A Quasi-Monte Carlo Algorithm for the Normal Inverse Gaussian Distribution and Valuation of Financial Derivatives

Benth, Fred Espen; Groth, Martin; Kettler, Paul C.

doi:10.1142/s0219024906003810

Cited by 14 publications

(9 citation statements)

References 84 publications

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“…The Heston model assumes diffusion dynamics for both the spot price and the volatility, but even though the spot price and volatility are jointly Markovian, we do not deal with a stationary, independent increment process. Therefore our situation is different to the majority of papers that apply QMC methods to finance problems [1,2,5,4,9,14,27,31,32,38,46,49] and, in particular, the Linear Transform (LT) method of Imai and Tan [31,32] (see also Leobacher [38]) does not seem applicable in our case of path-dependent options when the underlying follows either the Heston model or the SVJ model.…”

Section: Introductionmentioning

confidence: 71%

Quasi-Monte Carlo methods for the Kou model

Baldeaux¹

2008

Monte Carlo Methods and Applications

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In this paper, we discuss the application of quasi-Monte Carlo methods to the Heston model. We base our algorithms on the Broadie-Kaya algorithm, an exact simulation scheme for the Heston model. As the joint transition densities are not available in closed-form, the Linear Transformation method due to Imai and Tan, a popular and widely applicable method to improve the effectiveness of quasi-Monte Carlo methods, cannot be employed in the context of path-dependent options when the underlying price process follows the Heston model. Consequently, we tailor quasi-Monte Carlo methods directly to the Heston model. The contributions of the paper are threefold: We firstly show how to apply quasi-Monte Carlo methods in the context of the Heston model and the SVJ model, secondly that quasi-Monte Carlo methods improve on Monte Carlo methods, and thirdly how to improve the effectiveness of quasi-Monte Carlo methods by using bridge constructions tailored to the Heston and SVJ models. Finally, we provide some extensions for computing greeks, barrier options, multidimensional and multi-asset pricing, and the 3/2 model.2010 Mathematics Subject Classification. 65C05, 65D30, 91G20, 91G60.

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Section: Introductionmentioning

confidence: 71%

Quasi-Monte Carlo methods for the Kou model

Baldeaux¹

2008

Monte Carlo Methods and Applications

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“…For example, one can resort to Monte Carlo techniques to simulate sample paths for the asset. Averaging a sufficiently large number of realized payoffs yields the required price, see for example [6,23]. One can also attempt to derive a partial differential equation for pricing which can be solved using numerical methods [48].…”

Section: Pricing With Characteristic Functionmentioning

confidence: 99%

Pricing European options and risk measurement under exponential Lévy models — a practical guide

Salhi

2017

J. Finan. Eng.

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This paper provides a thorough survey of the European option pricing, with new trends in the risk measurement, under exponential Lévy models. We develop all steps of pricing from equivalent martingale measures construction to numerical valuation of the option price under these measures. We then construct an algorithm, based on Rockafellar and Uryasev representation and fast Fourier transform, to compute Risk indicators, like the VaR and the CVaR of derivatives. The results are illustrated with an example of each exponential Lévy class. The main contribution of this paper is to build a comprehensive study from the theoretical point of view to practical numerical illustration and to give a complete characterization of the studied equivalent martingale measures by discussing their similarity and their applicability in practice. Furthermore, this work proposes applications to the Fourier inversion technique in risk measurement.

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“…In Benth et al . (2006), a quasi‐Monte Carlo method is developed based on this simulation algorithm.…”

Section: The Normal Inverse Gaussian Distributionmentioning

confidence: 99%

“…In order to make the univariate dynamics risk neutral, we perform an Esscher transform which entails finding a parameter θ solving (5) in the one‐dimensional case ( d = 1). In practice this implies a modification of the skewness analogous to the multivariate case discussed above (see Benth et al ., 2006, for more details). In the simulation studies below we match using the terminal distribution at time t instead of the log‐returns.…”

Section: Approximation Using a Univariate Nig Distributionmentioning

confidence: 99%

Pricing of basket options using univariate normal inverse Gaussian approximations

Benth

Henriksen

2010

J. Forecast.

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In this paper we study the approximation of a sum of assets having marginal log-returns being multivariate normal inverse Gaussian distributed. We analyse the choice of a univariate exponential NIG distribution, where the approximation is based on matching of moments. Probability densities and European basket call option prices of the two-asset and univariate approximations are studied and analysed in two cases, each case consisting of nine scenarios of different volatilities and correlations, to assess the accuracy of the approximation. We fi nd that the sum can be well approximated, failing, however, to match the tails for some extreme parameter choices. The approximated option prices are close to the true ones, although becoming signifi cantly underestimated for far out-of-the-money call options.Obviously, having a univariate log normal approximation is advantageous, since then we obtain a closed-form (Black and Scholes) formula for the price. This approach is very often used when pricing basket options. When variables are negatively correlated, it turns out that the approximation is not very good, and option prices may deviate signifi cantly from the true ones (see, for example, Henriksen (2008)).It is extensively documented in the fi nancial literature (starting with the seminal paper of Fama (1970)) that geometric Brownian motion is not a good model for fi nancial asset prices. The works of Rydberg (1997), Eberlein and Keller (1995) and Prause (1999) show that the class of generalized hyperbolic distributions provides a fl exible family of distributions fi tting fi nancial data very well. In particular, the normal inverse Gaussian (NIG) distribution turns out to be very suitable for both modelling and analytical purposes, and has gained a lot of attention in the literature (see, for example, Lillestøl, 2002Lillestøl, , 2006. Having a multivariate asset price dynamics with NIG distributed log-returns raises the question of valid univariate approximations of its sum which can be used for effi cient pricing of options. We approach this question by suggesting an approximation using the NIG distribution.The idea to approximate the logarithm of a linear combination of exponential NIG variables with a univariate NIG was fi rst proposed and applied to option pricing in electricity markets by Börger (2007). We extend his analysis, providing structured examples testing the validity of the approximation using different matching procedures in the bivariate case. In particular, we test the robustness of the approximations with respect to different correlation structures and tail heaviness. We discuss the approximation of a sum of two asset prices with bivariate NIG distributed log-returns, which fails for some extreme choices of parameters. The matching on log-price levels, that is, the matching of the logarithm of the sum with a univariate NIG variable, is more robust towards different assumptions, but may provide bad approximations. However, when applying our methods on options, it seems that prices are overall well a...

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A Quasi-Monte Carlo Algorithm for the Normal Inverse Gaussian Distribution and Valuation of Financial Derivatives

Cited by 14 publications

References 84 publications

Quasi-Monte Carlo methods for the Kou model

Quasi-Monte Carlo methods for the Kou model

Pricing European options and risk measurement under exponential Lévy models — a practical guide

Pricing of basket options using univariate normal inverse Gaussian approximations

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