|
Quasi-Monte Carlo Methods In Finance
High-dimensional integrals in hundreds or thousands of variables occur commonly in finance. These integrals have to be computed numerically to within a threshold \epsilon. If the integral is of dimension d then in the worst case, where one has a guarantee of error at most \epsilon, the computational complexity is typically of order \epsilon^. That is, the problem suffers the curse of dimensionality. In 1977 P. Boyle, University of Waterloo, proposed using Monte Carlo (MC) to evaluate options.Boyle, P. (1977), Options: a Monte Carlo approach, J. Financial Economics, 4, 323-338. Starting in early 1992, J. F. Traub, Columbia University, and a graduate student at the time, S. Paskov, used quasi-Monte Carlo (QMC) to price a Collateralized mortgage obligation with parameters specified by Goldman Sachs. Even though it was believed by the world's leading experts that QMC should not be used for high-dimensional integration, Paskov and Traub found that QMC beat MC by one to three orders of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curse Of Dimensionality
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. The expression was coined by Richard E. Bellman when considering problems in dynamic programming. Dimensionally cursed phenomena occur in domains such as numerical analysis, sampling, combinatorics, machine learning, data mining and databases. The common theme of these problems is that when the dimensionality increases, the volume of the space increases so fast that the available data become sparse. In order to obtain a reliable result, the amount of data needed often grows exponentially with the dimensionality. Also, organizing and searching data often relies on detecting areas where objects form groups with similar properties; in high dimensional data, however, all objects appear to be sparse and dissimilar in many ways, which prevents co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobol Sequence
Sobol sequences (also called LPτ sequences or (''t'', ''s'') sequences in base 2) are an example of quasi-random low-discrepancy sequences. They were first introduced by the Russian mathematician Ilya M. Sobol (Илья Меерович Соболь) in 1967.Sobol,I.M. (1967), "Distribution of points in a cube and approximate evaluation of integrals". ''Zh. Vych. Mat. Mat. Fiz.'' 7: 784–802 (in Russian); ''U.S.S.R Comput. Maths. Math. Phys.'' 7: 86–112 (in English). These sequences use a base of two to form successively finer uniform partitions of the unit interval and then reorder the coordinates in each dimension. Good distributions in the ''s''-dimensional unit hypercube Let ''Is'' = ,1sup>''s'' be the ''s''-dimensional unit hypercube, and ''f'' a real integrable function over ''Is''. The original motivation of Sobol was to construct a sequence ''xn'' in ''Is'' so that : \lim_ \frac \sum_^n f(x_i) = \int_ f and the convergence be as fast as possible. It is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Historical Simulation (finance)
Historical simulation in finance's value at risk (VaR) analysis is a procedure for predicting the value at risk by 'simulating' or constructing the cumulative distribution function (CDF) of assets returns over time. Unlike parametric VaR models, historical simulation does not assume a particular distribution of the asset returns. Also, it is relatively easy to implement. However, there are a couple of shortcomings of historical simulation. Historical simulation applies equal weight to all returns of the whole period; this is inconsistent with the diminishing predictability of data that are further away from the present. Weighted historical simulation Weighted historical simulation applies decreasing weights to returns that are further away from the present, which overcomes the inconsistency of historical simulation with diminishing predictability of data that are further away from the present. However, weighted historical simulation still assumes independent and identically dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Methods In Finance
Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes. This is usually done by help of stochastic asset models. The advantage of Monte Carlo methods over other techniques increases as the dimensions (sources of uncertainty) of the problem increase. Monte Carlo methods were first introduced to finance in 1964 by David B. Hertz through his ''Harvard Business Review'' article, discussing their application in Corporate Finance. In 1977, Phelim Boyle pioneered the use of simulation in derivative valuation in his seminal ''Journal of Financial Economics'' paper. This article discusses typical financial problems in which Monte Carlo methods are used. It also touches on the use of so-called "quasi-random" methods such as the use of Sobo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russel E
Russel is an alternate spelling of Russell. Russel may also refer to: People *Russel Arnold (born 1973), Sri Lankan cricketer *Russel Crouse (1893–1966), American playwright *Russel Farnham (1784–1832), American frontiersman * Russel Honoré (born 1947), American general * Russel Mthembu (born 1947), South African singer * Russel Mwafulirwa (born 1983), Malawian soccer player *Russel Norman (born 1967), New Zealand politician *Russel Walder (born 1959), American jazz musician *Alfred Russel Wallace (1823–1913), British naturalist *Russel Ward (1914–1995), Australian historian *Russel Wong (born 1961), Singaporean photographer *Russel Wright (1904–1976), American industrial designer * Andrew Russel (1856–1934), American politician *Tony Russel (1925–2017), American actor Fiction *Russel Hobbs, fictional drummer character in the virtual band ''Gorillaz'' *Wataru Sanzu (also known as Russel Walk in America Version), fictional character in ''Inazuma Eleven'' Oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barry Arthur Cipra
Barry Arthur Cipra, an American mathematician and freelance writer, regularly contributes to ''Science'' magazine and ''SIAM New''s, a monthly publication of the Society for Industrial and Applied Mathematics. Along with Dana Mackenzie and Paul Zorn he is the author of several of the volumes in the American Mathematical Society series ''What's Happening in the Mathematical Sciences'', a collection of articles about recent results in pure and applied mathematics oriented towards the undergraduate mathematics major. Biography Cipra got his Ph.D. from University of Maryland College Park in 1980. He was an instructor at The Massachusetts Institute of Technology and at Ohio State University. He was an assistant professor of mathematics at St. Olaf College in Northfield, Minnesota. Cipra received the 1991 Merten M. Hasse Prize from the Mathematical Association of America for his work on the Ising model. In 2005 he received the JPBM Communications Award. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harald Niederreiter
Harald G. Niederreiter (born June 7, 1944) is an Austrian mathematician known for his work in discrepancy theory, algebraic geometry, quasi-Monte Carlo methods, and cryptography. Education and career Niederreiter was born on June 7, 1944, in Vienna, and grew up in Salzburg... He began studying mathematics at the University of Vienna in 1963, and finished his doctorate there in 1969, with a thesis on discrepancy in compact abelian groups supervised by Edmund Hlawka. He began his academic career as an assistant professor at the University of Vienna, but soon moved to Southern Illinois University. During this period he also visited the University of Illinois at Urbana-Champaign, Institute for Advanced Study, and University of California, Los Angeles. In 1978 he moved again, becoming the head of a new mathematics department at the University of the West Indies in Jamaica. In 1981 he returned to Austria for a post at the Austrian Academy of Sciences, where from 1989 to 2000 he served ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases). Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of ris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Finance
Computational finance is a branch of applied computer science that deals with problems of practical interest in finance.Rüdiger U. Seydel, '' tp://nozdr.ru/biblio/kolxo3/F/FN/Seydel%20R.U.%20Tools%20for%20Computational%20Finance%20(4ed.,%20Springer,%202009)(ISBN%203540929282)(O)(348s)_FN_.pdf Tools for Computational Finance', Springer; 3rd edition (May 11, 2006) 978-3540279235 Some slightly different definitions are the study of data and algorithms currently used in finance and the mathematics of computer programs that realize financial models or systems.Cornelis A. Los, ''Computational Finance'' World Scientific Pub Co Inc (December 2000) Computational finance emphasizes practical numerical methods rather than mathematical proofs and focuses on techniques that apply directly to economic analyses.Mario J. Miranda and Paul L. Fackler, ''Applied Computational Economics and Finance'', The MIT Press (September 16, 2002) It is an interdisciplinary field between mathematical finance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
