# Contingent claim pricing using a Normal Inverse Gaussian probability distortion operator.

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. In recent years, several non-Gaussian distributions have been proposed in order to better model asset prices. Recently, the Generalized Hyperbolic family of distributions has been successfully used in finance (Eberlein and Keller, 1995; Eberlein, 2001; Prause, 1999). This family has many interesting properties that better capture the semi-heavy tail feature, skewness, and kurtosis of financial returns. It is a very large family that contains the normal distribution as a limiting case. Among these distributions, we find the subclass of Normal Inverse Gaussian (NIG) distributions. This law is a natural generalization of the normal distribution and, with four parameters, is more flexible in terms of skewness and kurtosis. Just as the normal distribution is the cornerstone for the geometric Brownian motion, we can construct an exponential NIG Levy motion that is a natural generalization of the former. Based on models like the NIG process, a non-Gaussian financial theory is now readily available (see Schoutens, 2003, for an introduction). Under non-Gaussian assumptions for the asset returns, markets are incomplete and there are many equivalent martingale measures. This implies in turn that the arbitrage-free price of contingent claims is not unique. Among this family of equivalent martingale measures, one can find subclasses for which explicit formulas can be worked out. One such subclass is the so-called mean-correcting equivalent martingale measure. Within this class of equivalent measures, we have that a NIG-Levy processes remains within the same family of processes with only a change in the mean parameter. Moreover, a Black-Scholes type formula can be worked out in this case. This is an interesting analog to the situation found in the Brownian model for which the celebrated Black-Scholes formula for contingent claims was first developed.

In Hamada and Sherris (2003), they set out to illustrate the limitations of the operator of Wang under non-Gaussian assumptions. In this note, we propose a generalized version of this operator based on a NIG distribution. We show that this operator is compatible with standard non-Gaussian financial theory and in particular, we recuperate the option formula under the mean-correcting subclass of equivalent martingale measures. We also carry out a simulation analysis in order to illustrate how this new distortion operator improves upon the limitations of Wang's original distortion, in particular, those discussed in Wang (2004). Following Hamada and Sherris, we consider four asset price models: a geometric Brownian motion, an exponential NIG-Levy model, a jump diffusion model, and a constant elasticity of variance (CEV) model. We compare the performance of the normal-based operator (Wang, 2000), the asymmetric Student-t operator of Wang and our new NIG-based operator for all four models. We confirm again that Wang's normal operator does not perform very well under non-Gaussian assumptions. But we also show that the NIG-based operator is more robust under different models for the underlying even under different non-Gaussian assumptions.

The article is organized as follows. In "The Normal Inverse Gaussian Distribution and Non-Gaussian Black-Scholes Contingent Pricing" section, we present a brief summary of results about the NIG family of distributions and the corresponding non-Gaussian financial theory. We also introduce in this section several results that will be needed throughout out this article. In the "A New Class of Distortion Operators" section, we introduce a NIG-based distortion operator and discuss some of their properties and features. In the "Contingent Claims Pricing" section, we show how this new operator is consistent with standard non-Gaussian financial theory by recuperating the Black-Scholes type formula. In the "Simulation and Further Analysis" section, we present a simulation analysis of this operator for several examples. Finally, in the "Conclusions and Further Discussion" section, we conclude with a brief discussion some potential applications in risk management while hinting at interesting directions for future work.

THE NORMAL INVERSE GAUSSIAN DISTRIBUTION AND NON-GAUSSIAN BLACK-SCHOLES CONTINGENT PRICING

The NIG distribution is a member of the wider class of generalized hyperbolic distributions. This larger family was introduced in Barndorff-Nielsen and Halgreen (1977). It contains either directly or as a limiting case the inverse Gaussian, normal, Student-t, Cauchy, exponential, and gamma distributions. In the last decade, this family has been widely applied in finance (see Eberlein and Keller, 1995; Eberlein, 2001; Prause, 1999). 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. Recent studies propose the generalized hyperbolic family of distributions as a more adequate model for asset returns. They show that the medium-tailed generalized hyperbolic distribution fits well to stock returns and use it in a general option pricing model. In financial applications it is often preferred to [alpha]-stable distributions since their density is known and all of the moments exist. Moreover, it has been shown (Barndorff-Nielsen and Halgreen, 1977) that this family belongs to the infinitely divisible class of distributions that allows for the construction of a non-Gaussian Black-Scholes option pricing theory. In this note we focus on the NIG subclass of this large family, a thorough account of more general non-Gaussian distributions in finance can be found in Schoutens (2003).

In this section, we present some well-known facts and results about the NIG subclass and its applications in finance. The NIG is one of the only two subclasses being closed under convolutions (the other one being the variance-gamma distribution). Its density function is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [K.sub.[lambda]] is the modified Bessel function of the third kind with index [lambda] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and [[gamma].sup.2] = [[alpha].sup.2] - [[beta].sup.2]. The parameter domain is [delta] > 0, [alpha] [greater than or equal to] 0, [[alpha].sup.2] > [[beta].sup.2] and [mu] [member of] R. The parameter [alpha] > 0 determines the shape, [beta] with 0 [less than or equal to] |[beta]| < [alpha] the skewness and [mu] [member of] R the location. [delta] > 0 is a scaling parameter.

In this article, we denote the NIG distribution function by NIG, that is,

NIG(x; [alpha], [beta], [delta], [mu]) = [[integral].sup.x.sub.-[infinity]] nig(y; [alpha], [beta], [delta], [mu])dy.

The NIG survival function is denoted by [bar.NIG], that is,

[bar.NIG](x; [alpha], [beta], [delta], [mu]) = [[integral].sup.[infinity].sub.x] nig(y; [alpha], [beta], [delta], [mu])dy.

The mean and the variance are given, respectively, by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Interestingly enough, the normal distribution is a limiting case of the NIG distribution as stated in the following remark.

Remark 1: A normal distribution with mean [mu] + [beta][[sigma].sup.2] and variance [[sigma].sup.2] is obtained as a limiting case of the NIG distribution for [alpha], [delta] [right arrow] [infinity] and [delta]/[alpha] [right arrow] [[sigma].sup.2] (see Prause, 1999).

An interesting feature of the NIG density is that, unlike the normal density, it is not symmetric and that its asymmetry is determined by the parameter [beta] as it can be inferred from the density itself (1) where [beta] appears in the exponential term. A positive [beta] leads to right skewness whereas a negative [beta] has the opposite effect. Moreover, it is a straightforward exercise to verify that the NIG distribution has a symmetry property with respect to fl as stated in the following remark.

Remark 2: For a given parameter [beta] such that [[alpha].sup.2] > [[beta].sup.2], we have that

NIG(x; [alpha], [beta], [delta], [mu] = 0) = 1 - NIG(-x; [alpha], -[beta], [delta], [mu] = 0).

The Laplace transform of the NIG distribution is particularly simple:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [[gamma].sup.2] = [[alpha].sup.2] - [[beta].sup.2] and [[gamma].sup.2.sub.z] = [[alpha].sup.2] - [([beta] - z).sup.2].

This form of the Laplace transform yields an expression for the expectation of an exponential transformation of a NIG random variable. This is given in the following remark.

Remark 3: If X ~ nig([alpha], [beta], [delta], [mu]) we have that Y = [e.sup.X] is a LogNIG random variable with mean given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the form of the Laplace transform (2), we can see that the NIG distribution is closed under convolutions. If [X.sub.1] and [X.sub.2] are two independent random variables with NIG densities nig(x; [alpha], [beta], [[delta].sub.1], [[mu].sub.1]) and nig(x; [alpha], [beta], [[delta].sub.2], [[mu].sub.2], respectively, then [X.sub.1] + [X.sub.2] has density nig(x; [alpha], [beta], [[delta].sub.1], [[mu].sub.1] + [[mu].sub.2]). Moreover, the NIG is closed under affine transformations as stated in the following remark.

Remark 4: If X ~ nig([alpha], [beta], [delta], [mu]) we have that Y = aX + b, for a > 0 and b [member of] R, is such that

Y ~ nig([alpha]/a, [beta]/a, a[delta], a[mu] + b).

The NIG distribution was originally constructed in Barndorff-Nielsen (1977) as a normal variance-mean mixture where the mixing distribution is an inverse Gaussian. This is, if X is a NIG distributed random variable then, the conditional distribution given W = w is N([mu] + [beta]w, w) where W is inverse Gaussian distributed IG([delta], [square root of [[alpha].sup.2] - [[beta].sup.2])] (see Jorgensen, 1982, for a reference on inverse Gaussian distributions). This gives a simple way of simulating NIG random variables. Moreover, this property allows us to numerically evaluate the NIG distribution function through the following result that is now standard in the literature (see, e.g., Schoutens, 2003).

Proposition 1: The distribution function NIG is given by the expression

NIG(x; [alpha], [beta], [delta], [mu]) = [[integral].sup.[infinity].sub.-[infinity]] [PHI] ([x-[mu]-[beta]t] / [square root of t])ig(t; [delta], [square root of [a.sup.2] - [[beta].sup.2]])dt, (3)

where [PHI] is the standard normal distribution and the inverse Gaussian density ig is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Barndorff-Nielsen and Halgreen (1977) showed that the generalized hyperbolic family is infinitely divisible. Because of the infinite divisibility of NIG distributions, we can construct a NIG Levy process, that is, a process with independent and stationary NIG-distributed increments. Recall that this class of processes is in one-to-one correspondence with the class of infinitely divisible distributions. Every infinitely divisible distribution generates a Levy process and the increments of every Levy process are infinitely divisible distributed. We refer to Bertoin (1996) or Sato (1999) for comprehensive discussions on Levy processes and to Barndorff-Nielsen, Mikosh, and Resnick (2001) for recent applications.

The NIG process is an extension of the Brownian motion that allows for finite dimensional distributions with semi-heavy tails. In a way, it can be seen as a purely discontinuous version of the latter. Within the wide spectrum of Levy processes, it lies somewhere between the Brownian motion and the [alpha]-stable process. In finance, Brownian motion is at the heart of the Black-Scholes option pricing theory. In recent years, this arbitrage-free analysis has been extended to include more general Levy processes as the basic building block. In this section, we briefly mention some elements of this so-called non-Gaussian Black-Scholes option pricing. For a more extensive introduction we refer the reader to Schoutens (2003).

The NIG Levy processes can be defined as follows:

Definition 1: Let ([OMEGA], F, [([F.sub.t]).sub.[t [greater than or equal to] 0], P) be a filtered probability space. An adapted cadlag R-valued process X = {[F.sub.t]).sub.[t [greater than or equal to] 0]} with X(0) = 0 is a NIG Levy process if X(t) has independent and stationary increments distributed as nig(x; [alpha], [beta], [delta]t, [mu]t).

The NIG process is a pure jump process plus a drift term. The drift term is nothing but the expected value of X(1). In Figure 1 we can see different paths of NIG processes.

The NIG Levy process exhibits a diffusion-like feature along with a jump-driven structure. Despite the apparent continuity, its paths are composed by an infinite number of small jumps. We refer to Prause (1999) for a comprehensive discussion on the NIG Levy process.

In all, the NIG process has features that make it a natural model for asset returns. A general exponential model for asset prices in terms of a NIG process has been proposed and analyzed in Barndorff-Nielsen (1998). The so-called exponential NIG Levy model for asset prices is of the form

[S.sub.t] = [S.sub.0e][Z.sub.t], t > 0, (4)

where [Z.sub.t] is a ([F.sub.t], P)-NIG Levy process. A Black-Scholes analysis can still be carried out for derivatives defined on a more sophisticated market composed by a risky asset following (4) and a riskless asset of the form [B.sub.t] = [e.sup.rt]. The fundamental theorem of asset pricing still applies and the arbitrage-free price of a derivative is still given in terms of an expectation under a risk-neutral (equivalent) probability measure (Delbaen and Schachermayer, 1994). This is, an European-type contingent payment f([S.sub.T]) has an arbitrage-free price [C.sub.t] given by

[C.sub.t] = [E.sup.Q][[e.sup.-r(T - t)] f([S.sub.T]) | [F.sub.t]], (5)

where Q is an probability measure, equivalent to P, under which the discounted asset [e.sup.-rt][S.sub.t] is a ([F.sub.t], Q)-martingale. It turns out such markets are incomplete and therefore, unlike the geometric Brownian motion case, there exist an infinite number of such equivalent martingale measures (Eberlein and Jacod, 1997). Despite the lack of uniqueness of the equivalent martingale measure, if we restrict ourselves to a certain family of equivalent measures, closed-form formulas can be obtained. A common choice is the mean-correcting martingale measure (see Schoutens, 2003, for a thorough description). Under this change of measure, the original drift parameter [mu] becomes [mu] + [[theta].sup.*], therefore, the name mean correcting. Within this family of equivalent measures, it can be shown that a Black-Scholes-like formula can be obtained for a European-type contingent claim. This fact is stated in the following result.

[FIGURE 1 OMITTED]

Theorem 1: Let St be the exponential NIG-Levy price process defined in (4) with parameters [[alpha], [beta], [delta], [mu]]. One possible arbitrage-free price of a European-type contingent pay-off f([S.sub.T]) at time t is given by

[C.sub.t] = [E.sup.Q[theta]*][[e.sup.-r(T-t)] f([S.sub.T]) | [F.sub.t]], (6)

where [Q.sub.[theta]*] is an equivalent martingale measure under which St is an exponential NIG-Levy process with parameters [[alpha], [beta], [delta], [mu + [[theta].sup.*]] and the value of [[theta].sub.*] is given by

[[theta].sup.*] = r - [mu] + [delta][[square root of ([[alpha].sup.2] - [([beta] + 1]).sup.2])] - [square root of ([[alpha].sup.2] - [[beta].sup.2])]. (7)

Proof: This is a standard result in the literature. We refer to Schoutens (2003) for more details. Q.E.D.

We remark that in this setting the parameter [mu] plays the role of the drift in the Black-Scholes setting. We notice that for this equivalent measure to exist we need the condition [([beta] + 1).sup.2] < [[alpha].sup.2].

Theorem 2 immediately yields the following Black-Scholes-type formula for a European call with strike K (i.e., f([S.sub.T], K) = [([S.sub.T] - K).sub.+]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where [[theta].sup.*] is given in (7). We remark that, just like in the classical Black-Schole setting, the call option pricing Formula (8) does not depend in [mu]. Indeed, a closer look at Equation (8) shows that the parameter [mu] cancels out with the one appearing inside the parameter [[theta].sup.*].

Formula (8) has the same structure as the Black-Scholes formula and it is just as simple to compute. It only requires the evaluation of the distribution function of a NIG. This can be easily implemented through Equation (3).

For a comprehensive treatment of equivalent martingale measures and non-Gaussian option pricing we refer the reader to Prause (1999) and Schoutens (2003). In the following section, we discuss how we can define a distortion operator based on a NIG distribution.

A NEW CLASS OF DISTORTION OPERATORS

In Wang (2000), the author discusses an approach to price financial and insurance risks alike. This approach stems out of the dual theory of risk developed in the economic literature (Yaari, 1987). Under this approach, the risk premium (or price) of an insurance (or financial) position is given in terms of a distortion function g that somehow encapsulates the perceived risk aversion. Let X be a random variable representing a financial (insurance) risk and let [F.sub.X] and [S.sub.X] be its distribution and survival function respectively. The premium (price) associated with this position is

[PI](X) = [integral] g ([S.sub.X](x)) dx, (9)

where g is an increasing differentiable function with 0 [less than or equal to] g(x) [less than or equal to] 1 for all x. Notice that all integrals in this section have to be understood as integrals over the entire domain of the random variable X. Moreover, this function is such that g(0) = 0 and g(1) = 1. This function is a so-called distortion function. A close look at Equation (9) shows that the premium function 1-I can be seen as a corrected mean under a new density measure given by [eta](dx) = g'([S.sub.X](x))d[F.sub.X](x), that is,

[PI](X) = [integral] x g'([S.sub.X](x)) d[F.sub.X](x) = [E.sup.[eta]][X], (10)

where the [E.sup.[eta]] denotes expectation under the density measure [eta].

Wang (2000) proposes the following class of distortion function based on the normal distribution in order to price insurance and financial risks.

[g.sub.[alpha]](u) = [THETA]([[THETA].sup.-1](u) + [alpha]), (11)

where [THETA] is the standard normal cumulative distribution function. We refer to Wang (2000) for a thorough discussion of this class of distortions and their properties.

In a later article, Wang (2004) proposes a second class of distortion function based on a Student-t distribution.

[g.sub.[alpha]](u) = [[PSI].sub.k]([[THETA].sup.-1](u) + [alpha]), (12)

where [THETA] is the standard normal cumulative distribution function and [[PSI].sub.k] is a Student-t cumulative distribution with k degrees of freedom.

This distortion allows for asymmetric distortion of the tail probabilities. We refer to the original article Wang (2004) for a comprehensive discussion on these features.

Later in this section, we introduce a new class of distortion functions that improves upon the original distortion (11) in a similar fashion as the more recent class of operators in (12). In addition, a numerical comparison of all three distortions is carried out. For now, we focus on the consistency of financial pricing using distortion operators and the classical Black-Scholes framework.

In Hamada and Sherris (2003), they show that the distortion in (11) is consistent with the Black-Scholes formula. Let us consider the following pricing kernel associated to the distortion (11),

H[X = h(Z);[alpha]] = [integral] [g.sub.[alpha]]([S.sub.X](x)) dx,

where h is a continuous, positive and increasing function. It is a straightforward exercise to show that, for a normal random variable Z

H[X = h(Z);[alpha]] = E[h(Z + [alpha])].

In the standard Black-Scholes model, the risk position at time [X.sub.t] is modeled by a geometric Brownian motion, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where W is a standard Brownian motion.

If we consider the payoff of a European call option (with maturity T and strike price K), we can write

C([X.sub.T], K) = [([X.sub.T] - K).sub.+],

where XT is a log-normal random variable. If we now apply the pricing kernel to this payoff with [[alpha].sup.*] - ([mu] - r)[square root of (T)]/[sigma], we can show that

[e.sup.-rT] H[C([X.sub.T], K); - [[alpha].sup.*]] = [X.sub.0][THETA]([d.sub.1]) - [e.sup.-rT] K[THETA]([d.sub.2]),

where

[d.sub.1] = ln([X.sub.0]/K + (r + [[sigma].sup.2]/2)T/[sigma][square root of (T)], [d.sub.2] = [d.sub.1] - [sigma][square root of (T)],

and r the market risk-free rate. That is, we can recuperate the Black-Scholes formula through the distortion operator (11).

Along the same lines, we introduce a generalized version of the distortion (11) that is based on a NIG distribution rather than on a normal distribution. Moreover, we show that the mean-correcting non-Gaussian Black-Scholes formula can be recuperated from this new NIG-based distortion operator.

Definition 2: Let [[THETA].sup.NIG] denote the NIG cumulative distribution function NIG([square root of [alpha][sigma]], - [beta][square root of ([delta]/[alpha])], [square root of ([alpha][delta]]), 0). We define de NIG distortion as

[g.sub.[alpha], [beta], [sigma], [theta]] (u) = [[THETA].sup.NIG] ([[THETA].sup.NIG-1] (u) + [theta]). (13)

We notice that unlike the distortions in (11) and (12), this generalized distortion has four parameters that can be calibrated from data just like the [alpha] parameter in Wang's distortion (11) and the [alpha] and k parameters in (12). In fact, for computational purposes the calibration for these parameters can be reduced to calibrating only three using the following reparametrization [xi] = [square root of ([alpha][delta])] and [zeta] = - [beta][square root of ([delta]/[alpha])]. In all, these extra parameters give more flexibility to this family of distortions as we will illustrate in the "Simulation and Further Analysis" section. Indeed, the resulting risk-adjusted probability can be thought of as containing a risk premium for higher moments through these parameters. In particular, the skewness of any data set will be captured through the parameter [beta]. This is in fact an interesting feature of this new distortion. Since it is based on a skewed distribution, the underlying probabilities are distorted asymmetrically at the tails through the parameter [beta]. In Wang's distortion, both tails are distorted in the same way because of the symmetry of the normal distribution. In this respect, the NIG distortion (13) is somewhat similar to the asymmetric distortion (12).

We start our discussion with the following proposition that shows the effect of the NIG distortion on a NIG random variable.

Proposition 2: Consider the NIG distortion [g.sub.[alpha],[beta],[delta],0] defined in (13). Let Z be a random variable with distribution given by NIG([alpha], [beta], [delta], [mu]) and let X = h(Z) be a transformation through a continuous, positive and increasing function h. Then,

H[X;[THETA]] [equivalent to] [integral] [g.sub.[alpha],[beta],[delta],[theta]([S.sub.X](x)) dx = E[h(Z + [theta] [square root of ([delta]/alpha])]]. (14)

Proof: By definition we have that

H[X;[THETA]] [equivalent to] [integral] [g.sub.[alpha],[beta],[delta],[theta]([S.sub.X](x)) dx.

Now,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But from Remark 4, we have that

Z - [mu]/[square root of ([sigma]/[alpha])] ~ NIG([square root of ([alpha][delta])], [beta][square root of ([delta]/[alpha])], [square root of ([alpha][delta]), 0).

This implies that

[S.sub.X](t) = 1 - NIG [[h.sup.-1](t) -[mu]/[square root of ([delta]/[alpha])]; [square root of ([delta]/[alpha])], [square root of ([alpha][delta]) 0].

Because of the symmetry property of the parameter [beta] in Remark 2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[THETA].sup.NIG] denotes the cumulative distribution function NIG([square root of [alpha][delta])], [beta][square root of ([delta]/[alpha])], [square root of ([alpha][delta])], 0). If we now apply the distortion [g.sub.[alpha],[beta],[delta],[theta]] on [S.sub.X] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we use the symmetry property of the [beta] parameter again we have

where Y ~ NIG([square root of ([alpha][delta])], [beta] [square root of ([delta]/[alpha]], [square root of ([alpha][delta])], 0). Using Remark 4, we can see that [square root of ([delta]/[alpha])] Y + mu ~ NIG([alpha], [beta], [delta], [mu]), so that we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where Z ~ NIG([alpha], [beta], [delta], [mu]).

This implies that

[integral] [g.sub.[alpha],[beta],[delta],[theta]]([S.sub.X](x)) dx = E[h(Z + [theta] [square root of ([delta]/[alpha])]],

which concludes the proof. Q.E.D.

This result shows that under a [g.sub.[alpha],[beta],[delta],[theta]] distortion, a NIG random variable is translated by a factor [theta] [square root of ([delta]/[alpha]]. This generalizes the equivalent result found in Hamada and Sherris (2003).

We now have to study how this distortion affects an exponential Levy model for asset prices and in particular if there is a value of [theta] such that discounted asset prices behave like risk-neutral asset prices. Let us consider the following exponential NIG asset price model

[S.sub.t] = [S.sub.0][e.sup.Zt], t > 0, (15)

where [Z.sub.t] is a ([F.sub.t], P)-NIG Levy process with parameters [[alpha], [beta],[delta], [mu]]. Then, the ([F.sub.T], P)-random variable [S.sub.T] is the price of the security at time T and it can be written as [S.sub.T] = h([Z.sub.T]) for a function h(u) = [S.sub.0][e.sup.u] and a random variable [Z.sub.T] with distribution NIG([alpha], [beta], [delta]T, [mu]T).

If we apply Proposition 2 we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This comes from Remark 4, which implies that [Z.sub.T] - [theta] [square root of ([delta]T/[alpha])] ~ NIG([alpha], [beta], [delta]T, [mu]T -- [theta] ([delta]T/[alpha])] and from the form of the expectation of a LogNIG random variable give in Remark 3.

It can be easily seen that if we set

[theta] = [mu] - r - [delta][[square root of ([[alpha].sup.2] - [([beta] + 1).sup.2]] - [square root of ([[alpha].sup.2] - [[beta].sup.2]]/[square root of ([delta]/[alpha])] [square root of (T)], (16)

then the expression for H[[S.sub.T]; -[theta]] simplifies to

H[[S.sub.T]; - [theta]] = [S.sub.0][e.sup.rT].

In other words, under the NIG distortion with a value of [theta] given in (16), the price [S.sub.T] evolves like a risk-neutral asset.

CONTINGENT CLAIMS PRICING

In this section, we show how the NIG distortion operator defined in the "A New Class of Distortion Operators" section is consistent with the non-Gaussian option pricing theory. In particular, we recuperate the non-Gaussian Black-Scholes option pricing Formula (8).

Let us consider the NIG asset model in (15) and a standard European call option payoff at maturity T given by f([S.sub.T], K) = [([S.sub.T] - K).sub.+]. This is clearly a function of the ([F.sub.T], P)-random variable ST that is the price of the security at time T. It can be written as [S.sub.T] = h([Z.sub.T]) for a function h(u) = [([S.sub.0][e.sup.u] - K).sub.+] and a random variable [Z.sub.T] with distribution NIG([alpha], [beta], [delta]T, [mu]T).

If we apply Proposition 2 we have that the price of this standard European call payoff is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

This comes from Remark 4 and from the fact that [Z.sub.T] ~ NIG([alpha], [beta], [delta]T, [mu]T). The values of z for which [S.sub.0][e.sup.z] - K > 0 is the interval (In K/[S.sub.0], [infinity]), and the integral in (17) becomes,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

If we set the parameter [theta] to be the one that makes [S.sub.T] evolve like a risk-neutral asset, that is,

[theta] = [mu] - r - [delta][[square root of ([[delta].sup.2] - [([beta] + 1).sup.2]] - [square root of ([[alpha].sup.2] - [[beta].sup.2]/[square root of ([delta]/[alpha]] [square root of (T)],

then Equation (18) becomes,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where

[[theta].sup.*] = r - [mu] + [delta][[square root of ([[alpha].sup.2] - (([beta] + 1).sup.2]] - [[square root of ([[alpha].sup.2] - [[beta].sup.2]].

The first integral in (19) can be reduced to a much simpler form by directly using the expression in (1) for the NIG density. This yields,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that the price of a standard European pay-off evaluated with the pricing kernel associated to the NIG distortion (13) with a parameter 0* is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Equation (20) is the Black-Scholes-type formula in (8) at t = 0. This shows that the NIG-distorted pricing kernel in (13), with parameter [[theta].sup.*], reduces to the Black-Scholes-type formula under the mean-correcting martingale measure.

SIMULATION AND FURTHER ANALYSIS

In this section, we carry out an empirical analysis of the performance of a pricing kernel based on (13) under different types of data sets. Our aim is two-fold. First, we want to corroborate the findings in Hamada and Sherris (2003) according to which the Wang distortion operator (11) performs poorly when it is used outside a Gaussian setting. This is a major drawback since it is a well-documented fact that market log-returns are anything but Gaussian. Our second aim is to show how the NIG distortion operator in (13) performs better in a wide range of non-Gausian situations. This is an interesting feature of the NIG operator since one would expect to use such a pricing kernel on actual market asset prices and not on Gaussian examples.

In order to provide a controlled setting for our analysis, simulated data are used to test all three distortions discussed in this article, namely, the normal distortion (11), the Student-t distortion (12) and the NIG distortion (13). These pricing kernels are tested on four different simulated data sets. Asset prices are simulated according to four well-known models for which theoretical option price formulas are available: a geometric Brownian model, a log-normal model with jumps, a constant elasticity of variance (CEV) model and a NIG-Levy model. Both pricing kernels were tested on these data sets in order to compare how these two distortions perform when evaluating options on underlying assets that do not have Gaussian log-returns.

Let X = C([S.sub.T], K) be the European call pay-off with strike K. These options are evaluated through Equation (9) using all three distortions in (11), (12), and in (13) under the four different models.

WANG DISTORTIONS

First, for all four models, the performance of both Wang's distortion operators (11) and (12) are evaluated. Using the simulated data, the empirical survival function [[??].sub.X] is computed. The parameter [alpha] is then calibrated to verify the risk-neutral condition such that

H[[S.sub.T]; - [alpha]] = [S.sub.0][e.sup.rT].

The latter is then used in order to compute the right-hand side in (9) with both distortions (11) and (12), that is,

H[X; -[alpha]] = [[integral].sup.[infinity].sub.0] [g.sub.[alpha]]([??].sub.X](x)) dx. (21)

NIG DISTORTION

The performance of the NIG distortion is also tested on the four models. Using the simulated data, the empirical survival function [[??].sub.X] is computed. The latter is then used in order to compute the right-hand side in (9) with the NIG distortion in (13), that is,

H[X; -[theta]] = [integral] [g.sub.[alpha],[beta],[delta],[theta]]([[??].sub.X](x)) dx. (22)

Here, parameters [alpha], [beta], and [delta] have to be first estimated from log-returns of simulated data. The estimation is done by fitting the NIG distribution to the simulated log-returns using maximum likelihood estimation. This estimations is implemented using the nigFit tool in R. Alternatively, we can refer to an algorithm developed in Karlis (2002) that can also be used to perform this estimation. Once [alpha], [beta] and [delta] have been estimated, the parameter [theta] can be calibrated to verify the risk-neutral condition, that is,

H[[S.sub.T]; -[theta]] = [S.sub.0][e.sup.rT]. (23)

Alternatively, parameters [alpha], [beta], and [delta] can be calibrated from market option prices through (22) and the parameter [theta] can still be calibrated through the risk-neutral condition (23). Both approaches yield similar results yet pricing using maximum likelihood estimation is easier to implement.

It is worthwhile pointing out again that these parameters bring new flexibility to the distortion (13). When calibrated, the resulting distortion produces a risk-adjusted distribution that can be thought of as including premiums for higher moments. In particular for skewness through the parameter [beta].

GEOMETRIC BROWNIAN MOTION MODEL

The geometric Brownian model is used as a benchmark for Wang's distortion. Prices are modeled by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [W.sub.t] is a standard Brownian motion. It is well known that the price of a European call under this model is given by the celebrated Black-Scholes formula. In Hamada and Sherris (2003), there is evidence that the normal-based distortion of Wang (2000) can accurately recuperate option prices under the Black-Scholes model. In order to verify this, we simulate 1,000 log-normal prices [S.sub.T] with drift [mu] = 16 percent, volatility [sigma] = 20 percent, and an initial price of [S.sub.0] = 20. The maturity and the risk-free rate are respectively set at T = 0.5 and r = 5 percent. The option prices are calculated for a range of strike prices K going from 16 to 24.

In Table 1 the theoretical option prices are shown and compared with options prices obtained through procedures (21) and (22). The prices obtained with Wang's distortions (11) and (12) are numerically equivalent to the Black-Scholes theoretical price as expected. The prices obtained with the NIG distortion also reproduce the numerical values obtained through the Black-Scholes formula. This first test shows that all three distortions perform well under Gaussian conditions. For the sake of completeness, in Table 2 we present results for a put option. These results are computed using the call-put parity relation. We remark that both distortions (NIG and normal) produce similar results and, in fact, different simulation seeds produce results where Wang's distortion can be slightly better than the NIG distortion or viceversa. This is not unexpected since we know that Wang's distortion replicates the Black-Scholes formula under a geometric Brownian motion model.

JUMP DIFFUSION MODEL

The geometric Brownian motion produces continuous sample paths with probability one. This feature can be considered too restrictive when it comes to modeling certain types of assets, such as stocks (see Merton, 1976). In Merton (1976) we find a very simple model that incorporates jumps in sample paths of the asset price. In Merton's model, the price of the asset is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [mu], [sigma], [lambda], [mu]J, and [sigma]J are constants; [W.sub.t] is a standard Brownian motion; [N.sub.t] is a Poisson Process with parameter [lambda; and {[Y.sub.i]} is a sequence of Gaussian random variables with mean [mu]J and standard deviation [sigma]J. Variables [N.sub.t] and [Y.sub.i]s are assumed to be all independent. Notice that this is the parametrization used in Hull (2009).

Merton's jump diffusion model produces an incomplete market where the equivalent martingale measure is not unique. However, there exists a martingale measure which only shifts the drift parameter [mu] to r, and keeps the rest unchanged (in particular the distributions of the jump times and sizes are identical under this martingale measure). This mean-correcting martingale measure is used to calculate an arbitrage-free price for a call option which is given by Merton (1976) as

[C.sub.t] = [[infinity].summation over (i=0)][e.sup.-[bar.[lambda]](T - t) [(bar.[lambda](T - t)).sup.i]/i! [C.sub.BS]([[sigma].sub.i], [r.sub.i])

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here [C.sub.BS] ([[sigma].sub.i], [r.sub.i]) denotes the standard Black-Scholes price for [[sigma].sub.i] and [r.sub.i] and with same initial asset price, maturity and strike price (respectively, [S.sub.t], T - t, and K).

Once again, 1,000 asset prices [S.sub.T] are simulated with [mu] = 30 percent, [sigma = 15 percent, [lambda] = [mu]J [sigma]J = -20 percent, 0.1 = 20 percent, and an initial price of [S.sub.0] = 20. The maturity and the risk-free rate are respectively set at T = 0.5 and r = 5 percent. The option prices are calculated for a range of strike prices K going from 16 to 24.

In Table 3 the theoretical option prices are shown and compared to options prices obtained through procedures (21) and (22). It can be seen that the prices obtained with the NIG distortion (13) are much closer to the theoretical arbitrage-free prices than those obtained with Wang's distortion (11). Indeed, the relative errors are smaller for the prices given by the NIG distortion. We can also observe that it performs as well as the Student-t distortion (12). This is a first example that shows that Wang's distortion performs poorly in a non-Gaussian context, whereas the Student-t and NIG operators are flexible enough to price options accurately. For the sake of completeness, in Table 4 we present results for a put option. These results are computed using the call-put parity relation.

CEV MODEL

In the literature, there is evidence that assets log-returns are heteroskedastic; that is, the variance of the log-returns is not constant in time (Black, 1975). Because geometric Brownian motion produces homoskedastic log-returns, an alternative model has been proposed by Cox and Ross (1976) to capture this heteroskedasticy feature. It is called the CEV model. In this model, the asset price is given by the solution of the stochastic differential equation

d[S.sub.t] = [mu][S.sub.t]dt + [sigma] [S.sub.t.sup.[beta]/2]d [W.sub.t], 0 [less than or equal to] t [less than or equal to] T,

where [W.sub.t] is a standard Brownian motion. We can recuperate the Black-Scholes model from the CEV by posing [beta] = 2. If [beta] < 2, volatility is a decreasing function of the asset price, which causes a heavy left tail in the distribution of the asset. Conversely, when [beta] > 2, volatility is an increasing function of the asset price, which creates a heavy right taft. This feature can be useful to capture the asymmetry in the log-returns distribution. Motivated by the fact that there seems to exist an inverse relation between stock prices and their volatility (Beckers, 1980) we chose to illustrate this model with [beta] < 2. Under the CEV model, when [beta] < 2, the density formula of the asset price is known in closed form (Schroeder, 1989). The formula for the unique arbitrage-free price of the call option under the CEV model when [beta] < 2 is also found in Schroeder (1989) and is given by,

[C.sub.t] = [S.sub.t](1 - [chi-square](a,b + 2,c)) - [Ke.sup.-r(T-t)[chi- square](c,b,a),

where

a = [K.sup.2-[beta]] -r(T-t)(2-[beta])/[omega][(1-0.5[beta]).sup.2],

b = 2/2-[beta], c = [S.sup.2-[beta].sub.t]/[omega][(1 - 0.5[beta].sup.2],

[omega] = [[sigma].sup.2]/r([beta] - 2) ([e.sup.r([beta]-2)(T-t)-1).

This is the parametrization used in Hull (2009) with [beta] = 2[alpha]. Here, [chi-square] 2(x; y, z) denotes the noncentral chi-square cumulative distribution function evaluated at x where y is the number of degrees of freedom and z is the parameter of noncentrality. Schroder (1989) gives a method to compute the noncentral chi-square cumulative distribution. Alternatively, the function dchisq in R can also be used to readily compute this distribution.

We simulate 1,000 asset prices [S.sub.T] with [mu] = 15 percent, [sigma] = 30 percent, [beta] = 1.5, and an initial price of [S.sub.0] = 20. The maturity and the risk-free rate are respectively set at T = 0.5 and r = 5 percent. The simulation was made using Milstein's method (Higham, 2001), which is a simulation technique based on a discretization scheme used to approximate Ito stochastic differential equation solutions. The option prices have been calculated for a range of strike prices K going from 16 to 24.

In Table 5 the theoretical option prices are shown and compared with options prices obtained through procedures (21) and (22). It can be seen that for in-the-money calls, the prices obtained with the NIG distortion and Wang's distortion are close, and the relative errors are small. We can also observe that the Student-t and the NIG operators show an overall better performance than the normal distortion. Once again, this is another situation where the NIG operator is more efficient than Wang's distortion to do the call pricing. For the sake of completeness, in Table 6 we present results for a put option. These results are computed using the call-put parity relation.

EXPONENTIAL NIG-LEVY MODEL

The last model on which all three distortions are tested is the exponential NIG-Levy model previously described in (4). The arbitrage-free call price is given by (8). Once more, 1,000 asset prices [S.sub.T] are simulated with [alpha] = 9, [beta] = 7.8, [delta] = 0.5, [mu] = -0.7, and an initial price of [S.sub.0] = 20. The maturity and the risk-free rate are respectively set at T = 0.5 and r = 5 percent. The option prices are calculated for a range of strike prices K going from 16 to 24.

In Table 7 the theoretical option prices are shown and compared to options prices obtained through procedures (21) and (22). The results of the simulation show that the NIG distortion over-performs Wang's distortion for every strike price. The relative error for NIG distortion prices remain smaller than 6.5 percent, while it can climb up from 13 percent to 34 percent for Wang's distortion. We can also observe that the Student-t operator performs poorly as well when compared to the NIG distortion in this case. For the sake of completeness, in Table 8 we present results for a put option. These results are computed using the call-put parity relation.

CONCLUSIONS AND FURTHER DISCUSSION

In this article we propose a generalized version of the distortions proposed in Wang (2000, 2004). This generalization uses a NIG distribution instead of a standard normal. All three distortions have been tested using simulation. This empirical analysis using simulation attempts to replicate non-Gaussian conditions as they could be found in market data. It turns out that the NIG operator performs well for various non-Gaussian models for the underlying asset whereas Wang's operators can sometimes provide poor estimations of the theoretical price.

The NIG distribution is a skewed distribution that has been proven to effectively fit financial log-returns. It comes as no surprise that a distortion operator based on this distribution performs better than Wang's under non-Gaussian conditions. An interesting feature of this distortion is that since it is based on a skewed distribution, it distorts differently the right and the left tails of the underlying distribution just like the Student-t distortion in Wang (2004). The advantage is that this asymmetry can be controlled through the parameter [beta] and a greater variety of shapes are possible. In this article we set out to compare these three distortions in an option pricing context, further testing is needed in order to compare the performance of these distortions in different applications.

As for potential applications, this research contribution joins the discussion in the literature regarding the connection between risk measures and skewed fat-tailed distributions in finance. Indeed, there have been several attempts to incorporate these stylized features of asset returns into risk measures (e.g., Bali and Theodossiou, 2008). In this article, we proposed a distortion-based risk measure that also incorporates these features. The proposed NIG distortion produces a distortion-based risk measure that has the same potential applications as the normal-based risk measure in Wang (2000). In fact, there is a wide range of possible applications that are now open to further analysis and comparison. Concrete illustrations are needed in order to test the NIG distortion and its performance with respect to other risk measures. In this article, we limit ourselves to give a formal construction as well as an empirical study of the NIG-based distortion. We are yet to test its performance in those instances where the normal-based and Student-t-based distortions have been put to use. This will be the subject of future studies. Here, we only mention one of many possible applications that are worth looking into with our newly defined distortion.

In Wang (2004), the author carries out an empirical analysis illustrating how yield spreads in catastrophe bonds can be modeled with the distortions given by (11) and (12). Similar examples can be tested with our NIG distortion. It would be interesting to compare the performance of our proposed distortion in the same context instead of using simulated data. In Wang (2004), the author uses yield spreads from transaction data for catastrophe bonds. He then calibrates the parameters [alpha] and k in the distortion (12) to these data set. The resulting model seems to explain the observed yields in the transactions. One possible direction for a future empirical study would be to compare all three distortions (11), (12), and (13) with a similar catastrophe bonds data set. The aim would be to calibrate all four parameters to similar data sets and compare the performance of all three models in describing the observed yield spreads. Such a study will illustrate how risk-adjustment for higher moments is achieved through all four parameters in the NIG distortion.

In general, our distortion (13) produces a family of risk measures through (9) and as such it can be applied in different settings ranging from capital allocation to optimal reinsurance. An interesting direction to be explored in future research would then be to test the performance of these newly defined family of NIG-based risk measures in those applications where general distortion-based risk measures have been used. In particular, a potential application of interest could be in the problem of optimal reinsurance. In the literature, we recently find studies that address this issue using risk measures (see, e.g., Bernard and Tian, 2009). Other relevant references that give a review of existing distortion risk measures and some of their applications in insurance are Dowd and Blake (2006) and Balbas, Garrido, and Mayoral (2009). Future empirical research can focus on applying the NIG-based distortion risk measure in the context discussed in the above mentioned articles.

REFERENCES

Balbas A., J. Garrido, and S. Mayoral, 2009, Properties of Distortion Risk Measures, Methodology and Computing in Applied Probability, 11:385-399.

Bali, T. G., and P. Theodossiou, 2007, A Conditional-SGT-VaR Approach With Alternative GARCH Models, Annals of Operations Research, 151:241-267.

Bali, T. G., and P. Theodossiou, 2008, Risk Measurement Performance of Alternative Distribution Functions, Journal of Risk and Insurance, 75(2):411-437.

Bali, T. G., and D. Weinbaum, 2007, A Conditional Extreme Value Volatility Estimator Based on High-Frequency Returns, Journal of Economic Dynamics and Control, 31: 361-397.

Barndorff-Nielsen, O. E., 1977, Exponentially Decreasing Distributions for the Logarithm of Particle Size, Proceedings of the Royal Society of London A, 353: 401-419.

Barndorff-Nielsen, O. E., and C. Halgreen, 1977, Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distributions, Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete, 38:439-455.

Barndorff-Nielsen, O. E., 1998, Processes of Normal Inverse Gaussian Type, Finance and Stochastics, 2:41-68.

Barndorff-Nielsen, O. E., T. Mikosh, and S. Resnick, eds., 2001, Levy Processes-Theory and Applications (Birkhauser).

Bauwens, L., and S. Laurent, 2002, A New Class of Multivariate Skew Densities, With Application to GARCH Models, Journal of Business and Economic Statistics, 23: 346-354.

Beckers, S., 1980, The Constant Elasticity of Variance Model and Its Implications for Option Pricing, Journal of Finance, 35:661-673.

Bernard, C., and W. Tian, 2009, Optimal Reinsurance Arrangements Under Tail Risk Measures, Journal of Risk and Insurance, 76:709-725.

Bertoin, J., 1996, L6vy Processes. Cambridge Tracts in Mathematics, Vol. 121 (Cambridge, UK: Cambridge University Press).

Black, E, 1975, Fact and Fantasy in the Use of Options, Financial Analysts Journal, 31: 36-41, 61-72.

Bollerslev, T., 1987, A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return, Review of Economics and Statistics, 69:542-547.

Cox, J., and S. A. Ross, 1976, The Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics, 3:145-166.

Delbaen, F., and W. Schachermayer, 1994, A General Version of the Fundamental Theorem of Asset Pricing, Annals of Mathematics, 300:463-520.

Dowd, K., and D. Blake, 2006, After VaR: The Theory, Estimation, and Insurance Applications of Quantile-Based Risk Measures, Journal of Risk and Insurance, 73(2): 193-229.

Eberlein, E., and U. Keller, 1995, Hyperbolic Distributions in Finance, Bernouilli, 1: 281-299.

Eberlein, E., 2001, Application of Generalized Hyperbolic Levy Motions to Finance, in: Levy Processes: Theory and Applications (Birkhauser).

Eberlein, E., and J. Jacod, 1997, On the Range of Options Prices, Finance and Stochastics, 1:131-140.

Emanuel, D., and J. MacBeth, 1982, Further Results on the Constant Elasticity of Variance Call Option Pricing Model, Journal of Financial and Quantitative Analysis, 17:533-554.

Ghose, D., and K. F. Kroner, 1995, The Relationship Between GARCH and Symmetric Stable Processes: Finding the Source of Fat Tails in Financial Data, Journal of Empirical Finance, 2:225-251.

Hamada, M., and M. Sherris, 2003, Contingent Claim Pricing Using Probability Distortion Operators: Methods From Insurance Risk Pricing and Their Relationship to Financial Theory, Applied Mathematical Finance, 10:19-47.

Harvey, C. R., and A. Siddique, 1999, Autoregressive Conditional Skewness, Journal of Financial and Quantitative Analysis, 34:465-487.

Highman, D. J., 2001, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, 43:525-546.

Hsieh, D. A., 1989, Modeling Heteroscedasticity in Daily Foreign-Exchange Rates, Journal of Business and Economic Statistics, 7:307-317.

Hull, J. C., 2009, Options, Futures and Other Derivatives, 7th edition (Upper Saddle River, NJ: Prentice Hall).

Jondeau, E., and M. Rockinger, 2003, Conditional Volatility, Skewness, and Kurtosis: Existence, Persistence, and Comovements, Journal of Economic Dynamics and Control, 27:1699-1737.

Jorgensen, B., 1982, Statistical Properties of the Generalized Inverse Gaussian Distribution, Lecture Notes in Statistics, Vol. 9 (New York: Springer).

Karlis, D., 2002, An EM Type Algorithm for Maximum Likelihood Estimation for the Normal Inverse Gaussian Distribution, Statistics and Probability Letters, 57:43-52.

McCulloch, J. H., 1997, Financial Applications of Stable Distributions, in: G. Maddala and C. Rao, eds., Handbook of Statistics, Vol. 14 (Amsterdam: Elsevier).

McDonald, J. B., and W. K. Newey, 1988, Partially Adaptive Estimation of Regression Models via the Generalized t Distribution, Econometric Theory, 4:428-457.

Merton, R. C., 1976, Option Pricing When Underlying Stock Returns Are Discontinuous, Journal of Financial Economics, 3:125-144.

Nagahara, Y., 1999, The PDF and CF of Pearson Type IV Distributions and the ML Estimation of the Parameters, Statistics and Probability Letters, 43:251-264.

Pilsun, C., and K. Nam, 2008, Asymmetric and Leptokurtic Distribution for Heteroscedastic Asset Returns: The SU-Normal Distribution, Journal of Empirical Finance, 15(1):41-63.

Prause, K., 1999, The Generalized Hyperbolic Model: Estimation, Financial Derivatives and Risk Measures, Ph.D. Thesis, Albert-Ludwigs University of Freiburg, Germany.

Raible, S., 2000, Levy Processes in Finance: Theory, Numerics and Empirical Facts, Ph.D. Thesis, Albert-Ludwigs University of Freiburg, Germany.

Richardson, M. P., and T. Smith, 1993, A Test for Multivariate Normality in Stock Returns, Journal of Business, 66:295-321.

Rockinger, M., and E. Jondeau, 2002, Entropy Densities With an Application to Autoregressive Conditional Skewness and Kurtosis, Journal of Econometrics, 106:119-142.

Sato, K. I., 1999, Levy Processes and Infinitely Divisible Distributions (Cambridge, UK: Cambridge University Press).

Schoutens, W., 2003, Levy Processes in Finance: Pricing Financial Derivatives (West Sussex, England: Wiley).

Schroder, M., 1989, Computing the Constant Elasticity of Variance Option Pricing Formula, Journal of Finance, 44:211-219.

Theodossiou, P., 1998, Financial Data and the Skewed Generalized t Distribution, Management Science, 44:1650-1661.

Theodossiou, P., and L. Trigeorgis, 2003, Option Pricing When Log-Returns Are Skewed and Leptokurtic, Working Paper, Rutgers University, School of Business.

Wang, K., C. Fawson, C. B. Barrett, and J. B. McDonald, 2001, A Flexible Parametric GARCH Model With an Application to Exchange Rates, Journal of Applied Econometrics, 16:521-536.

Wang, S., 2000, A Class of Distortion Operators for Pricing Financial and Insurance Risks, Journal of Risk and Insurance, 36:15-36.

Wang, S., 2004, CAT Bond Pricing Using Probability Transforms, Geneva Papers: Etudes et Dossiers, 278:19-29.

Yaari, M. E., 1987, The Dual Theory of Choice Under Risk, Econometrica, 55:95-115.

DOI: 10.1111/j.1539-6975.2011.01445.x

Frederic Godin is with the University of Montreal. Silvia Mayoral is with the University Carlos III of Madrid. The research of this author was partially funded by the Ministerio de Ciencia e Innovacion (ECO2009-10796). Manuel Morales is at the Department of Mathematics and Statistics, University of Montreal. Manuel-Morales can be contacted by morales@dms.umontreal.ca. This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) operating grant RGPIN-311660 and by Le Fonds quebecois de la recherche sur la nature et les technologies (FQRNT) operating grant NC-113809
```TABLE 1
Numerical (Call) Option Prices From a Sample of Asset Prices
Simulated From a Black-Scholes Model

Arbitrage-    Wang       Wang       NIG        NIG
Strike   Free Price   Price    Rel. Error   Price    Rel. Error

16         4.4349     4.4470     0.0027     4.4423     0.0017
17         3.5305     3.5283     0.0006     3.5234     0.0020
18         2.6997     2.6922     0.0028     2.6882     0.0043
19         1.9745     1.9631     0.0058     1.9611     0.0068
20         1.3777     1.3642     0.0098     1.3649     0.0093
21         0.9163     0.9102     0.0067     0.9134     0.0032
22         0.5813     0.5911     0.0169     0.5960     0.0252
23         0.3523     0.3694     0.0486     0.3749     0.0641
24         0.2045     0.2210     0.0804     0.2262     0.1060

Student   Student Rel.
Strike    Price     Rel. Error

16       4.4766       0.0094
17       3.5639       0.0095
18       2.7328       0.0123
19       2.0067       0.0163
20       1.4090       0.0227
21       0.9547       0.0419
22       0.6337       0.0901
23       0.4086       0.1597
24       0.2553       0.2484

TABLE 2
Numerical (Put) Option Prices From a Sample of Asset Prices Simulated
From a Black-Scholes Model

Arbitrage-    Wang       Wang       NIG        NIG
Strike   Free Price   Price    Rel. Error   Price    Rel. Error

16         0.0399     0.0520     0.3033     0.0473     0.1855
17         0.1107     0.1085     0.0198     0.1037     0.0637
18         0.2553     0.2478     0.0293     0.2438     0.0450
19         0.5054     0.4940     0.0226     0.4920     0.0266
20         0.8839     0.8704     0.0153     0.8711     0.0146
21         1.3978     1.3917     0.0044     1.3949     0.0021
22         2.0381     2.0479     0.0048     2.0528     0.0072
23         2.7844     2.8016     0.0061     2.8070     0.0081
24         3.6120     3.6284     0.0046     3.6336     0.0060

Student    Student
Strike    Price    Rel. Error

16       0.0816      1.0458
17       0.1442      0.3020
18       0.2884      0.1297
19       0.5376      0.0637
20       0.9152      0.0354
21       1.4362      0.0274
22       2.0905      0.0257
23       2.8407      0.0202
24       3.6628      0.0141

TABLE 3
Numerical (Call) Option Prices From a Sample of Asset Prices
Simulated From Merton's Model

Arbitrage-    Wang       Wang       NIG        NIG
Strike   Free Price   Price    Rel. Error   Price    Rel. Error

16         4.7860     4.9979     0.0443     4.7822     0.0008
17         3.9713     4.2384     0.0673     3.9796     0.0021
18         3.1981     3.5250     0.1022     3.2270     0.0090
19         2.4779     2.8695     0.1581     2.5412     0.0256
20         1.8304     2.2729     0.2418     1.9267     0.0526
21         1.2796     1.7428     0.3620     1.3955     0.0906
22         0.8437     1.2773     0.5139     0.9465     0.1218
23         0.5258     0.8863     0.6855     0.5916     0.1252
24         0.3124     0.5787     0.8525     0.3380     0.0818

Student    Student
Strike    Price    Rel. Error

16       5.0446      0.0540
17       4.2895      0.0801
18       3.5795      0.1192
19       2.9264      0.1810
20       2.3311      0.2736
21       1.8016      0.4080
22       1.3358      0.5833
23       0.9435      0.7944
24       0.6334      1.0276

TABLE 4
Numerical (Put) Option Prices from a Sample of Asset Prices
Simulated From Merton's Model

Arbitrage-    Wang       Wang       NIG        NIG
Strike   Free Price   Price    Ret. Error   Price    Rel. Error

16         0.3910     0.6029     0.5421     0.3872     0.0096
17         0.5515     0.8187     0.4844     0.5599     0.0152
18         0.7537     1.0805     0.4337     0.7826     0.0384
19         1.0088     1.4004     0.3882     1.0721     0.0628
20         1.3366     1.7791     0.3311     1.4328     0.0720
21         1.7611     2.2244     0.2630     1.8770     0.0658
22         2.3005     2.7341     0.1885     2.4033     0.0447
23         2.9579     3.3184     0.1219     3.0238     0.0223
24         3.7198     3.9862     0.0716     3.7454     0.0069

Student    Student
Strike    Price    Rel. Error

16       0.6496      0.6615
17       0.8698      0.5770
18       1.1350      0.5060
19       1.4572      0.4446
20       1.8373      0.3747
21       2.2831      0.2964
22       2.7926      0.2139
23       3.3756      0.1412
24       4.0409      0.0863

TABLE 5
Numerical (Call) Option Prices From a Sample of Asset Prices
Simulated From CEV  Model

Arbitrage-    Wang       Wang       NIG         MG
Strike   Free Price   Price    Rel. Error   Price    Rel. Error

16         4.3999     4.3994     0.0001     4.3975     0.0005
17         3.4443     3.4457     0.0004     3.4422     0.0006
18         2.5343     2.5424     0.0032     2.5357     0.0006
19         1.7214     1.7430     0.0126     1.7324     0.0064
20         1.0615     1.0877     0.0247     1.0744     0.0122
21         0.5868     0.6083     0.0367     0.5949     0.0138
22         0.2886     0.3060     0.0605     0.2952     0.0228
23         0.1259     0.1380     0.0965     0.1308     0.0393
24         0.0487     0.0521     0.0698     0.0483     0.0096

Student    Student
Strike    Price    Rel. Error

16       4.4121      0.0028
17       3.4649      0.0060
18       2.5690      0.0137
19       1.7749      0.0311
20       1.1220      0.0570
21       0.6425      0.0949
22       0.3375      0.1696
23       0.1640      0.3028
24       0.0708      0.4531

TABLE 6
Numerical (Put) Option Prices From a Sample of Asset Prices Simulated
From CEV Model

Arbitrage-    Wang       Wang       NIG        NIG
Strike   Free Price   Price    Rel. Error   Price    Rel. Error

16         0.0049     0.0044     0.0963     0.0025     0.4861
17         0.0245     0.0260     0.0596     0.0225     0.0831
18         0.0899     0.0980     0.0897     0.0913     0.0157
19         0.2522     0.2739     0.0857     0.2633     0.0439
20         0.5677     0.5939     0.0462     0.5806     0.0228
21         1.0683     1.0898     0.0202     1.0764     0.0076
22         1.7454     1.7629     0.0100     1.7520     0.0038
23         2.5580     2.5702     0.0047     2.5630     0.0019
24         3.4562     3.4596     0.0010     3.4557     0.0001

Student    Student
Strike    Price    Rel. Error

16       0.0171      2.5134
17       0.0451      0.8411
18       0.1245      0.3854
19       0.3058      0.2122
20       0.6282      0.1066
21       1.1240      0.0522
22       1.7944      0.0280
23       2.5961      0.0149
24       3.4783      0.0064

TABLE 7
Numerical (Call) Option Prices From a Sample of Asset Prices
Simulated From NIG  Model

Arbftrage-    Wang       Wang       NIG        NIG
Strike   Free Price   Price    Rel. Error   Price    Rel. Error

16         6.3365     5.1613     0.1855     6.4967     0.0253
17         5.9810     4.6600     0.2209     6.1939     0.0356
18         5.6739     4.2549     0.2501     5.9379     0.0465
19         5.4059     3.9265     0.2737     5.7194     0.0580
20         5.1698     3.6589     0.2922     5.5319     0.0700
21         4.9601     3.4369     0.3071     5.3686     0.0823
22         4.7725     3.2569     0.3176     5.2286     0.0956
23         4.6034     3.1055     0.3254     5.1057     0.1091
24         4.4501     2.9736     0.3318     4.9946     0.1224

Student    Student
Strike    Price    Rel. Error

16       9.0724      0.4318
17       8.5713      0.4331
18       8.1651      0.4391
19       7.8345      0.4492
20       7.5637      0.4630
21       7.3374      0.4793
22       7.1521      0.4986
23       6.9949      0.5195
24       6.8566      0.5408

TABLE 8
Numerical (Put) Option Prices From a Sample of Asset Prices
Simulated From NIG Model

Arbitrage-    Wang       Wang       NIG        NIG
Strike   Free Price   Price    Rel. Error   Price    Rel. Error

16         1.9415     0.7663     0.6053     2.1017     0.0825
17         2.5612     1.2402     0.5158     2.7742     0.0831
18         3.2295     1.8105     0.4394     3.4934     0.0817
19         3.9368     2.4574     0.3758     4.2503     0.0796
20         4.6760     3.1651     0.3231     5.0381     0.0774
21         5.4416     3.9185     0.2799     5.8501     0.0751
22         6.2293     4.7137     0.2433     6.6855     0.0732
23         7.0355     5.5377     0.2129     7.5378     0.0714
24         7.8576     6.3810     0.1879     8.4021     0.0693

Student    Student
Strike    Price    Rel. Error

16       4.6774      1.4091
17       5.1515      1.0113
18       5.7207      0.7714
19       6.3654      0.6169
20       7.0699      0.5119
21       7.8189      0.4369
22       8.6090      0.3820
23       9.4270      0.3399
24       10.2640     0.3063
```
COPYRIGHT 2012 American Risk and Insurance Association, Inc.
No portion of this article can be reproduced without the express written permission from the copyright holder.