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Numéraire
The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradables are expressed. In a monetary economy, one of the functions of money is to act as the numéraire, i.e. to serve as a unit of account and therefore provide a common benchmark relative to which the value of various goods and services can be measured against. Using a numeraire, whether monetary or some consumable good, facilitates value comparisons when only the relative prices are relevant, as in general equilibrium theory. When economic analysis refers to a particular good as the numéraire, one says that all other prices are normalized by the price of that good. For example, if a unit of good ''g'' has twice the market value of a unit of the numeraire, then the (relative) price of ''g'' is 2. Since the value of one unit of the numeraire relative to one unit of itself is 1, the pric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Equilibrium Theory
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall general equilibrium. General equilibrium theory contrasts with the theory of ''partial'' equilibrium, which analyzes a specific part of an economy while its other factors are held constant. General equilibrium theory both studies economies using the model of equilibrium pricing and seeks to determine in which circumstances the assumptions of general equilibrium will hold. The theory dates to the 1870s, particularly the work of French economist Léon Walras in his pioneering 1874 work ''Elements of Pure Economics''. The theory reached its modern form with the work of Lionel W. McKenzie (Walrasian theory), Kenneth Arrow and Gérard Debreu (Hicksian theory) in the 1950s. Overview Broadly speaking, general equilibrium tries to give ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk-neutral Measure
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. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market, a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free. A risk-neutral measure is a probability measure The easiest way to remember what the risk-neutral measure is, or to explain it to a probability generalist who might not know much about finance, is to realize that it is: # The probability measure of a transformed random variable. Typically this transformation is the utility function of the payoff. The risk-neutral measure would be the measure co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Economics
Mathematical economics is the application of Mathematics, mathematical methods to represent theories and analyze problems in economics. Often, these Applied mathematics#Economics, applied methods are beyond simple geometry, and may include differential and integral calculus, Recurrence relation, difference and differential equations, Matrix (mathematics), matrix algebra, mathematical programming, or other Computational economics, computational methods.TOC. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity. Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vasicek Model
In Mathematical finance, finance, the Vasicek model is a mathematical model describing the evolution of interest rates. It is a type of one-factor short-rate model as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives, and has also been adapted for credit markets. It was introduced in 1977 by Oldřich Vašíček, and can be also seen as a stochastic investment model. Details The model specifies that the force of interest, instantaneous interest rate follows the stochastic differential equation: :dr_t= a(b-r_t)\, dt + \sigma \, dW_t where ''Wt'' is a Wiener process under the risk neutral framework modelling the random market risk factor, in that it models the continuous inflow of randomness into the system. The standard deviation parameter, \sigma, determines the Volatility (finance), volatility of the interest rate and in a way characterizes the amplitude of the instantaneous random ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stock Market
A stock market, equity market, or share market is the aggregation of buyers and sellers of stocks (also called shares), which represent ownership claims on businesses; these may include ''securities'' listed on a public stock exchange as well as stock that is only traded privately, such as shares of private companies that are sold to investors through equity crowdfunding platforms. Investments are usually made with an investment strategy in mind. Size of the market The total market capitalization of all publicly traded stocks worldwide rose from US$2.5 trillion in 1980 to US$111 trillion by the end of 2023. , there are 60 stock exchanges in the world. Of these, there are 16 exchanges with a market capitalization of $1 trillion or more, and they account for 87% of global market capitalization. Apart from the Australian Securities Exchange, these 16 exchanges are all in North America, Europe, or Asia. By country, the largest stock markets as of January 2022 are in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often with the help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios. French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Of Account
In economics, unit of account is one of the functions of money. A unit of account is a standard numerical monetary unit of measurement of the market value of goods, services, and other transactions. Also known as a "measure" or "standard" of relative worth and deferred payment, a unit of account is a necessary prerequisite for the formulation of commercial agreements that involve debt. Money acts as a standard measure and a common denomination of trade. It is thus a basis for quoting and bargaining of prices. It is necessary for developing efficient accounting systems. Economics Unit of account in economics allows a somewhat meaningful interpretation of prices, costs, and profits, so that an entity can monitor its own performance. It allows shareholders to make sense of its past performance and have an idea of its future profitability. The use of money, as a relatively stable unit of measure, can tend to drive market economies toward efficiency. Historically, prices were of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Price Index
A price index (''plural'': "price indices" or "price indexes") is a normalized average (typically a weighted average) of price relatives for a given class of goods or services in a specific region over a defined time period. It is a statistic designed to measure how these price relatives, as a whole, differ between time periods or geographical locations, often expressed relative to a base period set at 100. Price indices serve multiple purposes. Broad indices, like the Consumer price index, reflect the economy’s general price level or cost of living, while narrower ones, such as the Producer price index, assist producers with pricing and business planning. They can also guide investment decisions by tracking price trends. Types of price indices Some widely recognized price indices include: * Consumer price index – Measures retail price changes for consumer goods and services. * Producer price index – Tracks wholesale price changes for producers. * Wholesal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forward Measure
In finance, a ''T''-forward measure is a pricing measure equivalent to a risk-neutral measure, but rather than using the money market as numeraire, it uses a bond with maturity ''T''. The use of the forward measure was pioneered by Farshid Jamshidian (1987), and later used as a means of calculating the price of options on bonds. Mathematical definition Let : B(T) = \exp\left(\int_0^T r(u)\, du\right) be the bank account or money market account numeraire and : D(T) = 1/B(T) = \exp\left(-\int_0^T r(u)\, du\right) be the discount factor in the market at time 0 for maturity ''T''. If Q_* is the risk neutral measure, then the forward measure Q_T is defined via the Radon–Nikodym derivative given by :\frac = \frac = \frac. Note that this implies that the forward measure and the risk neutral measure coincide when interest rates are deterministic. Also, this is a particular form of the change of numeraire formula by changing the numeraire from the money market or bank accou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fabio Mercurio
Fabio Mercurio (born 26 September 1966) is an Italian mathematician, internationally known for a number of results in mathematical finance. Main results Mercurio worked during his Ph.D. on incomplete markets theory using dynamic mean-variance hedging techniques. With Damiano Brigo (2002–2003), he has shown how to construct stochastic differential equations consistent with mixture models, applying this to volatility smile modeling in the context of local volatility models. He is also one of the main authors in inflation modeling. Mercurio has also authored several publications in top journals and co-authored the book ''Interest rate models: theory and practice'' for Springer-Verlag, that quickly became an international reference for stochastic dynamic interest rate modeling. He is the recipient of the 2020 Risk quant-of-the-year award [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Damiano Brigo
Damiano Brigo (born Venice, Italy 1966) is a mathematician known for research in mathematical finance, filtering theory, stochastic analysis with differential geometry, probability theory and statistics, authoring more than 130 research publications and three monographs.Publications and citations page in From 2012 he serves as full professor with a chair in at the Department of Mathematics of |
