Mathematical finance

Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical or numerical models suggested by financial economics. 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 fair value of derivatives of the stock (see: Valuation of options).
In terms of practice, mathematical finance also overlaps heavily with the field of computational finance (also known as financial engineering). Arguably, these are largely synonymous, although the latter focuses on application, while the former focuses on modeling and derivation (see: Quantitative analyst). The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance. Many universities around the world now offer degree and research programs in mathematical finance; see Master of Mathematical Finance.

History
The history of mathematical finance starts with The Theory of Speculation (published 1900) by Louis Bachelier, which discussed the use of Brownian motion to evaluate stock options. However, it hardly caught any attention outside academia.
The first influential work of mathematical finance is the theory of portfolio optimization by Harry Markowitz on using mean-variance estimates of portfolios to judge investment strategies, causing 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 and bonds, an optimization strategy was used to choose a portfolio with largest mean return subject to acceptable levels of variance in the return. Simultaneously, William Sharpe developed the mathematics of determining the correlation between each stock and the market. 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 the “black art” of 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.^[1]

Main article: Black–Scholes

The next major revolution in mathematical finance came with the work of Fischer Black and Myron Scholes along with fundamental contributions by Robert C. Merton, by modeling financial markets with stochastic models. 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.
More sophisticated mathematical models and derivative pricing strategies were then developed but their credibility was damaged by the financial crisis of 2007–2010. Bodies such as the Institute for New Economic Thinking are now attempting to establish more effective theories and methods.^[2]

[edit] Criticism

Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by Nassim Nicholas Taleb in his book The Black Swan^[3] 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^[4] which addresses some of the most serious concerns

Mathematical finance articles

Mathematical tools

• Asymptotic analysis
• Calculus
• Copulas
• Differential equations
• Expected value
• Ergodic theory
• Feynman–Kac formula
• Fourier transform
• Gaussian copulas
• Girsanov's theorem
• Itô's lemma
• Martingale representation theorem
• Mathematical models
• Monte Carlo method
• Numerical analysis
• Real analysis
• Partial differential equations
• Probability
• Probability distributions
  • Binomial distribution
  • Log-normal distribution
• Quantile functions
  • Heat equation
• Radon–Nikodym derivative
• Risk-neutral measure
• Stochastic calculus
  • Brownian motion
  • Lévy process
• Stochastic differential equations
• Stochastic volatility
  • Numerical partial differential equations
    • Crank–Nicolson method
    • Finite difference method
• Value at risk
• Volatility
  • ARCH model
  • GARCH model

Derivatives pricing

• The Brownian Motion Model of Financial Markets
• Rational pricing assumptions
  • Risk neutral valuation
  • Arbitrage-free pricing
• Futures contract pricing
• Options
  • Put–call parity (Arbitrage relationships for options)
  • Intrinsic value, Time value
  • Moneyness
  • Pricing models
    • Black–Scholes model
    • Black model
    • Binomial options model
    • Monte Carlo option model
    • Implied volatility, Volatility smile
    • SABR Volatility Model
    • Markov Switching Multifractal
    • The Greeks
    • Finite difference methods for option pricing
    • Vanna Volga method
    • Trinomial tree
  • Optimal stopping (Pricing of American options)
• Interest rate derivatives
  • Short rate model
    • Hull–White model
    • Cox–Ingersoll–Ross model
    • Chen model
  • LIBOR Market Model
  • Heath–Jarrow–Morton framework

See also

• Computational finance
• Quantitative Behavioral Finance
• Derivative (finance), list of derivatives topics
• Modeling and analysis of financial markets
• 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

Great Mathematicians

Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus

Carl Friedrich Gauss, himself known as the "prince of mathematicians",[37] referred to mathematics as "the Queen of the Sciences".

Leonhard Euler, who created and popularized much of the mathematical notation used today

Greek mathematician Pythagoras (c.570-c.495 BC), commonly credited with discovering the Pythagorean theorem

Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens

Mathematics

Mathematics (from Greek μάθημα (máthēma) — knowledge, study, learning) is the study of quantity, structure, space, and change.

Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in mathematical logic. Since the pioneering work of Giuseppe Peano, David Hilbert, and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.

Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world in AD 800, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.

The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".

David Hilbert defined mathematics as follows: We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."