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Mathematical finance is a field of applied mathematics, concerned with financial markets. Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock (see: Valuation of options; Financial modeling). 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.
Mathematical finance also overlaps heavily with the field of computational finance (as well as financial engineering). The latter focuses on application, while the former focuses on modeling and derivation (see: Quantitative analyst), often by help of stochastic asset models. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk- and portfolio management on the other.
Many universities offer degree and research programs in mathematical finance; see Master of Mathematical Finance.
History: Q versus P
There exist 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, namely 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
|Goal||"extrapolate the present"|
|Tools||Itō calculus, PDE’s|
The goal of derivatives pricing is to determine the fair price of a given security in terms of more 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 plain vanilla and exotic 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 by Louis Bachelier in The Theory of Speculation (published 1900), with the introduction of the most basic and most influential of processes, the Brownian motion, and its applications to the pricing of options. 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. Bachelier's work, however, was largely unknown outside academia.
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 of his death in 1995.
The next important step was the fundamental theorem of asset pricing by Harrison and Pliska (1981), according to which the suitably normalized current price P0 of a security is arbitrage-free, and thus truly fair, only if there exists a stochastic process Pt with constant expected value which describes its future evolution:
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 the blackboard font letter "".
The relationship (1) must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time.
The 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 (1), a similar relationship is used to define the price of new derivatives.
Risk and portfolio management: the P world
|Goal||"model the future"|
Risk and portfolio management aims at modelling the 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.
The quantitative theory of risk and portfolio management started with the mean-variance framework of Harry Markowitz (1952), who caused a shift away from the concept of trying to identify the best individual stock for investment. Using a linear regression strategy to understand and quantify the risk (i.e. variance) and return (i.e. mean) of an entire portfolio of stocks, bonds, and other securities, an optimization strategy was used to choose a portfolio with largest mean return subject to acceptable levels of variance in the return. Next, breakthrough advances were made with the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT) developed by Treynor (1962), Mossin (1966), William Sharpe (1964), Lintner (1965) and Ross (1976).
For their pioneering work, Markowitz and 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-motion models, and the quadratic utility function implicit in mean–variance optimization was replaced by more general increasing, concave utility functions. Furthermore, in more 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 
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 trends give an indication of the future, at least in the short term. The claims of the technical analysts are disputed by many academics.
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 Nassim Nicholas Taleb, a professor of financial engineering at Polytechnic Institute of New York University, in his book The Black Swan and Paul Wilmott. 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 2008 which addresses some of the most serious concerns.
Bodies such as the Institute for New Economic Thinking are now attempting to establish more effective theories and methods.
In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate. In the 1960s it was discovered by Benoît Mandelbrot that changes in prices do not follow a Gaussian distribution, but are rather modeled better by Lévy alpha-stable distributions. The scale of change, or volatility, depends on the length of the time interval to a 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. See also Financial models with long-tailed distributions and volatility clustering.
Mathematical finance articles
- Computational finance
- Quantitative Behavioral Finance
- Derivative (finance), list of derivatives topics
- Modeling and analysis of financial markets
- Technical analysis
- International Swaps and Derivatives Association
- Fundamental financial concepts - topics
- Model (economics)
- List of finance topics
- List of economics topics, List of economists
- List of accounting topics
- Statistical Finance
- Brownian model of financial markets
- Master of Mathematical Finance
- Karatzas, Ioannis; Shreve, Steve (1998). Methods of Mathematical Finance. Secaucus, NJ, USA: Springer-Verlag New York, Incorporated. ISBN 9780387948393.
- Meucci, Attilio (2005). Risk and Asset Allocation. Springer. ISBN 9783642009648.
- Taleb, Nassim Nicholas (2007). The Black Swan: The Impact of the Highly Improbable. Random House Trade. ISBN 978-1-4000-6351-2.
- "Financial Modelers' Manifesto". Paul Wilmott's Blog. January 8, 2009. Retrieved June 1, 2012.
- Gillian Tett (April 15, 2010). "Mathematicians must get out of their ivory towers". Financial Times.
- Svetlozar T. Rachev, Frank J. Fabozzi, Christian Menn (2005). Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio Selection, and Option Pricing. John Wiley and Sons. ISBN 978-0471718864.
- Harold Markowitz, Portfolio Selection, Journal of Finance, 7, 1952, pp. 77–91
- William Sharpe, Investments, Prentice-Hall, 1985
- Attilio Meucci, P versus Q: Differences and Commonalities between the Two Areas of Quantitative Finance, GARP Risk Professional, February 2011, pp. 41-44
- Nicole El Karoui, The Future of Financial Mathematics, ParisTech Review, September 2013