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A **quantitative analyst** (or, in financial jargon, a **quant**) is a person who specializes in the application of mathematical and statistical methods – such as numerical or quantitative techniques – to financial and risk management problems. The occupation is similar to those in industrial mathematics in other industries.^{[1]}

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 mathematics in finance, including the buy side.^{[2]} Examples include statistical arbitrage, quantitative investment management, algorithmic trading, and electronic market making.

Quantitative finance started in 1900 with Louis Bachelier's doctoral thesis *Theory of Speculation*, which provided a model to price options under a Normal Distribution.

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 mathematics, statistics or specialized economics journals).^{[3]} 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 Itō calculus, management of risk in a quantifiable manner underlies much of the modern theory.

In 1965 Paul Samuelson introduced stochastic calculus into the study of finance.^{[4]}^{[5]} 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, 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, 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.^{[6]}

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.^{[7]}

Quantitative analysts often come from applied mathematics, physics or engineering backgrounds rather than economics-related fields, and quantitative analysis is a major source of employment for people with mathematics and physics PhD degrees, or with financial mathematics DEA degrees in the French education system. Typically, a quantitative analyst will also need extensive skills in computer programming, most commonly C, C++, Java, R, MATLAB, Mathematica, Python.

This demand for quantitative analysts has led to a resurgence in demand for actuarial qualifications as well as creation of specialized Masters and PhD courses in financial engineering, mathematical finance, computational finance, and/or financial reinsurance. In particular, Master's degrees in mathematical finance, financial engineering, operations research, computational statistics, machine learning, and financial analysis are becoming more popular with students and with employers. See Master of Quantitative Finance; Master of Financial Economics.

Data science and machine learning analysis and modelling methods are being increasingly employed in portfolio performance and portfolio risk modelling,^{[8]}^{[9]} and as such data science and machine learning Master's graduates are also in demand as quantitative analysts.

In sales & 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. In the field of algorithmic trading it has reached the point where there is little meaningful difference. 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.

Quantitative analysis is used extensively by asset managers. 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.

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 and C# 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 and finite difference methods, as well as the nature of the products being modeled.

Often the highest paid form of Quant, ATQs make use of methods taken from signal processing, game theory, gambling Kelly criterion, market microstructure, econometrics, and time series analysis. Algorithmic trading includes statistical arbitrage, 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.

This has grown in importance in recent years, as the credit crisis exposed holes in the mechanisms used to ensure that positions were correctly hedged, though in no bank does the pay in risk approach that in front office. A core technique is value at risk, and this is backed up with various forms of stress test (financial), economic capital analysis and direct analysis of the positions and models used by various bank's divisions.

In the aftermath of the financial crisis, 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 (MV) takes the models and methods developed by front office, library, and modeling quantitative analysts and determines their validity and correctness. 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. 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, this has changed. Regulators now typically talk directly to the quants in the middle office such as the model validators, and since profits highly depend of the regulatory infrastructure, model validation has gained in weight and importance with respect to the quants in the front office.

Quantitative developers 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 developer and quantitative analysts.

Because of their backgrounds, quantitative analysts draw from various forms of mathematics: statistics and probability, calculus centered around partial differential equations, linear algebra, discrete mathematics, 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 equations;
- Monte Carlo method – Also used to solve partial differential equations, but Monte Carlo simulation is also common in risk management;
- Ordinary least squares – 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, Newton, and Secant methods – used to find the roots, maxima and minima of functions (e.g. internal rate of return.)

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. The mindset 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.

One of the principal mathematical tools of quantitative finance is stochastic calculus.

- Society for Industrial and Applied Mathematics (SIAM) Journal on Financial Mathematics
- Quantitative Finance
^{[10]} - Risk Magazine
- Wilmott Magazine
- Finance and Stochastics
^{[11]}

- Trading strategy development
- Portfolio optimization
- Derivatives pricing 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 liability management
- Structured finance and securitization
- Asset pricing
- Portfolio management

- 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 - 1944 – Kiyosi Itô, "Stochastic Integral", Proceedings of the Imperial Academy, 20(8), pp. 519–524
- 1952 – Harry Markowitz,
*Portfolio Selection*, Modern portfolio theory - 1956 – John Kelly,
*A New Interpretation of Information Rate* - 1958 – Franco Modigliani and Merton Miller,
*The Cost of Capital, Corporation Finance and the Theory of Investment*, Modigliani–Miller theorem and Corporate finance - 1964 – William F. Sharpe,
*Capital asset prices: A theory of market equilibrium under conditions of risk*, Capital asset pricing model - 1965 – John Lintner,
*The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets*, Capital asset pricing model - 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 - 1973 – Fischer Black and Myron Scholes,
*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,
*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 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, 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 (June 1987). "Efficient analytic approximation of American option values". Journal of Finance. 42 (2): 301–20. Most widely used approximation for pricing American options.
- 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 - 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 JSTOR 4479577 Black-Litterman model
- 1994 – J.P. Morgan RiskMetrics Group, RiskMetrics Technical Document, 1996.
- 2002 – Patrick Hagan, Deep Kumar, Andrew Lesniewski, Diana Woodward,
*Managing Smile Risk', Wilmott Magazine, January 2002, SABR volatility model.* - 2004 – Emanuel Derman,
*My Life as a Quant: Reflections on Physics and Finance*

- List of quantitative analysts
- Mathematical finance
- Financial modeling
- Master of Quantitative Finance
- Certificate in Quantitative Finance

**^**See Definition in the Society for Applied and Industrial Mathematics [www.siam.org]**^**Derman, E. (2004). My life as a quant: reflections on physics and finance. John Wiley & Sons.**^**Markowitz, H. (1952). "Portfolio Selection".*Journal of Finance*.**7**(1): 77–91. doi:10.1111/j.1540-6261.1952.tb01525.x.**^**Samuelson, P. A. (1965). "Rational Theory of Warrant Pricing".*Industrial Management Review*.**6**(2): 13–32.**^**Henry McKean the co-founder of stochastic calculus (along with Kiyosi Itô) wrote the appendix: see McKean, H. P. Jr. (1965). "Appendix (to Samuelson): a free boundary problem for the heat equation arising from a problem of mathematical economics".*Industrial Management Review*.**6**(2): 32–39.**^**Harrison, J. Michael; Pliska, Stanley R. (1981). "Martingales and Stochastic Integrals in the Theory of Continuous Trading".*Stochastic Processes and their Applications*.**11**(3): 215–260. doi:10.1016/0304-4149(81)90026-0.**^**Derman, Emanuel (2004).*My Life as a Quant*. John Wiley and Sons.**^**"Machine Learning in Finance: Theory and Applications".*marketsmedia.com*. 22 January 2013. Retrieved 2 April 2018.**^**A Machine-Learning View of Quantitative Finance**^**[www.tandfonline.com]**^**[www.springer.com]

- 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*ISBN 0-470-19273-9 - Patterson, Scott D. (2010).
*The Quants: How a New Breed of Math Whizzes Conquered Wall Street and Nearly Destroyed It*. Crown Business, 352 pages. ISBN 0-307-45337-5 ISBN 978-0-307-45337-2. Amazon page for book via 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) ISBN 023027417X*

- [sqa-us.org] – Society of Quantitative Analysts
- [www.q-group.org] Institute for Quantitative Research in Finance
- [cqa.org] – CQA—Chicago Quantitative Alliance
- [qwafafew.org] – QWAFAFEW – Quantitative Work Alliance for Finance Education and Wisdom
- [prmia.org] – PRMIA—Professional Risk Managers Industry Association
- [iaqf.org] – International Association of Quantitative Finance
- [www.lqg.org.uk] – London Quant Group
- [quant.stackexchange.com] – question and answer site for quantitative finance