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Mathematical finance
 
 
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Mathematical finance is the branch of applied mathematics
Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains....
 concerned with the financial markets.

The subject has a close relationship with the discipline of financial economics
Financial economics

Financial economics is the branch of economics concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment" ....
, which is concerned with much of the underlying theory. Generally, mathematical finance will derive, and extend, the mathematical
Mathematical model

A mathematical model uses mathematics language to describe a system. Mathematical models are used not only in the natural sciences and engineering disciplines but also in the social sciences ; physicists, engineers, computer sciences, and economists use mathematical models most extensively....
 or numerical
Numerical analysis

Numerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square....
 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
Share price

A share price is the price of a single share of a company's stock. Once the stock is purchased, the owner becomes a Stock#Shareholder of the company that issued the share....
, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus
Stochastic calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes....
 to obtain the fair value of derivative
Derivative (finance)

Derivatives are financial contracts, or financial instruments, whose values are derived from the value of something else . The underlying on which a derivative is based can be an asset , an index , or other items ....
s of the stock
STOCK

Software for fixed assets management and stock control developed in 2004. Stocktaking process is carried using a hand-held mobile terminal equipped with barcode reader or RFID technology....
 (see: Valuation of options
Valuation of options

Because the values of Option contracts depend on a number of different variables in addition to the value of the underlying asset, they are complex to value....
).

In terms of practice, mathematical finance also overlaps heavily with the field of computational finance
Computational finance

Computational finance or financial engineering is a cross-disciplinary field which relies on computational intelligence, mathematical finance, Numerical analysis and computer simulations to make Trader , hedge and investment decisions, as well as facilitating the risk management of those decisions....
 (also known as financial engineering).






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Encyclopedia


Mathematical finance is the branch of applied mathematics
Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains....
 concerned with the financial markets.

The subject has a close relationship with the discipline of financial economics
Financial economics

Financial economics is the branch of economics concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment" ....
, which is concerned with much of the underlying theory. Generally, mathematical finance will derive, and extend, the mathematical
Mathematical model

A mathematical model uses mathematics language to describe a system. Mathematical models are used not only in the natural sciences and engineering disciplines but also in the social sciences ; physicists, engineers, computer sciences, and economists use mathematical models most extensively....
 or numerical
Numerical analysis

Numerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square....
 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
Share price

A share price is the price of a single share of a company's stock. Once the stock is purchased, the owner becomes a Stock#Shareholder of the company that issued the share....
, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus
Stochastic calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes....
 to obtain the fair value of derivative
Derivative (finance)

Derivatives are financial contracts, or financial instruments, whose values are derived from the value of something else . The underlying on which a derivative is based can be an asset , an index , or other items ....
s of the stock
STOCK

Software for fixed assets management and stock control developed in 2004. Stocktaking process is carried using a hand-held mobile terminal equipped with barcode reader or RFID technology....
 (see: Valuation of options
Valuation of options

Because the values of Option contracts depend on a number of different variables in addition to the value of the underlying asset, they are complex to value....
).

In terms of practice, mathematical finance also overlaps heavily with the field of computational finance
Computational finance

Computational finance or financial engineering is a cross-disciplinary field which relies on computational intelligence, mathematical finance, Numerical analysis and computer simulations to make Trader , hedge and investment decisions, as well as facilitating the risk management of those decisions....
 (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
Quantitative analyst

A quantitative analyst is a person who works in finance using numerical or quantitative techniques. Similar work is done in most other modern industries, but the work is not called quantitative analysis....
).

The fundamental theorem of arbitrage-free pricing
Fundamental theorem of arbitrage-free pricing

In a general sense, the fundamental theorem of arbitrage/finance is a way to relate arbitrage opportunities with risk neutral measures that are equivalent to the original probability measure....
 is one of the key theorems in mathematical finance.

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

Mathematical finance articles


Mathematical tools

  • Calculus
    Calculus

    Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
  • Differential equation
    Differential equation

    A differential equation is a mathematics equation for an unknown function of one or several variable that relates the values of the function itself and its derivatives of various orders....
  • Numerical analysis
    Numerical analysis

    Numerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square....
  • Real analysis
    Real analysis

    Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the Set of real numbers. In particular, it deals with the analytic function properties of real function and sequences, including convergence and limit s of sequences of real numbers, the calculus of the real numbers, and continu...
  • Probability
    Probability

    Probability, or wikt:chance, is a way of expressing knowledge or belief that an Event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about t...
  • Probability distribution
    Probability distribution

    In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
    • Binomial distribution
      Binomial distribution

      In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n statistical independence yes/no experiments, each of which yields success with probability p....
    • Log-normal distribution
      Log-normal distribution

      In probability and statistics, the log-normal distribution is the single-tailed probability distribution of any random variable whose logarithm is normal distribution....
  • Expected value
    Expected value

    In probability theory and statistics, the expected value of a random variable is the Lebesgue integral of the random variable with respect to its probability measure....
  • Value at risk
    Value at risk

    In financial mathematics and financial risk management, Value at Risk is a widely used measure of the market risk on a specific Portfolio of financial assets....
  • Risk-neutral measure
    Risk-neutral measure

    In mathematical finance, a risk-neutral measure is a probability measure that results when one assumes that the current value of all financial assets is equal to the expected value of the future payoff of the asset discounted at the risk-free rate....
  • Stochastic calculus
    Stochastic calculus

    Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes....
    • Brownian motion
      Wiener process

      In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called Brownian motion, after Robert Brown ....
    • Lévy process
      Lévy process

      In probability theory, a L?vy process, named after the French mathematician Paul Pierre L?vy, is any continuous-time stochastic process that starts at 0, admits c?dl?g modification and has "stationary independent increments" ? this phrase will be explained below....
  • Itô's lemma
    Ito's lemma

    In mathematics, Kiyoshi Ito's Lemma is used in Ito calculus to find the differential of a function of a particular type of stochastic process....
  • Fourier transform
    Fourier transform

    In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions....
  • Girsanov's theorem
  • Radon-Nikodym derivative
  • Monte Carlo method
    Monte Carlo method

    Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used when computer simulation physics and mathematics systems....
  • Quantile function
    Quantile function

    In probability theory, a quantile function of aprobability distribution is the inverse function F −1 of its cumulative distribution function F....
  • Partial differential equation
    Partial differential equation

    In mathematics, partial differential equations are a type of differential equation, i.e., a Relation involving an unknown Function of several independent variables and its partial derivatives with respect to those variables....
    s
    • Heat equation
      Heat equation

      The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time. For a function u of three spatial variables and the time variable t, the heat equation is...
  • Martingale representation theorem
    Martingale representation theorem

    In probability theory, the martingale representation theorem states that a random variable which is measurable with respect to the Filtration #Measure theory generated by a Brownian motion can be written in terms of an It? integral with respect to this Brownian motion....
  • Feynman Kac Formula
  • Stochastic differential equations
  • Volatility
    Volatility

    Volatility is the measure of the state of instability.*For volatility in chemistry, see Volatility .*For volatility in finance, see Volatility ....
    • ARCH model
      Autoregressive conditional heteroskedasticity

      In econometrics,an autoregressive conditional heteroscedasticity model considers the variance of the current error term to be a function of the variances of the previous time period's error terms....
    • GARCH model
      Autoregressive conditional heteroskedasticity

      In econometrics,an autoregressive conditional heteroscedasticity model considers the variance of the current error term to be a function of the variances of the previous time period's error terms....
  • Stochastic volatility
    Stochastic volatility

    Stochastic volatility models are used in the field of quantitative finance to evaluate derivative securities, such as option . The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the underlying, the tendency of volatility to revert to...
  • Mathematical model
    Mathematical model

    A mathematical model uses mathematics language to describe a system. Mathematical models are used not only in the natural sciences and engineering disciplines but also in the social sciences ; physicists, engineers, computer sciences, and economists use mathematical models most extensively....
  • Numerical method
    Numerical analysis

    Numerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square....
    • Numerical partial differential equations
      Numerical partial differential equations

      Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations ....
      • Crank-Nicolson method
        Crank-Nicolson method

        In numerical analysis, the Crank?Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations....
      • Finite difference method
        Finite difference

        A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient....


Derivatives pricing

  • Rational pricing
    Rational pricing

    Rational pricing is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away"....
     assumptions
    • Risk neutral valuation
      Risk-neutral measure

      In mathematical finance, a risk-neutral measure is a probability measure that results when one assumes that the current value of all financial assets is equal to the expected value of the future payoff of the asset discounted at the risk-free rate....
    • Arbitrage
      Arbitrage

      In economics and finance, arbitrage is the practice of taking advantage of a price differential between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices....
      -free pricing
  • Futures
    • Futures contract pricing
      Futures contract

      In finance, a futures contract is a standardized contract, traded on a futures exchange, to buy or sell a standardized quantity of a specified commodity of standardized quality at a certain date in the future, at a price determined by the instantaneous equilibrium between the forces of supply and demand among competing buy and sell orders...
  • Options
    • Put–call parity
      Put–call parity

      In financial mathematics, put-call parity defines a relationship between the price of a call option and a put option?both with the identical strike price and expiry....
       (Arbitrage relationships for options)
    • Intrinsic value
      Intrinsic value (finance)

      In finance, intrinsic value refers to the value of a Security which is intrinsic to or contained in the security itself. It is also frequently called fundamental value....
      , Time value
      Option time value

      In finance, the value of an option consists of two components, its intrinsic value and its time value. Time value is simply the difference between option value and intrinsic value....
    • Moneyness
      Moneyness

      In finance, moneyness is a measure of the degree to which a derivative is likely to have positive monetary value at its expiration, in the risk-neutral measure....
    • Pricing models
      Mathematical model

      A mathematical model uses mathematics language to describe a system. Mathematical models are used not only in the natural sciences and engineering disciplines but also in the social sciences ; physicists, engineers, computer sciences, and economists use mathematical models most extensively....
      • Black–Scholes model
      • Black model
        Black model

        The Black model is a variant of the Black-Scholes option pricing model. Its primary applications are for pricing bond options, interest rate caps / floors, and swaptions....
      • Binomial options model
        Binomial options pricing model

        In finance, the binomial options pricing model provides a generalizable Numerical analysis for the valuation of Option . The binomial model was first proposed by John C....
      • Monte Carlo option model
        Monte Carlo option model

        In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an Option with multiple sources of uncertainty or with complicated features....
      • Implied volatility
        Implied volatility

        In financial mathematics, the implied volatility of an option contract is the Volatility implied by the market price of the option based on an Valuation of options model....
        , Volatility smile
        Volatility Smile

        In finance, the volatility smile is a long-observed pattern in which at-the-money option tend to have lower Implied volatility than in- or out-of-the-money options....
      • SABR Volatility Model
        SABR Volatility Model

        In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets....
      • The Greeks
        Greeks (finance)

        In mathematical finance, the Greeks are the quantities representing the sensitivities of derivative such as option to a change in underlying parameters on which the value of an instrument or Portfolio of financial instruments is dependent....
    • Optimal stopping
      Optimal stopping

      The theory of optimal stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost....
       (Pricing of American options)
  • Interest rate derivative
    Interest rate derivative

    An interest rate derivative is a derivative where the underlying asset is the right to pay or receive a amount of money at a given interest rate....
    s
    • Short rate model
      Short rate model

      In the context of interest rate derivative , a short rate model is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate....
      • Hull-White model
        Hull-White model

        In financial mathematics, the Hull-White model is a mathematical model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates....
      • Cox-Ingersoll-Ross model
        Cox-Ingersoll-Ross model

        The Cox-Ingersoll-Ross model in Mathematical finance is a mathematical model describing the evolution of interest rates. It is a type of "one factor model" as describes interest rate movements as driven by only one source of market risk....
      • Chen model
        Chen model

        In finance, the Chen model is a mathematical model describing the evolution of interest rates. It is a type of "one-factor model" as it describes interest rate movements as driven by only one source of market risk....
    • LIBOR Market Model
      LIBOR Market Model

      The LIBOR market model, also known as the BGM Model , is a financial model of interest rates. It is used for pricing interest rate derivatives, especially exotic derivatives like Bermudan swaptions, ratchet caps and floors, target redemption notes, autocaps, zero coupon swaptions, constant maturity swaps and spread options, among many o...
    • Heath-Jarrow-Morton framework
      Heath-Jarrow-Morton framework

      The Heath-Jarrow-Morton framework is a general framework to model the evolution of interest rates - forward rates in particular - for risk management in general and asset liability management in particular....


See also

  • Computational finance
    Computational finance

    Computational finance or financial engineering is a cross-disciplinary field which relies on computational intelligence, mathematical finance, Numerical analysis and computer simulations to make Trader , hedge and investment decisions, as well as facilitating the risk management of those decisions....
  • Quantitative Behavioral Finance
    Quantitative behavioral finance

    Quantitative behavioral finance is a new discipline that uses mathematical and statistical methodology to understand behavioral biases in conjunction with valuation ....
  • Derivative (finance)
    Derivative (finance)

    Derivatives are financial contracts, or financial instruments, whose values are derived from the value of something else . The underlying on which a derivative is based can be an asset , an index , or other items ....
    , list of derivatives topics
    List of finance topics

    Topics in finance include:...
  • Modeling and analysis of financial markets
    Modeling and analysis of financial markets

    Much effort has gone into the study of financial markets and how prices vary with time. Charles Dow, one of the founders of Dow Jones & Company and The Wall Street Journal, enunciated a set of ideas on the subject which are now called Dow Theory....
  • International Swaps and Derivatives Association
    International Swaps and Derivatives Association

    The International Swaps and Derivatives Association is a trade organization of participants in the market for derivative #Over-the-counter derivatives....
  • Fundamental financial concepts - topics
    List of finance topics

    Topics in finance include:...
  • Model (economics)
    Model (economics)

    In economics, a model is a theory construct that represents economic Process by a set of variables and a set of logical and/or quantitative relationships between them....
  • List of finance topics
    List of finance topics

    Topics in finance include:...
  • List of economics topics
    List of economics topics

    This aims to be a complete article list of economics topics:...
    , List of economists
    List of economists

    This is an alphabetical list of notable economists, that is, experts in the social science of economics. There is also a separate list of politicians with economics training....
  • List of accounting topics
    List of accounting topics

    This page is a list of accounting topics.AAccounting Ethics- Accounting for risk- Accounting information system- Accounting methods...


External links

  • Quantitative Mathematics Glossary
  • , Prof. Mark Davis, Imperial College
  • , Prof. Campbell R. Harvey
  • , University of Oxford
  • at the University of Technology, Sydney
  • , London, Financial Mathematics
  • , study day with numerous speakers held at Gresham College
    Gresham College

    File:Gresham College, 1740.jpgGresham College is an unusual institution of higher learning off Holborn in central London. It enrolls no students and grants no academic degrees....
    , 25 April 2008
  • , London, Mathematical Finance.
  • , London, Financial Engineering.