Academic journal article Journal of Risk and Insurance

# Contingent Claim Pricing Using a Normal Inverse Gaussian Probability Distortion Operator

Academic journal article Journal of Risk and Insurance

# Contingent Claim Pricing Using a Normal Inverse Gaussian Probability Distortion Operator

## Article excerpt

ABSTRACT

We consider the problem of pricing contingent claims using distortion operators. This approach was first developed in (Wang, 2000) where the original distortion function was defined in terms of the normal distribution. Here, we introduce a new distortion based on the Normal Inverse Gaussian (NIG) distribution. The NIG is a generalization of the normal distribution that allows for heavier skewed tails. The resulting operator asymmetrically distorts the underlying distribution. Moreover, we show how we can recuperate non-Gaussian Black-Scholes formulas using distortion operators and we provide illustrations of their performance. We conclude with a brief discussion on risk management applications.

INTRODUCTION

In Wang (2000), the author proposes a form of insurance risk pricing based on a normal-based distortion risk measure. Distortion risk measures are quantile-based measures that have been developed in the actuarial literature and that are now part of the risk measurement tools inventory available for practitioners in finance and insurance (see Dowd and Blake, 2006, for an account on these and other risk measures). It turns out that this distortion-based pricing principle is consistent with the financial theory of Gaussian option pricing. In Hamada and Sherris (2003), it is shown that the celebrated Black-Scholes formula can be recuperated through the distortion operator of Wang (2000) under the assumption of a normal model for asset prices. Moreover, the authors carry out a numerical analysis of the normal-based distortion operator of Wang (2000) in order to assess its performance under a non-normal model for asset prices. They numerically illustrate the limitations of Wang's approach under non-Gaussian assumptions. Another downside of the normal distortion of Wang (2000) is its underlying symmetry that poses some constrains in applications. In Wang (2004), we find an application in catastrophe (CAT) bonds pricing where a Student-t distribution-based distortion is introduced. Unlike the normal operator of Wang (2000), this Student-t distortion allows for skewness that translates into large losses as well as large gains being inflated under the distorted probability. In this article, we address the concerns in Hamada and Sherris (2003) while using a distortion that brings skewness into the picture. Indeed, this new family of distortion, which is based on an asymmetric distribution, allows for similar applications as those discussed in Wang (2004).

It is a well-known fact that the returns of most financial assets have semi-heavy tails and the actual kurtosis is higher than that of a normal distribution. Indeed, a large body of literature has documented common features as skewness and excess kurtosis of asset returns (Bollerslev, 1987; Richardson and Smith, 1993; Ghose and Kroner, 1995; McCulloch, 1997; Theodossiou, 1998; Rockinger and Jondeau, 2002; Jondeau and Rockinger, 2003; Theodossiou and Trigeorgis, 2003; Bali and Theodossiou, 2007; Bali and Weinbaum, 2007; among others). In particular Bali (2003) provides this evidence using the extreme value distributions. Moreover, several studies have suggested different distributions to capture the fat tails of asset returns. These include the Student-t (Bollerslev, 1987; Hsieh, 1989), Generalized t distribution (McDonald and Neweys, 1988), skewed Generalized t (McDonald and Newey, 1988; Theodossiou, 1998; Bauwens and Laurent, 2002), non-central-t distribution by Harvey and Siddique (1999), SU-normal distribution (Pilsun and Nam, 2008), exponential generalized beta of the second kind (Wang et al., 2001), and Pearson Type IV (Nagahara, 1999).

Clearly, these stylized features of asset returns have to be taken into account in risk measurements. For instance, in Bali and Theodossiou (2008) we find a recent study where non-Gaussian distributions are used for VaR and TVaR estimation. Similarly, the pricing principle approach of Wang must be somehow modified in order for it to capture the non-Gaussian feature of market prices. …

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