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Heston Model
In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process. Mathematical formulation The Heston model assumes that ''St'', the price of the asset, is determined by a stochastic process, : dS_t = \mu S_t\,dt + \sqrt S_t\,dW^S_t, where the volatility \sqrt is given by a Feller square-root or CIR process, : d\nu_t = \kappa(\theta - \nu_t)\,dt + \xi \sqrt\,dW^_t, and W^S_t, W^_t are Wiener processes (i.e., continuous random walks) with correlation ρ. The value \nu_t, being the square of the volatility, is called the instantaneous variance. The model has five parameters: * \nu_0, the initial variance. * \theta, the long variance, or long-run average variance of the price; as ''t'' tends to infinity, the expected value of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steven L
Stephen or Steven is an English given name, first name. It is particularly significant to Christianity, Christians, as it belonged to Saint Stephen ( ), an early disciple and deacon who, according to the Book of Acts, was stoned to death; he is widely regarded as the first martyr (or "protomartyr") of the Christian Church. The name, in both the forms Stephen and Steven, is often shortened to Steve or Stevie (given name), Stevie. In English, the female version of the name is Stephanie. Many surnames are derived from the first name, including Template:Stephen-surname, Stephens, Stevens, Stephenson, and Stevenson, all of which mean "Stephen's (son)". In modern times the name has sometimes been given with intentionally non-standard spelling, such as Stevan or Stevon. A common variant of the name used in English is Stephan (given name), Stephan ( ); related names that have found some currency or significance in English include Stefan (given name), Stefan (pronounced or in English) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Squares
The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The method is widely used in areas such as regression analysis, curve fitting and data modeling. The least squares method can be categorized into linear and nonlinear forms, depending on the relationship between the model parameters and the observed data. The method was first proposed by Adrien-Marie Legendre in 1805 and further developed by Carl Friedrich Gauss. History Founding The method of least squares grew out of the fields of astronomy and geodesy, as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the Age of Discovery. The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilmott (magazine)
''Wilmott Magazine'' is a mathematical finance and risk management magazine, combining technical articles with humor pieces. Each copy of ''Wilmott'' is 11 inches square, runs about 100 pages, and is printed on glossy paper. The magazine has the highest subscription price of any magazine. ''Esquire''. 16 July 2007. Retrieved 4 March 2017. Content and contributors ''Wilmott'' has a section with technical articles on , but includes quantitative financial comic strips, and lighter articles. ''Wilmott'' magazine's regular contributors include[...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] |
Martingale (probability Theory)
In probability theory, a martingale is a stochastic process in which the expected value of the next observation, given all prior observations, is equal to the most recent value. In other words, the conditional expectation of the next value, given the past, is equal to the present value. Martingales are used to model fair games, where future expected winnings are equal to the current amount regardless of past outcomes. History Originally, ''martingale (betting system), martingale'' referred to a class of betting strategy, betting strategies that was popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, their pr ... [...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] |
Automatic Differentiation
In mathematics and computer algebra, automatic differentiation (auto-differentiation, autodiff, or AD), also called algorithmic differentiation, computational differentiation, and differentiation arithmetic Hend Dawood and Nefertiti Megahed (2023). Automatic differentiation of uncertainties: an interval computational differentiation for first and higher derivatives with implementation. PeerJ Computer Science 9:e1301 https://doi.org/10.7717/peerj-cs.1301. Hend Dawood and Nefertiti Megahed (2019). A Consistent and Categorical Axiomatization of Differentiation Arithmetic Applicable to First and Higher Order Derivatives. Punjab University Journal of Mathematics. 51(11). pp. 77-100. doi: 10.5281/zenodo.3479546. http://doi.org/10.5281/zenodo.3479546. is a set of techniques to evaluate the partial derivative of a function specified by a computer program. Automatic differentiation is a subtle and central tool to automatize the simultaneous computation of the numerical values of arbitrarily ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filon Quadrature
In numerical analysis, Filon quadrature or Filon's method is a technique for numerical integration of oscillatory integrals. It is named after English mathematician Louis Napoleon George Filon, who first described the method in 1934. Description The method is applied to oscillatory definite integrals in the form: :\int_a^b f(x) g(x) dx where f(x) is a relatively slowly-varying function and g(x) is either sine or cosine or a complex exponential that causes the rapid oscillation of the integrand, particularly for high frequencies. In Filon quadrature, the f(x) is divided into 2N subintervals of length h, which are then interpolated by parabolas. Since each subinterval is now converted into a Fourier integral of quadratic polynomials, these can be evaluated in closed-form by integration by parts. For the case of g(x)=\cos(kx), the integration formula is given as: :\int_a^b f(x) \cos(kx) dx \approx h ( \alpha \left f(b) \sin(kb)-f(a) \sin(ka)\right+ \beta C_ + \gamma C_ ) where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fabien Le Floc'h
Fabien is both a French given masculine name and a French surname. Notable people with the name include: People with the given name Fabien: * Fabien Audard (born 1978), French professional football (soccer) player * Fabien Barthez (born 1971), retired French football goalkeeper * Fabien Boudarène (born 1978), French footballer * Fabien Camus (born 1985), French football player * Fabien Chéreau (born 1980), French computer programmer * Fabien Cool (born 1972), former French football goalkeeper * Fabien Cordeau (1923-2007), politician in Quebec, Canada * Fabien Cousteau (born 1967), French aquatic filmmaker * Fabien Delrue (born 2000), French badminton player * Fabien Foret (born 1973), professional motorcycle racer * Fabien Frankel (born 1994), British actor * Fabien Galthié (born 1969), French rugby union coach and former player * Fabien Gilot (born 1984), French Olympic and world champion swimmer * Fabien Giroix (born 1960), French racing driver * Fabien Laurenti (born 1983), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volatility Smile
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices (and thus implied volatilities) than what is suggested by standard option pricing models. These options are said to be either deep in-the-money or out-of-the-money. Graphing implied volatilities against strike prices for a given expiry produces a skewed "smile" instead of the expected flat surface. The pattern differs across various markets. Equity options traded in American markets did not show a volatility smile before the Crash of 1987 but began showing one afterwards. It is believed that investor reassessments of the probabilities of fat-tail have led to higher prices for out-of-the-money options. This anomaly implies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barrier Options
A barrier option is an option whose payoff is conditional upon the underlying asset's price breaching a barrier level during the option's lifetime. Types Barrier options are path-dependent exotics that are similar in some ways to ordinary options. You can call or put in American, Bermudan, or European exercise style. But they become activated (or extinguished) only if the underlying breaches a predetermined level (the barrier). "In" options only become active in the event that a predetermined knock-in barrier price is breached: # If the barrier price is far from being breached, the knock-in option will be worth slightly more than zero. # If the barrier price is close to being breached, the knock-in option will be worth slightly less than the corresponding vanilla option. # If the barrier price has been breached, the knock-in option will trade at the exact same value as the corresponding vanilla option. "Out" options start their lives active and become null and void in the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
