √9 = 3, √1.2 = 1.0954; the same as the power with exponent 0.5. 1
• Squaring – Multiplying a number by itself: 3 squared is 9, 2.5 squared 6.25; the reverse is the square root. 2
• Stablecoin – Cryptocurrency meant to hold a fixed value, usually one dollar; TerraUSD lost its peg in May 2022, the EU has regulated stablecoins with MiCA since 30 June 2024. 7
• Standard deviation – Square root of the variance, measured in the unit of the quantity itself; for n independent trials with probability p it is √(n · p · (1 − p)) and grows only with the square root of n. 2
• Standard error – Standard deviation of an estimate, for instance of a mean of n cases: the standard deviation of one case divided by √n. 7
• Stochastic process – Family of random quantities ordered by time, such as a price, a temperature, a particle; mathematically tractable since Kolmogorov’s axioms of 1933. 3
• Stock exchange – Market on which shares, bonds and other securities are bought and sold every day. 2
• Stream of payments – Series of payments at different times, for instance the interest and the repayment of a bond; it is valued through its present value. 1
• Strike price – Fixed price at which the owner of an option may buy or sell. 5
• Sukuk – Islamic certificates that represent not a claim but a share in an asset; since February 2008 AAOIFI requires real ownership instead of a promise to repurchase at the original amount. 1
• Swap – Contract to exchange payments over years, for instance interest in different currencies; the first formal currency swap linked the World Bank and IBM in 1981. 5

T

• Term (of a bond) – Time until a loan or a bond is repaid. 1
• Term-structure model – Model describing the interest rate or the whole yield curve as a stochastic process in order to value bonds and interest-rate derivatives; Vašíček 1977, Heath, Jarrow and Morton 1992. 5
• Tranche – Layer of a package of many loans with a fixed rank: the lowest bears the first defaults, the top one only when all those below are used up. 6

U

• Usury – In the Middle Ages any interest, not only the excessive kind; forbidden by the church in 1179, narrowed by Calvin in 1545 to the “biting” loan to the needy. 1
• Utility – Value a person attaches to an amount of money, according to Daniel Bernoulli (1738) not proportional to the amount but diminishing, roughly logarithmic; basis of risk aversion and of every risk premium. 2

V

• Value at risk – Loss that over a period will not be exceeded with a set probability, say 99 per cent; since Weatherstone’s report of 1989 and RiskMetrics (1994) the banks’ risk number, with the weakness of saying nothing about the loss beyond the threshold. 6
• Variance – Expected value of the squared deviation of a random variable from its expected value; for independent random variables the variances add up. 2
• Volatility – Standard deviation of the return, measure of how far outcomes scatter around the expected value; in Markowitz the measure of risk, in the option formula the one quantity nobody knows. 3
• Volatility smile – Skewed curve of implied volatility over the strike price since October 1987: protection against crashes costs a premium the model does not know. 5

Y

• Yield curve – The interest rates for all terms, from one day to thirty years; Heath, Jarrow and Morton described the whole curve as random in 1992. 5
• Yield spread – Yield of a bond minus that of a safe bond of the same maturity, in percentage points; roughly default intensity times (1 − recovery rate). 6
• Yield to maturity – Rate at which the present value of all payments of a bond equals its price; above the coupon when the price is below face value. 2