Some New Bivariate IG and NIG-Distributions for Modelling Covariate Financial Returns

Lillestøl, Jostein

doi:10.2139/ssrn.971557

Cited by 1 publication

(1 citation statement)

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“…The works of Rydberg (1997), Eberlein and Keller (1995) and Prause (1999) show that the class of generalized hyperbolic distributions provides a flexible family of distributions fitting financial 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, 2002, 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 efficient pricing of options.…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%