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Constant Elasticity Of Variance Model
In mathematical finance, the CEV or constant elasticity of variance model is a stochastic volatility model that attempts to capture stochastic volatility and the leverage effect. The model is widely used by practitioners in the financial industry, especially for modelling equities and commodities. It was developed by John Cox in 1975. Dynamic The CEV model describes a process which evolves according to the following stochastic differential equation: :\mathrmS_t = \mu S_t \mathrmt + \sigma S_t ^ \gamma \mathrmW_t in which ''S'' is the spot price, ''t'' is time, and ''μ'' is a parameter characterising the drift, ''σ'' and ''γ'' are other parameters, and ''W'' is a Brownian motion. And so we have :\sigma(S_t, t)=\sigma S_t^ The constant parameters \sigma,\;\gamma satisfy the conditions \sigma\geq 0,\;\gamma\geq 0. The parameter \gamma controls the relationship between volatility and price, and is the central feature of the model. When \gamma 1,Geman, H, and Shih, YF. 2009. ... [...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 of financial markets. 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 by 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 mathematical fina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Volatility
In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. 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 security, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others. Stochastic volatility models are one approach to resolve a shortcoming of the Black–Scholes model. In particular, models based on Black-Scholes assume that the underlying volatility is constant over the life of the derivative, and unaffected by the changes in the price level of the underlying security. However, these models cannot explain long-observed features of the implied volatility surface such as volatility smile and skew, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leverage Effect
In finance, leverage (or gearing in the United Kingdom and Australia) is any technique involving borrowing funds to buy things, hoping that future profits will be many times more than the cost of borrowing. This technique is named after a lever in physics, which amplifies a small input force into a greater output force, because successful leverage amplifies the comparatively small amount of money needed for borrowing into large amounts of profit. However, the technique also involves the high risk of not being able to pay back a large loan. Normally, a lender will set a limit on how much risk it is prepared to take and will set a limit on how much leverage it will permit, and would require the acquired asset to be provided as collateral security for the loan. Leveraging enables gains to be multiplied.Brigham, Eugene F., ''Fundamentals of Financial Management'' (1995). On the other hand, losses are also multiplied, and there is a risk that leveraging will result in a loss if financi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equities
In finance, stock (also capital stock) consists of all the shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which ownership of a company is divided, or these shares considered together" "When a company issues shares or stocks ''especially AmE'', it makes them available for people to buy for the first time." (Especially in American English, the word "stocks" is also used to refer to shares.) A single share of the stock means fractional ownership of the corporation in proportion to the total number of shares. This typically entitles the shareholder (stockholder) to that fraction of the company's earnings, proceeds from liquidation of assets (after discharge of all senior claims such as secured and unsecured debt), or voting power, often dividing these up in proportion to the amount of money each stockholder has invested. Not all stock is necessarily equal, as certain classe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commodities
In economics, a commodity is an economic good, usually a resource, that has full or substantial fungibility: that is, the market treats instances of the good as equivalent or nearly so with no regard to who produced them. The price of a commodity good is typically determined as a function of its market as a whole: well-established physical commodities have actively traded spot and derivative markets. The wide availability of commodities typically leads to smaller profit margins and diminishes the importance of factors (such as brand name) other than price. Most commodities are raw materials, basic resources, agricultural, or mining products, such as iron ore, sugar, or grains like rice and wheat. Commodities can also be mass-produced unspecialized products such as chemicals and computer memory. Popular commodities include crude oil, corn, and gold. Other definitions of commodity include something useful or valued and an alternative term for an economic good or service avail ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Carrington Cox
John Carrington Cox is the Nomura Professor of Finance at the MIT Sloan School of Management. He is one of the world's leading experts on options theory and one of the inventors of the Cox–Ross–Rubinstein model for option pricing, as well as of the Cox–Ingersoll–Ross model for interest rate dynamics. He was named Financial Engineer of the Year by the International Association of Financial Engineers The International Association for Quantitative Finance (IAQF), formerly the International Association of Financial Engineers (IAFE), is a non-profit professional society dedicated to fostering the fields of quantitative finance and financial engin ... in 1998. References External links Webpage at MIT 1943 births Living people Academics from Houston Financial economists MIT Sloan School of Management faculty Fellows of the Econometric Society Economists from Texas 21st-century American economists {{US-economist-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Differential Equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. Random differential equations are conjugate to stochastic differential equations. Background Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Smoluchowski. These early examples were linear stochastic differential equations, also called 'Langevin' equations after French physicist Langevin, describing the motion of a harmonic oscillator subject to a random force. The mathematical theory of stochasti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis Bachelier
Louis Jean-Baptiste Alphonse Bachelier (; 11 March 1870 – 28 April 1946) 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, as part of his doctoral thesis ''The Theory of Speculation'' (''Théorie de la spéculation'', defended in 1900). Bachelier's doctoral thesis, which introduced the first mathematical model of Brownian motion and its use for valuing stock options, was the first paper to use advanced mathematics in the study of finance. His Bachelier model has been influential in the development of other widely used models, including the Black-Scholes model. Thus, Bachelier is considered as the forefather of mathematical finance and a pioneer in the study of stochastic processes. Early years Bachelier was born in Le Havre. His father was a wine merchant and amateur scientist, and the vice-consul of Venezuela at Le Havre. His mother was the daughter of an impor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volatility (finance)
In finance, volatility (usually denoted by ''σ'') is the degree of variation of a trading price series over time, usually measured by the standard deviation of logarithmic returns. Historic volatility measures a time series of past market prices. Implied volatility looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option). Volatility terminology Volatility as described here refers to the actual volatility, more specifically: * actual current volatility of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price. * actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past **near synonymous is realized volatility, the square root of the realized variance, in turn calculated using the sum of squ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Volatility
In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. 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 security, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others. Stochastic volatility models are one approach to resolve a shortcoming of the Black–Scholes model. In particular, models based on Black-Scholes assume that the underlying volatility is constant over the life of the derivative, and unaffected by the changes in the price level of the underlying security. However, these models cannot explain long-observed features of the implied volatility surface such as volatility smile and skew, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especially in the interest rate derivative markets. It was developed by Patrick S. Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward. Dynamics The SABR model describes a single forward F, such as a LIBOR forward rate, a forward swap rate, or a forward stock price. This is one of the standards in market used by market participants to quote volatilities. The volatility of the forward F is described by a parameter \sigma. SABR is a dynamic model in which both F and \sigma are represented by stochastic state variables whose time evolution is given by the following system of stochastic differential equations: :dF_t=\sigma_t \left(F_t\right)^\beta\, dW_t, :d\sigma_t=\a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
