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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Quasi-Monte Carlo methods in finance ： ウィキペディア英語版 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 $\backslash epsilon$. If the integral is of dimension $d$ then in the worst case, where one has a guarantee of error at most $\backslash epsilon$, the computational complexity is typically of order $\backslash epsilon^$. 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 magnitude and also enjoyed other desirable attributes. Their results were first published〔Paskov, S. H. and Traub, J. F. (1995), Faster evaluation of financial derivatives, J. Portfolio Management, 22(1), 113-120.〕 in 1995. Today QMC is widely used in the financial sector to value financial derivatives; see list of books below. QMC is not a panacea for all high-dimensional integrals. A number of explanations have been proposed for why QMC is so good for financial derivatives. This continues to be a very fruitful research area. ==Monte Carlo and quasi-Monte Carlo methods== Integrals in hundreds or thousands of variables are common in computational finance. These have to be approximated numerically to within an error threshold $\backslash epsilon$. It is well known that if a worst case guarantee of error at most $\backslash epsilon$ is required then the computational complexity of integration may be exponential in $d$, the dimension of the integrand; See 〔Traub, J. F and Werschulz, A. G. (1998), Complexity and Information, Cambridge University Press, Cambridge, UK.〕 Ch. 3 for details. To break this curse of dimensionality one can use the Monte Carlo (MC) method defined by :$\backslash varphi^^nf(x\_i),$ where the evaluation points $x\_i$ are randomly chosen. It is well known that the expected error of Monte Carlo is of order $n^$. Thus the cost of the algorithm that has error $\backslash epsilon$ is of order $\backslash epsilon^$ breaking the curse of dimensionality. Of course in computational practice pseudo-random points are used. Figure 1 shows the distribution of 500 pseudo-random points on the unit square. Note there are regions where there are no points and other regions where there are clusters of points. It would be desirable to sample the integrand at uniformly distributed points. A rectangular grid would be uniform but even if there were only 2 grid points in each Cartesian direction there would be $2^d$ points. So the desideratum should be as few points as possible chosen as uniform as possible. It turns out there is a well-developed part of number theory which deals exactly with this desideratum. Discrepancy is a measure of deviation from uniformity so what one wants are low discrepancy sequences (LDS).〔Niederreiter, H. (1992), Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.〕 Numerous LDS have been created named after their inventors, e.g. *Halton *Hammersley *Sobol *Faure *Niederreiter Figure 2. gives the distribution of 500 LDS points. The quasi-Monte Carlo (QMC) method is defined by :$\backslash varphi^^nf(x\_i),$ where the $x\_i$ belong to an LDS. The standard terminology quasi-Monte Carlo is somewhat unfortunate since MC is a randomized method whereas QMC is purely deterministic. The uniform distribution of LDS is desirable. But the worst case error of QMC is of order :$\backslash frac,$ where $n$ is the number of sample points. See 〔 for the theory of LDS and references to the literature. The rate of convergence of LDS may be contrasted with the expected rate of convergence of MC which is $n^$. For $d$ small the rate of convergence of QMC is faster than MC but for $d$ large the factor $(\backslash log\; n)^d$ is devastating. For example, if $d=360$, then even with $\backslash log\; n=2$ the QMC error is proportional to $2^$. Thus it was widely believed by the world's leading experts that QMC should not be used for high-dimensional integration. For example, in 1992 Bratley, Fox and Niederreiter〔Bratley, P., Fox, B. L. and Niederreiter, H. (1992), Implementation and tests of low-discrepancy sequences, ACM Transactions on Modelling and Computer Simulation, Vol. 2, No. 3, 195-213.〕 performed extensive testing on certain mathematical problems. They conclude "in high-dimensional problems (say $d\; >\; 12$), QMC seems to offer no practical advantage over MC". In 1993, Rensburg and Torrie〔van Rensburg, E. J. J. and Torrie, G. M. (1993), Estimation of multidimensional integrals: is Monte Carlo the best method? J. Phys. A: Math. Gen., 26(4), 943-953.〕 compared QMC with MC for the numerical estimation of high-dimensional integrals which occur in computing virial coefficients for the hard-sphere fluid. They conclude QMC is more effective than MC only if . As we shall see, tests on 360-dimensional integrals arising from a collateralized mortgage obligation (CMO) lead to very different conclusions. Woźniakowski's 1991 paper〔Woźniakowski, H. (1991), Average case complexity of multivariate integration, Bull. Amer. Math. Soc. (New Ser.), 24(1), 185-194.〕 showing the connection between average case complexity of integration and QMC led to new interest in QMC. Woźniakowski's result received considerable coverage in the scientific press〔Cipra, B. A. (1991), Multivariate Integration: It ain't so tough (on average), SIAM NEWS, 28 March.〕 .〔Traub, J. F. and Woźniakowski, H. (1994), Breaking intractability, Scientific American, 270(1), January, 102-107.〕 In early 1992, I. T. Vanderhoof, New York University, became aware of Woźniakowski's result and gave Woźniakowski's colleague J. F. Traub, Columbia University, a CMO with parameters set by Goldman Sachs. This CMO had 10 tranches each requiring the computation of a 360 dimensional integral. Traub asked a Ph.D. student, Spassimir Paskov, to compare QMC with MC for the CMO. In 1992 Paskov built a software system called FinDer and ran extensive tests. To the Columbia's research group's surprise and initial disbelief Paskov reported that QMC was always superior to MC in a number of ways. Details are given below. Preliminary results were presented by Paskov and Traub to a number of Wall Street firms in Fall 1993 and Spring 1994. The firms were initially skeptical of the claim that QMC was superior to MC for pricing financial derivatives. A January 1994 article in Scientific American by Traub and Woźniakowski〔 discussed the theoretical issues and reported that "Preliminary results obtained by testing certain finance problems suggests the superiority of the deterministic methods in practice". In Fall 1994 Paskov wrote a Columbia University Computer Science Report which appeared in slightly modified form in 1997.〔Paskov, S. H., New methodologies for valuing derivatives, 545-582, in Mathematics of Derivative Securities, S. Pliska and M. Dempster eds., Cambridge University Press, Cambridge.〕 In Fall 1995 Paskov and Traub published a paper in the "Journal of Portfolio Management".〔 They compared MC and two QMC methods. The two deterministic methods used Sobol and Halton points. Since better LDS were created later, no comparison will be made between Sobol and Halton sequences. The experiments drew the following conclusions regarding the performance of MC and QMC on the 10 tranche CMO: *QMC methods converge significantly faster than MC *MC is sensitive to the initial seed *The convergence of QMC is smoother than the convergence of MC. This makes automatic termination easier for QMC. To summarize, QMC beats MC for the CMO on accuracy, confidence level, and speed. This paper was followed by reports on tests by a number of researchers which also led to the conclusion the QMC is superior to MC for a variety of high-dimensional finance problems. This includes papers by Caflisch and Morokoff (1996),〔Caflisch, R. E. and Morokoff, W. (1996), Quasi-Monte Carlo computation of a finance problem, 15-30, in Proceedings Workshop on Quasi-Monte Carlo Methods and their Applications, 11 December 1995, K.-T. Fang and F. Hickernell eds., Hong Kong Baptist University.〕 Joy, Boyle, Tan (1996),〔Joy, C., Boyle, P. P. and Tang, K. S. (1996), Quasi-Monte Carlo methods in numerical finance, Management Science, 42(6), 926-938.〕 Ninomiya and Tezuka (1996),〔Ninomiya, S. and Tezuka, S. (1996), Toward real-time pricing of complex financial derivatives, Appl. Math. Finance, 3, 1-20.〕 Papageorgiou and Traub (1996),〔Papageorgiou, A. and Traub, J. F. (1996), Beating Monte Carlo, Risk, 9(6), 63-65.〕 Ackworth, Broadie and Glasserman (1997)〔Ackworth, P., Broadie, M. and Glasserman, P. (1997), A comparison of some Monte Carlo techniques for option pricing, 1-18, in Monte Carlo and Quasi-Monte Carlo Methods '96, H. Hellekalek, P. Larcher and G. Zinterhof eds., Springer Verlag, New York.〕 Further testing of the CMO〔 was carried out by Anargyros Papageorgiou, who developed an improved version of the FinDer software system. The new results include the following: *Small number of sample points: For the hardest CMO tranche QMC using the generalized Faure LDS due to S. Tezuka〔Tezuka, S., Uniform Random Numbers:Theory and Practice, Kluwer, Netherlands.〕 achieves accuracy $10^$ with just 170 points. MC requires 2700 points for the same accuracy. The significance of this is that due to future interest rates and prepayment rates being unknown, financial firms are content with accuracy of $10^$. *Large number of sample points: The advantage of QMC over MC is further amplified as the sample size and accuracy demands grow. In particular, QMC is 20 to 50 times faster than MC with moderate sample sizes, and can be up to 1000 times faster than MC〔 when high accuracy is desired QMC. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quasi-Monte Carlo methods in finance」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.