Mathematical finance, also known as quantitative finance and financial mathematics, is a field of

applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...

, concerned with mathematical modeling of

financial market A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities include stocks and bonds, raw materials and precious metals, which are known in the financial ma ...

s. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and

risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environm ...

and portfolio management on the other. Mathematical finance overlaps heavily with the fields of

computational finance Computational finance is a branch of applied computer science that deals with problems of practical interest in finance.Rüdiger U. Seydel, '' tp://nozdr.ru/biblio/kolxo3/F/FN/Seydel%20R.U.%20Tools%20for%20Computational%20Finance%20(4ed.,%20Sprin ...

and

financial engineering Financial engineering is a multidisciplinary field involving financial theory, methods of engineering, tools of mathematics and the practice of programming. It has also been defined as the application of technical methods, especially from mathem ...

. The latter focuses on applications and modeling, often by help of

stochastic asset model A stochastic investment model tries to forecast how returns and prices on different assets or asset classes, (e. g. equities or bonds) vary over time. Stochastic models are not applied for making point estimation rather interval estimation and t ...

s, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is

quantitative investing Quantitative analysis is the use of mathematical and statistical methods in finance and investment management. Those working in the field are quantitative analysts (quants). Quants tend to specialize in specific areas which may include derivat ...

, which relies on statistical and numerical models (and lately

machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...

) as opposed to traditional

fundamental analysis Fundamental analysis, in accounting and finance, is the analysis of a business's financial statements (usually to analyze the business's assets, liabilities, and earnings); health; and competitors and markets. It also considers the overall sta ...

when managing portfolios. French mathematician

Louis Bachelier Louis Jean-Baptiste Alphonse Bachelier (; 11 March 1870 – 28 April 1946) was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, as part ...

's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical finance. But mathematical finance emerged as a discipline in the 1970s, following the work of

Fischer Black Fischer Sheffey Black (January 11, 1938 – August 30, 1995) was an American economist, best known as one of the authors of the Black–Scholes equation. Background Fischer Sheffey Black was born on January 11, 1938. He graduated from Harvard ...

Myron Scholes Myron Samuel Scholes ( ; born July 1, 1941) is a Canadian-American financial economist. Scholes is the Frank E. Buck Professor of Finance, Emeritus, at the Stanford Graduate School of Business, Nobel Laureate in Economic Sciences, and co-origin ...

and Robert Merton on option pricing theory. Mathematical investing originated from the research of mathematician Edward Thorp who used statistical methods to first invent

card counting Card counting is a blackjack strategy used to determine whether the player or the dealer has an advantage on the next hand. Card counters are advantage players who try to overcome the casino house edge by keeping a running count of high and low ...

blackjack Blackjack (formerly Black Jack and Vingt-Un) is a casino banking game. The most widely played casino banking game in the world, it uses decks of 52 cards and descends from a global family of casino banking games known as Twenty-One. This fam ...

and then applied its principles to modern systematic investing. The subject has a close relationship with the discipline of

financial economics Financial economics, also known as finance, is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on ''both sides'' of a trade". William F. Sharpe"Financia ...

, which is concerned with much of the underlying theory that is involved in financial mathematics. While trained economists use complex

economic models In economics, a model is a theoretical construct representing economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified, often mathematical, framework design ...

that are built on observed empirical relationships, in contrast, mathematical finance analysis will derive and extend the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. See:

Valuation of options In finance, a price (premium) is paid or received for purchasing or selling options. This article discusses the calculation of this premium in general. For further detail, see: for discussion of the mathematics; Financial engineering for the impl ...

;

Financial modeling Financial modeling is the task of building an abstract representation (a model) of a real world financial situation. This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio ...

;

Asset pricing In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, but correspon ...

. The

fundamental theorem of arbitrage-free pricing The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both financial economics and mathematical finance, provide necessary and sufficient conditions for a market to be arbitrage-free, and for a market to be complete. An arb ...

is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results. Today many universities offer degree and research programs in mathematical finance.

History: Q versus P

There are two separate branches of finance that require advanced quantitative techniques: derivatives pricing, and risk and portfolio management. One of the main differences is that they use different probabilities such as the risk-neutral probability (or arbitrage-pricing probability), denoted by "Q", and the actual (or actuarial) probability, denoted by "P".

Derivatives pricing: the Q world

The goal of derivatives pricing is to determine the fair price of a given security in terms of more

liquid securities In business, economics or investment, market liquidity is a market's feature whereby an individual or firm can quickly purchase or sell an asset without causing a drastic change in the asset's price. Liquidity involves the trade-off between the ...

whose price is determined by the law of

supply and demand In microeconomics, supply and demand is an economic model of price determination in a Market (economics), market. It postulates that, Ceteris paribus, holding all else equal, in a perfect competition, competitive market, the unit price for a ...

. The meaning of "fair" depends, of course, on whether one considers buying or selling the security. Examples of securities being priced are

plain vanilla Plain vanilla is an adjective describing the simplest version of something, without any optional extras, basic or ordinary. In analogy with the common ice cream flavour vanilla, which became widely and cheaply available with the development of art ...

and

exotic option In finance, an exotic option is an option which has features making it more complex than commonly traded vanilla options. Like the more general exotic derivatives they may have several triggers relating to determination of payoff. An exotic op ...

convertible bond In finance, a convertible bond or convertible note or convertible debt (or a convertible debenture if it has a maturity of greater than 10 years) is a type of bond that the holder can convert into a specified number of shares of common stock ...

s, etc. Once a fair price has been determined, the sell-side trader can make a market on the security. Therefore, derivatives pricing is a complex "extrapolation" exercise to define the current market value of a security, which is then used by the sell-side community. Quantitative derivatives pricing was initiated by

in ''The Theory of Speculation'' ("Théorie de la spéculation", published 1900), with the introduction of the most basic and most influential of processes,

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

, and its applications to the pricing of options. Brownian motion is derived using the

Langevin equation In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Lang ...

and the discrete

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

. Bachelier modeled the

time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...

of changes in the

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...

of stock prices as a

in which the short-term changes had a finite

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...

. This causes longer-term changes to follow a

Gaussian distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

. The theory remained dormant until

and

, along with fundamental contributions by

Robert C. Merton Robert Cox Merton (born July 31, 1944) is an American economist, Nobel Memorial Prize in Economic Sciences laureate, and professor at the MIT Sloan School of Management, known for his pioneering contributions to continuous-time finance, especia ...

, applied the second most influential process, the

geometric Brownian motion A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It i ...

, to

option pricing In finance, a price (premium) is paid or received for purchasing or selling options. This article discusses the calculation of this premium in general. For further detail, see: for discussion of the mathematics; Financial engineering for the imple ...

. For this M. Scholes and R. Merton were awarded the 1997

Nobel Memorial Prize in Economic Sciences The Nobel Memorial Prize in Economic Sciences, officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel ( sv, Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne), is an economics award administered ...

. Black was ineligible for the prize because of his death in 1995. The next important step was the fundamental theorem of asset pricing by Harrison and Pliska (1981), according to which the suitably normalized current price ''P₀'' of a security is arbitrage-free, and thus truly fair only if there exists a

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...

''P_t'' with constant

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

which describes its future evolution: A process satisfying () is called a " martingale". A martingale does not reward risk. Thus the probability of the normalized security price process is called "risk-neutral" and is typically denoted by the blackboard font letter "

\mathbb

". The relationship () must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time. The quants who operate in the Q world of derivatives pricing are specialists with deep knowledge of the specific products they model. Securities are priced individually, and thus the problems in the Q world are low-dimensional in nature. Calibration is one of the main challenges of the Q world: once a continuous-time parametric process has been calibrated to a set of traded securities through a relationship such as (), a similar relationship is used to define the price of new derivatives. The main quantitative tools necessary to handle continuous-time Q-processes are Itô's stochastic calculus,

simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...

and

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...

s (PDEs).

Risk and portfolio management: the P world

Risk and portfolio management aims at modeling the statistically derived probability distribution of the market prices of all the securities at a given future investment horizon.
This "real" probability distribution of the market prices is typically denoted by the blackboard font letter "

\mathbb

", as opposed to the "risk-neutral" probability "

\mathbb

" used in derivatives pricing. Based on the P distribution, the buy-side community takes decisions on which securities to purchase in order to improve the prospective profit-and-loss profile of their positions considered as a portfolio. Increasingly, elements of this process are automated; see for a listing of relevant articles. For their pioneering work, Markowitz and Sharpe, along with

Merton Miller Merton Howard Miller (May 16, 1923 – June 3, 2000) was an American economist, and the co-author of the Modigliani–Miller theorem (1958), which proposed the irrelevance of debt-equity structure. He shared the Nobel Memorial Prize in Economic ...

, shared the 1990

, for the first time ever awarded for a work in finance. The portfolio-selection work of Markowitz and Sharpe introduced mathematics to

investment management Investment management is the professional asset management of various securities, including shareholdings, bonds, and other assets, such as real estate, to meet specified investment goals for the benefit of investors. Investors may be instit ...

. With time, the mathematics has become more sophisticated. Thanks to Robert Merton and Paul Samuelson, one-period models were replaced by continuous time, Brownian-motion models, and the quadratic utility function implicit in mean–variance optimization was replaced by more general increasing, concave utility functions. Furthermore, in recent years the focus shifted toward estimation risk, i.e., the dangers of incorrectly assuming that advanced time series analysis alone can provide completely accurate estimates of the market parameters. See . Much effort has gone into the study of financial markets and how prices vary with time.

Charles Dow Charles Henry Dow (; November 6, 1851 – December 4, 1902) was an American journalist who co-founded Dow Jones & Company with Edward Jones and Charles Bergstresser. Dow also co-founded ''The Wall Street Journal'', which has become one of th ...

, one of the founders of

Dow Jones & Company Dow Jones & Company, Inc. is an American publishing firm owned by News Corp and led by CEO Almar Latour. The company publishes ''The Wall Street Journal'', '' Barron's'', ''MarketWatch'', ''Mansion Global'', ''Financial News'' and ''Private ...

and

The Wall Street Journal ''The Wall Street Journal'' is an American business-focused, international daily newspaper based in New York City, with international editions also available in Chinese and Japanese. The ''Journal'', along with its Asian editions, is published ...

, enunciated a set of ideas on the subject which are now called

Dow Theory The Dow theory on stock price movement is a form of technical analysis that includes some aspects of sector rotation. The theory was derived from 255 editorials in '' The Wall Street Journal'' written by Charles H. Dow (1851–1902), journalist ...

. This is the basis of the so-called

technical analysis In finance, technical analysis is an analysis methodology for analysing and forecasting the direction of prices through the study of past market data, primarily price and volume. Behavioral economics and quantitative analysis use many of the sam ...

method of attempting to predict future changes. One of the tenets of "technical analysis" is that

market trend A market trend is a perceived tendency of financial markets to move in a particular direction over time. Analysts classify these trends as ''secular'' for long time-frames, ''primary'' for medium time-frames, and ''secondary'' for short time-fram ...

s give an indication of the future, at least in the short term. The claims of the technical analysts are disputed by many academics.

Criticism

The aftermath of the financial crisis of 2009 as well as the multiple Flash Crashes of the early 2010s resulted in social uproars in the general population and ethical malaises in the scientific community which triggered noticeable changes in Quantitative Finance (QF). More specifically, mathematical finance was instructed to change and become more realistic as opposed to more convenient. The concurrent rise of

Big data Though used sometimes loosely partly because of a lack of formal definition, the interpretation that seems to best describe Big data is the one associated with large body of information that we could not comprehend when used only in smaller am ...

and

Data Science Data science is an interdisciplinary field that uses scientific methods, processes, algorithms and systems to extract or extrapolate knowledge and insights from noisy, structured and unstructured data, and apply knowledge from data across a br ...

contributed to facilitating these changes. More specifically, in terms of defining new models, we saw a significant increase in the use of

Machine Learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...

overtaking traditional Mathematical Finance models. Over the years, increasingly sophisticated mathematical models and derivative pricing strategies have been developed, but their credibility was damaged by the

financial crisis of 2007–2010 Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fi ...

. Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by

Paul Wilmott Paul Wilmott (born 8 November 1959) is an English researcher, consultant and lecturer in quantitative finance.Nassim Nicholas Taleb Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist whose work concerns problems of randomness ...

, in his book ''The Black Swan''. Taleb claims that the prices of financial assets cannot be characterized by the simple models currently in use, rendering much of current practice at best irrelevant, and, at worst, dangerously misleading. Wilmott and

Emanuel Derman Emanuel Derman (born 1945) is a South African-born academic, businessman and writer. He is best known as a quantitative analyst, and author of the book ''My Life as a Quant: Reflections on Physics and Finance''. He is a co-author of Black–Derm ...

published the '' Financial Modelers' Manifesto'' in January 2009 which addresses some of the most serious concerns. Bodies such as the

Institute for New Economic Thinking The Institute for New Economic Thinking (INET) is a New York City–based nonprofit think tank. It was founded in October 2009 as a result of the 2007–2012 global financial crisis, and runs a variety of affiliated programs at major universiti ...

are now attempting to develop new theories and methods. In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate. In the 1960s it was discovered by

Benoit Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of p ...

that changes in prices do not follow a

, but are rather modeled better by Lévy alpha-

stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be sta ...

s. The scale of change, or volatility, depends on the length of the time interval to a

power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...

a bit more than 1/2. Large changes up or down are more likely than what one would calculate using a Gaussian distribution with an estimated

standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...

. But the problem is that it does not solve the problem as it makes parametrization much harder and risk control less reliable. Perhaps more fundamental: though mathematical finance models may generate a profit in the short-run, this type of modeling is often in conflict with a central tenet of modern macroeconomics, the

Lucas critique The Lucas critique, named for American economist Robert Lucas's work on macroeconomic policymaking, argues that it is naive to try to predict the effects of a change in economic policy entirely on the basis of relationships observed in historica ...

- or

rational expectations In economics, "rational expectations" are model-consistent expectations, in that agents inside the model are assumed to "know the model" and on average take the model's predictions as valid. Rational expectations ensure internal consistency i ...

- which states that observed relationships may not be structural in nature and thus may not be possible to exploit for public policy or for profit unless we have identified relationships using

causal analysis Causal analysis is the field of experimental design and statistics pertaining to establishing cause and effect. Typically it involves establishing four elements: correlation, sequence in time (that is, causes must occur before their proposed effec ...

and

econometrics Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. ...

. Mathematical finance models do not, therefore, incorporate complex elements of human psychology that are critical to modeling modern macroeconomic movements such as the self-fulfilling panic that motivates

bank runs A bank run or run on the bank occurs when many clients withdraw their money from a bank, because they believe the bank may cease to function in the near future. In other words, it is when, in a fractional-reserve banking system (where banks no ...

***Margrabe's formula **Pricing of American options ***Barone-Adesi and Whaley ***Bjerksund and Stensland ***Black's approximation ***Monte Carlo methods for option pricing#Least Square Monte Carlo, Least Square Monte Carlo ***Optimal stopping ***Roll-Geske-Whaley *Interest rate derivatives ** Black model ***Interest rate cap and floor#Black model, caps and floors ***Swaption#Valuation, swaptions ***Bond option#Valuation, Bond options **Short-rate models ***Rendleman–Bartter model ***Vasicek model ***Ho–Lee model ***Hull–White model ***Cox–Ingersoll–Ross model ***Black–Karasinski model ***Black–Derman–Toy model ***Kalotay–Williams–Fabozzi model ***Longstaff–Schwartz model ***Chen model **Forward rate-based models ***LIBOR market model (Brace–Gatarek–Musiela Model, BGM) ***Heath–Jarrow–Morton framework, Heath–Jarrow–Morton Model (HJM)

Portfolio modelling

Other

* Brownian model of financial markets *Computational finance *Derivative (finance), Outline of finance#Derivatives market, list of derivatives topics *Economic model *Econophysics *Financial economics *Financial engineering * *International Swaps and Derivatives Association *Index of accounting articles *List of economists *Master of Quantitative Finance *Outline of economics *Outline of finance *Physics of financial markets *Quantitative behavioral finance *Statistical finance *Technical analysis * XVA *Quantum finance

History: Q versus P

Derivatives pricing: the Q world

Risk and portfolio management: the P world

Criticism

See also

Mathematical tools

Derivatives pricing

Portfolio modelling

Other

Notes

Further reading