The construction of good lattice rules and polynomial lattice rules

Nuyens, Dirk

doi:10.1515/9783110317930.223

Cited by 46 publications

(62 citation statements)

References 48 publications

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“…A second type of deterministic quadrature that we test in this work is the randomized QMC method. Specifically, we use the lattice rules family of QMC [35,15,32]. The main input for the lattice rule is one integer vector with d component (d dimension of the integration problem).…”

Section: Quasi Monte Carlo (Qmc)mentioning

confidence: 99%

Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model

Bayer

Hammouda

Tempone

2020

Quantitative Finance

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The rough Bergomi (rBergomi) model, introduced recently in [4], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet exhibits remarkable fit to empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we design a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi Monte Carlo (QMC). Both techniques are coupled with Brownian bridge construction and Richardson extrapolation. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model. Motivated by the statistical analysis of realized volatility by Gatheral, Jaisson and Rosenbaum [20] and the theoretical results on implied volatility [2,18], rough stochastic volatility has emerged as a new paradigm in quantitative finance, overcoming the observed limitations of diffusive stochastic volatility models. In these models, the trajectories of the volatility have lower Hölder regularity than the trajectories of standard Brownian motion [4,20]. In fact, they are based on fractional Brownian motion (fBm), which is a centered Gaussian process, whose covariance structure depends on the so-called Hurst parameter, H (we refer to [26,16,11] for more details regarding the fBm processes). In the rough volatility case, where 0 < H < 1/2, the fBm has negatively correlated increments and rough sample paths. Gatheral, Jaisson, and Rosenbaum [20] empirically demonstrated the advantages of such models. For instance, they showed that the log-volatility in practice has a similar behavior to fBm with the Hurst exponent H ≈ 0.1 at any reasonable time scale (see also [19]). These results were confirmed by Bennedsen, Lunde and Pakkanen [8], who studied over a thousand individual US equities and showed that H lies in (0, 1/2) for each equity. Other works [8,4,20] showed further benefits of such rough volatility models over standard stochastic volatility models, in terms of explaining crucial phenomena observed in financial markets.The rough Bergomi (rBergomi) model, proposed by Bayer, Friz and Gatheral [4], was one of the first developed rough volatility models. This model, depending on only three parameters, shows remarkable fit to empirical implied volatility surfaces. The construction of the rB...

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Section: Quasi Monte Carlo (Qmc)mentioning

confidence: 99%

Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model

Bayer

Hammouda

Tempone

2020

Quantitative Finance

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“…5. Other types of digital net constructions can be found in [9,43,52] and the references given there.…”

Section: Digital Netsmentioning

confidence: 99%

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

L’Ecuyer

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We survey basic ideas and results on randomized quasi-Monte Carlo (RQMC) methods, discuss their practical aspects, and give numerical illustrations. RQMC can improve accuracy compared with standard Monte Carlo (MC) when estimating an integral interpreted as a mathematical expectation. RQMC estimators are unbiased and their variance converges at a faster rate (under certain conditions) than MC estimators, as a function of the sample size. Variants of RQMC also work for the simulation of Markov chains, for function approximation and optimization, for solving partial differential equations, etc. In this introductory survey, we look at how RQMC point sets and sequences are constructed, how we measure their uniformity, why they can work for high-dimensional integrals, and how can they work when simulating Markov chains over a large number of steps.

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“…Fast CBC constructions (using FFT) can produce generating vectors for an n-point rule in s dimensions in O(s n log n) operations in the case of product weights [66], and in O(s n log n + s 2 n) operations in the case of POD weights [54]. Note that these are considered to be pre-computation costs.…”

Section: Choosing the Weightsmentioning

confidence: 99%

“…There is a certain structure in some QMC methods that can allow for fast matrix-vector multiplication using FFT. This structure has been exploited in the fast CBC construction of lattice rules and polynomial lattice rules [66]. We now explain how this same structure can also be used in more general circumstances [11].…”

Section: Saving 3: Fast Qmc Matrix-vector Multiplicationmentioning

confidence: 99%

“…See e.g., the classical references [64,75] and the recent books [16,60,61]. One key element is the fast component-by-component construction [6,66,67,68] which provides parameters for first order or higher order QMC methods [13,16] for sufficiently smooth functions. Another key element is the careful selection of parameters called weights [77,78] to ensure that the worst case errors in an appropriately weighted function space are bounded independently of the dimension.…”

Section: Introductionmentioning

confidence: 99%

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Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

Kuo

Nuyens

2018

Springer Proceedings in Mathematics &Amp; Statistics

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This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube [0, 1] s and in R s , and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension s under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when s is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications. * Frances Y. Kuo ( ):

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The construction of good lattice rules and polynomial lattice rules

Cited by 46 publications

References 48 publications

Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model

Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

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