Faster Valuation of Financial Derivatives

Paskov, Spassimir H.; Traub, Joseph F.

doi:10.3905/jpm.1995.409541

Cited by 313 publications

(185 citation statements)

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“…Let us now consider a typical collateralized mortgage obligation problem, which involves several tranches which in turn derive their cash flows from an underlying pool of mortgages [6,26]. The problem is to estimate the expected value of the sum of present values of future cash flows for each tranche.…”

Section: Cmo Problem (256 Dimensions)mentioning

confidence: 99%

“…Let us assume that the pool of mortgages has a 21 1/3 year maturity and cash flows are obtained monthly. Then, the with a certain normalizing constant K 0 and an initial interest rate i 0 (for details see [6], first example, and [14,26]). Again the interest rates can either be discretized using a random walk or the Brownian bridge construction.…”

Section: Cmo Problem (256 Dimensions)mentioning

confidence: 99%

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Dimension?Adaptive Tensor?Product Quadrature

2003

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We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low-and moderatedimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm.The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.

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Section: Cmo Problem (256 Dimensions)mentioning

confidence: 99%

Section: Cmo Problem (256 Dimensions)mentioning

confidence: 99%

Dimension?Adaptive Tensor?Product Quadrature

2003

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“…For the technical definition of a low discrepancy sequence and a more detailed introduction to their properties and financial applications, see Boyle, Broadie and Glasserman (1997). See Birge (1994), Joy, Boyle and Tan (1996) and Paskov and Traub (1995) for some of these applications.…”

Section: B Low Discrepancy Sequencesmentioning

confidence: 99%

Pricing American Options: A Duality Approach

Haugh

Kogan

2004

Operations Research

353

112

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We develop a new method for pricing American options. The main practical contribution of this paper is a general algorithm for constructing upper and lower bounds on the true price of the option using any approximation to the option price. We show that our bounds are tight, so that if the initial approximation is close to the true price of the option, the bounds are also guaranteed to be close. We also explicitly characterize the worst-case performance of the pricing bounds. The computation of the lower bound is straightforward and relies on simulating the suboptimal exercise strategy implied by the approximate option price. The upper bound is also computed using Monte Carlo simulation. This is made feasible by the representation of the American option price as a solution of a properly defined dual minimization problem, which is the main theoretical result of this paper. Our algorithm proves to be accurate on a set of sample problems where we price call options on the maximum and the geometric mean of a collection of stocks. These numerical results suggest that our pricing method can be successfully applied to problems of practical interest.

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“…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%

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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Faster Valuation of Financial Derivatives

Cited by 313 publications

References 0 publications

Dimension?Adaptive Tensor?Product Quadrature

Dimension?Adaptive Tensor?Product Quadrature

Pricing American Options: A Duality Approach

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

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