Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition

In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first non-trivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or non-linear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. M\"uller-Gronbach, B. Niu, K. Ritter, J. Complexity 26 (2010), 229-254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.Comment: 31 pages, 0 figure

Pricing currency derivatives under the benchmark approach

This paper considers the realistic modelling of derivative contracts on exchange rates. We propose a stochastic volatility model that recovers not only the typically observed implied volatility smiles and skews for short dated vanilla foreign exchange options but allows one also to price payoffs in foreign currencies, lower than possible under classical risk neutral pricing, in particular, for long dated derivatives. The main reason for this important feature is the strict supermartingale property of benchmarked savings accounts under the real world probability measure, which the calibrated parameters identify under the proposed model. Using a real dataset on vanilla option quotes, we calibrate our model on a triangle of currencies and find that the risk neutral approach fails for the calibrated model, while the benchmark approach still works

Duality theory and propagation rules for higher order nets

AbstractHigher order nets and sequences are used in quasi-Monte Carlo rules for the approximation of high dimensional integrals over the unit cube. Hence one wants to have higher order nets and sequences of high quality.In this paper we introduce a duality theory for higher order nets whose construction is not necessarily based on linear algebra over finite fields. We use this duality theory to prove propagation rules for such nets. This way we can obtain new higher order nets (sometimes with improved quality) from existing ones. We also extend our approach to the construction of higher order sequences

Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base~2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets

Quasi-Monte Carlo for finance beyond Black--Scholes

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

Quasi-Monte Carlo for finance beyond Black--Scholes

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