A general control variate method for option pricing under Lévy processes

Dingec, Kemal Dincer; Hörmann, Wolfgang

doi:10.1016/j.ejor.2012.03.046

Cited by 30 publications

(14 citation statements)

References 32 publications

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“…3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table 1: Convergence w.r.t. different initial points (n = 10 7 , ρ = 10 −8 ) λ * [1] λ * [2] λ * [3] λ * [4] λ * [5] λ * [6] λ * [7] λ * [8] λ * [9] λ * [10] λ * [11] λ * [12] Steps λ * In practice, using very large n and very small ρ is computationally inefficient, so for option pricing, we use n = 1000, ρ = 10 −4 . Using Algorithm 2, Table 2 shows the estimated variance reduction of importance sampling, where "MC price" denotes the estimate price via classical Monte Carlo simulation, "IS price" denotes the estimate price via importance sampling, and "VR ratio" is the variance reduction ratio defined as the variance of classical Monte Carlo simulation divided by the variance of importance sampling.…”

Section: Numerical Examplementioning

confidence: 99%

“…Although Monte Carlo simulation can solve high-dimensional problems, variance reduction techniques are often needed to improve the computational efficiency; see Glasserman [4] for a survey. For Lévy processes models, variance reduction techniques used in derivatives pricing include importance sampling (Kawai [8]), control variates (Dingeç and Hörmann [9]), stratified sampling (Kawai [10]), etc.…”

Section: Introductionmentioning

confidence: 99%

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On sample average approximation algorithms for determining the optimal importance sampling parameters in pricing financial derivatives on Lévy processes

Jiang

2016

Operations Research Letters

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We formulate the problem of determining the optimal importance sampling measure change for pricing financial derivatives under Lévy processes as a parametric optimization problem, and propose a solution approach using sample average approximation (SAA) with Newton iteration to find the optimal parameters in the Esscher probability measure change. Theoretical results, such as convergence rate of the optimal solutions, are provided. A numerical example illustrates the effectiveness of the approach.

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Section: Numerical Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On sample average approximation algorithms for determining the optimal importance sampling parameters in pricing financial derivatives on Lévy processes

Jiang

2016

Operations Research Letters

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“…To demonstrate the efficiency of the auto-realignment method, variance reduction factors (VRFs) of using various PGMs in QMC with respect to the crude MC method are assessed and compared. The VRF measures the efficiency gain of QMC estimators as in many related literatures (see Dingeç and Hörmann, 2012). The VRF is defined by…”

Section: Numerical Experimentsmentioning

confidence: 99%

An auto-realignment method in quasi-Monte Carlo for pricing financial derivatives with jump structures

Weng

Wang

2016

European Journal of Operational Research

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Discontinuities are common in the pricing of financial derivatives and have a tremendous impact on the accuracy of quasi-Monte Carlo (QMC) method. While if the discontinuities are parallel to the axes, good efficiency of the QMC method can still be expected. By realigning the discontinuities to be axes-parallel, Wang and Tan (2013) succeeded in recovering the high efficiency of the QMC method for a special class of functions. Motivated by this work, we propose an auto-realignment method to deal with more general discontinuous functions. The k-means clustering algorithm, a classical algorithm of machine learning, is used to select the most representative normal vectors of the discontinuity surface. By applying this new method, the discontinuities of the resulting function are realigned to be friendly for the QMC method. Numerical experiments demonstrate that the proposed method significantly improves the performance of the QMC method.

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“…Moreover, closed‐form approximations may also allow for efficient computation of the option Greeks and implied parameters, which are of great importance for hedging purposes, as well as the pricing of new instruments. Finally, analytical approximations can also be useful in developing control variates to increase the efficiency of these Monte Carlo schemes; see the works of Kemna and Vorst, Fusai and Meucci, and Dingeç and Hörmann …”

Section: Introductionmentioning

confidence: 99%

Closed‐form approximations for spread options in Lévy markets

Belle

Vanduffel

Yao

2018

Appl Stoch Models Bus & Ind

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We provide new closed-form approximations for the pricing of spread options in three specific instances of exponential Lévy markets, ie, when log-returns are modeled as Brownian motions (Black-Scholes model), variance gamma processes (VG model), or normal inverse Gaussian processes (NIG model). For the specific case of exchange options (spread options with zero strike), we generalize the well-known Margrabe formula (1978) that is valid in a Black-Scholes model to the VG model under a homogeneity assumption.

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A general control variate method for option pricing under Lévy processes

Cited by 30 publications

References 32 publications

On sample average approximation algorithms for determining the optimal importance sampling parameters in pricing financial derivatives on Lévy processes

On sample average approximation algorithms for determining the optimal importance sampling parameters in pricing financial derivatives on Lévy processes

An auto-realignment method in quasi-Monte Carlo for pricing financial derivatives with jump structures

Closed‐form approximations for spread options in Lévy markets

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