The British-Russian Offshore Development Agency (Broda) has implemented Sobol' sequence quasi-random generation software up to and including dimension 370. Russian mathematical guru, Ilya Sobol' pioneered this sequence and modified direction numbers for it. Broda chief co-ordinator described Sobol' as: "A theorist who has always worked very closely with practitioners from the 1960s onwards and has been involved with many big projects - he has made incredible contributions to the field of applied quantitive mathematics." The quasi-random numbers are used for pricing derivatives and for value at risk in the form of highly efficient high-dimensional Monte Carlo simulations. This latest development from Broda has been described by the organization as evidence of their support for all areas of information systems. In an official statement it said: "This is further evidence of us bringing Russian mathematical expertise and scientific skills to the West."
BRODA has been for many years a leader in the development and distribution of the high dimensional Sobol' LDS generators. BRODA's generators over the years have become an industry standard in many fields of mathematics and finance. This is what Peter Jackel - the leading experts in the field of Monte Carlo methods in finance - says in his book about one of the early versions of BRODA's generators: "There is a commercial library module available from an organisation called BRODA that can generate Sobol' sequences in up to 370 dimensions. In a way, this module can claim to be a genuine Sobol' number generator since Professor Sobol' himself is behind the initialisation numbers that drive the sequence, and he is also linked to the company distributing the library." (Peter Jackel, "Monte Carlo methods in finance", John Wiley & Sons, 2002.) With constantly increasing complexity of problems there has been a need for very dimensional LDS generators. BRODA has recently developed a new 65000 dimensional Sobol' LDS generator SobolSeq65000 which has even better performance and efficiency than previously developed generators. Not only this generator has very high dimensionality and employs the super fast generation algorithm but the generated Sobol' sequences satisfy Property A in all dimensions and property A' for the adjacent dimensions. BRODA's SobolSeq generators outperform all other known generators both in speed and accuracy as shown in this paper: I. Sobol’, D. Asotsky, A. Kreinin, S. Kucherenko. Construction and Comparison of High-Dimensional Sobol’ Generators, Wilmott Journal, Nov, pp. 64-79, 2012), which can be downloaded here.
Global Sensitivity Analysis and Application of Global Sensitivity Analysis in Monte Carlo option Pricing
Global Sensitivity Analysis (GSA) quantifies the relative importance of input model parameters (variables) in determining the value of the output variable. GSA enables to identify key parameters whose uncertainly affects most of the output. It then can be used to rank variables, fix unessential variables or decrease dimensionality of the problem. I. Sobol' developed the most general GSA method introducing global sensitivity indices. This method is an ideal tool for the analysis of complex multidimensional nonlinear models. Review on global GSA (Sobol' I., Kucherenko S. "Global Sensitivity Indices for Nonlinear Mathematical Models. Review", Wilmott Magazine, 2005, Vol. 2) can be downloaded here. The most recent paper on application of GSA in Monte Carlo Option Pricing (Kucherenko S., Shah N. "The Importance of being Global. Application of Global Sensitivity Analysis in Monte Carlo option Pricing", Wilmott Magazine, 2007, Vol. 4) can be found here. See also a slide presentation on Application of GSA in Monte Carlo and Quasi Monte Carlo option Pricing here. Four different types of options and Greeks are considered. It is shown that the efficiency of Quasi Monte Carlo changes with the change of effective dimensions and it can be different not only for different payoffs but even for different Greeks for the same payoff. Methods and techniques of GSA and the difference between the nominal and effective dimensions, which is very important to understand the superior performance of Sobol' sequences even in very high dimensional problems are presented in this paper.
Many problems of financial mathematics and risk analysis can be formulated as global nonlinear optimization problems (GNOP). Despite of huge practical importance of GNOP, problem of developing robust and efficient numerical methods for GNOP is far from being solved. Existing methods have either limited applicability or are inefficient because of prohibitively large required CPU-time and therefore have limited practical usefulness. There are two kinds of commonly used techniques for solving GNOP: deterministic and stochastic. Deterministic methods guarantee convergence to a global solution. However, for large-scale problems these methods require prohibitively large CPU-time. And they are applicable only if the objective function and the constraints are twice continuously differentiable. Stochastic methods are much faster than deterministic methods. They can be used for any class of objective functions Stochastic search methods yield an asymptotic (in a limit N, where N are randomly chosen points) guarantee of convergence. But as in reality problems can be solved with limited sets of sample points N, convergence to a global solution is not guaranteed.
Broda developed a novel deterministic method, which combines advantages of deterministic and Stochastic methods. It is based on application of low-discrepancy sequences (LDS) and multi-level linkage methods. In comparison with Stochastic methods employing random numbers application of LDS significantly decreases number of points N required to achieve the same tolerance of finding the global minimum. At the same time it is a deterministic method. It has deterministic error bounds instead of probabilistic Monte Carlo error bounds. Many studies have shown that Sobol' LDS is superior to other known LDS and it is used in the developed method. To learn more, please download these papers.