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Quantitative analysis is the use of mathematical and statistical methods in
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
and
investment management Investment management is the professional asset management of various securities, including shareholdings, bonds, and other assets, such as real estate, to meet specified investment goals for the benefit of investors. Investors may be institut ...
. Those working in the field are quantitative analysts (quants). Quants tend to specialize in specific areas which may include
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
structuring or pricing, risk management,
investment management Investment management is the professional asset management of various securities, including shareholdings, bonds, and other assets, such as real estate, to meet specified investment goals for the benefit of investors. Investors may be institut ...
and other related finance occupations. The occupation is similar to those in
industrial mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical ...
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 or mean reversion). Although the original quantitative analysts were " sell side quants" from market maker 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 firms 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 common ...
. 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 models involving broadly diversified portfolios of securities (hundreds to thousan ...
, algorithmic trading and electronic trading. Some of the larger investment managers using quantitative analysis include
Renaissance Technologies Renaissance Technologies LLC, also known as RenTech or RenTec, is an American hedge fund based in East Setauket, New York, on Long Island, which specializes in systematic trading using quantitative models derived from mathematical and statisti ...
, D. E. Shaw & Co., and
AQR Capital Management AQR Capital Management (Applied Quantitative Research) is a global investment management firm based in Greenwich, Connecticut, United States. The firm, which was founded in 1998 by Cliff Asness, David Kabiller, John Liew, and Robert Krail, offer ...
.


History

Quantitative finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...
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: ...
"Theory of Speculation", which provided a model to price
options Option or Options may refer to: Computing *Option key, a key on Apple computer keyboards *Option type, a polymorphic data type in programming languages * Command-line option, an optional parameter to a command *OPTIONS, an HTTP request method ...
under a
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
. 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. 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 Edward Oakley Thorp (born August 14, 1932) is an American mathematics professor, author, hedge fund manager, and blackjack researcher. He pioneered the modern applications of probability theory, including the harnessing of very small correlat ...
, a mathematics professor at New Mexico State University (1961–1965) and University of California, Irvine (1965–1977). Considered the "Father of Quantitative Investing", Thorp sought to predict and simulate
blackjack Blackjack (formerly Black Jack and Vingt-Un) is a casino banking game. The most widely played casino banking game in the world, it uses decks of 52 cards and descends from a global family of casino banking games known as Twenty-One. This fami ...
, a card-game he played in Las Vegas casinos. He was able to create a system, known broadly as card counting, which used probability theory 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 Post-Modern Portfolio Theory (PMPT) is an extension of the traditional Modern Portfolio Theory (MPT), an application of mean-variance analysis (MVA). Both theories propose how rational investors can use diversification to optimize their portfolios. ...
, . In 1965 Paul Samuelson introduced stochastic calculus into the study of finance. In 1969 Robert Merton promoted continuous stochastic calculus and
continuous-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
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 Myron Samuel Scholes ( ; born July 1, 1941) is a Canadian-American financial economist. Scholes is the Frank E. Buck Professor of Finance, Emeritus, at the Stanford Graduate School of Business, Nobel Laureate in Economic Sciences, and co-origina ...
developed the Black–Scholes model, which was awarded the 1997 Nobel Memorial Prize in Economic Sciences. It provided a solution for a practical problem, that of finding a fair price for a
European call option In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options ...
, 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. 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 prin ...
and
interest rate derivatives In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of diff ...
. Similarly, and in parallel, models were developed for various other underpinnings and applications, including
credit derivatives In finance, a credit derivative refers to any one of "various instruments and techniques designed to separate and then transfer the ''credit risk''"The Economist ''Passing on the risks'' 2 November 1996 or the risk of an event of default of a cor ...
,
exotic derivatives An exotic derivative, in finance, is a 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 derivatives with a non-s ...
, real options, and employee stock options. Quants are thus involved in pricing and hedging a wide range of securities – asset-backed, government, and corporate – additional to classic derivatives; see contingent claim analysis. Emanuel Derman'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 financial crisis of 2007–2008, considerations re counterparty credit risk were incorporated into the modelling, previously performed in an entirely " risk neutral 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 models; (ii) The risk neutral value is adjusted for the impact of counter-party credit risk via a
credit valuation adjustment Credit valuation adjustments (CVAs) are accounting adjustments made to reserve a portion of profits on uncollateralized financial derivatives. They are charged by a bank to a risky (capable of default) counterparty to compensate the bank for taking ...
, or CVA, as well as various of the other
XVA An X-Value Adjustment (XVA, xVA) is an umbrella term referring to a number of different “valuation adjustments” that banks must make when assessing the value of derivative contracts that they have entered into. The purpose of these is twofold: ...
; (iii) For discounting, the OIS curve is used for the "risk free rate", as opposed to LIBOR as previously, and, relatedly, quants must model under a "
multi-curve framework In finance, an interest rate swap (IRS) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a "linear" IRD and one of the most liquid, benchmark products. It has associations with ...
" ( LIBOR is due to be phased out by the end of 2021, with replacements including SOFR and TONAR, necessitating technical changes to the latter framework, while the underlying logic is unaffected).


Education

Quantitative analysts often come from financial mathematics, financial engineering, applied mathematics, physics or engineering backgrounds, and quantitative analysis is a major source of employment for people with financial mathematics master's degrees, or with mathematics and physics
PhD degrees PHD or PhD may refer to: * Doctor of Philosophy (PhD), an academic qualification Entertainment * '' PhD: Phantasy Degree'', a Korean comic series * ''Piled Higher and Deeper'', a web comic * Ph.D. (band), a 1980s British group **Ph.D. (Ph.D. albu ...
. Typically, a quantitative analyst will also need extensive skills in computer programming, most commonly C, C++, Java, R, MATLAB,
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
, and Python.
Data science Data science is an interdisciplinary field that uses scientific methods, processes, algorithms and systems to extract or extrapolate knowledge and insights from noisy, structured and unstructured data, and apply knowledge from data across a br ...
and machine learning analysis and modelling 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. This demand for quantitative analysts has led to the creation of specialized Masters and PhD courses in financial engineering, mathematical finance, computational finance, and/or
financial reinsurance Financial Reinsurance (or fin re), is a form of reinsurance which is focused more on capital management than on risk transfer. In the non-life segment of the insurance industry this class of transactions is often referred to as finite reinsurance. ...
. In particular, Master's degrees in mathematical finance, financial engineering, operations research, computational statistics, applied mathematics, machine learning, and
financial analysis Financial analysis (also known as financial statement analysis, accounting analysis, or analysis of finance) refers to an assessment of the viability, stability, and profitability of a business, sub-business or project. It is performed by profes ...
are becoming more popular with students and with employers. See
Master of Quantitative Finance A master's degree in quantitative finance concerns the application of mathematical methods to the solution of problems in financial economics. There are several like-titled degrees which may further focus on financial engineering, computational fin ...
for general discussion. This has in parallel led to a resurgence in demand for actuarial qualifications, as well as commercial certifications such as the CQF. The more general Master of Finance (and
Master of Financial Economics A master's degree in Financial Economics provides a Mathematical rigor, rigorous understanding of theory, theoretical finance and the economics, economic framework upon which that theory is based. The degree is postgraduate, and usually incorporates ...
) increasingly includes a significant technical component.


Types


Front office quantitative analyst

In sales and trading, quantitative analysts work to determine prices, manage risk, and identify profitable opportunities. Historically this was a distinct activity from trading but the boundary between a desk 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. See also structurer.


Quantitative investment management

:''See , for related articles.'' Quantitative analysis is used extensively by
asset managers Asset management is 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 may apply both to tangible assets (physical objects such as buildings ...
. Some, such as FQ, AQR or Barclays, rely almost exclusively on quantitative strategies while others, such as PIMCO, Blackrock or Citadel 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 ( ; , Spanish for 'Holy Faith'; tew, Oghá P'o'oge, Tewa for 'white shell water place'; tiw, Hulp'ó'ona, label=Tiwa language, Northern Tiwa; nv, Yootó, Navajo for 'bead + water place') is the capital of the U.S. state of New Mexico. ...
and began trading in 1991 under the name
Prediction Company Prediction Company was founded in Santa Fe, New Mexico, USA, in March 1991 by J. Doyne Farmer, Norman Packard, and James McGill. The company used forecasting techniques to build black-box trading systems for financial markets, mainly employing st ...
. 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 models involving broadly diversified portfolios of securities (hundreds to thousan ...
to secure investment returns, along with three other funds at the time,
Renaissance Technologies Renaissance Technologies LLC, also known as RenTech or RenTec, is an American hedge fund based in East Setauket, New York, on Long Island, which specializes in systematic trading using quantitative models derived from mathematical and statisti ...
and D. E. Shaw & Co, both based in New York. Prediction hired scientists and computer programmers from the neighboring Los Alamos National Laboratory to create sophisticated statistical models using "industrial-strength computers" in order to " uildthe
Supercollider A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle ...
of Finance".


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 ExCeL London (an abbreviation for Exhibition Centre London) is an exhibition centre, international convention centre and former hospital in the Custom House area of Newham, East London. It is situated on a site on the northern quay of the ...
is very rare, with most development being in C++, though Java, 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 repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
and
finite difference methods 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 dom ...
, 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,
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, gambling
Kelly criterion In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet), is a formula that determines the optimal theoretical size for a bet. It is valid when the expected returns are known. The Kelly bet size is found by maximizing the expec ...
, market microstructure, econometrics, and time series analysis.
Algorithmic trading Algorithmic trading is a method of executing orders using automated pre-programmed trading instructions accounting for variables such as time, price, and volume. This type of trading attempts to leverage the speed and computational resources of ...
includes
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 models involving broadly diversified portfolios of securities (hundreds to thousan ...
, but includes techniques largely based upon speed of response, to the extent that some ATQs modify hardware and Linux kernels to achieve ultra low latency.


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 The Fundamental Review of the Trading Book (FRTB), is a set of proposals by the Basel Committee on Banking Supervision for a new market risk-related capital requirement for banks. Background The reform, which is part of Basel III, is one of th ...
, . A core technique continues to be value at risk - applying both the parametric and "Historical" approaches, as well as
Conditional value at risk Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the wor ...
and Extreme value theory - 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 r ...
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

In the aftermath of the financial crisis hich one?/sup>, 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 In finance, model risk is the risk of loss resulting from using insufficiently accurate models to make decisions, originally and frequently in the context of valuing financial securities. However, model risk is more and more prevalent in activitie ...
. 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 In finance, model risk is the risk of loss resulting from using insufficiently accurate models to make decisions, originally and frequently in the context of valuing financial securities. However, model risk is more and more prevalent in activitie ...
, 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 systematic engineering approach to software development. A software engineer is a person who applies the principles of software engineering to design, develop, maintain, test, and evaluate computer software. The term '' ...
and quantitative analysts. The term is also sometimes used outside the finance industry to refer to those working at the intersection of software engineering 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 philosop ...
.


Mathematical and statistical approaches

Because of their backgrounds, quantitative analysts draw from various forms of mathematics:
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
and probability, calculus centered around
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, linear algebra,
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. Some on the buy side may use machine learning. 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. Commonly used numerical methods are: * Finite difference method – used to solve
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; * Monte Carlo method – Also used to solve
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, but
Monte Carlo simulation 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 ...
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 model (with fixed level-one effects of a linear function of a set of explanatory variables) by the prin ...
– 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, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
,
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * 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 focusing ...
, maxima and minima 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. 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.


Academic and technical field journals

*
Society for Industrial and Applied Mathematics Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific socie ...
(SIAM) ''Journal on Financial Mathematics'' * ''
The Journal of Portfolio Management ''The Journal of Portfolio Management'' (also known as JPM) is a quarterly academic journal for finance and investing, covering topics such as asset allocation, performance measurement, market trends, risk management, and portfolio optimization. ...
'' * ''Quantitative Finance'' * ''Risk Magazine'' * ''Wilmott Magazine'' * ''Finance and Stochastics'' * ''Mathematical Finance''


Areas of work

* Trading strategy development * Portfolio management and
Portfolio optimization Portfolio optimization is the process of selecting the best portfolio (asset distribution), out of the set of all portfolios being considered, according to some objective. The objective typically maximizes factors such as expected return, and minimi ...
*
Derivatives pricing In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be u ...
and hedging: involves software development, advanced numerical techniques, and stochastic calculus. * Risk management: 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 and risk. Strategies may involve legal and corporate restructuring, off balance sheet accounting, or the use of financial instruments. Securitization ...
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 selling ...
*
Asset pricing In financial economics, asset pricing refers to a formal treatment and development of two main Price, pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, but cor ...


Seminal publications

* 1900 – Louis Bachelier, ''Théorie de la spéculation'' * 1938 –
Frederick Macaulay Frederick Robertson Macaulay (August 12, 1882 – March 1970) was a Canadian economist of the Institutionalist School. He is known for introducing the concept of bond duration. Macaulay's contributions also include a mammoth empirical stud ...
, ''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, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield, dur ...
* 1944 –
Kiyosi Itô was a Japanese mathematician who made fundamental contributions to probability theory, in particular, the theory of stochastic processes. He invented the concept of stochastic integral and stochastic differential equation, and is known as the fo ...
, "Stochastic Integral", Proceedings of the Imperial Academy, 20(8), pp. 519–524 * 1952 – Harry Markowitz, ''Portfolio Selection'', Modern portfolio theory * 1956 –
John Kelly John or Jack Kelly may refer to: People Academics and scientists * John Kelly (engineer), Irish professor, former Registrar of University College Dublin *John Kelly (scholar) (1750–1809), at Douglas, Isle of Man *John Forrest Kelly (1859–1922) ...
, ''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 Merton Howard Miller (May 16, 1923 – June 3, 2000) was an American economist, and the co-author of the Modigliani–Miller theorem (1958), which proposed the irrelevance of debt-equity structure. He shared the Nobel Memorial Prize in Economic ...
, ''The Cost of Capital, Corporation Finance and the Theory of Investment'', Modigliani–Miller theorem and
Corporate finance Corporate finance is the area of finance that deals with the sources of funding, the capital structure of corporations, the actions that managers take to increase the Value investing, value of the firm to the shareholders, and the tools and anal ...
* 1964 –
William F. Sharpe William Forsyth Sharpe (born June 16, 1934) is an American economist. He is the STANCO 25 Professor of Finance, Emeritus at Stanford University's Graduate School of Business, and the winner of the 1990 Nobel Memorial Prize in Economic Sciences. ...
, ''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 well-diversified portfolio. The model takes into accou ...
* 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 (1965 a, b) of the capital asset pricing model. For a time, much confusion was created because the ...
, ''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 well-diversified portfolio. The model takes into accou ...
* 1967 –
Edward O. Thorp Edward Oakley Thorp (born August 14, 1932) is an American mathematics professor, author, hedge fund manager, and blackjack researcher. He pioneered the modern applications of probability theory, including the harnessing of very small correlatio ...
and Sheen Kassouf, ''Beat the Market'' * 1972 – Eugene Fama and
Merton Miller Merton Howard Miller (May 16, 1923 – June 3, 2000) was an American economist, and the co-author of the Modigliani–Miller theorem (1958), which proposed the irrelevance of debt-equity structure. He shared the Nobel Memorial Prize in Economic ...
, ''Theory of Finance'' * 1972 –
Martin L. Leibowitz Martin L. Leibowitz is a financial researcher, business leader, and a managing director of Morgan Stanley. Career Before joining Morgan Stanley, Leibowitz was vice chairman and chief investment officer of TIAA-CREF from 1995 to 2004. Previously he ...
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 Myron Samuel Scholes ( ; born July 1, 1941) is a Canadian-American financial economist. Scholes is the Frank E. Buck Professor of Finance, Emeritus, at the Stanford Graduate School of Business, Nobel Laureate in Economic Sciences, and co-origina ...
, ''The Pricing of Options and Corporate Liabilities'' and Robert C. Merton, ''Theory of Rational Option Pricing'', Black–Scholes * 1976 – Fischer Black, ''The pricing of commodity contracts'', Black model * 1977 –
Phelim Boyle Phelim P. Boyle (born 1941), is an Irish economist and distinguished professor and actuary, and a pioneer of quantitative finance. He is best known for initiating the use of Monte Carlo methods in option pricing. Biography Born on a farm in L ...
, ''Options: A Monte Carlo Approach'',
Monte Carlo methods for option pricing In mathematical finance, a Monte Carlo option model uses Monte Carlo methodsAlthough the term 'Monte Carlo method' was coined by Stanislaw Ulam in the 1940s, some trace such methods to the 18th century French naturalist Buffon, and a question he as ...
* 1977 –
Oldřich Vašíček Oldřich Alfons Vašíček (; born 1942) is a Czech mathematician and quantitative analyst, best known for his pioneering work on interest rate modelling; see Vasicek model. Vašíček received his master's degree in math from the Czech Technical U ...
, ''An equilibrium characterisation of the term structure'',
Vasicek model In finance, the Vasicek model is a mathematical model describing the evolution of interest rates. It is a type of one-factor short-rate model as it describes interest rate movements as driven by only one source of market risk. The model can be u ...
* 1979 –
John Carrington Cox John Carrington Cox is the Nomura Professor of Finance at the MIT Sloan School of Management. He is one of the world's leading experts on options theory and one of the inventors of the Cox–Ross–Rubinstein model for option pricing, as well as ...
; 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 f ...
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, ''Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of U.K. Inflation,'' Seminal paper in ARCH family of models GARCH * 1985 –
John C. Cox John Carrington Cox is the Nomura Professor of Finance at the MIT Sloan School of Management. He is one of the world's leading experts on options theory and one of the inventors of the Cox–Ross–Rubinstein model for option pricing, as well as ...
,
Jonathan E. Ingersoll Jonathan Edwards "Jon" Ingersoll, Jr. is an American economist. He is the Adrian C. Israel Professor of International Trade and Finance at Yale School of Management. Prior to coming to Yale he was on the faculty at the Graduate School of Busin ...
and Stephen Ross, ''A theory of the term structure of interest rates'',
Cox–Ingersoll–Ross model In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes the evolution of interest rates. It is a type of "one factor model" ( short-rate model) as it describes interest rate movements as driven by only one source of mark ...
* 1987 – Giovanni Barone-Adesi and
Robert Whaley Robert Antawon Whaley (born April 16, 1982) is an American former professional basketball player. High school and college career Whaley graduated from Benton Harbor High School in 2001. He was a leading contender for Mr. Basketball of Michigan, ...
, ''Efficient analytic approximation of American option values''. Journal of Finance. 42 (2): 301–20. Barone-Adesi and Whaley method for pricing
American options In finance, the style or family of an option (finance), option is the class into which the option falls, usually defined by the dates on which the option may be Exercise (options), exercised. The vast majority of options are either European or Amer ...
. * 1987 – David Heath,
Robert A. Jarrow __NOTOC__ Robert Alan Jarrow is the Ronald P. and Susan E. Lynch Professor of Investment Management at the Johnson Graduate School of Management, Cornell University. Professor Jarrow is a co-creator of the Heath–Jarrow–Morton framework for ...
, and Andrew Morton ''Bond pricing and the term structure of interest rates: a new methodology'' (1987), Heath–Jarrow–Morton framework for interest rates * 1990 – Fischer Black, Emanuel Derman and William Toy, ''A One-Factor Model of Interest Rates and Its Application to Treasury Bond'',
Black–Derman–Toy model In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see . It is a one-factor model; that is, a single stochastic factor—t ...
* 1990 –
John Hull John Hull may refer to: Politicians *John Hull (MP for Hythe), MP for Hythe *John Hull (MP for Exeter) (died 1549), English politician *John A. T. Hull (1841–1928), American politician *John C. Hull (politician) (1870–1947), Speaker of the Mas ...
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 Steven Eugene Shreve is a mathematician and currently the Orion Hoch Professor of Mathematical Sciences at Carnegie Mellon University and the author of several major books on the mathematics of financial derivatives. His first degree, awarded in 1 ...
. ''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 JP may refer to: Arts and media * ''JP'' (album), 2001, by American singer Jesse Powell * ''Jp'' (magazine), an American Jeep magazine * ''Jönköpings-Posten'', a Swedish newspaper * Judas Priest, an English heavy metal band * ''Jurassic Park ...
RiskMetrics 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 In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR ...
. * 2004 – Emanuel Derman, ''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 :''See for a listing of relevant articles.'' An Investment, investme ...
* Financial modeling *
Black–Scholes equation In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE ...
*
Financial signal processing Financial signal processing is a branch of signal processing technologies which applies to signals within financial markets. They are often used by quantitative analysts to make best estimation of the movement of financial markets, such as stock ...
* Financial analyst * Technical analysis * Fundamental analysis * Financial economics *
Mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...
*
Alpha generation platform An alpha generation platform is a technology used in algorithmic trading to develop quantitative financial models, or trading strategies, that generate consistent alpha, or absolute returns. The process of alpha generation refers to generating exc ...


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, Feb. 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

* http://sqa-us.org – Society of Quantitative Analysts * http://www.q-group.org/ — Q-Group Institute for Quantitative Research in Finance * http://cqa.org – CQA—Chicago Quantitative Alliance * http://qwafafew.org/ – QWAFAFEW – Quantitative Work Alliance for Finance Education and Wisdom * http://prmia.org – PRMIA—Professional Risk Managers Industry Association * http://iaqf.org – International Association of Quantitative Finance * http://www.lqg.org.uk/ – London Quant Group * http://quant.stackexchange.com – question and answer site for quantitative finance {{stock market Valuation (finance) Mathematical finance Financial analysts