source: http://en.wikipedia.org/wiki/Mathematical_finance
The discipline of financial economics, is concerned with economics underlying theory.
Mathematical finance will derive, and extend, the mathematical or numerical models suggested by financial economics.
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.
Computational finance (also known as financial engineering) focuses on application, while mathematical finance focuses on modeling and derivation.
The Theory of Speculation (published 1900) by Louis Bachelier discussed the use of Brownian motion to evaluate stock options.
Theory of portfolio optimization by Harry Markowitz use mean-variance estimates of portfolios to judge investment strategies.
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.
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 Prize in economics, for the first time ever awarded for a work in finance.
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 .
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 Prize in economics. Black was ineligible for the prize because of his death in 1995.
Since then, many more sophisticated mathematical models and derivative pricing strategies have been developed.
Mathematical tools
Asymptotic analysis
Calculus
Copulas
Differential equations
Ergodic theory
Gaussian copulas
Numerical analysis
Real analysis
Probability
Probability distributions
Binomial distribution
Log-normal distribution
Expected value
Value at risk
Risk-neutral measure
Stochastic calculus
Brownian motion
Lévy process
Itô's lemma
Fourier transform
Girsanov's theorem
Radon–Nikodym derivative
Monte Carlo method
Quantile functions
Partial differential equations
Heat equation
Martingale representation theorem
Feynman–Kac formula
Stochastic differential equations
Volatility
ARCH model
GARCH model
Stochastic volatility
Mathematical models
Numerical methods
Numerical partial differential equations
Crank–Nicolson method
Finite difference method
Derivatives pricing
The Brownian Motion Model of Financial Markets
Rational pricing assumptions
Risk neutral valuation
Arbitrage-free pricing
Futures
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
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
