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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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and be ...
, concerned with mathematical modeling of
financial market A financial market is a market (economics), market in which people trade financial Security (finance), securities and derivative (finance), derivatives at low transaction costs. Some of the securities include stocks and Bond (finance), bonds, ra ...
s. See
Quantitative analyst Quantitative may refer to: * Quantitative research Quantitative research is a research strategy that focuses on quantifying the collection and analysis of data. It is formed from a deductive Deductive reasoning, also deductive logic, is the ...
. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and
risk-
risk-
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 Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. ...
and
financial engineering Financial engineering is a multidisciplinary field involving financial theory, methods of engineering, tools of mathematics and the practice of Mathematical programming, programming. It has also been defined as the application of technical methods ...

financial engineering
. The latter focuses on applications and modeling, often by help of
stochastic asset model A stochastic investment model tries to forecast how rate of return, 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 es ...
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 finance, 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 ma ...
, which relies on statistical and numerical models (and lately
machine learning Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data ...

machine learning
) 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, Liability (financial accounting), liabilities, and earnings); health; and Competition, competitors an ...
when managing portfolios. French mathematician
Louis Bachelier Louis Jean-Baptiste Alphonse Bachelier (; 11 March 1870 – 28 April 1946) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topi ...
is considered the author of the first scholarly work on mathematical finance, published in 1900. 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 American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of ...
,
Myron Scholes Myron Samuel Scholes ( ; born July 1, 1941) is a Canadians, Canadian-Americans, American financial economist. Scholes is the Frank E. Buck Professor of Finance, Emeritus, at the Stanford Graduate School of Business, Nobel Laureate in Economic Sci ...
and
Robert Merton
Robert Merton
on option pricing theory. Mathematical investing originated from the research of mathematician
Edward Thorp Edward Oakley Thorp (born August 14, 1932) is an American mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
who used statistical methods to first invent
card counting Card counting is a casino card game strategy used primarily in the blackjack family of casino games to determine whether the next hand is likely to give a probable advantage to the player or to the dealer. Card counters are a class of advantage pl ...
in
blackjack Blackjack, formerly also Black Jack and Vingt-Un, is the American member of a global family of banking game The following is a glossary of terms used in card games. Besides the terms listed here, there are thousands of common and uncommon sla ...

blackjack
and then applied its principles to modern systematic investing. The subject has a close relationship with the discipline of
financial economics Financial economics is the branch of economics Economics () is a social science Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among i ...
, which is concerned with much of the underlying theory that is involved in financial mathematics. Generally, mathematical finance will derive and extend the
mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while a
financial economist Financial economics 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"Financial Economics", in Its co ...
might study the structural reasons why a company may have a certain
share price A share price is the price of a single share Share may refer to: * Share, to make joint use of a resource (such as food, money, or space); see Sharing * Share (finance), a stock or other financial security (such as a mutual fund) * Share, Kwara, a ...
, a financial mathematician may take the share price as a given, and attempt to use
stochastic calculus Stochastic calculus is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
to obtain the corresponding value of
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
s of the
stock In finance, stock (also capital stock) consists of all of the shares In financial markets A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities i ...

stock
. See:
Valuation of options In finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in t ...
;
Financial modeling Financial modeling is the task of building an abstraction, abstract representation (a mathematical model, model) of a real world finance, financial situation. This is a mathematical model designed to represent (a simplified version of) the perfor ...
;
Asset pricing :'' This article is theory focused: for the corporate finance Corporate finance is the area of finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned wit ...
. 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, arbitrage free, and for a market to be Compl ...
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 of
supply and demand In microeconomics Microeconomics is a branch of that studies the behavior of individuals and in making decisions regarding the allocation of and the interactions among these individuals and firms. Microeconomics focuses on the study ...

supply and demand
. 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 Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in t ...
s,
convertible bond In finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topi ...
in ''The Theory of Speculation'' ("Théorie de la spéculation", published 1900), with the introduction of the most basic and most influential of processes, the
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physica ...

Brownian motion
, and its applications to the pricing of options. The 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
. Bachelier modeled the
time series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

time series
of changes in the
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

logarithm
of stock prices as a
random walk In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
in which the short-term changes had a finite
variance In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

variance
. This causes longer-term changes to follow a
Gaussian distribution In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spac ...

Gaussian distribution
. The theory remained dormant until
Fischer Black Fischer Sheffey Black (January 11, 1938 – August 30, 1995) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of ...
and
Myron Scholes Myron Samuel Scholes ( ; born July 1, 1941) is a Canadians, Canadian-Americans, American financial economist. Scholes is the Frank E. Buck Professor of Finance, Emeritus, at the Stanford Graduate School of Business, Nobel Laureate in Economic Sci ...
, along with fundamental contributions by
Robert C. Merton
Robert C. Merton
, 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 stochastic d ...
, to
option pricing In finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in t ...
. 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 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, arbitrage free, and for a market to be Compl ...
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 Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and phen ...

stochastic process
''Pt'' with constant
expected value In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...
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
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 models; the model represents the key characteristics or behaviors of the selected system or process, whereas the simulat ...
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 calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s (PDE's).


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 theoremThe Modigliani–Miller theorem (of Franco Modigliani, Merton Miller) is an influential element of economic theor ...
, 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 Asset management refers to a systematic approach to the governance and realization of value from the things that a group or entity is responsible for, over their whole life cycles. It ...
. 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. 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 Dow Jones & Company, Inc. is an American publishing firm owned by News Corp and led by CEO Almar Latour. The company ...
, 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 (newspaper), Barron's'', ''MarketWatch'', ''Mansion Global'', ''Financial Ne ...
and
The Wall Street Journal ''The Wall Street Journal'', also known as ''The Journal'', is an American business-focused, English-language international daily newspaper A newspaper is a periodical Periodical literature (also called a periodical publication or sim ...

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 In finance, technical analysis is an analysis Analysis is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to ...
. This is the basis of the so-called
technical analysis In finance, technical analysis is an analysis methodology for forecasting the direction of prices through the study of past market data, primarily price and volume. Behavioral economics and Quantitative analysis (finance), quantitative analysis us ...

technical analysis
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. These trends are classified as ''secular'' for long time frames, ''primary'' for medium time frames, and ''secondary'' for short time frames. ...
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

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 a term for the management, creation, and study of money Image:National-Debt-Gillray.jpeg, In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms w ...
. 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 people, English researcher, consultant and lecturer in quantitative finance.
, and by
Nassim Nicholas Taleb Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist whose work concerns problems of randomness In co ...
, 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 Quantitative may refer to: * Quantitative research, scientific investigation of quantitative properties * Quantitati ...
published the ''
Financial Modelers' Manifesto The Financial Modelers' Manifesto was a proposal for more responsibility in risk management and quantitative finance written by quantitative finance, financial engineers Emanuel Derman and Paul Wilmott. The manifesto includes a Modelers' Hippocratic ...
'' 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 New York, often called New York City to distinguish it from New York State New York is a state State may refer to: Arts, entertainment, and media Literature * ''Stat ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study ...
that changes in prices do not follow a
Gaussian distribution In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spac ...

Gaussian distribution
, but are rather modeled better by Lévy alpha-
stable distribution In probability theory, a probability distribution, distribution is said to be stable if a linear combination of two Independence (probability theory), independent random variables with this distribution has the same distribution, up to location ...
s. B. Mandelbrot
"The variation of certain Speculative Prices"
''The Journal of Business'' 1963
The scale of change, or volatility, depends on the length of the time interval to a
power Power most often refers to: * Power (physics) In physics, power is the amount of energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, bu ...
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 statistical dispersion, 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 v ...

standard deviation
. But the problem is that it does not solve the problem as it makes parametrization much harder and risk control less reliable.


See also


Mathematical tools

*
Asymptotic analysis In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathem ...
*
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 generalizations ...

Calculus
* Copulas, including Gaussian *
Differential equation In mathematics, a differential equation is an equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

Differential equation
s *
Expected value In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...
*
Ergodic theory Ergodic theory (Ancient Greek, Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties wh ...
* Feynman–Kac formula *
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
*
Girsanov theorem In probability theory, the Girsanov theorem (named after Igor Vladimirovich Girsanov) describes how the dynamics of stochastic processes change when the original measure (probability), measure is changed to an Equivalence (measure theory), equiva ...
*
Itô's lemma In mathematics, Itô's lemma is an Identity_(mathematics), identity used in Itô calculus to find the differential (calculus), differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of t ...
*
Martingale representation theorem In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the Filtration (probability theory), filtration generated by a Wiener process, Brownian motion can be written in terms of ...
*
Mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environm ...
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. Optimization problems of sorts arise i ...
**
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 A mathematical model is a description of a system A system is a group of ...
**
Nonlinear programming In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

Nonlinear programming
**
Quadratic programming Quadratic programming (QP) is the process of solving certain mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some ...
*
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 determini ...
*
Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
**
Gaussian quadrature In numerical analysis, a quadrature rule is an approximation of the integral, definite integral of a function (mathematics), function, usually stated as a weighted sum of function values at specified points within the domain of integration. (Se ...
*
Real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

Real analysis
*
Partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s **
Heat equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
**
Numerical partial differential equations Numerical may refer to: * Number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally define ...
***
Crank–Nicolson method In numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... Numerical analysis is the study of ...
***
Finite difference method #REDIRECT Finite difference method#REDIRECT Finite difference method In numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 ...
*
Probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

Probability
*
Probability distribution In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
s **
Binomial distribution In probability theory and statistics, the Binomial coefficient, binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' statistical independence, indep ...

Binomial distribution
** Johnson's SU-distribution **
Log-normal distribution In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces i ...
**
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 Expected value, mean of a Normal distribution, norm ...
*
Quantile function In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equa ...
s * Radon–Nikodym derivative *
Risk-neutral measure In mathematical finance 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 ...
*
Scenario optimization #REDIRECT Scenario optimizationThe scenario approach or scenario optimization approach is a technique for obtaining solutions to robust optimizationRobust optimization is a field of optimization (mathematics), optimization theory that deals with opt ...
*
Stochastic calculus Stochastic calculus is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
**
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physica ...
**
Lévy process In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in ...
*
Stochastic differential equation A stochastic differential equation (SDE) is a differential equation In mathematics, a differential equation is an equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas an ...
*
Stochastic optimization Stochastic optimization (SO) methods are optimization (mathematics), optimization iterative method, methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem ...
*
Stochastic volatility In statistics, stochastic volatility models are those in which the variance In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic ...
*
Survival analysis Survival analysis is a branch of statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, ...
*
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 f ...

Value at risk
* Volatility ** ARCH model ** GARCH model


Derivatives pricing

* The Brownian model of financial markets * Rational pricing assumptions **Risk-neutral measure, Risk neutral valuation **Arbitrage-free pricing *Valuation adjustments **Credit valuation adjustment **XVA *Yield curve modelling **Multi-curve framework **Bootstrapping (finance), Bootstrapping **Yield curve#Construction of the full yield curve from market data, Construction from market data **Fixed-income attribution #Modeling the yield curve, Fixed-income attribution **Nelson-Siegel **Principal component analysis#Quantitative finance, Principal component analysis *Forward price#Forward Price Formula, Forward Price Formula *Futures contract#Pricing, Futures contract pricing *Swap (finance)#Valuation and 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 (finance), Intrinsic value, Option time value, Time value **Moneyness **Pricing Mathematical model, models ***Black–Scholes model ***Black model ***Binomial options pricing model, Binomial options model ****Implied binomial tree ****Edgeworth binomial tree ***Monte Carlo option model ***Implied volatility, Volatility smile *** Local volatility ***
Stochastic volatility In statistics, stochastic volatility models are those in which the variance In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic ...
**** Constant elasticity of variance model **** Heston model ***** Stochastic volatility jump **** SABR volatility model ***Markov switching multifractal ***Greeks (finance), 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 ***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


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"
''Global Association of Risk Professionals, GARP Risk Professional'', February 2011, pp. 41–44 * William F. Sharpe, ''Investments'', Prentice-Hall, 1985 {{Authority control Mathematical finance, Applied statistics