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