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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] |
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] |
Derivatives (finance)
The derivative of a function is the rate of change of the function's output relative to its input value. Derivative may also refer to: In mathematics and economics * Brzozowski derivative in the theory of formal languages *Covariant derivative, a way of specifying a derivative along tangent vectors of a manifold with a connection. * Exterior derivative, an extension of the concept of the differential of a function to differential forms of higher degree. *Formal derivative, an operation on elements of a polynomial ring which mimics the form of the derivative from calculus * Fréchet derivative, a derivative defined on normed spaces. * Gateaux derivative, a generalization of the concept of directional derivative in differential calculus. * Lie derivative, the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. * Radon–Nikodym derivative in measure theory * Derivative (set theory), a concept app ... [...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] |
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, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volatility (finance)
In finance, volatility (usually denoted by "sigma, σ") is the Variability (statistics), 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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Futures Markets
''Journal of Futures Markets'' is a monthly peer-reviewed academic journal covers developments in financial futures and derivatives. The editor-in-chief is Bart Frijns. The journal covers subjects including: futures, derivatives, risk management and control, financial engineering, new financial instruments, hedging strategies, analysis of trading systems, legal, accounting, and regulatory issues, and portfolio optimization. It is published by Wiley-Blackwell. According to the ''Journal Citation Reports'', its 2022 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... is 1.9. External links * {{Official website, 1=http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1096-9934/homepage/ProductInformation.html Wiley-Blackwell academic journals Academic journals esta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milstein Method
In mathematics, the Milstein method is a technique for the approximate numerical analysis, numerical solution of a stochastic differential equation. It is named after Grigori Milstein who first published it in 1974. Description Consider the autonomous equation, autonomous Itō calculus, Itō stochastic differential equation: \mathrm X_t = a(X_t) \, \mathrm t + b(X_t) \, \mathrm W_t with initial condition X_ = x_, where W_ denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time [0,T]. Then the Milstein approximation to the true solution X is the Markov chain Y defined as follows: * Partition the interval [0,T] into N equal subintervals of width \Delta t>0: 0 = \tau_0 < \tau_1 < \dots < \tau_N = T\text\tau_n:=n\Delta t\text\Delta t = \frac * Set * Recursively define for by: |
Euler–Maruyama Method
In Itô calculus, the Euler–Maruyama method (also simply called the Euler method) is a method for the approximate numerical analysis, numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations named after Leonhard Euler and Gisiro Maruyama. The same generalization cannot be done for any arbitrary deterministic method. Definition Consider the stochastic differential equation (see Itô calculus) :\mathrm X_t = a(X_t, t) \, \mathrm t + b(X_t, t) \, \mathrm W_t, with initial condition ''X''0 = ''x''0, where ''W''''t'' denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time [0, ''T'']. Then the Euler–Maruyama approximation to the true solution ''X'' is the Markov chain ''Y'' defined as follows: * Partition the interval [0, ''T''] into ''N'' equal subintervals of width \Delta t>0: ::0 = \tau_ < \tau_ < \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Brownian Motion
A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. 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. Technical definition: the SDE A stochastic process ''S''''t'' is said to follow a GBM if it satisfies the following stochastic differential equation (SDE): : dS_t = \mu S_t\,dt + \sigma S_t\,dW_t where W_t is a Wiener process or Brownian motion, and \mu ('the percentage drift') and \sigma ('the percentage volatility') are constants. The former parameter is used to model deterministic trends, while the latter parameter models unpredictable events occurring during the motion. Solving the SDE For an arbitrary initial value ''S' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Journal Of Theoretical And Applied Finance
The ''International Journal of Theoretical and Applied Finance'' (IJTAF) was founded in 1998 and is published by World Scientific. The journal spans a wide range of topics focussing on the use of quantitative tools in finance, including articles on development and implementation of mathematical models, their industrial usage, and application of modern stochastic methods. The Editor-in-Chief of IJTAF is Matheus R. Grasselli of McMaster University, Ontario. Before Grasselli assumed this role the Editor-in-Chief was Lane P. Hughston of Goldsmiths, University of London who served in that capacity from 2007 until 2022. Grasselli and Hughston served jointly as co-Editors over the year 2022. The disciplines and topics covered by IJTAF include: mathematical finance; financial engineering; applications of modern probability theory and stochastic analysis in finance; models for interest rate, equity, foreign exchange, commodity, credit and inflation-linked products; portfolio theory and l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative Interest Rates
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, the compounding frequency, and the length of time over which it is lent, deposited, or borrowed. The annual interest rate is the rate over a period of one year. Other interest rates apply over different periods, such as a month or a day, but they are usually annualized. The interest rate has been characterized as "an index of the preference . . . for a dollar of present ncomeover a dollar of future income". The borrower wants, or needs, to have money sooner, and is willing to pay a fee—the interest rate—for that privilege. Influencing factors Interest rates vary according to: * the government's directives to the central bank to accomplish the government's goals * the currency of the principal sum lent or borrowed * the term to mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
