Mathematical finance comprises the branches of applied mathematics Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory ; and applied probability. These areas of mathematics were intimately tied to the development of Newtonian physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the concerned with the financial markets In economics, a financial market is a mechanism that allows people to buy and sell financial securities (such as stocks and bonds), commodities (such as precious metals or agricultural goods), and other fungible items of value at low transaction costs and at prices that reflect the efficient-market hypothesis.

The subject has a close relationship with the discipline of 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". It is additionally characterised by its "concentration on monetary activities", in which "money of one type or another is likely to appear on both, which is concerned with much of the underlying theory. Generally, mathematical finance will derive, and extend, the mathematical A mathematical model uses mathematical language to describe a system. The process of developing a mathematical model is termed mathematical modelling . Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines, but also in the social sciences (such as economics, or numerical Numerical analysis is the study of algorithms that use numerical approximation for the problems of continuous mathematics (as distinguished from discrete mathematics) 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 In the United States, a share must be priced at $1 or more to be covered by NASDAQ. If the share price falls below that level the stock is "delisted", and becomes an OTC . A stock must have a price of $1 or more for 10 consecutive trading days during each month to remain listed, a financial mathematician may take the share price as a given, and attempt to use 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. It is used to model systems that behave randomly to obtain the fair value of derivatives A derivative, in non-financial-expert terms, is an agreement or contract that is not based on a real, or true, exchange, i.e.: There is nothing tangible like money, or a product, that is being exchanged. For example, a person goes to the grocery store, exchanges a currency for a commodity (say, an apple). The exchange is complete, both parties of the stock The stock or capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors. Stock is distinct from the property and the assets of a business which may fluctuate in (see: 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. There are many pricing models in use, although all essentially incorporate the concepts of rational pricing, Moneyness, Option time value and Put-call parity).

In terms of practice, mathematical finance also overlaps heavily with the field of computational finance Computational finance or financial engineering is a cross-disciplinary field which relies on computational intelligence, mathematical finance, numerical methods and computer simulations to make trading, hedging and investment decisions, as well as facilitating the risk management of those decisions. Utilising various methods, practitioners of (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 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. In the investment industry, people who perform quantitative analysis are frequently called quants. See List of quantitative analysts).

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 A masters degree in quantitative or mathematical finance concerns the application of mathematical methods to the solution of problems in financial economics . There are several "like-titled degrees" which may further focus on financial engineering, financial risk management, computational finance and/or mathematical finance. In general,.

Contents

History

The history of mathematical finance starts with The Theory of Speculation (published 1900) by Louis Bachelier Louis Jean-Baptiste Alphonse Bachelier was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, which was part of his PhD thesis The Theory of Speculation, (published 1900), which discussed the use of Brownian motion Brownian motion or pedesis is the seemingly random movement of particles suspended in a fluid (i.e. a liquid such as water or air) or the mathematical model used to describe such random movements, often called a particle theory 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 Harry Max Markowitz is an American economist and a recipient of the John von Neumann Theory Prize and the Nobel Memorial Prize in Economic Sciences 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 In statistics, linear regression includes any approach to modeling the relationship between a scalar variable y and one or more variables denoted X, such that the model depends linearly on the unknown parameters to be estimated from the data. Such a model is called a “linear model.” Most commonly, linear regression refers to a model in which strategy to understand and quantify the risk Risk concerns the deviation of one or more results of one or more future events from their expected value. Technically, the value of those results may be positive or negative. However, general usage tends to focus only on potential harm that may arise from a future event, which may accrue either from incurring a cost or by failing to attain some (i.e. variance) and return (i.e. mean) of an entire portfolio of stocks Stocks are devices used in the medieval times for torture, public humiliation, and corporal punishment. The stocks partially immobilized its victims and they were often exposed in a public place such as the site of a market to the scorn of those who passed by. Since the purpose was to punish offenders against the standards of conduct of the time and bonds In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest and/or to repay the principal at a later date, termed maturity. A bond is a formal contract to repay borrowed money with interest at fixed intervals, 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 William Forsyth Sharpe is the STANCO 25 Professor of Finance, Emeritus at Stanford University's Graduate School of Business and the winner of the 1990 Nobel Memorial Prize in Economic Sciences developed the mathematics of determining the correlation between each stock and the market A market is any one of a variety of different systems, institutions, procedures, social relations and infrastructures whereby persons trade, and goods and services are exchanged, forming part of the economy. It is an arrangement that allows buyers and sellers to exchange things. Competition is essential in markets, and separates market from trade. For their pioneering work, Markowitz and Sharpe, along with Merton Miller, shared the 1990 Nobel Memorial Prize in Economic Sciences The Nobel Memorial Prize in Economic Sciences, commonly referred to as the Nobel Prize in Economics , is an award for outstanding contributions to the science of economics and is generally considered one of the most prestigious awards for that science. The official name is the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel (, 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 The Brownian motion models for financial markets are based on the work of Robert C. Merton and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William Sharpe, and are concerned with defining the concepts of financial assets and markets, portfolios, gains and wealth in terms of continuous-time stochastic, and the quadratic utility function implicit in mean–variance optimization was replaced by more general increasing, concave utility functions [1].

Main article: Black–Scholes

The next major revolution in mathematical finance came with the work of Fischer Black Fischer Sheffey Black was an American economist, best known as one of the authors of the famous Black–Scholes equation and Myron Scholes Myron Samuel Scholes is one of the authors of the Black–Scholes equation. In 1997 he was awarded the Nobel Memorial Prize in Economic Sciences for a new method to determine the value of derivatives. The model provides the fundamental conceptual framework for valuing options, such as calls or puts, and is referred to as the Black-Scholes model, along with fundamental contributions by Robert C. Merton Robert C. Merton is an American economist, university professor and Nobel laureate in economics , by modeling financial markets with stochastic Stochastic means random. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E. Nelson, any kind of time development (be it deterministic or essentially probabilistic) which is models. For this M. Scholes and R. Merton were awarded the 1997 Nobel Memorial Prize in Economic Sciences. 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

Derivatives pricing

See also

Book:Finance
Books are collections of articles which can be downloaded or ordered in print.

Notes

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

References

External links

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