Quasi-Monte Carlo for finance beyond Black--Scholes

Abstract

Quasi-Monte Carlo methods are used to approximate integrals of high dimensionality. However, if the problem under consideration is of unbounded dimensionality, it is not obvious if one can apply quasi-Monte Carlo methods at all. We introduce a hybrid approach combining quasi-Monte Carlo and Monte Carlo methods and apply it to a finance problem of unbounded dimensionality. We find that this hybrid approach improves on a Monte Carlo approach. References A. Kyprianou, W. Schoutens, and P. Wilmott. {Exotic option pricing and advanced {L}evy models}. Wiley, Chichester, 2005. P. L'Ecuyer and C. Lemieux. Recent advances in randomized quasi-{M}onte {C}arlo methods. In {Modeling uncertainty}, volume 46 of {Internat. Ser. Oper. Res. Management Sci.}, pages 419--474. Kluwer Acad. Publ., Boston, MA, 2002. {http://www.iro.umontreal.ca/ lecuyer/myftp/papers/survey01.ps}. S. M. Ross. {Introduction to probability models}. Harcourt/Academic Press, Burlington, MA, sixth edition, 1997. R. E. Caflisch, W. J. Morokoff, and A. B. Owen. Valuation of mortgage backed securities using {B}rownian {Br}idges to reduce effective dimension. {J. Comp. Finance}, 1:27--46, 1997. {http://www-stat.stanford.edu/ owen/reports/cmo.ps}. R. Cont and P. Tankov. {Financial {M}odelling with {J}ump {P}rocesses}. Chapman and Hall/CRC, Boca Raton, London, New York, Washington, D. C., 2004. P. Glasserman. {Monte {C}arlo {M}ethods in {F}inancial {E}ngineering}. Springer, New York, Berlin, Heidelberg, Hong Kong, London, Milan, Paris, Tokyo, 2004. S.G. Kou and H. Wang. Option pricing under a double exponential jump diffusion model. {Management Science}, 50:1178--1192, 2004