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Mathematical finance comprises the branches 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 ( 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 Quantitative 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 Prize in economics, 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 .

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 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 finance articles

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

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

Notes

1. ^ Karatzas, I., Methods of Mathematical Finance , Secaucus, NJ, USA: Springer-Verlag New York, Incorporated, 1998

References

• Harold Markowitz, Portfolio Selection , Journal of Finance, 7, 1952, pp.77-91
• William Sharpe, Investments , Prentice-Hall, 1985

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