Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension

Caflisch, Russel E.; Morokoff, William J.; Owen, Art B.

doi:10.21314/jcf.1997.005

Cited by 458 publications

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“…This number is referred to a limited number of input variables (less than tens). 22 Recently, Sobol and Kucherenko [23,24] have proposed the so-called Derivative- 23 based Global Sensitivity Measures (DGSM), which can be seen as a kind of general- 24 ization of the Morris screening method. DGSM seem computationally more tractable 25 than variance-based measures, specially for high-dimensional models.…”

mentioning

confidence: 99%

Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests

Lamboni

Iooss

Popelin

et al. 2013

Mathematics and Computers in Simulation

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The estimation of variance-based importance measures (called Sobol' indices) of the input variables of a numerical model can require a large number of model evaluations. It turns to be unacceptable for high-dimensional model involving a large number of input variables (typically more than ten). Recently, Sobol and Kucherenko have proposed the Derivative-based Global Sensitivity Measures (DGSM), defined as the integral of the squared derivatives of the model output, showing that it can help to solve the problem of dimensionality in some cases. We provide a general inequality link between DGSM and total Sobol' indices for input variables belonging to the class of Boltzmann probability measures, thus extending the previous results of Sobol and Kucherenko for uniform and normal measures. The special case of log-concave measures is also described. This link provides a DGSM-based maximal bound for the total Sobol indices. Numerical tests show the performance of the bound and its usefulness in practice.

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mentioning

confidence: 99%

Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests

Lamboni

Iooss

Popelin

et al. 2013

Mathematics and Computers in Simulation

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show abstract

“…This is confirmed in our findings, see Section 4.2. It has been proved in many works [KFSM11,CMO97,Owe03] that QMC outperforms MC regardless of the nominal dimension whenever the effective dimension is low in one or more senses. Hence, in the case of type A and type B functions (we assume that functions are sufficiently smooth), QMC always outperform MC, while for type C functions the two methods are expected to have similar efficiency.…”

Section: Global Sensitivity Analysis and Effective Dimensionsmentioning

confidence: 99%

“…The initial coordinates of Sobol' LDS are much better distributed than the later high dimensional coordinates [Gla03,CMO97]. The BBD changes the order in which inputs (linked with time steps) are sampled.…”

Section: Global Sensitivity Analysis For Prices and Greeksmentioning

confidence: 99%

“…However, many financial applications have been reported where Quasi Monte Carlo outperforms standard Monte Carlo even in the presence of very high dimensions [PT95,PP99,CMO97,KMRZ98a,KMRZ98b,KS07,SAKK12]. This fact is usually explained by a reduced effective dimension of the problem, with respect to its nominal dimension.…”

Section: Introductionmentioning

confidence: 99%

“…The notions of effective dimension in the truncation and superposition sense were introduced in [CMO97]. Further, Owen added the notion of "average dimension" which is more practical from the computational point of view [LO06].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pricing and Risk Management with High-Dimensional Quasi-Monte Carlo and Global Sensitivity Analysis

Bianchetti¹,

Kucherenko

Scoleri³

2015

Wilmott

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We review and apply Quasi Monte Carlo (QMC) and Global Sensitivity Analysis (GSA) techniques to pricing and risk management (greeks) of representative financial instruments of increasing complexity. We compare QMC vs standard Monte Carlo (MC) results in great detail, using high-dimensional Sobol' low discrepancy sequences, different discretization methods, and specific analyses of convergence, performance, speed up, stability, and error optimisation for finite differences greeks. We find that our QMC outperforms MC in most cases, including the highest-dimensional simulations and greeks calculations, showing faster and more stable convergence to exact or almost exact results. Using GSA, we are able to fully explain our findings in terms of reduced effective dimension of our QMC simulation, allowed in most cases, but not always, by Brownian bridge discretization. We conclude that, beyond pricing, QMC is a very promising technique also for computing risk figures, greeks in particular, as it allows to reduce the computational effort of high-dimensional Monte Carlo simulations typical of modern risk management.

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Quasi‐MonteCarlo Methods

Niederreiter¹

2010

Encyclopedia of Quantitative Finance

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This article presents a survey of quasi‐Monte Carlo methods for multidimensional numerical integration and their applications to computational finance. Quasi‐Monte Carlo methods are deterministic versions of Monte Carlo methods, which outperform Monte Carlo methods for many types of integrals. First, a general background on quasi‐Monte Carlo methods is given. Then we describe principles for the construction of low‐discrepancy sequences, which are basic ingredients of any implementation of quasi‐Monte Carlo methods. Special emphasis is placed on the currently most powerful constructions based on the digital method and the theory of ( T , s )‐sequences. Next, the important concept of effective dimension is discussed. One section is devoted to scrambling schemes for low‐discrepancy sequences. A review of applications to pricing problems, in particular for mortgage‐backed securities and options, concludes the article.

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Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension

Cited by 458 publications

References 0 publications

Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests

Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests

Pricing and Risk Management with High-Dimensional Quasi-Monte Carlo and Global Sensitivity Analysis

Quasi‐MonteCarlo Methods

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