Mathematical finance

Mathematical finance is the branch of applied mathematics concerned with the 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.

In terms of practice, mathematical finance also overlaps heavily with the fields of financial engineering and computational finance. Arguably, all three are largely synonymous, although the latter two focus on application, while the former focuses on modelling and derivation; see Quantitative analyst.

Many universities around the world now offer degree and research programs in mathematical finance.

Mathematical tools

 * Calculus
 * Differential equation
 * Numerical analysis
 * Real analysis
 * Probability
 * Probability distribution
 * 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
 * Partial differential equations
 * Heat equation
 * Martingale representation theorem
 * Feynman Kac Formula
 * Dynkin formula
 * Stochastic differential equations
 * Volatility
 * ARCH model
 * GARCH model
 * Stochastic volatility
 * Mathematical model
 * Numerical method
 * Numerical partial differential equations
 * Crank-Nicolson method
 * Finite difference method

Derivatives pricing

 * Rational pricing assumptions
 * Risk neutral valuation
 * Arbitrage-free pricing
 * Futures
 * Futures contract pricing
 * Options
 * Put-call parity (Arbitrage relationships for options)
 * Moneyness
 * Option time value
 * Pricing models
 * Black-Scholes
 * Black model
 * Binomial options model
 * Monte Carlo option model
 * Implied volatility
 * Volatility smile
 * The Greeks
 * Interest rate derivatives
 * Short rate model
 * Hull-White model
 * Cox-Ingersoll-Ross model
 * Chen model
 * LIBOR Market Model
 * Heath-Jarrow-Morton framework

Theory

 * Mathematics of Financial Markets, Prof. Mark Davis, Imperial College
 * Option Valuation, Prof. Campbell R. Harvey

Research

 * Oxford-Man Institute
 * Quantitative Finance Research Papers at the University of Technology, Sydney

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