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

_{0}'' of a security is arbitrage-free, and thus truly fair only if there exists a _{t}'' with constant

This "real" probability distribution of the market prices is typically denoted by the blackboard font letter "$\backslash mathbb$", as opposed to the "risk-neutral" probability "$\backslash mathbb$" 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

"The future of financial mathematics"

'' ParisTech Review'', 6 September 2013 * Harold Markowitz, "Portfolio Selection", '' The Journal of Finance'', 7, 1952, pp. 77–91 * Attilio Meucci

" 'P Versus Q': Differences and Commonalities between the Two Areas of Quantitative Finance"

'' GARP Risk Professional'', February 2011, pp. 41–44 * William F. Sharpe, ''Investments'', Prentice-Hall, 1985 * Pierre Henry Labordere (2017). “Model-Free Hedging A Martingale Optimal Transport Viewpoint”. Chapman & Hall/ CRC. {{Authority control Applied statistics

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. 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 when managing portfolios.
French mathematician Louis Bachelier'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, 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 in 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 that are built on observed empirical relationships, in contrast, mathematical finance analysis will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input.
See: Valuation of options; Financial modeling; Asset pricing.
The fundamental theorem of arbitrage-free pricing 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 whose price is determined by the law ofsupply 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 and exotic options, 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 Louis Bachelier 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, 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 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 ...

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 Fischer Black and 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 ...

, along with fundamental contributions by Robert C. Merton, applied the second most influential process, the geometric Brownian motion, to option pricing. 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 ''Pstochastic 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 ...

''Pexpected 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 "$\backslash mathbb$".
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 equations (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 "$\backslash mathbb$", as opposed to the "risk-neutral" probability "$\backslash mathbb$" 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 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 ...

, for the first time ever awarded for a work in finance.
The portfolio-selection work of Markowitz and Sharpe introduced mathematics to investment management. 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, one of the founders of Dow Jones & Company 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. This is the basis of the so-called technical analysis method of attempting to predict future changes. One of the tenets of "technical analysis" is that market trends 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 ofBig 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 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.
Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by Paul Wilmott, and by 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 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 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 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 ...

, but are rather modeled better by Lévy alpha- stable distributions. The scale of change, or volatility, depends on the length of the time interval to a power 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. 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 - or rational expectations - 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.
See also

Mathematical tools

*Asymptotic analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as bec ...

*Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

* Copulas, including Gaussian
*Differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

s
*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 ...

* Ergodic theory
* Feynman–Kac formula
*
*Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...

* Girsanov theorem
* Itô's lemma
*Martingale representation theorem
In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian m ...

*Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...

s
*Mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

**Linear programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is ...

**Nonlinear programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or ...

** Quadratic programming
*Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...

*Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

** Gaussian quadrature
* Real analysis
* Partial differential equations
** Heat equation
** Numerical partial differential equations
***Crank–Nicolson method
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit in time, can be writ ...

*** Finite difference method
*Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

*Probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

s
** Binomial distribution
** Johnson's SU-distribution
**Log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...

**Student's t-distribution
In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situ ...

* Quantile functions
* Radon–Nikodym derivative
* Risk-neutral measure
* Scenario optimization
* Stochastic calculus
** Brownian motion
** Lévy process
* Stochastic differential equation
*Stochastic optimization
Stochastic optimization (SO) methods are optimization methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involves random objective functi ...

* Stochastic volatility
* Survival analysis
* Value at risk
* Volatility
** ARCH model
** GARCH model
Derivatives pricing

* The Brownian model of financial markets * Rational pricing assumptions ** Risk neutral valuation **Arbitrage
In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between t ...

-free pricing
*Valuation adjustments
**Credit valuation adjustment Credit valuation adjustments (CVAs) are accounting adjustments made to reserve a portion of profits on uncollateralized financial derivatives. They are charged by a bank to a risky (capable of default) counterparty to compensate the bank for taking ...

** XVA
* Yield curve modelling
**Multi-curve framework
In finance, an interest rate swap (IRS) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a "linear" IRD and one of the most liquid, benchmark products. It has associations wi ...

** Bootstrapping
** Construction from market data
** Fixed-income attribution
** Nelson-Siegel
** Principal component analysis
* Forward Price Formula
* Futures contract pricing
* Swap valuation
** Currency swap#Valuation and Pricing
** Interest rate swap#Valuation and pricing
***Multi-curve framework
In finance, an interest rate swap (IRS) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a "linear" IRD and one of the most liquid, benchmark products. It has associations wi ...

** Variance swap#Pricing and valuation
** Asset swap #Computing the asset swap spread
** Credit default swap #Pricing and valuation
* Options
** Put–call parity (Arbitrage relationships for options)
** Intrinsic value, Time value
** Moneyness
**Pricing models
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...

*** Black–Scholes model
*** Black model
*** Binomial options model
**** Implied binomial tree
**** Edgeworth binomial tree
*** Monte Carlo option model
*** Implied volatility, Volatility smile
*** Local volatility
*** Stochastic volatility
**** Constant elasticity of variance model
**** Heston model
***** Stochastic volatility jump
**** SABR volatility model
*** Markov switching multifractal
*** The Greeks
*** Finite difference methods for option pricing
*** Vanna–Volga pricing
*** Trinomial tree
**** Implied trinomial tree
*** Garman-Kohlhagen model
*** Lattice model (finance)
*** Margrabe's formula
**Pricing of American options
*** Barone-Adesi and Whaley
*** Bjerksund and Stensland
*** Black's approximation
*** Least Square Monte Carlo
*** Optimal stopping
*** Roll-Geske-Whaley
* Interest rate derivatives
** Black model
*** caps and floors
***swaptions
A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swaps.
Types o ...

*** Bond options
**Short-rate model
A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,.
The short rate
Under a s ...

s
***Rendleman–Bartter model
The Rendleman–Bartter model (Richard J. Rendleman, Jr. and Brit J. Bartter) in finance is a short-rate model describing the evolution of interest rates. It is a "one factor model" as it describes interest rate movements as driven by only one s ...

***Vasicek model
In finance, the Vasicek model is a mathematical model describing the evolution of interest rates. It is a type of one-factor short-rate model as it describes interest rate movements as driven by only one source of market risk. The model can be use ...

*** Ho–Lee model
*** Hull–White model
***Cox–Ingersoll–Ross model
In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes the evolution of interest rates. It is a type of "one factor model" ( short-rate model) as it describes interest rate movements as driven by only one source of mark ...

*** Black–Karasinski model
***Black–Derman–Toy model
In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see . It is a one-factor model; that is, a single stochastic factor—t ...

*** Kalotay–Williams–Fabozzi model
***Longstaff–Schwartz model
A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,.
The short rate
Under a s ...

***Chen model
In finance, the Chen model is a mathematical model describing the evolution of interest rates. It is a type of "three-factor model" (short-rate model) as it describes interest rate movements as driven by three sources of market risk. It was the fir ...

**Forward rate
The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a ''forward rate''..
Forward rate calculation
To extract the forward rate, we n ...

-based models
***LIBOR market model The LIBOR market model, also known as the BGM Model (Brace Gatarek Musiela Model, in reference to the names of some of the inventors) is a financial model of interest rates. It is used for pricing interest rate derivatives, especially exotic deriv ...

(Brace–Gatarek–Musiela Model, BGM)
*** Heath–Jarrow–Morton Model (HJM)
Portfolio modelling

Other

* Brownian model of financial markets *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 ...

* Derivative (finance), list of derivatives topics
* Economic model
* Econophysics
*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 ...

* Financial engineering
*
* International Swaps and Derivatives Association
* Index of accounting articles
*List of economists
This is an incomplete alphabetical list by surname of notable economists, experts in the social science of economics, past and present. For a history of economics, see the article History of economic thought. Only economists with biographical arti ...

*Master of Quantitative Finance A master's degree in quantitative finance concerns the application of mathematical methods to the solution of problems in financial economics. There are several like-titled degrees which may further focus on financial engineering, computational fin ...

* Outline of economics
*Outline of finance
The following outline is provided as an overview of and topical guide to finance:
Finance – addresses the ways in which individuals and organizations raise and allocate monetary resources over time, taking into account the risks entailed ...

* Physics of financial markets
*Quantitative behavioral finance Quantitative behavioral finance is a new discipline that uses mathematical and statistical methodology to understand behavioral biases in conjunction with valuation.
The research can be grouped into the following areas:
# Empirical studies that d ...

*Statistical finance Statistical finance, is the application of econophysics to financial markets. Instead of the normative roots of finance, it uses a positivist framework. It includes exemplars from statistical physics with an emphasis on emergent or collective p ...

* Technical analysis
* XVA
* Quantum finance
Notes

Further reading

* Nicole El Karoui"The future of financial mathematics"

'' ParisTech Review'', 6 September 2013 * Harold Markowitz, "Portfolio Selection", '' The Journal of Finance'', 7, 1952, pp. 77–91 * Attilio Meucci

" 'P Versus Q': Differences and Commonalities between the Two Areas of Quantitative Finance"

'' GARP Risk Professional'', February 2011, pp. 41–44 * William F. Sharpe, ''Investments'', Prentice-Hall, 1985 * Pierre Henry Labordere (2017). “Model-Free Hedging A Martingale Optimal Transport Viewpoint”. Chapman & Hall/ CRC. {{Authority control Applied statistics