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 Merton Miller, shared the 1990

"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 mathematica ...

, concerned with mathematical modeling of financial markets.
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 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 mathe ...

. The latter focuses on applications and modeling, often by help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models.
Also related is quantitative investing, 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 s ...

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 pa ...

'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-origi ...

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 l ...

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 fa ...

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"Financi ...

, 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
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 option (finance), options. This article discusses the calculation of this premium in general. For further detail, see: for discussion of the mathematics; Financial enginee ...

; 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 o ...

; 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 correspo ...

.
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 ...

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. It postulates that, holding all else equal, in a competitive market, the unit price for a particular good, or other traded item such as labor or ...

. 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 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 ...

s, 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
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 pa ...

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 in ...

, 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 Lange ...

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 ...

. 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
In mathematics, a sequence is an enumerated collection of objects in which repetitions are al ...

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 ...

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 ...

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 numbers ...

. This causes longer-term changes to follow a Gaussian distribution.
The theory remained dormant until 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 ...

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-origi ...

, along with fundamental contributions by Robert C. Merton, applied the second most influential process, the geometric Brownian motion, 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 impl ...

. 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 appea ...

''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 models; the model represents the key characteristics or behaviors of the selected system or process, whereas the ...

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 ...

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 "$\backslash mathbb$", 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, 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
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 insti ...

. 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 the ...

, 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 '' Priva ...

and The Wall Street Journal, 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 s ...

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-fra ...

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 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
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 ...

.
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 The Financial Modelers' Manifesto was a proposal for more responsibility in risk management and quantitative finance written by financial engineers Emanuel Derman and Paul Wilmott. The manifesto includes a Modelers' Hippocratic Oath. The struct ...

'' 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 that changes in prices do not follow a Gaussian distribution, 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 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 - 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 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 mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of ari ...

* 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
In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which de ...

*Itô's lemma
In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serve ...

* Martingale representation theorem
*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 s ...

**Quadratic programming
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear const ...

*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 determ ...

*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
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conve ...

*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 ...

s
**Heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for ...

** Numerical partial differential equations
*** Crank–Nicolson method
*** 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 phenomeno ...

s
**Binomial distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no ques ...

** 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 normal ...

**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 si ...

* Quantile functions
* Radon–Nikodym derivative
* Risk-neutral measure
* Scenario optimization
*Stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created ...

**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 in ...

**Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which di ...

*Stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...

*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
Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysi ...

*Value at risk
Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by ...

* Volatility
** ARCH model
** GARCH model
Derivatives pricing

* The Brownian model of financial markets *Rational pricing
Rational pricing is the assumption in financial economics that asset prices - and hence asset pricing models - will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is us ...

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 th ...

-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
An X-Value Adjustment (XVA, xVA) is an umbrella term referring to a number of different “valuation adjustments” that banks must make when assessing the value of derivative contracts that they have entered into. The purpose of these is twofold: ...

*Yield curve
In finance, the yield curve is a graph which depicts how the yields on debt instruments - such as bonds - vary as a function of their years remaining to maturity. Typically, the graph's horizontal or x-axis is a time line of months or ye ...

modelling
** Multi-curve framework
** Bootstrapping
** Construction from market data
** Fixed-income attribution
** Nelson-Siegel
**Principal component analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...

* Forward Price Formula
* Futures contract pricing
* Swap valuation
** Currency swap#Valuation and Pricing
** Interest rate swap#Valuation and pricing
*** Multi-curve framework
** 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
In finance, moneyness is the relative position of the current price (or future price) of an underlying asset (e.g., a stock) with respect to the strike price of a derivative, most commonly a call option or a put option. Moneyness is firstly a th ...

**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
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Bl ...

*** Black model
*** Binomial options model
**** Implied binomial tree
**** Edgeworth binomial tree
*** Monte Carlo option model
***Implied volatility In financial mathematics, the implied volatility (IV) of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model (such as Black–Scholes), will return a theoretical value equ ...

, Volatility smile
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given exp ...

*** 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)
In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is ...

*** 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 derivative
In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of diff ...

s
** Black model
*** caps and floors
*** swaptions
*** 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
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 (Brace–Gatarek–Musiela Model, BGM)
*** Heath–Jarrow–Morton Model (HJM)
Portfolio modelling

Other

* Brownian model of financial markets * Computational finance *Derivative (finance)
In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be u ...

, list of derivatives topics
*Economic model
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 de ...

*Econophysics
Econophysics is a heterodox interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic processes and nonlinear dyn ...

*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"Financi ...

*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 mathe ...

*
*International Swaps and Derivatives Association
International is an adjective (also used as a noun) meaning "between nations".
International may also refer to:
Music Albums
* ''International'' (Kevin Michael album), 2011
* ''International'' (New Order album), 2002
* ''International'' (The T ...

* 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 art ...

* Master of Quantitative Finance
* Outline of economics
* Outline of finance
* Physics of financial markets
* Quantitative behavioral finance
* Statistical finance
*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 s ...

*XVA
An X-Value Adjustment (XVA, xVA) is an umbrella term referring to a number of different “valuation adjustments” that banks must make when assessing the value of derivative contracts that they have entered into. The purpose of these is twofold: ...

* 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