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Quantitative analysis is the use of
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and statistical methods in
finance Finance refers to monetary resources and to the study and Academic discipline, discipline of money, currency, assets and Liability (financial accounting), liabilities. As a subject of study, is a field of Business administration, Business Admin ...
and
investment management Investment management (sometimes referred to more generally as financial asset management) is the professional asset management of various Security (finance), securities, including shareholdings, Bond (finance), bonds, and other assets, such as r ...
. Those working in the field are quantitative analysts (quants). Quants tend to specialize in specific areas which may include
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
structuring or pricing,
risk management Risk management is the identification, evaluation, and prioritization of risks, followed by the minimization, monitoring, and control of the impact or probability of those risks occurring. Risks can come from various sources (i.e, Threat (sec ...
,
investment management Investment management (sometimes referred to more generally as financial asset management) is the professional asset management of various Security (finance), securities, including shareholdings, Bond (finance), bonds, and other assets, such as r ...
and other related finance occupations. The occupation is similar to those in industrial mathematics in other industries. The process usually consists of searching vast databases for patterns, such as correlations among liquid assets or price-movement patterns (
trend following Trend following or trend trading is a trading strategy according to which one should buy an asset when its price trend goes up, and sell when its trend goes down, expecting price movements to continue. There are a number of different techniques, ...
or reversion). Although the original quantitative analysts were "
sell side Sell side is a term used in the financial services industry to mean providing services to sell securities. Firms or institutions on this side include investment banks, brokerages and market makers, who facilitate offering securities to investors, ...
quants" from
market maker A market maker or liquidity provider is a company or an individual that quotes both a buy and a sell price in a tradable asset held in inventory, hoping to make a profit on the difference, which is called the ''bid–ask spread'' or ''turn.'' Thi ...
firms, concerned with derivatives pricing and risk management, the meaning of the term has expanded over time to include those individuals involved in almost any application of mathematical finance, including the
buy side Buy-side is a term used in investment banking to refer to advising institutions concerned with buying investment services. Private equity funds, mutual funds, life insurance companies, unit trusts, hedge funds, and pension funds are the most c ...
. Applied quantitative analysis is commonly associated with quantitative investment management which includes a variety of methods such as
statistical arbitrage In finance, statistical arbitrage (often abbreviated as Stat Arb or StatArb) is a class of short-term financial trading strategies that employ Mean reversion (finance), mean reversion models involving broadly diversified portfolios of securities (h ...
, algorithmic trading and electronic trading. Some of the larger investment managers using quantitative analysis include Renaissance Technologies, D. E. Shaw & Co., and AQR Capital Management.


History

Quantitative finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field. In general, there exist two separate branches of finance that requ ...
started in 1900 with Louis Bachelier's doctoral
thesis A thesis (: theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: D ...
"Theory of Speculation", which provided a model to price options under a
normal distribution In probability theory and 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 ...
. Jules Regnault had posited already in 1863 that stock prices can be modelled as a
random walk In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space. An elementary example of a rand ...
, suggesting "in a more literary form, the conceptual setting for the application of probability to stockmarket operations". It was, however, only in the years 1960-1970 that the "merit of hesewas recognized" L. Carraro and P. Crépel (N.D.)
Bachelier, Louis
Encyclopedia of Mathematics
as options pricing theory was developed. Harry Markowitz's 1952 doctoral thesis "Portfolio Selection" and its published version was one of the first efforts in economics journals to formally adapt mathematical concepts to finance (mathematics was until then confined to specialized economics journals). Markowitz formalized a notion of mean return and covariances for common stocks which allowed him to quantify the concept of "diversification" in a market. He showed how to compute the mean return and variance for a given portfolio and argued that investors should hold only those portfolios whose variance is minimal among all portfolios with a given mean return. Thus, although the language of finance now involves Itô calculus, management of risk in a quantifiable manner underlies much of the modern theory. Modern quantitative investment management was first introduced from the research of Edward Thorp, a mathematics professor at
New Mexico State University New Mexico State University (NMSU or NM State) is a public, land-grant, research university in Las Cruces, New Mexico, United States. Founded in 1888, it is the state's oldest public institution of higher education, and was the original land-g ...
(1961–1965) and
University of California, Irvine The University of California, Irvine (UCI or UC Irvine) is a Public university, public Land-grant university, land-grant research university in Irvine, California, United States. One of the ten campuses of the University of California system, U ...
(1965–1977). Considered the "Father of Quantitative Investing", Thorp sought to predict and simulate blackjack, a card-game he played in Las Vegas casinos. He was able to create a system, known broadly as
card counting Card counting is a blackjack betting strategy, strategy used to determine whether the player or the dealer has an advantage on the next hand. Card counters try to overcome the casino house edge by keeping a running count of high and low valued c ...
, which used
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and statistical analysis to successfully win blackjack games. His research was subsequently used during the 1980s and 1990s by investment management firms seeking to generate systematic and consistent returns in the U.S. stock market. The field has grown to incorporate numerous approaches and techniques; see , Post-modern portfolio theory, . In 1965,
Paul Samuelson Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he "h ...
introduced
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 an ...
into the study of finance. In 1969, Robert Merton promoted continuous stochastic calculus and continuous-time processes. Merton was motivated by the desire to understand how prices are set in financial markets, which is the classical economics question of "equilibrium", and in later papers he used the machinery of stochastic calculus to begin investigation of this issue. At the same time as Merton's work and with Merton's assistance, Fischer Black and Myron Scholes developed the
Black–Scholes model The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
, which was 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 (), commonly referred to as the Nobel Prize in Economics(), is an award in the field of economic sciences adminis ...
. It provided a solution for a practical problem, that of finding a fair price for a European call option, i.e., the right to buy one share of a given stock at a specified price and time. Such options are frequently purchased by investors as a risk-hedging device. In 1981, Harrison and Pliska used the general theory of continuous-time stochastic processes to put the Black–Scholes model on a solid theoretical basis, and showed how to price numerous other derivative securities, laying the groundwork for the development of 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-free, and for a market to be complete. An a ...
. The various
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 sh ...
s (beginning with Vasicek in 1977), and the more general HJM Framework (1987), relatedly allowed for an extension to
fixed income Fixed income refers to any type of investment under which the borrower or issuer is obliged to make payments of a fixed amount on a fixed schedule. For example, the borrower may have to pay interest at a fixed rate once a year and repay the pr ...
and interest rate derivatives. Similarly, and in parallel, models were developed for various other underpinnings and applications, including credit derivatives,
exotic derivatives An exotic derivative, in finance, is a derivative (finance), derivative which is more complex than commonly traded "vanilla" products. This complexity usually relates to determination of payoff; see option style. The category may also include de ...
,
real options Real options valuation, also often termed real options analysis,Adam Borison (Stanford University)''Real Options Analysis: Where are the Emperor's Clothes?'' (ROV or ROA) applies option (finance), option Valuation of options, valuation technique ...
, and employee stock options. Quants are thus involved in pricing and hedging a wide range of securities – asset-backed,
government A government is the system or group of people governing an organized community, generally a State (polity), state. In the case of its broad associative definition, government normally consists of legislature, executive (government), execu ...
, and
corporate A corporation or body corporate is an individual or a group of people, such as an association or company, that has been authorized by the state to act as a single entity (a legal entity recognized by private and public law as "born out of s ...
– additional to classic derivatives; see contingent claim analysis.
Emanuel Derman Emanuel Derman (born 1945) is a South African-born academic, businessman and writer. He is best known as a quantitative analyst, and author of the book ''My Life as a Quant: Reflections on Physics and Finance''. He is a co-author of Black–D ...
's 2004 book ''My Life as a Quant'' helped to both make the role of a quantitative analyst better known outside of finance, and to popularize the abbreviation "quant" for a quantitative analyst. After the
2008 financial crisis The 2008 financial crisis, also known as the global financial crisis (GFC), was a major worldwide financial crisis centered in the United States. The causes of the 2008 crisis included excessive speculation on housing values by both homeowners ...
, considerations regarding
counterparty credit risk Credit risk is the chance that a borrower does not repay a loan or fulfill a loan obligation. For lenders the risk includes late or lost interest and principal payment, leading to disrupted cash flows and increased collection costs. The loss ...
were incorporated into the modelling, previously performed in an entirely "
risk neutral In economics and finance, risk neutral preferences are preference (economics), preferences that are neither risk aversion, risk averse nor risk seeking. A risk neutral party's decisions are not affected by the degree of uncertainty in a set of out ...
world", entailing three major developments; see : (i) Option pricing and hedging inhere the relevant volatility surface - to some extent, equity-option prices have incorporated the volatility smile since the 1987 crash - and banks then apply "surface aware" local- or
stochastic volatility In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name ...
models; (ii) The risk neutral value is adjusted for the impact of counter-party credit risk via a
credit valuation adjustment A Credit valuation adjustment (CVA), in financial mathematics, is an "adjustment" to a derivative's price, as charged by a bank to a counterparty to compensate it for taking on the credit risk of that counterparty during the life of the tran ...
, or CVA, as well as various of the other
XVA X-Value Adjustment (XVA, xVA) is an hyponymy and hypernymy, umbrella term referring to a number of different "valuation adjustments" that banks must make when assessing the value of derivative (finance), derivative contracts that they have entered ...
; (iii) For discounting, the OIS curve is used for the "risk free rate", as opposed to
LIBOR The London Inter-Bank Offered Rate (Libor ) was an interest rate average calculated from estimates submitted by the leading Bank, banks in London. Each bank estimated what it would be charged were it to borrow from other banks. It was the prim ...
as previously, and, relatedly, quants must model under a "
multi-curve framework In finance, an interest rate swap (finance), swap (IRS) is an interest rate derivative, interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a Interest rate derivative#Linear and non-linear ...
" ( LIBOR is being phased out, with replacements including
SOFR Secured Overnight Financing Rate (SOFR) is a secured overnight rate, overnight interest rate. SOFR is a reference rate (that is, a rate used by parties in commercial contracts that is outside their direct control) established as an alternative to L ...
and TONAR, necessitating technical changes to the latter framework, while the underlying logic is unaffected).


Types


Front office quantitative analyst

In
sales and trading Sales and trading is one of the primary front-office divisions of major investment banks. The term is typically reserved for the trading activities done by sell-side investment banks who are primarily engaged in making markets for institutional cl ...
, quantitative analysts work to determine prices, manage risk, and identify profitable opportunities. Historically this was a distinct activity from
trading Trade involves the transfer of goods and services from one person or entity to another, often in exchange for money. Economists refer to a system or network that allows trade as a market (economics), market. Traders generally negotiate throu ...
but the boundary between a
desk A desk or bureau is a piece of furniture with a flat table (furniture), table-style work surface used in a school, office, home or the like for academic, professional or domestic activities such as reading (activity), reading, writing, or using ...
quantitative analyst and a quantitative trader is increasingly blurred, and it is now difficult to enter trading as a profession without at least some quantitative analysis education. Front office work favours a higher speed to quality ratio, with a greater emphasis on solutions to specific problems than detailed modeling. FOQs typically are significantly better paid than those in back office, risk, and model validation. Although highly skilled analysts, FOQs frequently lack software engineering experience or formal training, and bound by time constraints and business pressures, tactical solutions are often adopted. Increasingly, quants are attached to specific desks. Two cases are: XVA specialists, responsible for managing
counterparty risk Credit risk is the chance that a borrower does not repay a loan or fulfill a loan obligation. For lenders the risk includes late or lost interest and principal payment, leading to disrupted cash flows and increased collection costs. The loss ...
as well as (minimizing) the capital requirements under
Basel III Basel III is the third of three Basel Accords, a framework that sets international standards and minimums for bank capital requirements, Stress test (financial), stress tests, liquidity regulations, and Leverage (finance), leverage, with the goa ...
; and
structurer In investment banking, a structurer Joris Luyendijk (2012)Interview: Head of Structuring equity-derivatives ''theguardian.com'' is the finance professional responsible for designing structured products. Their solution will typically deliver ...
s, tasked with the design and manufacture of client specific solutions.


Quantitative investment management

Quantitative analysis is used extensively by asset managers. Some, such as FQ, AQR or
Barclays Barclays PLC (, occasionally ) is a British multinational universal bank, headquartered in London, England. Barclays operates as two divisions, Barclays UK and Barclays International, supported by a service company, Barclays Execution Services ...
, rely almost exclusively on quantitative strategies while others, such as
PIMCO Pacific Investment Management Company LLC (PIMCO) is an American investment management firm. While it has a specific focus on active fixed income management worldwide, it manages investments in many asset classes, including fixed income, share ca ...
,
BlackRock BlackRock, Inc. is an American Multinational corporation, multinational investment company. Founded in 1988, initially as an enterprise risk management and fixed income institutional asset manager, BlackRock is the world's largest asset manager ...
or
Citadel A citadel is the most fortified area of a town or city. It may be a castle, fortress, or fortified center. The term is a diminutive of ''city'', meaning "little city", because it is a smaller part of the city of which it is the defensive core. ...
use a mix of quantitative and fundamental methods. One of the first quantitative investment funds to launch was based in
Santa Fe, New Mexico Santa Fe ( ; , literal translation, lit. "Holy Faith") is the capital city, capital of the U.S. state of New Mexico, and the county seat of Santa Fe County. With over 89,000 residents, Santa Fe is the List of municipalities in New Mexico, fourt ...
and began trading in 1991 under the name Prediction Company. By the late-1990s, Prediction Company began using
statistical arbitrage In finance, statistical arbitrage (often abbreviated as Stat Arb or StatArb) is a class of short-term financial trading strategies that employ Mean reversion (finance), mean reversion models involving broadly diversified portfolios of securities (h ...
to secure investment returns, along with three other funds at the time, Renaissance Technologies and D. E. Shaw & Co, both based in New York. Prediction hired scientists and computer programmers from the neighboring
Los Alamos National Laboratory Los Alamos National Laboratory (often shortened as Los Alamos and LANL) is one of the sixteen research and development Laboratory, laboratories of the United States Department of Energy National Laboratories, United States Department of Energy ...
to create sophisticated statistical models using "industrial-strength computers" in order to " uildthe Supercollider of Finance". Machine learning models are now capable of identifying complex patterns in financial market data. With the aid of artificial intelligence, investors are increasingly turning to deep learning techniques to forecast and analyze trends in stock and foreign exchange markets. See .


Library quantitative analysis

Major firms invest large sums in an attempt to produce standard methods of evaluating prices and risk. These differ from front office tools in that Excel is very rare, with most development being in C++, though
Java Java is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea (a part of Pacific Ocean) to the north. With a population of 156.9 million people (including Madura) in mid 2024, proje ...
, C# and Python are sometimes used in non-performance critical tasks. LQs spend more time modeling ensuring the analytics are both efficient and correct, though there is tension between LQs and FOQs on the validity of their results. LQs are required to understand techniques such as
Monte Carlo methods Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on Resampling (statistics), repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve pr ...
and finite difference methods, as well as the nature of the products being modeled.


Algorithmic trading quantitative analyst

Often the highest paid form of Quant, ATQs make use of methods taken from
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
,
game theory Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...
, gambling Kelly criterion,
market microstructure Market microstructure is a branch of finance concerned with the details of how exchange occurs in markets. While the theory of market microstructure applies to the exchange of real or financial assets, more evidence is available on the microstruct ...
,
econometrics Econometrics is an 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. 8 ...
, and
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. ...
analysis.


Risk management

This area has grown in importance in recent years, as the credit crisis exposed holes in the mechanisms used to ensure that positions were correctly hedged; see FRTB, . A core technique continues to be value at risk - applying both the parametric and "Historical" approaches, as well as Conditional value at risk and
Extreme value theory Extreme value theory or extreme value analysis (EVA) is the study of extremes in statistical distributions. It is widely used in many disciplines, such as structural engineering, finance, economics, earth sciences, traffic prediction, and Engin ...
- while this is supplemented with various forms of stress test, expected shortfall methodologies,
economic capital In finance, mainly for financial services firms, economic capital (ecap) is the amount of risk capital, assessed on a realistic basis, which a firm requires to cover the risks that it is running or collecting as a going concern, such as market ...
analysis, direct analysis of the positions at the desk level, and, as below, assessment of the models used by the bank's various divisions.


Innovation

After the
2008 financial crisis The 2008 financial crisis, also known as the global financial crisis (GFC), was a major worldwide financial crisis centered in the United States. The causes of the 2008 crisis included excessive speculation on housing values by both homeowners ...
, there surfaced the recognition that quantitative valuation methods were generally too narrow in their approach. An agreed upon fix adopted by numerous financial institutions has been to improve collaboration.


Model validation

Model validation (MV) takes the models and methods developed by front office, library, and modeling quantitative analysts and determines their validity and correctness; see model risk. The MV group might well be seen as a superset of the quantitative operations in a financial institution, since it must deal with new and advanced models and trading techniques from across the firm. Post crisis, regulators now typically talk directly to the quants in the middle office - such as the model validators - and since profits highly depend on the regulatory infrastructure, model validation has gained in weight and importance with respect to the quants in the front office. Before the crisis however, the pay structure in all firms was such that MV groups struggle to attract and retain adequate staff, often with talented quantitative analysts leaving at the first opportunity. This gravely impacted corporate ability to manage model risk, or to ensure that the positions being held were correctly valued. An MV quantitative analyst would typically earn a fraction of quantitative analysts in other groups with similar length of experience. In the years following the crisis, as mentioned, this has changed.


Quantitative developer

Quantitative developers, sometimes called quantitative software engineers, or quantitative engineers, are computer specialists that assist, implement and maintain the quantitative models. They tend to be highly specialised language technicians that bridge the gap between
software engineers Software engineering is a branch of both computer science and engineering focused on designing, developing, testing, and maintaining software applications. It involves applying engineering principles and computer programming expertise to develop ...
and quantitative analysts. The term is also sometimes used outside the finance industry to refer to those working at the intersection of
software engineering Software engineering is a branch of both computer science and engineering focused on designing, developing, testing, and maintaining Application software, software applications. It involves applying engineering design process, engineering principl ...
and
quantitative research Quantitative research is a research strategy that focuses on quantifying the collection and analysis of data. It is formed from a deductive approach where emphasis is placed on the testing of theory, shaped by empiricist and positivist philoso ...
.


Hypothesis of non-ergodicity of financial markets

The nonergodicity of financial markets and the time dependence of returns are central issues in modern approaches to quantitative trading. Financial markets are complex systems in which traditional assumptions, such as independence and normal distribution of returns, are frequently challenged by empirical evidence. Thus, under the non-ergodicity hypothesis, the future returns about an investment strategy, which operates on a non-stationary system, depend on the ability of the algorithm itself to predict the future evolutions to which the system is subject. As discussed by Ole Peters in 2011, ergodicity is a crucial element in understanding economic dynamics, especially in non-stationary contexts. Identifying and developing methodologies to estimate this ability represents one of the main challenges of modern quantitative trading. In this perspective, it becomes fundamental to shift the focus from the result of individual financial operations to the individual evolutions of the system. Operationally, this implies that clusters of trades oriented in the same direction offer little value in evaluating the strategy. On the contrary, sequences of trades with alternating buy and sell are much more significant. Since they indicate that the strategy is actually predicting a statistically significant number of evolutions of the system.


Mathematical and statistical approaches

Because of their backgrounds, quantitative analysts draw from various forms of mathematics:
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
and
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
,
calculus Calculus 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 of arithmetic operations. Originally called infinitesimal calculus or "the ...
centered around
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s,
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
,
discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...
, and
econometrics Econometrics is an 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. 8 ...
. Some on the buy side may use
machine learning Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of Computational statistics, statistical algorithms that can learn from data and generalise to unseen data, and thus perform Task ( ...
. The majority of quantitative analysts have received little formal education in mainstream economics, and often apply a mindset drawn from the physical sciences. Quants use mathematical skills learned from diverse fields such as computer science, physics and engineering. These skills include (but are not limited to) advanced statistics, linear algebra and partial differential equations as well as solutions to these based upon
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). It is the study of ...
. Commonly used numerical methods are: *
Finite difference method In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating Derivative, derivatives with Finite difference approximation, finite differences. Both the spatial doma ...
– used to solve
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s; *
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 ...
– Also used to solve
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s, but Monte Carlo simulation is also common in risk management; *
Ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression In statistics, linear regression is a statistical model, model that estimates the relationship ...
– used to estimate parameters in statistical regression analysis; * Spline interpolation – used to interpolate values from spot and forward interest rates curves, and volatility smiles; *
Bisection In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''s ...
, Newton, and Secant methods – used to find the
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...
,
maxima and minima In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...
of functions (e.g.
internal rate of return Internal rate of return (IRR) is a method of calculating an investment's rate of return. The term ''internal'' refers to the fact that the calculation excludes external factors, such as the risk-free rate, inflation, the cost of capital, or fin ...
, interest rate curve-building.)


Techniques

A typical problem for a mathematically oriented quantitative analyst would be to develop a model for pricing, hedging, and risk-managing a complex derivative product. These quantitative analysts tend to rely more on numerical analysis than statistics and econometrics. One of the principal mathematical tools of quantitative finance is
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 an ...
. The mindset, however, is to prefer a deterministically "correct" answer, as once there is agreement on input values and market variable dynamics, there is only one correct price for any given security (which can be demonstrated, albeit often inefficiently, through a large volume of Monte Carlo simulations). A typical problem for a statistically oriented quantitative analyst would be to develop a model for deciding which stocks are relatively expensive and which stocks are relatively cheap. The model might include a company's book value to price ratio, its trailing earnings to price ratio, and other accounting factors. An investment manager might implement this analysis by buying the underpriced stocks, selling the overpriced stocks, or both. Statistically oriented quantitative analysts tend to have more of a reliance on statistics and econometrics, and less of a reliance on sophisticated numerical techniques and object-oriented programming. These quantitative analysts tend to be of the psychology that enjoys trying to find the best approach to modeling data, and can accept that there is no "right answer" until time has passed and we can retrospectively see how the model performed. Both types of quantitative analysts demand a strong knowledge of sophisticated mathematics and computer programming proficiency.


Education

Quantitative analysts often come from
applied mathematics Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
,
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
or
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
backgrounds, learning finance " on the job". Quantitative analysis is a then major source of employment for those with mathematics and physics PhD degrees. Typically, a quantitative analyst will also need
Emanuel Derman Emanuel Derman (born 1945) is a South African-born academic, businessman and writer. He is best known as a quantitative analyst, and author of the book ''My Life as a Quant: Reflections on Physics and Finance''. He is a co-author of Black–D ...
(2004)
"Finding a job in finance"
''
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 environ ...
''
extensive skills in computer programming, most commonly C, C++ and
Java Java is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea (a part of Pacific Ocean) to the north. With a population of 156.9 million people (including Madura) in mid 2024, proje ...
, and lately R,
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
,
Mathematica Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...
, and Python.
Data science Data science is an interdisciplinary academic field that uses statistics, scientific computing, scientific methods, processing, scientific visualization, algorithms and systems to extract or extrapolate knowledge from potentially noisy, stru ...
and
machine learning Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of Computational statistics, statistical algorithms that can learn from data and generalise to unseen data, and thus perform Task ( ...
analysis and methods are being increasingly employed in portfolio performance and portfolio risk modelling, and as such data science and machine learning Master's graduates are also hired as quantitative analysts. The demand for quantitative skills has led to the creation of specialized Masters International Association of Financial Engineers (2007)
"Student FAQ"
/ref> and PhD courses in
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 ...
,
mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field. In general, there exist two separate branches of finance that req ...
and
computational finance Computational finance is a branch of applied computer science that deals with problems of practical interest in finance.Rüdiger U. Seydel, ''Tools for Computational Finance'', Springer; 3rd edition (May 11, 2006) 978-3540279235 Some slightly diff ...
(as well as in specific topics such as financial reinsurance). In particular, the Master of Quantitative Finance, Master of Financial Mathematics, Master of Computational Finance and Master of Financial Engineering are becoming popular with students and with employers. See . This has, in parallel, led to a resurgence in demand for
actuarial Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in insurance, pension, finance, investment and other industries and professions. Actuaries are professionals trained in this discipline. In m ...
qualifications, as well as commercial certifications such as the CQF. Similarly, the more general Master of Finance (and Master of Financial Economics) increasingly Lindsey Gerdes (2009
"Master's of the Financial Universe"
''
Businessweek ''Bloomberg Businessweek'', previously known as ''BusinessWeek'' (and before that ''Business Week'' and ''The Business Week''), is an American monthly business magazine published 12 times a year. The magazine debuted in New York City in Septembe ...
''
includes a significant technical component. Likewise, masters programs in
operations research Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods to improve management and ...
,
computational statistics Computational statistics, or statistical computing, is the study which is the intersection of statistics and computer science, and refers to the statistical methods that are enabled by using computational methods. It is the area of computational ...
,
applied mathematics Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
and
industrial engineering Industrial engineering (IE) is concerned with the design, improvement and installation of integrated systems of people, materials, information, equipment and energy. It draws upon specialized knowledge and skill in the mathematical, physical, an ...
may offer a quantitative finance specialization.


Academic and technical field journals

* Society for Industrial and Applied Mathematics (SIAM) ''Journal on Financial Mathematics'' * '' The Journal of Portfolio Management'' * ''Quantitative Finance'' * ''Risk Magazine'' * ''Wilmott Magazine'' * ''Finance and Stochastics'' * ''Mathematical Finance''


Areas of work

*
Trading strategy In finance, a trading strategy is a fixed plan that is designed to achieve a profitable return by going long or short in markets. The difference between short trading and long-term investing is in the opposite approach and principles. Going shor ...
development * Portfolio management and Portfolio optimization * Derivatives pricing and hedging: involves software development, advanced numerical techniques, and stochastic calculus. *
Risk management Risk management is the identification, evaluation, and prioritization of risks, followed by the minimization, monitoring, and control of the impact or probability of those risks occurring. Risks can come from various sources (i.e, Threat (sec ...
: involves a lot of time series analysis, calibration, and backtesting. * Credit analysis * Asset and liability management *
Structured finance Structured finance is a sector of finance — specifically financial law — that manages Leverage (finance), leverage and Financial risk, risk. Strategies may involve legal and corporate restructuring, off balance sheet accounting, or the use of ...
and
securitization Securitization is the financial practice of pooling various types of contractual debt such as residential mortgages, commercial mortgages, auto loans, or credit card debt obligations (or other non-debt assets which generate receivables) and sellin ...
*
Asset pricing In financial economics, asset pricing refers to a formal treatment and development of two interrelated Price, pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, ...


Seminal publications

* 1900 – Louis Bachelier, ''Théorie de la spéculation'' * 1938 – Frederick Macaulay, ''The Movements of Interest Rates. Bond Yields and Stock Prices in the United States since 1856'', pp. 44–53,
Bond duration In finance, the duration of a financial asset that consists of fixed cash flows, such as a Bond (finance), bond, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a functio ...
* 1944 – Kiyosi Itô, "Stochastic Integral", Proceedings of the Imperial Academy, 20(8), pp. 519–524 * 1952 – Harry Markowitz, ''Portfolio Selection'',
Modern portfolio theory Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of Diversificatio ...
* 1956 – John Kelly, ''A New Interpretation of Information Rate'' * 1958 –
Franco Modigliani Franco Modigliani (; ; 18 June 1918 – 25 September 2003) was an Italian-American economist and the recipient of the 1985 Nobel Memorial Prize in Economics. He was a professor at University of Illinois at Urbana–Champaign, Carnegie Mellon Uni ...
and Merton Miller, ''The Cost of Capital, Corporation Finance and the Theory of Investment'', Modigliani–Miller theorem and
Corporate finance Corporate finance is an area of finance that deals with the sources of funding, and the capital structure of businesses, the actions that managers take to increase the Value investing, value of the firm to the shareholders, and the tools and analy ...
* 1964 – William F. Sharpe, ''Capital asset prices: A theory of market equilibrium under conditions of risk'',
Capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a Diversification (finance), well-diversified Portfolio (f ...
* 1965 –
John Lintner John Virgil Lintner Jr. (February 9, 1916 – June 8, 1983) was a professor at the Harvard Business School in the 1960s and one of the co-creators of the capital asset pricing model. For a time, much confusion was created because the various econ ...
, ''The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets'',
Capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a Diversification (finance), well-diversified Portfolio (f ...
* 1967 – Edward O. Thorp and Sheen Kassouf, ''Beat the Market'' * 1972 – Eugene Fama and Merton Miller, ''Theory of Finance'' * 1972 – Martin L. Leibowitz and Sydney Homer, '' Inside the Yield Book'',
Fixed income analysis Fixed income analysis is the process of determining the value of a debt security based on an assessment of its risk profile, which can include interest rate risk, risk of the issuer failing to repay the debt, market supply and demand for the secu ...
* 1973 – Fischer Black and Myron Scholes, ''The Pricing of Options and Corporate Liabilities'' and
Robert C. Merton Robert Cox Merton (born July 31, 1944) is an American economist, Nobel Memorial Prize in Economic Sciences laureate, and professor at the MIT Sloan School of Management, known for his pioneering contributions to continuous-time finance, especia ...
, ''Theory of Rational Option Pricing'', Black–Scholes * 1976 – Fischer Black, ''The pricing of commodity contracts'',
Black model The Black model (sometimes known as the Black-76 model) is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond options, interest rate cap and floors, and swaptions. ...
* 1977 – Phelim Boyle, ''Options: A Monte Carlo Approach'', Monte Carlo methods for option pricing * 1977 – Oldřich Vašíček, ''An equilibrium characterisation of the term structure'', Vasicek model * 1979 – John Carrington Cox; Stephen Ross; Mark Rubinstein, ''Option pricing: A simplified approach'',
Binomial options pricing model In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" ( lattice based) model of the varying price over time of the underlying fin ...
and Lattice model * 1980 – Lawrence G. McMillan, ''Options as a Strategic Investment'' * 1982 – Barr Rosenberg and Andrew Rudd, ''Factor-Related and Specific Returns of Common Stocks: Serial Correlation and Market Inefficiency'', Journal of Finance, May 1982 V. 37: #2 * 1982 –
Robert Engle Robert Fry Engle III (born November 10, 1942) is an American economist and statistician. He won the 2003 Nobel Memorial Prize in Economic Sciences, sharing the award with Clive Granger, "for methods of analyzing economic time series with time-va ...
, ''Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of U.K. Inflation,'' Seminal paper in ARCH family of models GARCH * 1985 – John C. Cox, Jonathan E. Ingersoll and Stephen Ross, ''A theory of the term structure of interest rates'', Cox–Ingersoll–Ross model * 1987 – Giovanni Barone-Adesi and Robert Whaley, ''Efficient analytic approximation of American option values''. Journal of Finance. 42 (2): 301–20. Barone-Adesi and Whaley method for pricing American options. * 1987 – David Heath,
Robert A. Jarrow __NOTOC__ Robert Alan Jarrow is the Ronald P. and Susan E. Lynch Professor of Investment Management at the Cornell Johnson Graduate School of Management. Professor Jarrow is a co-creator of the Heath–Jarrow–Morton framework for pricing inte ...
, and Andrew Morton ''Bond pricing and the term structure of interest rates: a new methodology'' (1987),
Heath–Jarrow–Morton framework The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curves – instantaneous forward rate curves in particular (as opposed to simple forward rates). When the volatility and drift of the ...
for interest rates * 1990 – Fischer Black,
Emanuel Derman Emanuel Derman (born 1945) is a South African-born academic, businessman and writer. He is best known as a quantitative analyst, and author of the book ''My Life as a Quant: Reflections on Physics and Finance''. He is a co-author of Black–D ...
and William Toy, ''A One-Factor Model of Interest Rates and Its Application to Treasury Bond'', Black–Derman–Toy model * 1990 – John Hull and Alan White, "Pricing interest-rate derivative securities", The Review of Financial Studies, Vol 3, No. 4 (1990) Hull-White model * 1991 – Ioannis Karatzas & Steven E. Shreve. ''Brownian motion and stochastic calculus''. * 1992 – Fischer Black and Robert Litterman: Global Portfolio Optimization, Financial Analysts Journal, September 1992, pp. 28–43 Black–Litterman model * 1994 – J.P. Morgan
RiskMetrics The RiskMetrics variance model (also known as exponential smoother) was first established in 1989, when Sir Dennis Weatherstone, the new chairman of J.P. Morgan, asked for a daily report measuring and explaining the risks of his firm. Nearly ...
Group
RiskMetrics Technical Document
1996, RiskMetrics model and framework * 2002 – Patrick Hagan, Deep Kumar, Andrew Lesniewski, Diana Woodward, ''Managing Smile Risk'', Wilmott Magazine, January 2002, SABR volatility model. * 2004 –
Emanuel Derman Emanuel Derman (born 1945) is a South African-born academic, businessman and writer. He is best known as a quantitative analyst, and author of the book ''My Life as a Quant: Reflections on Physics and Finance''. He is a co-author of Black–D ...
, ''My Life as a Quant: Reflections on Physics and Finance''


See also

* List of quantitative analysts *
Quantitative fund A quantitative fund is an investment fund that uses Quantitative analysis (finance), quantitative investment management instead of fundamental human analysis. Investment process An Investment, investment process is classified as Quantitative ana ...
*
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 ...
* Black–Scholes equation * Financial signal processing *
Financial analyst A financial analyst is a professional undertaking financial analysis for external or internal clients as a core feature of the job. 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. As a type of active management, it stands in contradiction to ...
*
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; Competition, competitors and Ma ...
*
Financial economics 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 ...
*
Mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field. In general, there exist two separate branches of finance that req ...
* Alpha generation platform * Rocket science (finance)


References


Further reading

* Bernstein, Peter L. (1992) ''Capital Ideas: The Improbable Origins of Modern Wall Street'' * Bernstein, Peter L. (2007) ''Capital Ideas Evolving'' * Derman, Emanuel (2007) ''My Life as a Quant'' * Patterson, Scott D. (2010). '' The Quants: How a New Breed of Math Whizzes Conquered Wall Street and Nearly Destroyed It''. Crown Business, 352 pages.
Amazon page for book
vi
Patterson and Thorp interview
on Fresh Air, February 1, 2010, including excerpt "Chapter 2: The Godfather: Ed Thorp". Also
an excerpt
from "Chapter 10: The August Factor", in the January 23, 2010 ''Wall Street Journal''. * Read, Colin (2012) ''Rise of the Quants'' (Great Minds in Finance Series)
Analysing Quantitative Data for Business and Management Students


External links


Society of Quantitative Analysts

Q-Group Institute for Quantitative Research in Finance

CQA—Chicago Quantitative Alliance

Quantitative Work Alliance for Finance Education and Wisdom (QWAFAFEW)

Professional Risk Managers Industry Association (PRMIA)

International Association of Quantitative Finance

London Quant Group

Quantitative Finance
at Stack Exchange – question and answer site for quantitative finance {{stock market Financial analysts Mathematical finance Valuation (finance)