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Local Volatility
A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility as a function of both the current asset level S_t and of time t . As such, it is a generalisation of the Black–Scholes model, where the volatility is a constant (i.e. a trivial function of S_t and t ). Formulation In mathematical finance, the asset ''S''''t'' that underlies a financial derivative is typically assumed to follow a stochastic differential equation of the form : dS_t = (r_t-d_t) S_t\,dt + \sigma_t S_t\,dW_t , under the risk neutral measure, where r_t is the instantaneous risk free rate, giving an average local direction to the dynamics, and W_t is a Wiener process, representing the inflow of randomness into the dynamics. The amplitude of this randomness is measured by the instant volatility \sigma_t. In the simplest model i.e. the Black–Scholes model, \sigma_t is assumed to be constant; in reality, the realised volatility of an un ... [...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 fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iraj Kani
Iraj ( fa, ایرج - ʾīraj; Pahlavi: ērič; from Avestan: 𐬀𐬌𐬭𐬌𐬌𐬀 airiia, literally "Aryan") is the seventh Shah of the Pishdadian dynasty, depicted in the ''Shahnameh''. Based on Iranian mythology, he is the youngest son of Fereydun. In the Avestan legends, Pahlavi literature, Sasanian-based Persian sources, some Arabic sources, and particularly in ''Shahnameh'', he is considered the name-giver of the Iranian nation, the ancestor of their royal houses, and a paragon of those slain in defense of just causes. File:Firdawsi - The Murder of Iraj - Walters W60230B - Full Page.jpg, A page from ''Shahnameh'', in Walters Art Museum, showing the murder of Iraj by his brothers File:Salm and Tur murder Iraj.jpg, Murder scene of Iraj by his brothers, Salm and Tur, from National Library of Russia, St Petersburg – The calligraphy in the margins are Nastaliq ''Nastaliq'' (; fa, , ), also romanized as ''Nastaʿlīq'', is one of the main calligraphic hands used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cliquet Option
A cliquet option or ratchet option is an exotic option consisting of a series of consecutive forward start options. The first is active immediately. The second becomes active when the first expires, etc. Each option is struck at-the-money when it becomes active. A cliquet is, therefore, a series of at-the-money options but where the total premium is determined in advance. A cliquet can be thought of as a series of "pre-purchased" at-the-money options. The payout on each option can either be paid at the final maturity, or at the end of each reset period.http://docs.fincad.com/support/developerFunc/mathref/cliquet.htm FiNCAD - Cliquet options] Example * A three-year cliquet with reset dates each year would have three payoffs. The first would pay off at the end of the first year and has the same payoff as a normal ATM option. * The second year's payoff has the same payoff as a one-year option, but with the strike price In finance, the strike price (or exercise price) of an op ... [...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 an applied mathematician and Chair in Mathematical Finance at Imperial College London. He is known for research in filtering theory and mathematical finance. Main results Brigo started his work with the development, with Bernard Hanzon and Francois Le Gland (1998), of the projection filters, a family of approximate nonlinear filters based on the differential geometry approach to statistics, also related to information geometry. With Fabio Mercurio (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. With Aurelien Alfonsi (2005), Brigo introduced new families of multivariate distributions in statistics through the periodic copula function concept. Since 2002, Brigo contributed to credit derivatives modeling and counterparty risk valuation, showing with Pallavicini and Torresetti (2007) h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volatility Surface
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 def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martingale Pricing
Martingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. options, futures, interest rate derivatives, credit derivatives, etc. In contrast to the PDE approach to pricing, martingale pricing formulae are in the form of expectations which can be efficiently solved numerically using a Monte Carlo approach. As such, Martingale pricing is preferred when valuing high-dimensional contracts such as a basket of options. On the other hand, valuing American-style contracts is troublesome and requires discretizing the problem (making it like a Bermudan option) and only in 2001 F. A. Longstaff and E. S. Schwartz developed a practical Monte Carlo method for pricing American options. Measure theory representation Suppose the state of the market can be represented by the filtered probability space,(\Omega,(\math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fokker–Planck Equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck, who described it in 1914 and 1917. It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered it in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski), and in this context it is equivalent to the convection–diffusion equation. The case with zero diffusion is the continuity equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion. The first consistent microscopic derivation of the Fokker–Planck equation in the sing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decisions In Economics And Finance
Decision may refer to: Law and politics *Judgment (law), as the outcome of a legal case *Landmark decision, the outcome of a case that sets a legal precedent * ''Per curiam'' decision, by a court with multiple judges Books * ''Decision'' (novel), a 1983 political novel by Allen Drury * ''The Decision'' (novel), a 1998 book in the ''Animorphs'' series Sports *Decision (baseball), a statistical credit earned by a baseball pitcher * Decisions in combat sports *Decisions (professional wrestling), by which a wrestler scores a point against his opponent Film and TV * ''Decision'' (TV series), an American anthology TV series * ''The Decision'' (play), by the 20th-century German dramatist Bertolt Brecht * ''The Decision'' (TV special), in which NBA player LeBron James announced that he would switch teams * "The Decision" (song), by English indie rock band Young Knives Music Albums * ''Decisions'' (George Adams and Don Pullen album), 1984 * ''Decisions'' (The Winans album), 1987 Songs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implied Trinomial Tree
In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times (any time) before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par. The method is also used for valuing certain exotic options, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, though methods now exist for solving this problem. Equity and commodity derivatives In general the approach is to div ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neil Chriss
Neil A. Chriss is a mathematician, academic, hedge fund manager, philanthropist and a founding board member of the charity organization "Math for America" which seeks to improve math education in the United States. Chriss also serves on the board of trustees of the Institute for Advanced Study. Early career Chriss learned programming at the age of 11. He developed a videogame called D' Fuse and sold it to Tymac when he was a sophomore in high school. The game quickly faded when the Commodore 64 with 64K of memory and much better graphics appeared. Chriss went to the University of Chicago, where he majored in mathematics. Following his junior year in college, he worked at Fermilab with Myron Campbell and Bruce Denby; he developed a neural network to find b-quark jets. He then earned his master's degree in applied mathematics at Caltech. Chriss studied pure mathematics at the University of Chicago, working in the Langlands Program. He received a Ph.D. in 1993, with the thesis ''A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implied Binomial Tree
In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times (any time) before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par. The method is also used for valuing certain exotic options, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, though methods now exist for solving this problem. Equity and commodity derivatives In general the approach is to div ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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