On Properties of the Hyperbolic Distribution

Abstract

This paper is set to analytically describe properties of the hyperbolic distribution. This law, along with the variance-gamma distribution, is one of the most popular normal mean–variance mixtures from the point of view of various applications. We have found closed form expressions for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. The obtained formulas use the values of the Humbert confluent hypergeometric and Whittaker special functions. The results are applied to the problem of European option pricing in the related Lévy model of financial market. The research demonstrates that the discussed normal mean–variance mixture is analytically tractable.

1. Introduction

This work continues the list of papers where the properties and applications of the hyperbolic distribution are considered. The hyperbolic distribution was introduced as a member of the family of generalized hyperbolic distributions by Barndorff-Nielsen [1]. In that paper, the hyperbolic law was suggested to be applied to the grain size analysis.

Barndorff-Nielsen [2,3] employed the hyperbolic distribution for the modeling of the turbulence effects and in statistical physics, respectively. The hyperbolic distribution was discussed in a study of the particle size distribution of sand by Barndorff-Nielsen et al. [4] and Barndorff-Nielsen and Christiansen [5]. An application of this law to the grain size analysis of coastal sediments was presented in Hartmann and Bowman [6]. To estimate the parameters of the hyperbolic distribution, Hartmann and Bowman [6] used the method of moments implemented in a special statistical program. Bhatia and Durst [7] and Xu et al. [8] used the hyperbolic distribution to model the drop size in sprays. The use of the hyperbolic law in biology was discussed in Blaesild [9]. The hyperbolic distribution was considered for the modeling of aspects of the lava magnetization by Kristjansson and McDougall [10].

Investigations in the above-mentioned areas of research based on the hyperbolic law were proceeded in more recent works. Characteristics of fluids were examined in Babinsky and Sojka [11]. Sandy systems were studied by Ghoshal et al. [12] and Hajek et al. [13]. Similarly to Barndorff-Nielsen [1], the estimation of the parameters of the hyperbolic law was made in Hajek et al. [13], applying the likeness function instead of the likelihood one. A part of modern scientific literature relates to the applications of the hyperbolic distribution to the stochastic modeling of financial market data.

It was shown that the hyperbolic distribution fits well to the empirical distributions of stock returns first by Eberlein and Keller [14]. The authors focused on the shares of the Frankfurt Stock Exchange. To assess the parameters of the hyperbolic distribution, Eberlein and Keller [14] employed maximum likelihood estimation. Bibby and Sørensen [15] used the hyperbolic law to model the volatility of the stocks of Danish companies. Eberlein et al. [16] affirmed that the hyperbolic distribution approximates the US stock market data quite well, including the NYSE composite. Proceeding the research of Eberlein and Keller [14], Küchler et al. [17] confirmed that the hyperbolic distribution satisfactorily describes not only the DAX, but also the FAZ index. To fit the parameters of the hyperbolic law, Küchler et al. [17] applied a procedure of minimizing the Kullback–Leibler distance between the probability distribution and the observed frequencies over intervals. Bauer [18] successfully tested the hyperbolic model using the data of German stocks, Dow Jones, and Nikkei indices. Daskalaki and Katris [19] showed that the hyperbolic law distinguishes well the dynamics of the CSE financial index. They provided maximum likelihood estimates of the parameters using an iterative algorithm. Calibrating portfolios of the US stock indices with multidimensional models, Luciano et al. [20] found that the hyperbolic model fits very well with both the marginal distributions and long–short investment strategies. The marginal return parameters were calibrated in Luciano et al. [20] by maximum likelihood estimation. The correlation coefficients in the multivariate case were assessed by minimizing the Frobenius distance between model and empirical correlations. It was shown in Baciu [21] that the hyperbolic distribution is very close to the empirical one in the central part for the index of the Romanian market. Cryptocurrency returns were modeled with the hyperbolic law by Sheraz and Dedu [22].

Mathematically, the hyperbolic distribution was defined in Barndorff-Nielsen [1] as the normal mean–variance mixture with the hyperbolic inverse Gaussian mixing distribution. Primary properties of the hyperbolic distribution were summarized in Barndorff-Nielsen [1] and Blaesild [23]. The Lévy measure of hyperbolic distribution was found by Eberlein and Keller [14]. The absolute moments were derived in Barndorff-Nielsen and Stelzer [24]. Special statistical techniques based on the hyperbolic law were developed in Leobacher and Pillichshammer [25] and Fonseca et al. [26]. Specifically, Fonseca et al. [26] compared the Bayesian and maximum likelihood estimators in the hyperbolic model. Following the objectives of the financial modeling including the option pricing, Eberlein and Keller [14] and Bingham and Kiesel [27] discussed the equivalent martingale change of measure for the exponent of hyperbolic distribution. Eberlein et al. [16] and Bauer [18] estimated the value at risk in the hyperbolic model. The employment of the hyperbolic law to the pension fund assessment was considered by Kabašinskas et al. [28]. An extension of the hyperbolic inverse Gaussian distribution which does not belong to the family of generalized inverse Gaussian distributions was discussed in Tan et al. [29].

As analytic representations for the cumulative distribution function, partial moments and partial-moment-generating functions of the hyperbolic distribution are not known; the computations in the hyperbolic model have to be elaborated using the Fourier transform method (Eberlein [30]) or employing a suitable simulation algorithm (Ch. XII of Asmussen and Glynn [31]). But if a distribution is considered as the substitution for the normal one in stochastic modeling, then it is desirable to describe the characteristics of this distribution in closed forms as it is made for the Gaussian one. In this paper, we obtain analytic expressions for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. The established formulas afford to obtain in a closed form the price of the European call option.

The rest of the paper is organized as follows. Section 2 recalls the properties of hyperbolic law which are useful in the research. Section 3 formulates the main results of the work. Applications to the option pricing are given in Section 4. Section 5 numerically analyzes the difference between the normal and hyperbolic models. The method of proofs develops the ideas of Madan et al. [32], Ano and Ivanov [33], and Ivanov [34]. Detailed proofs are placed in Appendix A and Appendix B.

2. Materials and Methods

In this section, we confirm that the hyperbolic distribution can be defined as the normal mean–variance mixture where the mixing density has the hyperbolic inverse Gaussian distribution. Then, we recall the formulas for the probability density and moment-generating functions of the hyperbolic law.

2.1. Hyperbolic Inverse Gaussian Distribution

According to Eberlein and Keller [14] or Barndorff-Nielsen et al. [35], the hyperbolic inverse Gaussian distribution $\mathrm{HIG}=\mathrm{HIG}(\eta ,\delta )$ has two parameters $\eta ,\delta >0$ and the probability density function

$$\begin{array}{c}\hfill f_{\mathrm{HIG}}\left(x\right)=C_{\mathrm{HIG}}{e}^{-{\displaystyle \frac{{\delta }^2}{2x}}-{\displaystyle \frac{{\eta }^{2}x}{2}}}\end{array}$$

(1)

with

$$\begin{array}{c}\hfill C_{\mathrm{HIG}}={\displaystyle \frac{\eta }{2\delta {\mathrm{K}}_1\left(\delta \eta \right)}},\end{array}$$

(2)

where ${\mathrm{K}}_{{\varsigma }_1}\left({\varsigma }_2\right)$ is the modified Bessel function of the second kind.

Next, formula 3.471.9 of Gradshteyn and Ryzhik [36] includes the identity

$$\begin{array}{c}\hfill {\int }_0^{\infty }{x}^{{\varsigma }_1-1}{e}^{-{\displaystyle \frac{{\varsigma }_2}{x}}-{\varsigma }_{3}x}dx=2{\left({\displaystyle \frac{{\varsigma }_2}{{\varsigma }_3}}\right)}^{{\displaystyle \frac{{\varsigma }_1}{2}}}{\mathrm{K}}_{{\varsigma }_1}\left(2\sqrt{{\varsigma }_2{\varsigma }_3}\right)\end{array}$$

(3)

for $\mathrm{Re}{\varsigma }_2>0$ and $\mathrm{Re}{\varsigma }_3>0$. It is easy to find from (3) that the $\varsigma$th moment of the HIG distribution for $\varsigma \ge 0$ is

$${\mathrm{M}}_{\mathrm{HIG},\varsigma }=\mathrm{E}{\left(\mathrm{HIG}\right)}^{\varsigma }={\displaystyle \frac{{\delta }^{\varsigma }{\mathrm{K}}_{1+\varsigma }\left(\delta \eta \right)}{{\eta }^{\varsigma }{\mathrm{K}}_1\left(\delta \eta \right)}}.$$

(4)

Once again, applying (3), one can find that the characteristic function of the HIG distribution is

$${\phi }_{\mathrm{HIG}}\left(\lambda \right)=\mathrm{E}{e}^{i\lambda \mathrm{HIG}}={\displaystyle \frac{\eta {\mathrm{K}}_1\left(\delta \sqrt{{\eta }^2-2i\lambda }\right)}{{\mathrm{K}}_1\left(\delta \eta \right)\sqrt{{\eta }^2-2i\lambda }}}.$$

If $\lambda <{\displaystyle \frac{{\eta }^2}{2}}$, then the moment-generating function of the HIG law exists and

$${\mathrm{mgf}}_{\mathrm{HIG}}\left(\lambda \right)=\mathrm{E}{e}^{\lambda \mathrm{HIG}}={\displaystyle \frac{\eta {\mathrm{K}}_1\left(\delta \sqrt{{\eta }^2-2\lambda }\right)}{{\mathrm{K}}_1\left(\delta \eta \right)\sqrt{{\eta }^2-2\lambda }}}.$$

2.2. Hyperbolic Distribution

2.2.1. Definition and Moments

The hyperbolic distribution $\mathrm{H}$ is defined (see Eberlein and Keller [14] or Ch. III.1d of Shiryaev [37]) as

$$\begin{array}{c}\hfill \mathrm{H}=\mu +\beta \mathrm{HIG}+\sigma \mathrm{N}\sqrt{\mathrm{HIG}},\end{array}$$

(5)

where $\mu ,\beta \in \mathbb{R}$, $\sigma >0$ are constants, $\mathrm{N}=\mathrm{N}(0,1)$ is the standard normal distribution, and $\mathrm{HIG}$ is the hyperbolic inverse Gaussian distribution with the density (1) independent with the normal one. Both distributions are assumed to be defined at a probability space $(\Omega ,\mathcal{F},\mathrm{P})$.

Employing the binomial expansion two times, we obtain that the moments of the hyperbolic distribution for $n\in \mathbb{N}\cup \left\{0\right\}$ are

$$\begin{array}{cc}\hfill {\mathrm{M}}_{\mathrm{H},n}& =\mathrm{E}{\left(\mu +\beta \mathrm{HIG}+\sigma \mathrm{N}\sqrt{\mathrm{HIG}}\right)}^n=\mathrm{E}\sum _{k=0}^n{\mathrm{C}}_n^k{\left(\beta \mathrm{HIG}+\sigma \mathrm{N}\sqrt{\mathrm{HIG}}\right)}^k{\mu }^{n-k}\hfill \\ \hfill & =\sum _{k=0}^n{\mathrm{C}}_n^k{\mu }^{n-k}\mathrm{E}\sum _{m=0}^k{\mathrm{C}}_k^m{\sigma }^m{\beta }^{k-m}{\left(\mathrm{HIG}\right)}^{k-{\displaystyle \frac{m}{2}}}{\mathrm{N}}^m\hfill \\ \hfill & =\sum _{k=0}^n{\mathrm{C}}_n^k{\mu }^{n-k}\sum _{l=0}^{[k/2]}{\mathrm{C}}_k^{2l}{\sigma }^{2l}{\beta }^{k-2l}{\mathrm{M}}_{\mathrm{HIG},k-l}(max\{2l-1,1\})!!,\hfill \end{array}$$

where $\left[\varsigma \right]$ denotes the integer part of $\varsigma$, $\varsigma !!$ is the double factorial of $\varsigma$, and ${\mathrm{C}}_{{\varsigma }_2}^{{\varsigma }_1}={\displaystyle \frac{{\varsigma }_2!}{{\varsigma }_1!({\varsigma }_2-{\varsigma }_1)!}}$ are the binomial coefficients.

2.2.2. Probability Density Function

Let $\alpha =\sqrt{{\eta }^2+{\beta }^2}$. Then we have from (3) and (5) that the probability density function of the hyperbolic distribution $\mathrm{H}=\mathrm{H}(\mu ,\sigma ,\alpha ,\beta ,\delta )$ is

$$\begin{array}{cc}\hfill & f_{\mathrm{H}}\left(x\right)={\int }_0^{\infty }{\displaystyle \frac{C_{\mathrm{HIG}}}{\sigma \sqrt{2\pi y}}}{e}^{-{\displaystyle \frac{{\left(x-\mu -\beta y\right)}^2}{2{\sigma }^{2}y}}-{\displaystyle \frac{{\delta }^2}{2y}}-{\displaystyle \frac{({\alpha }^2-{\beta }^2)y}{2}}}dy\hfill \\ \hfill =& {\displaystyle \frac{C_{\mathrm{HIG}}{e}^{\frac{\beta (x-\mu )}{{\sigma }^2}}}{\sqrt{2\pi }}}{\int }_0^{\infty }{y}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{{(x-\mu )}^2+{\sigma }^2{\delta }^2}{2{\sigma }^{2}y}}-{\displaystyle \frac{{\beta }^2+{\sigma }^2\left({\alpha }^2-{\beta }^2\right)}{2{\sigma }^2}}y}dy\hfill \\ \hfill =& {\displaystyle \frac{C_{\mathrm{HIG}}\sqrt{2}{e}^{\frac{\beta (x-\mu )}{{\sigma }^2}}}{\sqrt{\pi }}}{\left({\displaystyle \frac{{(x-\mu )}^2+{\sigma }^2{\delta }^2}{{\beta }^2+{\sigma }^2\left({\alpha }^2-{\beta }^2\right)}}\right)}^{{\displaystyle \frac{1}{4}}}\hfill \\ \hfill \times & {\mathrm{K}}_{{\displaystyle \frac{1}{2}}}\left({\displaystyle \frac{{\left({(x-\mu )}^2+{\sigma }^2{\delta }^2\right)}^{\frac{1}{2}}{\left({\beta }^2+{\sigma }^2\left({\alpha }^2-{\beta }^2\right)\right)}^{\frac{1}{2}}}{{\sigma }^2}}\right).\hfill \end{array}$$

Since

$${\mathrm{K}}_{\pm {\displaystyle \frac{1}{2}}}\left(\varsigma \right)=\sqrt{{\displaystyle \frac{\pi }{2\varsigma }}}{e}^{-\varsigma }$$

(6)

with respect to formula 8.469.3 of Gradshteyn and Ryzhik [36], we find that

$$\begin{array}{cc}\hfill f_{\mathrm{H}}\left(x\right)& {\displaystyle \frac{C_{\mathrm{HIG}}\sigma }{\sqrt{{\beta }^2+{\sigma }^2\left({\alpha }^2-{\beta }^2\right)}}}{e}^{{\displaystyle \frac{\beta (x-\mu )-\sqrt{\left({\beta }^2+{\sigma }^2\left({\alpha }^2-{\beta }^2\right)\right)\left({(x-\mu )}^2+{\sigma }^2{\delta }^2\right)}}{{\sigma }^2}}}\hfill \\ \hfill =& {\displaystyle \frac{C_{\mathrm{HIG}}\sigma }{\sqrt{{\beta }^2+{\sigma }^2\left({\alpha }^2-{\beta }^2\right)}}}{e}^{\widehat{\beta }(x-\mu )-\widehat{\alpha }\sqrt{{(x-\mu )}^2+{\sigma }^2{\delta }^2}}\hfill \\ \hfill =& {\widehat{C}}_{\mathrm{HIG}}{e}^{\widehat{\beta }(x-\mu )-\widehat{\alpha }\sqrt{{(x-\mu )}^2+{\widehat{\delta }}^2}}\hfill \end{array}$$

(7)

with

$$\widehat{\beta }={\displaystyle \frac{\beta }{{\sigma }^2}},\widehat{\delta }=\sigma \delta ,\widehat{\alpha }={\displaystyle \frac{\sqrt{{\beta }^2+{\sigma }^2\left({\alpha }^2-{\beta }^2\right)}}{{\sigma }^2}}$$

and

$${\widehat{C}}_{\mathrm{HIG}}={\displaystyle \frac{\sqrt{{\alpha }^2-{\beta }^2}}{2\widehat{\alpha }\widehat{\delta }{\mathrm{K}}_1\left(\delta \sqrt{{\alpha }^2-{\beta }^2}\right)}}.$$

When $\sigma =1$, we see from (7) that

$$f_{\mathrm{H}}\left(x\right)={\displaystyle \frac{\sqrt{{\alpha }^2-{\beta }^2}}{2\alpha \delta {\mathrm{K}}_1\left(\delta \sqrt{{\alpha }^2-{\beta }^2}\right)}}{e}^{\beta (x-\mu )-\alpha \sqrt{{(x-\mu )}^2+{\delta }^2}},$$

(8)

same as it is indicated by (9) of Eberlein and Keller [14].

2.2.3. Moment-Generating Function

So far as the characteristic function of the hyperbolic distribution

$${\phi }_{\mathrm{H}}\left(\lambda \right)=\mathrm{E}\left(\mathrm{E}\left(e^{i\lambda \mathrm{H}}|\mathrm{HIG}\right)\right)$$

and

$$\mathrm{E}\left(e^{i\lambda \mathrm{H}}|\mathrm{HIG}\right)=e^{i\lambda \mu }{\phi }_{\mathrm{N}\left(\beta \mathrm{HIG},\sigma \sqrt{\mathrm{HIG}}\right)}=e^{i\lambda \left(\mu +\beta \mathrm{HIG}\right)-{\displaystyle \frac{1}{2}}{\sigma }^2{\lambda }^2\mathrm{HIG}},$$

we find that

$$\begin{array}{cc}\hfill {\phi }_{\mathrm{H}}\left(\lambda \right)=& e^{i\lambda \mu }\mathrm{E}{e}^{i\lambda \beta \mathrm{HIG}-{\displaystyle \frac{1}{2}}{\sigma }^2{\lambda }^2\mathrm{HIG}}=C_{\mathrm{HIG}}{e}^{i\lambda \mu }{\int }_0^{\infty }{e}^{-{\displaystyle \frac{{\delta }^2}{2x}}-{\displaystyle \frac{\left({\alpha }^2+{\sigma }^2{\lambda }^2-{\beta }^2-2i\lambda \beta \right)x}{2}}}\hfill \\ \hfill =& e^{i\lambda \mu }{\displaystyle \frac{{\left({\alpha }^2-{\beta }^2\right)}^{\frac{1}{2}}{\mathrm{K}}_1\left(\delta \sqrt{{\alpha }^2+{\sigma }^2{\lambda }^2-{\beta }^2-2i\lambda \beta }\right)}{{\left({\alpha }^2+{\sigma }^2{\lambda }^2-{\beta }^2-2i\lambda \beta \right)}^{\frac{1}{2}}{\mathrm{K}}_1\left(\delta \sqrt{{\alpha }^2-{\beta }^2}\right)}}\hfill \end{array}$$

(9)

in accordance with (3). The moment-generating function of H exists if

$${\alpha }^2-{\sigma }^2{\lambda }^2-{\beta }^2-2\lambda \beta >0\iff$$

$$\iff {\left(\widehat{\beta }+\lambda \right)}^2<{\displaystyle \frac{{\alpha }^2-{\beta }^2}{{\sigma }^2}}+{\widehat{\beta }}^2={\widehat{\alpha }}^2.$$

Therefore, we note that

$${\mathrm{mgf}}_{\mathrm{H}}\left(\lambda \right)=e^{\mu \lambda }{\displaystyle \frac{{\left({\alpha }^2-{\beta }^2\right)}^{\frac{1}{2}}{\mathrm{K}}_1\left(\delta \sqrt{{\alpha }^2-{\sigma }^2{\lambda }^2-{\beta }^2-2\lambda \beta }\right)}{{\left({\alpha }^2-{\sigma }^2{\lambda }^2-{\beta }^2-2\lambda \beta \right)}^{\frac{1}{2}}{\mathrm{K}}_1\left(\delta \sqrt{{\alpha }^2-{\beta }^2}\right)}}<\infty$$

(10)

if

$$-\widehat{\alpha }<\widehat{\beta }+\lambda <\widehat{\alpha }.$$

(11)

3. Results

Throughout this section, we introduce the analytical formulas for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. To formulate the results, we need to enter extra constants and functions.

Let us set

$$\varphi =\widehat{\alpha }\widehat{\delta }$$

(12)

and define for $\left|c\right|<1$ the function

$$\begin{array}{cc}\hfill & {\mathfrak{I}}_1(x,c)={\displaystyle \frac{1}{c^2-1}}({\displaystyle \frac{c}{\varphi }}{e}^{-\varphi \left(\sqrt{1+x^2}+c\left|x\right|\right)}\hfill \\ \hfill -& {\displaystyle \frac{\sqrt{\pi }}{2\varphi }}{\mathrm{W}}_{-{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}}\left(2\varphi \sqrt{1-c^2}\right)-\sqrt{1-c^2}{\mathrm{K}}_0\left(\varphi \sqrt{1-c^2}\right)),\hfill \end{array}$$

(13)

where ${\mathrm{W}}_{{\varsigma }_1,{\varsigma }_2}\left({\varsigma }_3\right)$ is the Whittaker function (see Ch. XVI of Whittaker and Watson [38]). Furthermore, let

$$\begin{array}{cc}\hfill & {\mathfrak{I}}_2(x,c,j)\hfill \\ \hfill =& {\displaystyle \frac{x^{\frac{1}{2}+j}{e}^{-\varphi \sqrt{1-c^2}}\sqrt{2}}{{\left(1-c^2\right)}^{\frac{3}{4}+\frac{j}{2}}}}{\Phi }_1\left({\displaystyle \frac{1}{2}}+j,{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}}+j;-{\displaystyle \frac{x}{2\sqrt{1-c^2}}},-\varphi x\right),j\in \mathbb{N}\cup \left\{0\right\},\hfill \end{array}$$

(14)

for $\left|c\right|<1$, where ${\Phi }_1({\varsigma }_1,{\varsigma }_2,{\varsigma }_3;{\varsigma }_4,{\varsigma }_5)$ is the Humbert confluent hypergeometric series in two variables. It is defined for $|{\varsigma }_4|<1$ as the sum (see (16) in Section 1.3 of Srivastava and Karlsson [39])

$$\begin{array}{c}\hfill {\Phi }_1({\varsigma }_1,{\varsigma }_2,{\varsigma }_3;{\varsigma }_4,{\varsigma }_5)=\sum _{m=0,n=0}^{\infty }{\displaystyle \frac{{\left({\varsigma }_1\right)}_{m+n}{\left({\varsigma }_2\right)}_m}{m!n!{\left({\varsigma }_3\right)}_{m+n}}}{\varsigma }_4^m{\varsigma }_5^n,\end{array}$$

where ${\left(\varsigma \right)}_l$, $l\in \mathbb{N}\cup \left\{0\right\}$ is the Pochhammer’s symbol. The Humbert confluent hypergeometric function can be established for other values of ${\varsigma }_4$ by an analytic continuation (see Section 5.11 of Bateman and Erdélyi [40]).

The results of the paper are provided in the terms of the auxiliary functions ${\mathfrak{I}}_1$ and ${\mathfrak{I}}_2$.

Theorem 1. 

The cumulative distribution function $F_{\mathrm{H}}\left(u\right)$ of the hyperbolic distribution $\mathrm{H}=\mathrm{H}(\mu ,\sigma ,\alpha ,\beta ,\delta )$ for $\beta \ge 0$ and $u\le \mu$ is computed by the identity

$$\begin{array}{c}\hfill F_{\mathrm{H}}\left(u\right)={\widehat{C}}_{\mathrm{HIG}}\widehat{\delta }({\mathfrak{I}}_1(u_1,c)-({\mathfrak{I}}_2(u_2,c,0)+{\displaystyle \frac{{\mathfrak{I}}_2(u_2,c,1)}{3}}\left)I_{\left\{u_2>0\right\}}\right)\end{array}$$

(15)

with

$$u_1={\displaystyle \frac{u-\mu }{\widehat{\delta }}},c={\displaystyle \frac{\widehat{\beta }}{\widehat{\alpha }}},u_2=\sqrt{1+u_1^2}+c\left|u_1\right|-\sqrt{1-c^2}.$$

Proof Sketch.

The proof consists of four stages.

Stage 1. We pass to the computation of the integral

$${\int }_{{\varsigma }_1}^{\infty }{y}^{{\varsigma }_2}{\left(y+2\sqrt{1-c^2}\right)}^{-{\displaystyle \frac{1}{2}}}{e}^{-\varphi y}dy$$

(16)

for ${\varsigma }_1\ge 0$, ${\varsigma }_2\in \{-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\}$ and $\varphi$ defined in (12).

Stage 2. The integral (16) is calculated for ${\varsigma }_1=0$ using the Whittaker function.

Stage 3. The integral (16) is computed for ${\varsigma }_1>0$ employing the result of Stage 2 and leveraging the Humbert series.

Stage 4. To obtain the final formula, the functions (13) and (14) are applied to the expressions found at Stages 2 and 3. □

The complete proof of Theorem 1 can be found in Appendix A. The next remark supplements the result of Theorem 1. We have a twin formula for $\beta \le 0$ and $u\ge \mu$.

Remark 1. 

Since

$$\begin{array}{cc}\hfill F_{\mathrm{H}}\left(u\right)=& \mathrm{P}\left(\mu +\beta \mathrm{HIG}+\sigma \mathrm{N}\sqrt{\mathrm{HIG}}\le u\right)\hfill \\ \hfill =& \mathrm{P}\left(-\mu -\beta \mathrm{HIG}+\sigma \mathrm{N}\sqrt{\mathrm{HIG}}\ge -u\right)\hfill \\ \hfill =& 1-\mathrm{P}\left(-\mu -\beta \mathrm{HIG}+\sigma \mathrm{N}\sqrt{\mathrm{HIG}}\le -u\right),\hfill \end{array}$$

we see from Theorem 1 the similar result for $\beta \le 0$ and $u\ge \mu$. Indeed, let $\tilde{\mu }=-\mu$, $\tilde{\beta }=-\beta$, $\tilde{u}=-u$, and $\tilde{\mathrm{H}}=\tilde{\mu }+\tilde{\beta }\mathrm{HIG}+\sigma \mathrm{N}\sqrt{\mathrm{HIG}}$, where

$$\mathrm{HIG}=\mathrm{HIG}\left(\sqrt{{\alpha }^2-{\beta }^2},\delta \right)=\mathrm{HIG}\left(\sqrt{{\alpha }^2-{\tilde{\beta }}^2},\delta \right).$$

We have that $\tilde{\beta }\ge 0$ since $\beta \le 0$ and $\tilde{u}\le \tilde{\mu }$ as $u\ge \mu$. Therefore, the conditions of Theorem 1 are satisfied for $\tilde{\mathrm{H}}=\tilde{\mathrm{H}}(\tilde{\mu },\sigma ,\alpha ,\tilde{\beta },\delta )$ and we have that 

$$\begin{array}{c}\hfill F_{\mathrm{H}}\left(u\right)=1-F_{\tilde{\mathrm{H}}}\left(\tilde{u}\right)=1-{\widehat{C}}_{\mathrm{HIG}}\widehat{\delta }({\mathfrak{I}}_1({\tilde{u}}_1,\tilde{c})-({\mathfrak{I}}_2({\tilde{u}}_2,\tilde{c},0)+{\displaystyle \frac{{\mathfrak{I}}_2({\tilde{u}}_2,\tilde{c},1)}{3}}\left)I_{\left\{{\tilde{u}}_2>0\right\}}\right)\end{array}$$

with

$${\tilde{u}}_1={\displaystyle \frac{\mu -u}{\widehat{\delta }}},\tilde{c}=-{\displaystyle \frac{\widehat{\beta }}{\widehat{\alpha }}},{\tilde{u}}_2=\sqrt{1+{\tilde{u}}_1^2}+\tilde{c}\left|{\tilde{u}}_1\right|-\sqrt{1-{\tilde{c}}^2}.$$

Onwards, we consider the partial (or truncated) moment-generating functions of the hyperbolic distribution. The lower and upper partial-moment-generating functions of $\mathrm{H}$ are defined as

$${\mathrm{mgf}}_{\mathrm{H}}^{lo}(\lambda ,u)=\mathrm{E}{e}^{\lambda \mathrm{H}}{I}_{\left\{\mathrm{H}\le u\right\}}\mathrm{and}{\mathrm{mgf}}_{\mathrm{H}}^{up}(\lambda ,u)=\mathrm{E}{e}^{\lambda \mathrm{H}}{I}_{\left\{\mathrm{H}\ge u\right\}},$$

respectively, for $u\in \mathbb{R}$. The corollary below gives us a formula for the lower partial-moment-generating function of the hyperbolic distribution under the assumption that the moment-generating function of $\mathrm{H}$ exists.

Corollary 1. 

Let

$$0\le \widehat{\beta }+\lambda <\widehat{\alpha }.$$

(17)

Then, the lower partial-moment-generating function ${\mathrm{mgf}}_{\mathrm{H}}^{lo}(\lambda ,u)$ is determined for $u\le \mu$ by the equality

$$\begin{array}{cc}\hfill & {\mathrm{mgf}}_{\mathrm{H}}^{lo}(\lambda ,u)\hfill \\ \hfill =& {\widehat{C}}_{\mathrm{HIG}}\widehat{\delta }{e}^{\lambda \mu }({\mathfrak{I}}_1(u_1,c_{\lambda })-({\mathfrak{I}}_2(u_2^{\lambda },c_{\lambda },0)+{\displaystyle \frac{{\mathfrak{I}}_2(u_2^{\lambda },c_{\lambda },1)}{3}}\left)I_{\left\{u_2^{\lambda }>0\right\}}\right),\hfill \end{array}$$

(18)

where

$$u_1={\displaystyle \frac{u-\mu }{\widehat{\delta }}},c_{\lambda }={\displaystyle \frac{\widehat{\beta }+\lambda }{\widehat{\alpha }}},u_2^{\lambda }=\sqrt{1+u_1^2}+c_{\lambda }\left|u_1\right|-\sqrt{1-c_{\lambda }^2}.$$

A proof of Corollary 1 is placed in Appendix A. Same as it is got for the cumulative distribution function, we have here a cognate result for $-\widehat{\alpha }<\widehat{\beta }+\lambda \le 0$ and $u\ge \mu$.

Remark 2. 

So far as

$$\begin{array}{cc}\hfill {\mathrm{mgf}}_{\mathrm{H}}^{lo}(\lambda ,u)=& {\mathrm{mgf}}_{\mathrm{H}}\left(\lambda \right)-{\mathrm{mgf}}_{\mathrm{H}}^{up}(\lambda ,u)={\mathrm{mgf}}_{\mathrm{H}}\left(\lambda \right)-{\mathrm{mgf}}_{-\mathrm{H}}^{lo}(-\lambda ,-u)\hfill \\ \hfill =& {\mathrm{mgf}}_{\tilde{\mathrm{H}}}(-\lambda )-{\mathrm{mgf}}_{\tilde{\mathrm{H}}}^{lo}(-\lambda ,-u),\hfill \end{array}$$

where $\tilde{\mathrm{H}}=-\mu -\beta \mathrm{HIG}+\sigma \mathrm{N}\sqrt{\mathrm{HIG}}$, we find an analogous to (18) formula for $-\widehat{\alpha }<\widehat{\beta }+\lambda \le 0$ and $u\ge \mu$. Actually, set $\tilde{\lambda }=-\lambda$, $\tilde{\mu }=-\mu$, and $\tilde{\beta }=-\beta$. Then $\tilde{\mathrm{H}}=\tilde{\mathrm{H}}(\tilde{\mu },\sigma ,\alpha ,\tilde{\beta },\delta )$ and $0\le \widehat{\tilde{\beta }}+\tilde{\lambda }<\widehat{\alpha }$. Hence, the condition (11) holds for $\tilde{\mathrm{H}}$ and $\tilde{\lambda }$, and ${\mathrm{mgf}}_{\tilde{\mathrm{H}}}(-\lambda )$ exists. Moreover, the condition (17) is satisfied for $\tilde{\beta }$ and $\tilde{\lambda }$. Let $\tilde{u}=-u$. Then $\tilde{u}\le \tilde{\mu }$ and Corollary 1 can be applied. Using (10) and (18), we gather that

$$\begin{array}{cc}\hfill & {\mathrm{mgf}}_{\mathrm{H}}^{lo}(\lambda ,u)=e^{\mu \lambda }{\displaystyle \frac{{\left({\alpha }^2-{\beta }^2\right)}^{\frac{1}{2}}{\mathrm{K}}_1\left(\delta \sqrt{{\alpha }^2-{\sigma }^2{\lambda }^2-{\beta }^2-2\lambda \beta }\right)}{{\left({\alpha }^2-{\sigma }^2{\lambda }^2-{\beta }^2-2\lambda \beta \right)}^{\frac{1}{2}}{\mathrm{K}}_1\left(\delta \sqrt{{\alpha }^2-{\beta }^2}\right)}}\hfill \\ \hfill -& {\widehat{C}}_{\mathrm{HIG}}\widehat{\delta }{e}^{\lambda \mu }({\mathfrak{I}}_1({\tilde{u}}_1,{\tilde{c}}_{\lambda })-({\mathfrak{I}}_2({\tilde{u}}_2^{\lambda },{\tilde{c}}_{\lambda },0)+{\displaystyle \frac{{\mathfrak{I}}_2({\tilde{u}}_2^{\lambda },{\tilde{c}}_{\lambda },1)}{3}}\left)I_{\left\{{\tilde{u}}_2^{\lambda }>0\right\}}\right)\hfill \end{array}$$

with

$${\tilde{u}}_1={\displaystyle \frac{\mu -u}{\widehat{\delta }}},{\tilde{c}}_{\lambda }=-{\displaystyle \frac{\widehat{\beta }+\lambda }{\widehat{\alpha }}},{\tilde{u}}_2^{\lambda }=\sqrt{1+{\tilde{u}}_1^2}+{\tilde{c}}_{\lambda }\left|{\tilde{u}}_1\right|-\sqrt{1-{\tilde{c}}_{\lambda }^2}.$$

Summarizing, we deduce that the cumulative distribution function of the hyperbolic law is computed either if $\beta \ge 0$ and $u\le \mu$ or when $\beta \le 0$ and $u\ge \mu$. Its partial-moment-generating functions are found either if $0\le \widehat{\beta }+\lambda <\widehat{\alpha }$ and $u\le \mu$ or when $-\widehat{\alpha }<\widehat{\beta }+\lambda \le 0$ and $u\ge \mu$.

4. Applications

We discuss here the traditional stochastic model of financial markets with an underlying security with price dynamics ${\left({\mathrm{S}}_t\right)}_{t\le 1}$ and a saving account with the evolution process ${\left({\mathrm{R}}_t\right)}_{t\le 1}$ (see Ch. VII.1–2 of Shiryaev [37] and Section 1.8 of Musiela and Rutkowski [41]). It is assumed that

$${\mathrm{R}}_t={\mathrm{R}}_0{e}^{rt}$$

with some constants $R_0>0$ and $r\ge 0$.

Let ${\left({\mathrm{H}}_t\right)}_{t\ge 0}$ be the Lévy process generated by the hyperbolic distribution $\mathrm{H}$, that is

$$Law({\mathrm{H}}_1,\mathrm{P})=Law(\mathrm{H},\mathrm{P}).$$

This process exists since $\mathrm{H}$ has an infinitely divisible distribution, see for details Section 4 of Eberlein and Keller [14] and Ch. III.1 of Shiryaev [37]. Following, in particular, Section 3 of Carr et al. [42], we define the risk-neutral security price process ${\left({\mathrm{S}}_t\right)}_{t\ge 0}$ as

$${\mathrm{S}}_t={\mathrm{S}}_0{e}^{rt}{\displaystyle \frac{e^{{\mathrm{H}}_t}}{\mathrm{E}{e}^{{\mathrm{H}}_t}}}.$$

(19)

Since ${\mathrm{H}}_t$ is a process with independent increments, the discounted price process ${\mathrm{R}}_0^{-1}{e}^{-rt}{\mathrm{S}}_t$ is a martingale with respect to the measure $\mathrm{P}$. Indeed, it is enough to verify the property

$$\mathrm{E}{\mathrm{R}}_t^{-1}{\mathrm{S}}_t={\mathrm{R}}_0^{-1}{\mathrm{S}}_0$$

in this case. We have from (19) that

$$\mathrm{E}{\mathrm{R}}_t^{-1}{\mathrm{S}}_t={\mathrm{R}}_0^{-1}{\mathrm{S}}_0{\displaystyle \frac{\mathrm{E}{e}^{{\mathrm{H}}_t}}{\mathrm{E}{e}^{{\mathrm{H}}_t}}}={\mathrm{R}}_0^{-1}{\mathrm{S}}_0$$

if $\mathrm{E}{e}^{{\mathrm{H}}_t}<\infty$. Let us notice that one may pass to the process (19) from the historical stock price applying the Esscher equivalent martingale change of measure (Eberlein et al. [43]).

Note that

$$0\le \widehat{\beta }+1<\widehat{\alpha }.$$

(20)

Then, with respect to (10) and (11) we see that

$$\mathrm{E}{e}^{{\mathrm{H}}_1}=\mathrm{E}{e}^{\mathrm{H}}={\mathrm{mgf}}_{\mathrm{H}}\left(1\right)={\displaystyle \frac{e^{\mu }{\left({\alpha }^2-{\beta }^2\right)}^{\frac{1}{2}}{\mathrm{K}}_1\left(\delta \sqrt{{\alpha }^2-{\sigma }^2-{\beta }^2-2\beta }\right)}{{\left({\alpha }^2-{\sigma }^2-{\beta }^2-2\beta \right)}^{\frac{1}{2}}{\mathrm{K}}_1\left(\delta \sqrt{{\alpha }^2-{\beta }^2}\right)}}$$

and

$$\mathrm{E}{e}^{{\mathrm{H}}_t}={\left({\mathrm{mgf}}_{\mathrm{H}}\left(1\right)\right)}^t<\infty$$

because ${\mathrm{H}}_t$ is a Lévy process.

Throughout this section, we discuss cash-or-nothing and asset-or-nothing digital options at time intervals $[0,1]$. For the terminology, we refer to Section 6.5 of Musiela and Rutkowski [41]. The cash-or-nothing and asset-or-nothing digital call options have the payoffs at expiry

$$\tilde{\mathcal{K}}{I}_{\left\{{\mathrm{S}}_1>\mathcal{K}\right\}}\mathrm{and}{S}_1{I}_{\left\{{\mathrm{S}}_1>\mathcal{K}\right\}},$$

respectively. So far as the initial probability measure is martingale, these options have the risk-neutral prices

$${\mathbb{D}}_C={\mathrm{R}}_0^{-1}{e}^{-r}\tilde{\mathcal{K}}\mathrm{P}\left({\mathrm{S}}_1>\mathcal{K}\right)\mathrm{and}{\mathbb{D}}_A={\mathrm{R}}_0^{-1}{e}^{-r}\mathrm{E}{\mathrm{S}}_1{I}_{\left\{{\mathrm{S}}_1>\mathcal{K}\right\}}.$$

(21)

If $\tilde{\mathcal{K}}=\mathcal{K}$, then the price of the European call option

$$\mathbb{C}={\mathrm{R}}_0^{-1}{e}^{-r}\mathrm{E}{\left({\mathrm{S}}_1-\mathcal{K}\right)}^{+}={\mathbb{D}}_A-{\mathbb{D}}_C.$$

(22)

The corollary below computes the prices of the digital options in accordance with the expectations (21).

Corollary 2. 

Assume that $log\left({\displaystyle \frac{\mathcal{K}{\mathrm{mgf}}_{\mathrm{H}}\left(1\right)}{{\mathrm{S}}_0{e}^r}}\right)\le \mu$. Then the cash-or-nothing digital call price

$${\mathbb{D}}_C={\mathrm{R}}_0^{-1}{e}^{-r}\tilde{\mathcal{K}}\left[1-F_{\mathrm{H}}\left(log\left({\displaystyle \frac{\mathcal{K}{\mathrm{mgf}}_{\mathrm{H}}\left(1\right)}{{\mathrm{S}}_0{e}^r}}\right)\right)\right]$$

(23)

if $1\le \widehat{\beta }+1<\widehat{\alpha }$ and the asset-or-nothing digital call price

$${\mathbb{D}}_A={\mathrm{R}}_0^{-1}{e}^{-r}\left[1-{\mathrm{mgf}}_{\mathrm{H}}^{-1}\left(1\right){\mathrm{mgf}}_{\mathrm{H}}^{lo}\left(1,log\left({\displaystyle \frac{\mathcal{K}{\mathrm{mgf}}_{\mathrm{H}}\left(1\right)}{{\mathrm{S}}_0{e}^r}}\right)\right)\right]$$

(24)

when $0\le \widehat{\beta }+1<\widehat{\alpha }$.

A proof of Corollary 2 is set in Appendix B. To calculate the prices (21) and (22), we now can use the formulas of Section 3.

5. Numerical Analysis

In this section, we compare the hyperbolic law with the Gaussian distribution which has the same mean. Section 5.1 is set to select the appropriate parameters of the hyperbolic distribution. Section 5.2 presents the computation of the hyperbolic distribution values and shows the discrepancy between the normal and hyperbolic models.

5.1. Selection of the Parameters

We consider the normal distribution with mean

$$\mathrm{W}=\mathrm{N}+0.5,$$

(25)

where $\mathrm{N}=\mathrm{N}(0,1)$ is the standard Gaussian distribution. We compare it with the hyperbolic distribution

$${\mathrm{H}}^{*}=\mathrm{N}\sqrt{{\mathrm{HIG}}^{*}}+0.5{\mathrm{HIG}}^{*},$$

where the hyperbolic inverse-Gaussian distribution ${\mathrm{HIG}}^{*}={\mathrm{HIG}}^{*}({\alpha }^{*},{\beta }^{*},{\delta }^{*})$ has the parameter ${\beta }^{*}=0.5$. So far as we want the condition

$$\mathrm{E}{\mathrm{H}}^{*}=0.5\mathrm{E}\left({\mathrm{HIG}}^{*}\right)=\mathrm{E}\mathrm{W}=0.5$$

to be fulfilled, the ${\mathrm{HIG}}^{*}$ distribution should satisfy the property

$$\mathrm{E}\left({\mathrm{HIG}}^{*}\right)=1.$$

(26)

Set

$${\eta }^{*}=\sqrt{{\left({\alpha }^{*}\right)}^2-{\left({\beta }^{*}\right)}^2}$$

and let ${\eta }^{*}=2$. Then,

$${\alpha }^{*}\approx 2.06$$

since ${\beta }^{*}=0.5$. It follows from (4) that

$$\mathrm{E}\left({\mathrm{HIG}}^{*}\right)={\mathrm{M}}_{{\mathrm{HIG}}^{*},1}={\displaystyle \frac{{\delta }^{*}{\mathrm{K}}_2\left({\delta }^{*}{\eta }^{*}\right)}{{\eta }^{*}{\mathrm{K}}_1\left({\delta }^{*}{\eta }^{*}\right)}}.$$

Because (26) holds and ${\eta }^{*}=2$, ${\delta }^{*}$ should fit the equality

$${\delta }^{*}={\displaystyle \frac{2{\mathrm{K}}_1\left(2{\delta }^{*}\right)}{{\mathrm{K}}_2\left(2{\delta }^{*}\right)}}.$$

(27)

Using the tables of the values of the modified Bessel function of the second kind, it is easy to find from (27) that

$${\delta }^{*}\approx 1.2.$$

Now we conclude that

$${\mathrm{H}}^{*}\approx {\mathrm{H}}^{*}(0,1,2.06,0.5,1.2).$$

(28)

The plots of the probability density functions

$$f_{\mathrm{W}}\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{\left(x-0.5\right)}^2}{2}}}.$$

and

$$f_{{\mathrm{H}}^{*}}\left(x\right)\approx 4.816e^{0.5x-2.06\sqrt{x^2+1.44}}$$

are given at Figure 1.

5.2. Computations

Keeping in mind the values of the parameters ${\alpha }^{*}$, ${\beta }^{*}$, and ${\delta }^{*}$ which are found in Section 5.1, we see in the notations of Theorem 1 that

$$c^{*}={\displaystyle \frac{{\beta }^{*}}{{\alpha }^{*}}}\approx 0.243,{\varphi }^{*}={\alpha }^{*}{\delta }^{*}\approx 2.472,u_1\approx 0.833u,$$

and

$$u_2\approx \sqrt{1+0.694u^2}-0.202\left|u\right|-0.97.$$

Next,

$${\widehat{C}}_{\mathrm{HIG}}\approx 4.816,{\mathrm{W}}_{-{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}}}\left(2{\varphi }^{*}\sqrt{1-{\left(c^{*}\right)}^2}\right)\approx 0.037,$$

$${\mathfrak{I}}_1(u_1,c^{*})\approx 0.087-0.104e^{-2.472\left(\sqrt{1+0.694u^2}+0.202\left|u\right|\right)},$$

(29)

$${\mathfrak{I}}_2(u_2,c^{*},0)\approx 0.135u_2^{{\displaystyle \frac{1}{2}}}{\Phi }_1\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}},-0.515u_2,-2.472u_2\right),$$

(30)

and

$${\mathfrak{I}}_2(u_2,c^{*},1)\approx 0.139u_2^{{\displaystyle \frac{3}{2}}}{\Phi }_1\left({\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{5}{2}},-0.515u_2,-2.472u_2\right).$$

(31)

It follows from formula 4.3.24 of Erdélyi et al. [44] that

$${\Phi }_1\left({\varsigma }_1,{\varsigma }_2,{\varsigma }_1+1,{\varsigma }_3,{\varsigma }_4\right)={\mathrm{B}}^{-1}\left({\varsigma }_1,1\right){\int }_0^1{x}^{{\varsigma }_1-1}{\left(1-{\varsigma }_{3}x\right)}^{-{\varsigma }_2}{e}^{{\varsigma }_{4}x}dx$$

for ${\varsigma }_1>0$ and ${\varsigma }_3\le 1$. Hence we have that

$${\mathfrak{I}}_2(u_2,c^{*},0)\approx 0.068u_2^{{\displaystyle \frac{1}{2}}}{\int }_0^1{x}^{-0.5}{\left(1+0.515u_{2}x\right)}^{-0.5}{e}^{-2.472u_{2}x}dx,$$

$${\mathfrak{I}}_2(u_2,c^{*},1)\approx 0.209u_2^{{\displaystyle \frac{3}{2}}}{\int }_0^1{x}^{0.5}{\left(1+0.515u_{2}x\right)}^{-0.5}{e}^{-2.472u_{2}x}dx.$$

We find from (15) that

$$\begin{array}{c}\hfill F_{\mathrm{H}}\left(u\right)\approx 5.779({\mathfrak{I}}_1(u_1,c^{*})-({\mathfrak{I}}_2(u_2,c^{*},0)+{\displaystyle \frac{{\mathfrak{I}}_2(u_2,c^{*},1)}{3}}\left)I_{\left\{u_2>0\right\}}\right),\end{array}$$

where the values of the functions ${\mathfrak{I}}_0$, ${\mathfrak{I}}_1$, and ${\mathfrak{I}}_2$ are computed by (29)–(31), respectively.

Table 1. The comparison of the cumulative normal $F_{\mathrm{W}}$ and hyperbolic $F_{{\mathrm{H}}^{*}}$ distribution functions.

u	−2	−1	−0.5	−0.2	0
$F_{\mathrm{W}}\left(u\right)$	0.006	0.067	0.159	0.242	0.309
$F_{{\mathrm{H}}^{*}}\left(u\right)$	0.006	0.058	0.151	0.241	0.319

presents the values of the cumulative distribution functions of the normal distribution (25) and the hyperbolic distribution (28). The plots of the probability density functions of the distributions at the negative axis are given at Figure 2.

To consider the difference between the normal and hyperbolic models, let us discuss the double digital option with maturity $T=1$. It has two strike prices: the upper strike L and the lower strike K. The stock price ${\mathrm{S}}_1$ becomes the exercise price if $K<S_1<L$.

Let the double digital option have a cash pay $\tilde{K}>0$. Then it has the payoffs at $T=1$

$$\tilde{K}{I}_{\left\{K<S_1<L\right\}}$$

and the value which is the difference between the values of two digital options. The price of this double digital option in the hyperbolic model is

$$\mathbb{DD}\left({\mathrm{H}}^{*}\right)=\tilde{K}\mathrm{P}\left[log\left(K\mathrm{E}{e}^{{\mathrm{H}}^{*}}\right)<{\mathrm{H}}^{*}<log\left(L\mathrm{E}{e}^{{\mathrm{H}}^{*}}\right)\right].$$

Respectively, we see in the normal model that

$$\mathbb{DD}\left(\mathrm{W}\right)=\tilde{K}\mathrm{P}\left[log\left(K\mathrm{E}{e}^{\mathrm{W}}\right)<\mathrm{W}<log\left(L\mathrm{E}{e}^{\mathrm{W}}\right)\right]$$

and

$$log\left(L\mathrm{E}{e}^{\mathrm{W}}\right)-log\left(K\mathrm{E}{e}^{\mathrm{W}}\right)=log\left(L\mathrm{E}{e}^{{\mathrm{H}}^{*}}\right)-log\left(K\mathrm{E}{e}^{{\mathrm{H}}^{*}}\right).$$

(32)

Since

$$\mathrm{E}{e}^{\mathrm{W}}\approx 2.72\mathrm{and}\mathrm{E}{e}^{{\mathrm{H}}^{*}}\approx 3.55,$$

we have that

$$log\left(K\mathrm{E}{e}^{\mathrm{W}}\right)<log\left(K\mathrm{E}{e}^{{\mathrm{H}}^{*}}\right)\mathrm{and}log\left(L\mathrm{E}{e}^{\mathrm{W}}\right)<log\left(L\mathrm{E}{e}^{{\mathrm{H}}^{*}}\right).$$

Set

$$log\widehat{K}=-0.5-log\left(\mathrm{E}{e}^{\mathrm{W}}\right)\approx -1.5$$

and

$$log\widehat{L}=0-log\left(\mathrm{E}{e}^{{\mathrm{H}}^{*}}\right)\approx -1.267.$$

Then, it issues from (32), the fact that the functions $f_{\mathrm{W}}\left(x\right)$, $f_{{\mathrm{H}}^{*}}\left(x\right)-f_{\mathrm{W}}\left(x\right)$ both are increasing at $[-0.5,0]$ and the values of Table 1 for K and L are such that

$$\widehat{K}\approx 0.223<K<L\le \widehat{L}\approx 0.282$$

the risk-neutral option prices may vary by more than 1.12 times in the hyperbolic and normal models.

6. Discussion

So far as the hyperbolic distribution is widely used for the applied modeling in physics and finance, the study of its properties arises as an important scientific task. In this paper, we first find the analytical solutions for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. The similar formulas for other normal mean–variance mixtures, including the variance-gamma distribution (Madan et al. [32], Ano and Ivanov [33], Ivanov [45]), skew Student’s t distribution (Ivanov and Temnov [46], Ivanov [47]) and semi-hyperbolic law (Ivanov [34]), depend on values of the modified Bessel function of the second kind and the Humbert confluent hypergeometric function. The obtained closed form expressions for the hyperbolic distribution use values of these two special functions as well. But in addition, they depend also on values of the Whittaker function.

The established theoretical results are applied to the problem of digital and European call option pricing in the hyperbolic Lévy model. The numerical experiments are made for the double digital options. It is shown that the risk-neutral option prices can differ at more than 12% in the hyperbolic and normal models with the same common to both models’ characteristics.

Because we have received the formulas for half of the axis, the evident proximate problem is to obtain complete analytical expressions for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. The computation of partial moments of the hyperbolic distribution is also an actual target since it is substantial for the aim of risk measurement behind the value at risk (Cont et al. [48], Ivanov [49], Nawrocki [50]). The problem of the exotic option pricing (Ano and Ivanov [33], Section 6 of Musiela and Rutkowski [41], Schoutens et al. [51]) in the hyperbolic model should be considered as well. Taking into account the results of both this paper and Ivanov [34] for the semi-hyperbolic distribution, we may look forward for an analytical characterization of the whole subfamily with the integer second shape parameter (Perreault et al. [52]) of the family of generalized hyperbolic distributions.

7. Conclusions

Based on the discussion and results of the sections above, we can make the following conclusions.

• The hyperbolic distribution is very popular in many applications and the examination of its properties is a direction of research of current interest.
• In comparison with the formulas for other normal mean–variance mixtures, the closed form expressions for the cumulative distribution and partial-moment-generating functions of the hyperbolic law depend also on the values of the Whittaker function.
• The theoretical results can be applied to the problem of option valuation in the hyperbolic model of financial markets. It is shown that the prices of double digital options can differ by more than 12% from the prices of the same options in the normal model.
• Future investigations should relate to the complete analytical specification of the cumulative distribution, partial-moment-generating functions, and partial moments of the hyperbolic law. The research in the area of applications to finance should be continued as well.