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{{Finance sidebar}}{{Short description|Application of mathematical and statistical methods in finance}}
'''Mathematical finance''', also known as '''quantitative finance''' and '''financial mathematics''', is a field of [[applied mathematics]], concerned with mathematical modeling in the [[Finance#Quantitative_finance|financial field]].

In general, there exist two separate branches of finance that require advanced quantitative techniques: [[Derivative (finance)|derivatives]] pricing on the one hand, and [[risk management|risk]] and [[Investment management#Investment managers and portfolio structures|portfolio management]] on the other.<ref>{{cite web|title=Quantitative Finance|url=http://finance.laws.com/quantitative-finance|publisher=About.com|access-date=28 March 2014}}</ref>
Mathematical finance overlaps heavily with the fields of [[computational finance]] and [[financial engineering]]. The latter focuses on applications and modeling, often with the help of [[stochastic asset model]]s, while the former focuses, in addition to analysis, on building tools of implementation for the models. 
Also related is [[quantitative investing]], which relies on statistical and numerical models (and lately [[machine learning]]) as opposed to traditional [[fundamental analysis]] when [[investment management|managing portfolios]].

French mathematician [[Louis Bachelier]]'s doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical finance. But mathematical finance emerged as a discipline in the 1970s, following the work of [[Fischer Black]], [[Myron Scholes]] and [[Robert C. Merton|Robert Merton]] on option pricing theory. [[Quantitative_analysis_(finance)#Quantitative_investment_management)|Mathematical investing]] originated from the research of mathematician [[Edward O. Thorp|Edward Thorp]] who used statistical methods to first invent [[card counting]] in [[blackjack]] and then applied its principles to modern systematic investing.<ref>{{Cite web|last=Lam|first=Leslie P. Norton and Dan|title=Why Edward Thorp Owns Only Berkshire Hathaway|url=https://www.barrons.com/articles/why-edward-thorp-only-owns-berkshire-hathaway-1521547200|access-date=2021-06-06|website=www.barrons.com|language=en-US}}</ref>

The subject has a close relationship with the discipline of [[financial economics]], which is concerned with much of the underlying theory that is involved in financial mathematics. While trained economists use complex [[economic models]] that are built on observed empirical relationships, in contrast, mathematical finance analysis will derive and extend the [[Mathematical model|mathematical]] or [[Numerical analysis|numerical]] models without necessarily establishing a link to financial theory, taking observed market prices as input.
See: [[Valuation of options]]; [[Financial modeling#Quantitative finance|Financial modeling]]; [[Asset pricing]]. 
The [[fundamental theorem of arbitrage-free pricing]] is one of the key theorems in mathematical finance, while the [[Black–Scholes]] equation and formula are amongst the key results.<ref>{{cite web|last=Johnson|first=Tim|title=What is financial mathematics?|url=https://plus.maths.org/content/what-financial-mathematics|work=+Plus Magazine|access-date=1 March 2021|date=1 September 2009}}</ref> 

Today many universities offer degree and research programs in mathematical finance.

==History: Q versus P==
There are two separate branches of finance that require advanced quantitative techniques: derivatives pricing, and risk and portfolio management. One of the main differences is that they use different probabilities such as the risk-neutral probability (or arbitrage-pricing probability), denoted by "Q", and the actual (or actuarial) probability, denoted by "P".

=== Derivatives pricing: the Q world ===
{| class="wikitable floatright"
|+ '''The Q world''' 
|-
|Goal
|"extrapolate the present"
|-
|Environment
|risk-neutral probability <math>\mathbb{Q}</math>
|-
|Processes
|continuous-time martingales
|-
|Dimension
|low
|-
|Tools
|Itō calculus, PDEs
|-
|Challenges
|calibration
|-
|Business
|sell-side
|}
{{main|Risk-neutral measure}}
{{further|Black–Scholes model|Brownian model of financial markets|Martingale pricing|Quantitative analysis (finance)#History}}
The goal of derivatives pricing is to determine the fair price of a given security in terms of more [[market liquidity|liquid securities]] whose price is determined by the law of [[supply and demand]]. The meaning of "fair" depends, of course, on whether one considers buying or selling the security. Examples of securities being priced are [[option (finance)|plain vanilla]] and [[exotic option]]s, [[convertible bond]]s, etc.

Once a fair price has been determined, the sell-side trader can make a market on the security. Therefore, derivatives pricing is a complex "extrapolation" exercise to define the current market value of a security, which is then used by the sell-side community. 
Quantitative derivatives pricing was initiated by [[Louis Bachelier]] in ''The Theory of Speculation'' ("Théorie de la spéculation", published 1900), with the introduction of the most basic and most influential of processes, [[Brownian motion]], and its applications to the pricing of options.<ref>{{Cite book|title=Stochastic calculus for finance|last=E.|first=Shreve, Steven|date=2004|publisher=Springer|isbn=9780387401003|location=New York|oclc=53289874}}</ref><ref>{{Cite book|title=Introduction to Quantitative Finance.|last=Stephen.|first=Blyth|date=2013|publisher=Oxford University Press, USA|isbn=9780199666591|pages=157|oclc=868286679}}</ref> Brownian motion is derived using the [[Langevin equation]] and the discrete [[random walk]].<ref>{{Cite book|title=Quantitative finance for physicists : an introduction|last=B.|first=Schmidt, Anatoly|date=2005|publisher=Elsevier Academic Press|isbn=9780080492209|location=San Diego, Calif.|oclc=57743436}}</ref> Bachelier modeled the [[time series]] of changes in the [[logarithm]] of stock prices as a [[random walk]] in which the short-term changes had a finite [[variance]]. This causes longer-term changes to follow a [[Gaussian distribution]].<ref>{{cite web|last=Bachelir|first=Louis|title=The Theory of Speculation|url=https://docs.google.com/file/d/0B5LLDy7-d3SKNGI0M2E0NGItYzFlMS00NGU2LWE2ZDAtODc3MDY3MzdiNmY0/edit?hl=en_GB&pli=1|access-date=28 March 2014}}</ref>

The theory remained dormant until [[Fischer Black]] and [[Myron Scholes]], along with fundamental contributions by [[Robert C. Merton]], applied the second most influential process, the [[geometric Brownian motion]], to [[option pricing]]. For this M. Scholes and R. Merton were awarded the 1997 [[Nobel Memorial Prize in Economic Sciences]]. Black was ineligible for the prize because he died in 1995.<ref>{{cite web|last=Lindbeck|first=Assar|title=The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1969-2007|url=https://www.nobelprize.org/nobel_prizes/themes/economic-sciences/lindbeck/index.html|publisher=Nobel Prize|access-date=28 March 2014}}</ref>

The next important step was the [[fundamental theorem of asset pricing]] by Harrison and Pliska (1981), according to which the suitably normalized current price ''P<sub>0</sub>'' of security is arbitrage-free, and thus truly fair only if there exists a [[stochastic process]] ''P<sub>t</sub>'' with constant [[expected value]] which describes its future evolution:<ref>{{cite news|last=Brown|first=Angus|title=A risky business: How to price derivatives|url=http://plus.maths.org/content/risky-business-how-price-derivatives|access-date=28 March 2014|newspaper=Price+ Magazine|date=1 Dec 2008}}</ref>
{{NumBlk|:|<math>P_{0} = \mathbf{E}_{0} (P_{t}) </math>|{{EquationRef|1}} }}
A process satisfying ({{EquationNote|1}}) is called a "[[Martingale (probability theory)|martingale]]". A martingale does not reward risk. Thus the probability of the normalized security price process is called "risk-neutral" and is typically denoted by the [[Blackboard bold|blackboard font]] letter "<math>\mathbb{Q}</math>".

The relationship ({{EquationNote|1}}) must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time.

The [[Quantitative analyst|quants]] who operate in the Q world of derivatives pricing are specialists with deep knowledge of the specific products they model.

Securities are priced individually, and thus the problems in the Q world are low-dimensional in nature. Calibration is one of the main challenges of the Q world: once a continuous-time parametric process has been calibrated to a set of traded securities through a relationship such as ({{EquationNote|1}}), a similar relationship is used to define the price of new derivatives.

The main quantitative tools necessary to handle continuous-time Q-processes are [[Itô calculus|Itô's stochastic calculus]], [[Monte Carlo methods in finance|simulation]] and [[partial differential equation]]s (PDEs).<ref>For a survey, see [https://catalogimages.wiley.com/images/db/pdf/9781118487716.excerpt.pdf "Financial Models"], from Michael Mastro (2013). ''Financial Derivative and Energy Market Valuation'', John Wiley & Sons. {{ISBN| 978-1118487716}}.</ref>

=== Risk and portfolio management: the P world ===
{| class="wikitable floatright"
|+ '''The P world''' 
|-
|Goal
|"model the future"
|-
|Environment
|real-world probability <math>\mathbb{P}</math>
|-
|Processes
|discrete-time series
|-
|Dimension
|large
|-
|Tools
|multivariate statistics
|-
|Challenges
|estimation
|-
|Business
|buy-side
|}

Risk and portfolio management aims to model the statistically derived probability distribution of the market prices of all the securities at a given future investment horizon. This "real" probability distribution of the market prices is typically denoted by the blackboard font letter "<math>\mathbb{P}</math>", as opposed to the "risk-neutral" probability "<math>\mathbb{Q}</math>" used in derivatives pricing. Based on the P distribution, the buy-side community takes decisions on which securities to purchase in order to improve the prospective profit-and-loss profile of their positions considered as a portfolio. Increasingly, elements of this process are automated; see {{section link|Outline_of_finance#Quantitative investing}} for a listing of relevant articles.

For their pioneering work, [[Harry Markowitz|Markowitz]] and [[William F. Sharpe|Sharpe]], along with [[Merton Miller]], shared the 1990 [[Nobel Memorial Prize in Economic Sciences]], for the first time ever awarded for a work in finance.

The portfolio-selection work of Markowitz and Sharpe introduced mathematics to [[investment management]]. With time, the mathematics has become more sophisticated. Thanks to Robert Merton and [[Paul Samuelson]], one-period models were replaced by continuous time, [[Brownian Model of Financial Markets|Brownian-motion models]], and the quadratic [[Utility#Utility_function|utility function]] implicit in [[Modern portfolio theory|mean–variance optimization]] was replaced by more general increasing, concave utility functions.<ref>{{cite book|last1=Karatzas|first1=Ioannis|last2=Shreve|first2=Steve|title=Methods of Mathematical Finance|location=Secaucus, New Jersey, US|publisher=Springer-Verlag New York, Incorporated|year=1998|isbn=9780387948393}}</ref> Furthermore, in recent years the focus shifted toward estimation risk, i.e., the dangers of incorrectly assuming that advanced time series analysis alone can provide completely accurate estimates of the market parameters.<ref>{{cite book|last=Meucci|first=Attilio|author-link=Attilio Meucci|title=Risk and Asset Allocation|publisher=Springer|year=2005|isbn=9783642009648}}</ref>
See {{slink|Financial risk management#Investment management}}.

Much effort has gone into the study of financial markets and how prices vary with time. 
[[Charles Dow]], one of the founders of [[Dow Jones & Company]] and [[The Wall Street Journal]], enunciated a set of ideas on the subject which are now called [[Dow Theory]]. This is the basis of the so-called [[technical analysis]] method of attempting to predict future changes. One of the tenets of "technical analysis" is that [[market trend]]s give an indication of the future, at least in the short term. The claims of the technical analysts are disputed by many academics.{{citation needed|date=March 2021}} While numerous empirical studies have examined the effectiveness of technical analysis, there remains no definitive consensus on its usefulness in forecasting financial markets.<ref>{{cite journal |last1=Park |first1=C. H. |last2=Irwin |first2=S. H. |year=2007 |title=What Do We Know About the Profitability of Technical Analysis? |journal=Journal of Economic Surveys |volume=21 |issue=4 |pages=786–826 |url=https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-6419.2007.00519.x}}</ref>

== Criticism ==
{{more| Financial economics#Challenges and criticism|Financial engineering#Criticisms}}
{{See also|Financial models with long-tailed distributions and volatility clustering}}

Over the years, increasingly sophisticated mathematical models and derivative pricing strategies have been developed, but their credibility was damaged by the [[financial crisis of 2007–2010]].
Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by [[Paul Wilmott]], and by [[Nassim Nicholas Taleb]], in his book [[The Black Swan (Taleb book)|''The Black Swan'']].<ref name="Black Swan">{{cite book|last1=Taleb|first1=Nassim Nicholas|author-link=Nassim Nicholas Taleb|year=2007|title=The Black Swan: The Impact of the Highly Improbable|publisher=Random House Trade|isbn=978-1-4000-6351-2|url-access=registration|url=https://archive.org/details/blackswanimpacto00tale}}</ref> Taleb claims that the prices of financial assets cannot be characterized by the simple models currently in use, rendering much of current practice at best irrelevant, and, at worst, dangerously misleading. Wilmott and [[Emanuel Derman]] published the ''[[Financial Modelers' Manifesto]]'' in January 2009<ref>{{cite web|url=http://www.wilmott.com/blogs/paul/index.cfm/2009/1/8/Financial-Modelers-Manifesto|publisher=Paul Wilmott's Blog|title=Financial Modelers' Manifesto|date=January 8, 2009|access-date=June 1, 2012|archive-url=https://web.archive.org/web/20140908100545/http://www.wilmott.com/blogs/paul/index.cfm/2009/1/8/Financial-Modelers-Manifesto|archive-date=September 8, 2014|url-status=dead}}</ref> which addresses some of the most serious concerns.
Bodies such as the [[Institute for New Economic Thinking]] are now attempting to develop new theories and methods.<ref>{{cite news |url=http://www.ft.com/cms/s/0/cfb9c43a-48b7-11df-8af4-00144feab49a.html |title=Mathematicians must get out of their ivory towers |author=Gillian Tett |newspaper=[[Financial Times]] |date=April 15, 2010}}</ref>

In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate.<ref>{{cite book|author1=Svetlozar T. Rachev|author2-link=Frank J. Fabozzi|author2=Frank J. Fabozzi|author3=Christian Menn|year=2005|title=Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio Selection, and Option Pricing |publisher=[[John Wiley and Sons]] |isbn=978-0471718864 }}</ref> In the 1960s it was discovered by [[Benoit Mandelbrot]] that changes in prices do not follow a [[Gaussian distribution]], but are rather modeled better by Lévy alpha-[[stable distribution]]s.<ref>[[Benoit Mandelbrot|B. Mandelbrot]], [http://web.williams.edu/Mathematics/sjmiller/public_html/341Fa09/econ/Mandelbroit_VariationCertainSpeculativePrices.pdf "The variation of certain Speculative Prices"], ''The Journal of Business'' 1963</ref> The scale of change, or volatility, depends on the length of the time interval to a [[power law|power]] a bit more than 1/2. Large changes up or down are more likely than what one would calculate using a Gaussian distribution with an estimated [[standard deviation]]. But the problem is that it does not solve the problem as it makes parametrization much harder and risk control less reliable.<ref name="Black Swan" />

Perhaps more fundamental: though mathematical finance models may generate a profit in the short-run, this type of modeling is often in conflict with a central tenet of modern macroeconomics, the [[Lucas critique]] - or [[rational expectations]] -  which states that observed relationships may not be structural in nature and thus may not be possible to exploit for public policy or for profit unless we have identified relationships using [[causal analysis]] and [[econometrics]].<ref>{{Cite web|last=Lucas|first=Bob|title=ECONOMETRIC POEICY EVALUATION: A CRITIQUE|url=https://web.sgh.waw.pl/~atoroj/makroekonomia_zaawansowana/lucas76.pdf|access-date=2022-08-05|language=en-US}}</ref> Mathematical finance models do not, therefore, incorporate complex elements of human psychology that are critical to modeling modern macroeconomic movements such as the [[self-fulfilling prophecy|self-fulfilling]] panic that motivates [[bank runs]].

==See also==
{{Seealso|Outline of finance#Financial mathematics|Outline of finance#Mathematical tools|Outline of finance#Derivatives pricing|Outline of corporate finance}}

===Mathematical tools===
{{div col|colwidth=20em}}
*[[Asymptotic analysis]]
*[[Backward stochastic differential equation]]
*[[Calculus]]
*[[Copula (probability theory)|Copulas]], including Gaussian
*[[Differential equation]]s
*[[Expected value]]
*[[Ergodic theory]]
*[[Feynman&ndash;Kac formula]]
*{{slink|Finance#Quantitative finance}}
*[[Fourier transform]]
*[[Girsanov theorem]]
*[[Itô's lemma]]
*[[Martingale representation theorem]]
*[[Mathematical model]]s
*[[Mathematical optimization]]
**[[Linear programming]]
**[[Nonlinear programming]]
**[[Quadratic programming]]
*[[Monte Carlo method]]
*[[Numerical analysis]]
**[[Gaussian quadrature]]
*[[Real analysis]]
*[[Partial differential equation]]s
**[[Heat equation]]
**[[Numerical partial differential equations]]
***[[Crank–Nicolson method]]
***[[Finite difference#Numerical analysis|Finite difference method]]
*[[Probability]]
*[[Probability distribution]]s
**[[Binomial distribution]]
**[[Johnson's SU-distribution]]
**[[Log-normal distribution]]
**[[Student's t-distribution]]
*[[Quantile function]]s
*[[Radon–Nikodym derivative]]
*[[Risk-neutral measure]]
*[[Scenario optimization]]
*[[Stochastic calculus]]
**[[Wiener process|Brownian motion]]
**[[Lévy process]]
*[[Stochastic differential equation]]
*[[Stochastic optimization]]
*[[Stochastic volatility]]
*[[Survival analysis]]
*[[Value at risk]]
*[[Volatility (finance)|Volatility]]
**[[Autoregressive conditional heteroskedasticity|ARCH model]]
**[[Autoregressive conditional heteroskedasticity|GARCH model]]
{{div col end}}

===Derivatives pricing===
{{div col|colwidth=20em}}
* The [[Brownian model of financial markets]]
* [[Rational pricing]] assumptions
**[[Risk-neutral measure|Risk neutral valuation]]
**[[Arbitrage]]-free pricing
*Valuation adjustments
**[[Credit valuation adjustment]]
**[[XVA]]
*[[Yield curve]] modelling
**[[Multi-curve framework]]
**[[Bootstrapping (finance)|Bootstrapping]]
**[[Yield curve#Construction of the full yield curve from market data|Construction from market data]]
**[[Fixed-income attribution #Modeling the yield curve|Fixed-income attribution]]
**[[Nelson-Siegel]]
**[[Principal component analysis#Quantitative finance|Principal component analysis]]
*[[Forward price#Forward Price Formula|Forward Price Formula]]
*[[Futures contract#Pricing|Futures contract pricing]]
*[[Swap (finance)#Valuation and Pricing|Swap valuation]]
**[[Currency swap#Valuation and Pricing]]
**[[Interest rate swap#Valuation and pricing]]
***[[Multi-curve framework]]
**[[Variance swap#Pricing and valuation]]
**[[Asset swap #Computing the asset swap spread]]
**[[Credit default swap #Pricing and valuation]]
* Options
**[[Put–call parity]] (Arbitrage relationships for options)
**[[Intrinsic value (finance)|Intrinsic value]], [[Option time value|Time value]]
**[[Moneyness]]
**Pricing [[Mathematical model|models]]
***[[Black–Scholes model]]
***[[Black model]]
***[[Binomial options pricing model|Binomial options model]]
****[[Implied binomial tree]]
****[[Edgeworth binomial tree]]
***[[Monte Carlo option model]]
***[[Implied volatility]], [[Volatility smile]]
*** [[Local volatility]]
*** [[Stochastic volatility]]
**** [[Constant elasticity of variance model]]
**** [[Heston model]]
***** [[Stochastic volatility jump]]
**** [[SABR volatility model]]
***[[Markov switching multifractal]]
***[[Greeks (finance)|The Greeks]]
***[[Finite difference methods for option pricing]]
***[[Vanna–Volga pricing]]
***[[Trinomial tree]]
****[[Implied trinomial tree]]
***[[Garman-Kohlhagen model]]
***[[Lattice model (finance)]]
***[[Margrabe's formula]]
***[[Carr–Madan formula]]
**Pricing of American options
***[[Barone-Adesi and Whaley]]
***[[Bjerksund and Stensland]]
***[[Black's approximation]]
***[[Monte Carlo methods for option pricing#Least Square Monte Carlo|Least Square Monte Carlo]]
***[[Optimal stopping]]
***[[Roll-Geske-Whaley]]
*[[Interest rate derivative]]s
**[[Black model]]
***[[Interest rate cap and floor#Black model|caps and floors]]
***[[Swaption#Valuation|swaptions]]
***[[Bond option#Valuation|Bond options]]
**[[Short-rate model]]s
***[[Rendleman–Bartter model]]
***[[Vasicek model]]
***[[Ho–Lee model]]
***[[Hull–White model]]
***[[Cox–Ingersoll–Ross model]]
***[[Black–Karasinski model]]
***[[Black–Derman–Toy model]]
***[[Kalotay–Williams–Fabozzi model]]
***[[Longstaff–Schwartz model]]
***[[Chen model]]
**[[Forward rate]]-based models 
***[[LIBOR market model]] (Brace–Gatarek–Musiela Model, BGM)
***[[Heath–Jarrow–Morton framework|Heath–Jarrow–Morton Model]] (HJM)
{{col div end}}

===Portfolio modelling===
{{see|Outline of finance#Portfolio theory|Outline of finance#Quantitative investing|Outline of finance#Portfolio mathematics}}

===Other===
{{div col|colwidth=20em}}
*[[Computational finance]]
*[[Derivative (finance)]], [[Outline of finance#Derivatives market|list of derivatives topics]]
*[[Economic model]]
*[[Econophysics]]
*[[Financial economics]]
*[[Financial engineering]]
*{{Section link|Financial modeling|Quantitative finance}}
*[[International Association for Quantitative Finance]]
*[[International Swaps and Derivatives Association]]
*[[Index of accounting articles]]
*[[List of economists]]
*[[Master of Quantitative Finance]]
*[[Outline of economics]]
*[[Outline of finance]]
*[[Physics of financial markets]]
*[[Quantitative behavioral finance]]
*[[Quantum finance]]
*[[Rocket science (finance)]]
*[[Statistical finance]]
*[[Technical analysis]]
*[[XVA]]
{{div col end}}

==Notes==
{{reflist}}

==Further reading==
* [[Nicole El Karoui]], [https://web.archive.org/web/20170101051554/http://www.paristechreview.com/2013/09/06/future-financial-mathematics/ "The future of financial mathematics"], ''[[ParisTech]] Review'', 6 September 2013
* [[Harold Markowitz]], "Portfolio Selection", ''[[The Journal of Finance]]'', 7, 1952, pp.&nbsp;77–91
* [[William F. Sharpe]], ''Investments'', Prentice-Hall, 1985
* Pierre Henry Labordere (2017). “Model-Free Hedging A Martingale Optimal Transport Viewpoint”. Chapman & Hall/ CRC. 

{{Areas of mathematics}}
{{Finance}}
{{Financial risk}}
{{Economics}}

{{Authority control}}

[[Category:Mathematical finance| ]]
[[Category:Applied statistics]]