Classifications

• • G—PHYSICS
  • G06—COMPUTING; CALCULATING OR COUNTING
  • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
  • G06Q40/00—Finance; Insurance; Tax strategies; Processing of corporate or income taxes
  • G06Q40/02—Banking, e.g. interest calculation or account maintenance

Definitions

• the invention relates to financial securities trading such as, e.g., trading in stocks, bonds and financial derivative instruments, including futures, options and collateralized mortgage obligations.
• the value of a security may be estimated, e.g., based on expected future cash flow.
• cash flow may depend on variable interest rates, for example, and these and other relevant variables may be viewed as stochastic variables.
• CMO collateralized mortgage obligations
• instruments or securities variously called tranches, shares, participations, classes or contracts have cash flows which are determined by dividing and distributing the cash flow of an underlying collection or pool of mortgages on a monthly basis according to pre-specified rules.
• the present value of a tranche can be estimated on the basis of the expected monthly cash flows over the remaining term of the tranche, and
• SUBSTTTUTESHEET(RULE26) an estimate of the present value of a tranche can be represented as a multi-dimensional integral whose dimension is the number of payment periods of the tranche. For a typical instrument with a 30-year term and with monthly payments, this dimension is 360.
• such a high-dimensional integral can be evaluated only approximately, by numerical integration.
• Well known for numerical integration in securities trading is the so- called Monte Carlo method in which points in the domain of integration are generated at random. With integrands arising in financial securities trading, the computational work in combining the sampled values is negligible as compared with producing the integrand values.
• numerical integration methods in securities trading may be compared based on the number of samples required for obtaining a sufficiently accurate approximation to the integral.
• a preferred method for estimating the value of a financial security involves numerical integration unlike Monte Carlo integration in that an integrand is sampled at deterministic points having a low- discrepancy property. As compared with the Monte Carlo method, significant advantages are realized with respect to speed, accuracy, and dependability.
• Fig. 1 is a schematic of a programmed computer system in accordance with a preferred embodiment of the invention.
• FIG. 2 is a graphic representation of performance data obtained in computer trial runs with an exemplary embodiment of the invention as compared with two Monte Carlo computations. Further included is an Appendix with two computer algorithms in "C" source language, respectively for computing Sobol points and Halton points.
• C For a description of C, see B.W. Kernighan et al., The Programming Language C. Prentice-Hall, 1978.
• Fig. 1 shows a stored-program computer 11 con ⁇ nected to input means 12, e.g., a keyboard, for enter ⁇ ing financial securities data, and connected to output means 13, e.g., a visual display device, for displaying an estimated value of the financial security.
• the computer 11 includes a working memory M, a low- discrepancy deterministic point generator P, an inte ⁇ grand evaluator E, and an integrand-value combiner C.
• a multivariate integrand is sampled at points corresponding to a low-discrepancy deterministic sequence of points in the multivariate unit cube as defined below.
• the points of the low- discrepancy deterministic sequence can be used as sample points directly.
• sample points correspond to points of a low-discrepancy deterministic sequence in the multivariate unit cube via a suitable transformation or mapping.
• an approximation of the integral is obtained by suitably combining the computed values, e.g., by averaging or weighted averaging.
• Vj ai Vi., ⁇ a 2 v ; . 2 ⁇ ... ⁇ a ⁇ v ⁇ , ⁇ v lake ⁇ (v,. n /2 n ) , where ⁇ denotes a bit-by-bit "exclusive or" operation.
• X k b,v, ⁇ b 2 v 2 ⁇ ... ⁇ b w v w , k ⁇ 0
• P d the first d primitive polynomials P,, P 2 , ..., P d are used.
• ⁇ x k 'J k ⁇ * denotes the one-dimensional Sobol sequence generated by the polynomial P ;
• CMO FN collateralized mortgage obligation
• PAC tranches 23-A, 23-B, 23-C, 23-D, 23-E supporting tranches 23-G, 23-H, 23-J residual tranche 23-R accrual tranche 23-Z The monthly cash flow is divided and distributed according to pre-specified rules which are included in a formal prospectus. Some of the basic rules may be stated as follows:
• SUBSTTTUTESHEET(RULE26) Principal Distribution Amount is used for repayment of the principal.
• the Principal Amount Prior to a fixed date in the future, the Principal Amount will be allocated sequentially to the tranches 23-G, 23-H, 23-J and 23-Z. After that date, the Principal Distribution Amount will be allocated sequentially to the tranches 23-A, 23-B, 23- C, 23-D and 23-E according to a planned schedule. Any excess amount of the Principal Distribution Amount over the planned schedule wil be allocated sequentially to the tranches 23-G, 23-H, 23-J and 23-Z. A distribution of principal of the tranche 23-R will be made only after all other tranches have been retired.
• the remaining amount of principal borrowed is C-a k .
• ⁇ 0.0004 is chosen.
• E(PV T ) E(PV ⁇ (f lf ... f f 3 ⁇ )) f upon a change of variables
• a 360-variate integrand has to be integrated over the 360-dimensional unit cube, .
• Fig. 2 shows results from trial runs for CMO FN, 89-23 with a preferred method using Sobol points generated by the corresponding computer algorithm given in the Appendix, as compared with Monte Carlo integration.
• Two Monte Carlo computations were carried out, with different "seeds" or starting values of a congruential pseudo-random number generator known as RAN2; see W. Press et al., Numerical Recipes in C. Cambridge University Press, 1992. It is apparent that the preferred method reaches a steady value more rapidly.
• RAN2 congruential pseudo-random number generator
• Halton points were used as generated by the corresponding computer algorithm given in the Appendix. It is felt that Sobol points may be preferred over Halton points for integrals of high dimension. However, this preference may not apply in the case of lower-dimensional integrals, e.g., with dimension up to 5 or so.
• a computation as described may be terminated after a predetermined number of function evaluations.
• a current approximation may be compared with one or several preceding approximations, for termination once a suitable condition depending on the difference between approximations is met.
• Such ter ⁇ mination criteria may be called automatic.
• Automatic termination is particularly reliable where a sequence of approximations settles down smoothly; see, e.g., the curve in Fig. 2 corresponding to Sobol points.
• a cluster or network of multiple parallel processors or workstations can be used.
• the compu ⁇ tation can be speeded up in proportion to the number of processors used.
• the function sobseq from Numerical Recipes, 1992 was used as a basis, but changed significantly to acco odate the parallel distributed approach. The unction sobol can generate Sobol points skipping an initial part of the sequence.
• the constant MAXDIM see Numerical Recipes, is extended to 360 which required adding more initializing data to the arrays ip (the imitive polynomials) , deg (their degrees) , and iv (the initial direction numbers) .
• the polynomial x is used to generate the first coordinate of Sobol points.
• the first coordinate is set to MAXBIT, but it is never used */ static unsigned long ip[MAXDIM+l] - ⁇ 0,MAXBIT,0,1,1,2,1,4,

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• 1995
  • 1995-08-15 EP EP95928840A patent/EP0845123A4/de not_active Withdrawn
  • 1995-08-15 CA CA002229144A patent/CA2229144C/en not_active Expired - Fee Related
  • 1995-08-15 WO PCT/US1995/010363 patent/WO1997007475A1/en not_active Application Discontinuation

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