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Faster Valuation of Financial Derivatives

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  • Spassimir H. Paskov
  • Joseph F. Traub
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    Abstract

    High-dimensional integrals are usually solved with Monte Carlo algorithms although theory suggests that low-discrepancy algorithms are sometimes superior. We report on numerical testing which compares low-discrepancy and Monte Carlo algorithms on the evaluation of financial derivatives. The testing is performed on a Collateralized Mortgage Obligation (CMO) which is formulated as the computation of ten integrals of dimension up to 360. We tested two low-discrepancy algorithms (Sobol and Halton) and two randomized algorithms (classical Monte Carlo and Monte Carlo combined with antithetic variables). We conclude that for this CMO the Sobol algorithm is always superior to the other algorithms. We believe that it will be advantageous to use the Sobol algorithm for many other types of financial derivatives. Our conclusion regarding the superiority of the Sobol algorithm also holds when a rather small number of sample points are used, an important case in practice. We have built a software system called FINDER for computing high-dimensional integrals. FINDER runs on a heterogeneous network of workstations under PVM 3.2 (Parallel Virtual Machine). Since workstations are ubiquitous, this is a cost-effect way to do large computations fast. The measured speedup is at least .9N for $N$ workstations, $N$ less than or equal to 25. The software can also be used to compute high-dimensional integrals on a single workstation. A routine for generating Sobol points may be found, for example, in "Numerical Recipes in C" by Press et al. However, we incorporated major improvements in FINDER and we stress that the results reported in this paper were obtained using FINDER. One of the improvements was developing the table of primitive polynomials and initial direction numbers for dimensions up to 360.

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    Bibliographic Info

    Paper provided by Santa Fe Institute in its series Working Papers with number 95-03-034.

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    Date of creation: Mar 1995
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    Handle: RePEc:wop:safiwp:95-03-034

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    Cited by:
    1. Nelson Areal & Artur Rodrigues & Manuel Armada, 2008. "On improving the least squares Monte Carlo option valuation method," Review of Derivatives Research, Springer, vol. 11(1), pages 119-151, March.
    2. John Rust & Joseph Traub & Henryk Wozniakowski, 1999. "No Curse of Dimensionality for Contraction Fixed Points Even in the Worst Case," Computational Economics 9902001, EconWPA.
    3. Riccardo Rebonato & Ian Cooper, 1998. "Coupling backward induction with Monte Carlo simulations: a fast Fourier transform (FFT) approach," Applied Mathematical Finance, Taylor & Francis Journals, vol. 5(2), pages 131-141.
    4. Boyle, Phelim & Broadie, Mark & Glasserman, Paul, 1997. "Monte Carlo methods for security pricing," Journal of Economic Dynamics and Control, Elsevier, vol. 21(8-9), pages 1267-1321, June.
    5. Boyle, Phelim & Imai, Junichi & Tan, Ken Seng, 2008. "Computation of optimal portfolios using simulation-based dimension reduction," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 327-338, December.
    6. Siegl, Thomas & F. Tichy, Robert, 2000. "Ruin theory with risk proportional to the free reserve and securitization," Insurance: Mathematics and Economics, Elsevier, vol. 26(1), pages 59-73, February.
    7. Tan, Ken Seng & Boyle, Phelim P., 2000. "Applications of randomized low discrepancy sequences to the valuation of complex securities," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1747-1782, October.
    8. Gerstner, Thomas & Griebel, Michael & Holtz, Markus, 2009. "Efficient deterministic numerical simulation of stochastic asset-liability management models in life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 434-446, June.
    9. Broadie, Mark & Glasserman, Paul, 1997. "Pricing American-style securities using simulation," Journal of Economic Dynamics and Control, Elsevier, vol. 21(8-9), pages 1323-1352, June.
    10. Okten, Giray & Eastman, Warren, 2004. "Randomized quasi-Monte Carlo methods in pricing securities," Journal of Economic Dynamics and Control, Elsevier, vol. 28(12), pages 2399-2426, December.
    11. Gonzalez, Jorge & Tuerlinckx, Francis & De Boeck, Paul & Cools, Ronald, 2006. "Numerical integration in logistic-normal models," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1535-1548, December.
    12. S. Corsaro & P. De Angelis & Z. Marino & F. Perla, 2011. "Participating life insurance policies: an accurate and efficient parallel software for COTS clusters," Computational Management Science, Springer, vol. 8(3), pages 219-236, August.
    13. Raymond Ross, 1998. "Good point methods for computing prices and sensitivities of multi-asset European style options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 5(2), pages 83-106.
    14. Sobol, I.M. & Shukhman, B.V., 2007. "Quasi-random points keep their distance," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 75(3), pages 80-86.
    15. Chi, H. & Mascagni, M. & Warnock, T., 2005. "On the optimal Halton sequence," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 70(1), pages 9-21.
    16. John Rust, 1996. "Dealing with the Complexity of Economic Calculations," Computational Economics 9610002, EconWPA, revised 21 Oct 1997.

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