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Mathematical finance, also known asquantitative finance andfinancial mathematics, is a field ofapplied mathematics, concerned with mathematical modeling in thefinancial field.
In general, there exist two separate branches of finance that require advanced quantitative techniques:derivatives pricing on the one hand, andrisk andportfolio management on the other.[1]Mathematical finance overlaps heavily with the fields ofcomputational finance andfinancial engineering. The latter focuses on applications and modeling, often with the help ofstochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related isquantitative investing, which relies on statistical and numerical models (and latelymachine learning) as opposed to traditionalfundamental analysis whenmanaging portfolios.
French mathematicianLouis 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 ofFischer Black,Myron Scholes andRobert Merton on option pricing theory.Mathematical investing originated from the research of mathematicianEdward Thorp who used statistical methods to first inventcard counting inblackjack and then applied its principles to modern systematic investing.[2]
The subject has a close relationship with the discipline offinancial economics, which is concerned with much of the underlying theory that is involved in financial mathematics. While trained economists use complexeconomic models that are built on observed empirical relationships, in contrast, mathematical finance analysis will derive and extend themathematical ornumerical models without necessarily establishing a link to financial theory, taking observed market prices as input.See:Valuation of options;Financial modeling;Asset pricing. Thefundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while theBlack–Scholes equation and formula are amongst the key results.[3]
Today many universities offer degree and research programs in mathematical finance.
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".
| Goal | "extrapolate the present" |
| Environment | risk-neutral probability |
| Processes | continuous-time martingales |
| Dimension | low |
| Tools | Itō calculus, PDEs |
| Challenges | calibration |
| Business | sell-side |
The goal of derivatives pricing is to determine the fair price of a given security in terms of moreliquid securities whose price is determined by the law ofsupply and demand. The meaning of "fair" depends, of course, on whether one considers buying or selling the security. Examples of securities being priced areplain vanilla andexotic options,convertible bonds, 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 byLouis Bachelier inThe 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.[4][5] Brownian motion is derived using theLangevin equation and the discreterandom walk.[6] Bachelier modeled thetime series of changes in thelogarithm of stock prices as arandom walk in which the short-term changes had a finitevariance. This causes longer-term changes to follow aGaussian distribution.[7]
The theory remained dormant untilFischer Black andMyron Scholes, along with fundamental contributions byRobert C. Merton, applied the second most influential process, thegeometric Brownian motion, tooption pricing. For this M. Scholes and R. Merton were awarded the 1997Nobel Memorial Prize in Economic Sciences. Black was ineligible for the prize because he died in 1995.[8]
The next important step was thefundamental theorem of asset pricing by Harrison and Pliska (1981), according to which the suitably normalized current priceP0 of security is arbitrage-free, and thus truly fair only if there exists astochastic processPt with constantexpected value which describes its future evolution:[9]
| 1 |
A process satisfying (1) is called a "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 theblackboard font letter "".
The relationship (1) must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time.
Thequants 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 (1), a similar relationship is used to define the price of new derivatives.
The main quantitative tools necessary to handle continuous-time Q-processes areItô's stochastic calculus,simulation andpartial differential equations (PDEs).[10]
| Goal | "model the future" |
| Environment | real-world probability |
| 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 "", as opposed to the "risk-neutral" probability "" 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; seeOutline of finance § Quantitative investing for a listing of relevant articles.
For their pioneering work,Markowitz andSharpe, along withMerton Miller, shared the 1990Nobel 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 toinvestment management. With time, the mathematics has become more sophisticated. Thanks to Robert Merton andPaul Samuelson, one-period models were replaced by continuous time,Brownian-motion models, and the quadraticutility function implicit inmean–variance optimization was replaced by more general increasing, concave utility functions.[11] 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.[12]SeeFinancial 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 ofDow Jones & Company andThe Wall Street Journal, enunciated a set of ideas on the subject which are now calledDow Theory. This is the basis of the so-calledtechnical analysis method of attempting to predict future changes. One of the tenets of "technical analysis" is thatmarket trends 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] While numerous empirical studies have examined the effectiveness of technical analysis, there remains no definitive consensus on its usefulness in forecasting financial markets.[13]
Over the years, increasingly sophisticated mathematical models and derivative pricing strategies have been developed, but their credibility was damaged by the2008 financial crisis.
Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably byPaul Wilmott, and byNassim Nicholas Taleb, in his bookThe Black Swan.[14] 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 andEmanuel Derman published theFinancial Modelers' Manifesto in January 2009[15] which addresses some of the most serious concerns.Bodies such as theInstitute for New Economic Thinking are now attempting to develop new theories and methods.[16]
In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate.[17] In the 1960s it was discovered byBenoit Mandelbrot that changes in prices do not follow aGaussian distribution, but are rather modeled better by Lévy alpha-stable distributions.[18] The scale of change, or volatility, depends on the length of the time interval to apower 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 estimatedstandard deviation. But the problem is that it does not solve the problem as it makes parametrization much harder and risk control less reliable.[14]
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, theLucas critique - orrational 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 usingcausal analysis andeconometrics.[19] Mathematical finance models do not, therefore, incorporate complex elements of human psychology that are critical to modeling modern macroeconomic movements such as theself-fulfilling panic that motivatesbank runs.
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