# Mathematical finance

financial mathematicsquantitative financequantitative tradingquantitative investingquantitative investmentderivative pricingfinancefinancialquantitativequantitative financial
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.wikipedia
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### Financial modeling

financial modelfinancial modellingFinancial models
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 corresponding value of derivatives of the stock (see: Valuation of options; Financial modeling; Asset pricing).
At the same time, "financial modeling" is a general term that means different things to different users; the reference usually relates either to accounting and corporate finance applications, or to quantitative finance applications.

### Quantitative analyst

quantquantitative analysisquants
The latter focuses on applications and modeling, often by help of stochastic asset models (see: Quantitative analyst), while the former focuses, in addition to analysis, on building tools of implementation for the models.
A quantitative analyst (or, in financial jargon, a quant) is a person who specializes in the application of mathematical and statistical methods (mathematical finance) in finance.

### Financial engineering

financial engineerFinancial technologyfinancial engineers
Mathematical finance also overlaps heavily with the fields of computational finance and financial engineering.
It has also been defined as the application of technical methods, especially from mathematical finance and computational finance, in the practice of finance.

### Applied mathematics

applied mathematicianappliedapplications of mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
Quantitative finance is now taught in mathematics departments across universities and mathematical finance is considered a full branch of applied mathematics.

### Louis Bachelier

BachelierBachelier, Louis Jean-Baptiste Alphonse
French mathematician Louis Bachelier is considered the author of the first scholarly work on mathematical finance, published in 1900.
Thus, Bachelier is considered as the forefather of mathematical finance and a pioneer in the study of stochastic processes.

### Time series

time series analysistime-seriestime-series analysis
Bachelier modeled the time series of changes in the logarithm of stock prices as a random walk in which the short-term changes had a finite variance.
Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.

### Computational finance

Financial ComputingFinancial Engineering
Mathematical finance also overlaps heavily with the fields of computational finance and financial engineering.
It is an interdisciplinary field between mathematical finance and numerical methods.

### Monte Carlo methods in finance

Monte Carlo simulationsimulationsimulation techniques
The main quantitative tools necessary to handle continuous-time Q-processes are Itō's stochastic calculus, simulation and partial differential equations (PDE's).
Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes.

### Itô calculus

Itô integralItō calculusItō process
The main quantitative tools necessary to handle continuous-time Q-processes are Itō's stochastic calculus, simulation and partial differential equations (PDE's).
It has important applications in mathematical finance and stochastic differential equations.

### Stochastic calculus

stochastic analysisstochastic integralstochastic integration
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 corresponding value of derivatives of the stock (see: Valuation of options; Financial modeling; Asset pricing).
Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates.

### Paul Wilmott

Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by Paul Wilmott, and by Nassim Nicholas Taleb, in his book The Black Swan.
Paul Wilmott (born 8 November 1959) is an English researcher, consultant and lecturer in quantitative finance.

### Nassim Nicholas Taleb

Nassim TalebIncertoNicholas Taleb
Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by Paul Wilmott, and by Nassim Nicholas Taleb, in his book The Black Swan.
He has also been a practitioner of mathematical finance, a hedge fund manager, and a derivatives trader, and is currently listed as a scientific adviser at Universa Investments.

### Geometric Brownian motion

The theory remained dormant until Fischer Black and Myron Scholes, along with fundamental contributions by Robert C. Merton, applied the second most influential process, the geometric Brownian motion, to option pricing.
It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.

### Financial Modelers' Manifesto

Wilmott and Emanuel Derman published the Financial Modelers' Manifesto in January 2009 which addresses some of the most serious concerns.
The Financial Modelers' Manifesto was a proposal for more responsibility in risk management and quantitative finance written by financial engineers Emanuel Derman and Paul Wilmott.

### Heat equation

heat diffusionheatanalytic theory of heat
In financial mathematics it is used to solve the Black–Scholes partial differential equation.

### Girsanov theorem

Girsanov's theoremchange of measureGirsanov
The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure, which describes the probability that an underlying instrument (such as a share price or interest rate) will take a particular value or values, to the risk-neutral measure which is a very useful tool for pricing derivatives on the underlying instrument.

### Risk-neutral measure

martingale measureequivalent martingale measurerisk neutrality
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure.

### Derivative (finance)

derivativesderivativefinancial derivatives
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 corresponding value of derivatives of the stock (see: Valuation of options; Financial modeling; Asset pricing).
Arbitrage-free pricing is a central topic of financial mathematics.

### Itô's lemma

Itō's lemmaIto's lemmaItō's formula
The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values.

### Wiener process

Brownian motionWiener measurestandard Brownian motion
It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.

### Mathematical model

mathematical modelingmodelmathematical models
Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input.

### Stochastic volatility

random volatilitystochasticallyVolatility
They are used in the field of mathematical finance to evaluate derivative securities, such as options.

### Financial market

financial marketsmarketmarkets
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.

### Put–call parity

put-call parityPut call parityarbitrage bounds on option prices
In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short put option is equivalent to (and hence has the same value as) a single forward contract at this strike price and expiry.

### Fischer Black

BlackBlack, FischerFischer Black Prize
The theory remained dormant until Fischer Black and Myron Scholes, along with fundamental contributions by Robert C. Merton, applied the second most influential process, the geometric Brownian motion, to option pricing. But mathematical finance emerged as a discipline in the 1970s, following the work of Fischer Black, Myron Scholes and Robert Merton on option pricing theory.