Quantitative analysis is the use ofmathematical andstatistical methods infinance andinvestment management. Those working in the field arequantitative analysts (quants). Quants tend to specialize in specific areas which may includederivative structuring or pricing,risk management,investment management and other related finance occupations. The occupation is similar to those inindustrial mathematics in other industries.^[1] The process usually consists of searching vast databases for patterns, such as correlations among liquid assets or price-movement patterns (trend following orreversion).

Although the original quantitative analysts were "sell side quants" frommarket 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 thebuy side.^[2] Applied quantitative analysis is commonly associated withquantitative investment management which includes a variety of methods such asstatistical arbitrage,algorithmic trading andelectronic trading.

Some of the larger investment managers using quantitative analysis includeRenaissance Technologies,D. E. Shaw & Co., andAQR Capital Management.^[3]

Quantitative finance started in 1900 withLouis Bachelier's doctoralthesis "Theory of Speculation", which provided a model to priceoptions under anormal distribution.Jules Regnault had posited already in 1863 that stock prices can be modelled as arandom walk, suggesting "in a more literary form, the conceptual setting for the application of probability to stockmarket operations".^[4]It was, however, only in the years 1960-1970 that the "merit of [these] was recognized"^[4]asoptions 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).^[5] Markowitz formalized a notion ofmean return and covariances for common stocks which allowed him to quantify the concept of "diversification" in a market. He showedhow 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 involvesItô calculus, management of risk in a quantifiable manner underlies much of the modern theory.

Modernquantitative investment management was first introduced from the research ofEdward Thorp, a mathematics professor atNew Mexico State University (1961–1965) andUniversity of California, Irvine (1965–1977).^[6] Considered the "Father of Quantitative Investing",^[6] Thorp sought to predict and simulateblackjack, a card-game he played in Las Vegas casinos.^[7] He was able to create a system, known broadly ascard counting, which usedprobability theory and statistical analysis to successfully win blackjack games.^[7] 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.^[7]The field has grown to incorporate numerous approaches and techniques; seeOutline of finance § Quantitative investing,Post-modern portfolio theory,Financial economics § Portfolio theory.

In 1965,Paul Samuelson introducedstochastic calculus into the study of finance.^[8][9] In 1969,Robert Merton promoted continuous stochastic calculus andcontinuous-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 andMyron Scholes developed theBlack–Scholes model, which was awarded the 1997Nobel Memorial Prize in Economic Sciences. It provided a solution for a practical problem, that of finding a fair price for aEuropean 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 thefundamental theorem of asset pricing.^[10] The variousshort-rate models (beginning withVasicek in 1977), and the more generalHJM Framework (1987), relatedly allowed for an extension tofixed income andinterest rate derivatives. Similarly, and in parallel, models were developed for various other underpinnings and applications, includingcredit derivatives,exotic derivatives,real options, andemployee stock options. Quants are thus involved in pricing and hedging a wide range of securities –asset-backed,government, andcorporate – additional to classic derivatives; seecontingent claim analysis.Emanuel Derman's 2004 bookMy 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.^[11]

After thefinancial crisis of 2007–2008, considerations regardingcounterparty credit risk were incorporated into the modelling, previously performed in an entirely "risk neutral world", entailing three major developments; seeValuation of options § Post crisis: (i) Option pricing and hedging inhere the relevantvolatility surface - to some extent, equity-option prices have incorporated thevolatility smile since the1987 crash - and banks then apply "surface aware"local- orstochastic volatility models;(ii) The risk neutral value is adjusted for the impact of counter-party credit risk via acredit valuation adjustment, or CVA, as well as various of the otherXVA;(iii) For discounting, theOIS curve is used for the "risk free rate", as opposed toLIBOR as previously, and, relatedly, quants must model under a "multi-curve framework"(LIBOR is being phased out, with replacements includingSOFR andTONAR, necessitating technical changes to the latter framework, while the underlying logic is unaffected).

Insales and trading, quantitative analysts work to determine prices, manage risk, and identify profitable opportunities. Historically this was a distinct activity fromtrading but the boundary between adesk 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 managingcounterparty risk as well as (minimizing) the capital requirements underBasel III; andstructurers, tasked withthe design and manufacture of client specific solutions.

Quantitative analysis is used extensively byasset managers. Some, such as FQ,AQR orBarclays, rely almost exclusively on quantitative strategies while others, such asPIMCO,BlackRock orCitadel use a mix of quantitative andfundamental methods.

One of the first quantitative investment funds to launch was based inSanta Fe, New Mexico and began trading in 1991 under the namePrediction Company.^[7][12] By the late-1990s, Prediction Company began usingstatistical arbitrage to secure investment returns, along with three other funds at the time,Renaissance Technologies andD. E. Shaw & Co, both based in New York.^[7] Prediction hired scientists and computer programmers from the neighboringLos Alamos National Laboratory to create sophisticated statistical models using "industrial-strength computers" in order to "[build] theSupercollider of Finance".^[13][14]

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.^[15]SeeApplications of artificial intelligence § Trading and investment.

Major firms invest large sums in an attempt to produce standard methods of evaluating prices and risk. These differ from front office tools in thatExcel is very rare, with most development being inC++, thoughJava,C# andPython 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 asMonte Carlo methods andfinite 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 fromsignal processing,game theory, gamblingKelly criterion,market microstructure,econometrics, andtime series analysis.

This area has grown in importance in recent years, as the credit crisis exposed holes in the mechanisms used to ensure that positions were correctlyhedged; seeFRTB,Tail risk § Role of the global financial crisis (2007-2008).A core technique continues to bevalue at risk - applying boththe parametric and"Historical" approaches, as well asConditional value at risk andExtreme value theory - while this is supplemented with various forms ofstress test,expected shortfall methodologies,economic capital analysis,direct analysis of the positions at thedesk level, and,as below, assessment of the models used by the bank's various divisions.

In the aftermath of the financial crisis in 2008, 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; seemodel 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 managemodel 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 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 betweensoftware engineers and quantitative analysts. The term is also sometimes used outside the finance industry to refer to those working at the intersection ofsoftware engineering andquantitative research.

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.^[16][17] 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,^[18] especially in non-stationary contexts. Identifying and developing methodologies to estimate this ability represents one of the main challenges of modern quantitative trading.^[19][20] 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.

Because of their backgrounds, quantitative analysts draw from various forms of mathematics:statistics andprobability,calculus centered aroundpartial differential equations,linear algebra,discrete mathematics, andeconometrics. Some on the buy side may usemachine learning. Themajority 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 uponnumerical analysis.

Commonly used numerical methods are:

• Finite difference method – used to solvepartial differential equations;
• Monte Carlo method – Also used to solvepartial differential equations, butMonte Carlo simulation is also common in risk management;
• Ordinary least squares – used to estimate parameters instatistical regression analysis;
• Spline interpolation – used to interpolate values fromspot and forward interest rates curves, andvolatility smiles;
• Bisection,Newton, andSecant methods – used to find theroots,maxima and minima of functions (e.g.internal rate of return,interest rate curve-building.)

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 isstochastic 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 onlyone correct price for any given security (which can be demonstrated, albeit often inefficiently, through a large volume ofMonte 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.

Quantitative analysts often come fromapplied mathematics,physics orengineering backgrounds,^[21] learning finance "on the job". Quantitative analysis is a then major source of employment for those with mathematics and physicsPhD degrees.^[21]

Typically, a quantitative analyst will also need^[21][22] extensive skills in computer programming, most commonlyC,C++ andJava, and latelyR,MATLAB,Mathematica, andPython.Data science andmachine learning analysis and methods are being increasingly employed in portfolio performance and portfolio risk modelling,^[23][24] 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^[21] the creation of specialized Masters^[22] and PhD courses infinancial engineering,mathematical finance andcomputational finance (as well as in specific topics such asfinancial reinsurance). In particular, theMaster of Quantitative Finance,Master of Financial Mathematics,Master of Computational Finance andMaster of Financial Engineering are becoming popular with students and with employers.^[22][25]SeeMaster of Quantitative Finance § History.

This has, in parallel, led to a resurgence in demand foractuarial qualifications, as well ascommercial certifications such as theCQF.Similarly, the more generalMaster of Finance (andMaster of Financial Economics) increasingly^[25]includes a significant technical component.Likewise, masters programs inoperations research,computational statistics,applied mathematics andindustrial engineering may offer a quantitative finance specialization.

• Society for Industrial and Applied Mathematics (SIAM)Journal on Financial Mathematics
• The Journal of Portfolio Management^[26]
• Quantitative Finance^[27]
• Risk Magazine
• Wilmott Magazine
• Finance and Stochastics^[28]
• Mathematical Finance

• Trading strategy development
• Portfolio management andPortfolio optimization
• Derivatives pricing and hedging: involves software development, advanced numerical techniques, and stochastic calculus.
• Risk management: involves a lot of time series analysis, calibration, andbacktesting.
• Credit analysis
• Asset and liability management
• Structured finance andsecuritization
• Asset pricing

• 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 andMerton Miller,The Cost of Capital, Corporation Finance and the Theory of Investment,Modigliani–Miller theorem andCorporate 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 andMerton Miller,Theory of Finance
• 1972 –Martin L. Leibowitz and Sydney Homer,Inside the Yield Book,Fixed income analysis
• 1973 –Fischer Black andMyron Scholes,The Pricing of Options and Corporate Liabilities andRobert 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 andLattice 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 modelsGARCH
• 1985 –John C. Cox,Jonathan E. Ingersoll andStephen Ross,A theory of the term structure of interest rates,Cox–Ingersoll–Ross model
• 1987 – Giovanni Barone-Adesi andRobert Whaley,Efficient analytic approximation of American option values. Journal of Finance. 42 (2): 301–20.Barone-Adesi and Whaley method for pricingAmerican options.
• 1987 –David Heath,Robert A. Jarrow, and Andrew MortonBond 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
• 1990 –John Hull andAlan 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. 28–43JSTOR 4479577Black–Litterman model
• 1994 –J.P. MorganRiskMetrics 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,My Life as a Quant: Reflections on Physics and Finance

• List of quantitative analysts
• Quantitative fund
• Financial modeling
• Black–Scholes equation
• Financial signal processing
• Financial analyst
• Technical analysis
• Fundamental analysis
• Financial economics
• Mathematical finance
• Alpha generation platform
• Rocket science (finance)

1. ^See Definition in the Society for Applied and Industrial Mathematicshttps://web.archive.org/web/20060430115935/http://siam.org/about/pdf/brochure.pdf
2. ^Derman, E. (2004). My life as a quant: reflections on physics and finance. John Wiley & Sons.
3. ^"Top Quantitative Hedge Funds".Street of Walls.
4. ^^abL. Carraro and P. Crépel (N.D.).Bachelier, Louis,Encyclopedia of Mathematics
5. ^Markowitz, H. (1952). "Portfolio Selection".Journal of Finance.7 (1):77–91.doi:10.1111/j.1540-6261.1952.tb01525.x.S2CID 7492997.
6. ^^abLam, Leslie P. Norton and Dan."Why Edward Thorp Owns Only Berkshire Hathaway".barrons.com. Retrieved2021-06-06.
7. ^^abcdePatterson, Scott (2010-02-02).The Quants: How a New Breed of Math Whizzes Conquered Wall Street and Nearly Destroyed It. Crown.ISBN 978-0-307-45339-6.
8. ^Samuelson, P. A. (1965). "Rational Theory of Warrant Pricing".Industrial Management Review.6 (2):13–32.
9. ^Henry McKean the co-founder of stochastic calculus (along withKiyosi Itô) wrote the appendix: seeMcKean, 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.
10. ^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.
11. ^Derman, Emanuel (2004).My Life as a Quant. John Wiley and Sons.
12. ^Rothschild, John (November 7, 1999)."The Gnomes of Santa Fe".archive.nytimes.com.Archived from the original on Jun 6, 2021. RetrievedMay 6, 2021.
13. ^Kelly, Kevin (July 1, 1994)."Cracking Wall Street".Wired.ISSN 1059-1028. RetrievedMay 6, 2021.
14. ^Beilselki, Vincent (September 6, 2018)."Millennium Shuts Down Pioneering Quant Hedge Fund".Bloomberg.com. RetrievedMay 6, 2021.
15. ^Sahu, Santosh Kumar; Mokhade, Anil; Bokde, Neeraj Dhanraj (January 2023)."An Overview of Machine Learning, Deep Learning, and Reinforcement Learning-Based Techniques in Quantitative Finance: Recent Progress and Challenges".Applied Sciences.13 (3): 1956.doi:10.3390/app13031956.ISSN 2076-3417.
16. ^[2]Mandelbrot, B. (1963). "The Variation of Certain Speculative Prices." The Journal of Business, 36(4), 394-419
17. ^[8]Fama, E. F. (1965). "The Behavior of Stock-Market Prices." The Journal of Business, 38(1), 34-105.
18. ^[12]Peters, O. (2011). The Ergodicity Problem in Economics. Nature Physics, 7(10), 830–835. DOI: 10.1038/nphys2190
19. ^[9]Lo, A. W. (2004). "The Adaptive Markets Hypothesis: Market Efficiency from an Evolutionary Perspective." The Journal of Portfolio Management, 30(5), 15-29.
20. ^[11]Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. Springer.
21. ^^abcdEmanuel Derman (2004)."Finding a job in finance",Risk
22. ^^abcInternational Association of Financial Engineers (2007)."Student FAQ"
23. ^"Machine Learning in Finance: Theory and Applications".markets media.com. 22 April 2013. Retrieved2 April 2018.
24. ^"A Machine-Learning View of Quantitative Finance"(PDF).appliededucationpsychology.org.
25. ^^abLindsey Gerdes (2009)"Master's of the Financial Universe".Businessweek
26. ^"The Journal of Portfolio Management".jpm.iijournals.com. Retrieved2019-02-02.
27. ^"Quantitative Finance".Taylor & Francis.
28. ^"Finance and Stochastics – incl. Option to publish open access". 2025.

• 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 QuantISBN 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 0-307-45337-5ISBN 978-0-307-45337-2.Amazon page for book viaPatterson and Thorp interview onFresh 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, 2010Wall Street Journal.
• Read, Colin (2012)Rise of the Quants (Great Minds in Finance Series)ISBN 023027417X
• Analysing Quantitative Data for Business and Management Students

• 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