In recent years, more realistic models than the classic Brownian motion for the specification of financial markets were suggested and investigated. The generalized hyperbolic distributions were introduced in Barndorff-Nielsen [3]. These distributions are infinitely divisible and hence generate a particular class of Lévy processes which can be represented as time-changed Brownian motions, see the monographs by Barndorff-Nielsen and Shiryaev [6] or Cont and Tankov [9] for details. Another important class is the generalized tempered stable distributions which were firstly introduced in Koponen [23] and then investigated in particular by Bianchi et al. [8] and Rosinski [37]. The generalized tempered stable processes are not always time-changed Brownian motions (see Küchler and Tappe [24] or Küchler and Tappe [25] on the bilateral gamma processes), although for example the variance-gamma process and CGMY process can be decomposed in this way. We refer on these facts to Madan et al. [33] and Madan and Yor [32], respectively.

The variance-gamma process is the one of the most popular examples of the generalized tempered stable processes. The variance-gamma distribution was firstly proposed as a model for financial market data in Madan and Seneta [31] and Madan and Milne [30]. They discussed the symmetric case of the distribution. The properties of the variance-gamma process defined as the time-changed by gamma subordinator Brownian motion with drift were considered in Madan et al. [33]. Also, Madan et al. [33] gave the analytical expression for the European call option price in the variance-gamma model together with the definition of the process as the difference of two gamma ones. Further, a number of papers confirmed statistically the idea of using the variance-gamma process for the modeling financial indexes. Daal and Madan [10] and Finlay and Seneta [14] approved the variance-gamma model for the currency option pricing and the exchange rate modeling. Linders and Stassen [26], Moosbrucker [34] and Rathgeber et al. [36] simulated by the variance-gamma distribution the Dow Jones index returns. Mozumder et al. [35] considered the S&P500 index options in the variance-gamma model. Luciano and Schoutens [27] modeled the S&P500, the Nikkei225 and the Eurostoxx50 financial indexes by the variance-gamma process. Luciano et al. [29] and Wallmeier and Diethelm [43] confirmed the using of variance-gamma distribution for the modeling of the US and the Swiss stock markets, respectively.

The normal-inverse Gaussian distribution was introduced in Barndorff-Nielsen [3] to model some facts in geology as a member of the class of generalized hyperbolic distributions. Financial market data, including the Danish and the German ones, was specified then by the normal-inverse Gaussian process in Barndorff-Nielsen [4] and Rydberg [38]. Properties of the normal-inverse Gaussian process discussed as the time-changed by inverse-Gaussian subordinator Brownian motion were considered in Barndorff-Nielsen [5] and Shiryaev [41]. The normal-inverse Gaussian distribution in the context of risk modeling was discussed in Aas et al. [1] and Ivanov and Temnov [21]. Figueroa et al. [13] showed that the normal-inverse Gaussian distribution specifies well a high frequency data from the US equity markets. Teneng [42] proved that the normal-inverse Gaussian process fits to the dynamics of many various foreign exchange rates. Göncü et al. [15] confirmed that this distribution also relates to the statistics of emerging market stock indexes. The modeling of Bloomberg closing prices by the variance-gamma and the normal-inverse Gaussian distributions was discussed in Luciano and Semeraro [28].

If we discuss the problem of computing in Lévy models, the basic method is the Fourier transform one, see for details the review paper by Eberlein [11]. However, it puts some restrictions on the properties of the process or the type of the derivative payoffs. In particular, it can be shown that this method cannot be applied to the pricing of digital options in the volatile variance-gamma model or in the normal-inverse Gaussian one. The method of closed form solutions which had been introduced by Madan et al. [33] was proceeded then in the papers by Ivanov and Ano [20], Ivanov [18] and Ivanov [19] for the variance-gamma distribution and by Ivanov [17] and Ivanov and Temnov [21] for the normal-inverse Gaussian one. This paper continues the elements of the research by Madan et al. [33]. We discuss the problem of multi-asset digital option pricing in the variance-gamma model in the presence of extra downside compound Poisson jumps. These jumps reflect the influence of events which can evoke dramatic drops of assets on financial markets. The examples are the terror attack of 9/11, the Subprime mortgage crisis of 2007, the collapse of Lehman Brothers or the recent deep fall of oil prices. In Sections 3 and 4 the variance-gamma and the normal inverse-Gaussian models are considered, respectively. The obtained formulas give the option prices under different types of dependencies between the asset dynamics.

We suggest that the risky asset log-returns Htj=logStj , j=1,2,3 , t≤T , follow the sums of time-changed Brownian motions and independent compound Poisson processes which are supposed to be mutually independent, too. That is, (1) Htj=μjt+βjϑtj+σjBϑtjj−Ztj,H0j=0, where μj,βj∈R , σj≥0 , (Btj)t≥0 are the Wiener processes correlated with coefficients ρjl , (ϑtj)t≤T are independent with the Wiener processes subordinators and Ztj=∑l=0Ntjξjl , ξj0≡0 , where (Ntj)t≤T , N0j=0 , are the Poisson processes with intensities λj and ξjl≥0 , l=1,2,… , are independent arbitrary identically distributed for every j random variables, where j is the number of asset. Throughout this paper, the problem of pricing of digital asset-or-nothing call option in foreign currency, namely which has the payoff function (2) DCT=ST3ST2I{ST1≥K},K>0, 0,\]]]> is discussed. The dynamics St3 relates here to the exchange rate between the domestic and the foreign currencies. The stock prices St1 and St2 are measured in the domestic currency. It is supposed that the non-risky assets (bank accounts) in domestic and foreign currencies Rtd and Rtf , t≤T , have fixed interest rates rd,r≥0 and Rtd=erdt , Rtf=ert .

It is easy to observe that the problem of pricing the options with payoffs (2) includes the same problem for digital asset-or-nothing and cash-or-nothing call options with payoffs ST1I{ST1≥K} and K˜I{ST1≥K} , K˜>0 0$]]>, for the options in foreign currency with payoffs ST3(ST1−K)+ and for many other options. Indeed, if we discuss for example the payoffs ST3(ST1−K)+ , we just suppose in (1)–(2) that μ2=β2=σ2=0 and ξ2l≡0 .

Next, it is suggested in our model that the stock prices satisfy the inequality (3) E(ST3ST2)<∞. Let (4) Xtj=μjt+βjϑtj+σjBϑtjj. Then E(eXT2+XT3|ϑT2,ϑT3)=e∑j=23(μjT+βjϑTj)+∑j=23σj2ϑTj+2ρ23σ2σ3ϑT2ϑT32 and hence (3) is equivalent to (5) E(e∑j=23βjϑTj+∑j=23σj2ϑTj+2ρ23σ2σ3ϑT2ϑT32)<∞.

Since our model is not the classical two-asset financial market model (see for example the book by Shiryaev [41]), we need to consider at first the question of hedging of the option with payoffs (2). There are four hedging instruments in our situation. Namely, the bank account in foreign currency Rtf , the bank account in domestic currency transferred in foreign currency with the dynamics St3Rtd and the two stocks in foreign currency St3St1 and St3St2 . Leaving aside a well-investigated in literature question of change of measure (see for example Eberlein et al. [12], Kallsen and Shiryaev [22], Madan and Milne [30], Ch. VII.3 of Shiryaev [41] and Ch. 6 of Schoutens [39]), let us assume that the all four assets discounted with respect to the bank account in foreign currency (i.e., the processes Rtf/Rtf≡1 , St3Rtd/Rtf , St3St1/Rtf , St3St2/Rtf ) are martingales with respect to the initial probability measure. Then the price of the option with payoffs (2) is (6) DC=e−rTE(DCT)=e−rTE(ST3ST2I{ST1≥K}).

Next, we introduce some necessary notations. We denote as N(u),u∈R,Γ(u),u>0,B(u1,u2),u1>0,u2>0 0,\hspace{2em}\mathrm{B}({u_{1}},{u_{2}}),\hspace{1em}{u_{1}}>0,{u_{2}}>0\]]]> and Mu1(u2),u1∈R,u2>0, 0,\]]]> the normal distribution function, the gamma function, the beta function and the MacDonald function (the modified Bessel function of the second kind), respectively. The hypergeometric Gauss function is denoted as G(u1,u2,u3;u4),u1,u2,u3∈R,u4<1. Also, the degenerate Appell function (or the Humbert series) which is the double sum A(u1,u2,u3;u4,u5)= ∑m=0∞ ∑n=0∞(u1)m+n(u2)mm!n!(u3)m+nu4mu5n with u1,u2,u3,u5∈R and |u4|<1 , where (u)l , l∈N∪{0} , is the Pochhammer’s symbol, is exploited. For more information on the special mathematical functions above, see Bateman and Erdélyi [7], Gradshteyn and Ryzhik [16], Whittaker and Watson [44].

The gamma process γt=γt(a,b) , a>0,b>0 0,b>0$]]>, is a purely discontinuous Lévy process with gamma-distributed increments and γ0=0 . It is the subordinator with the probability density function f(γt,x)=batxat−1e−bxΓ(at),x>0. 0.\]]]> The gamma process has mean at/b and variance at/b2 . If u<b , the moment-generating function of the gamma process is (7) Eeuγt=(bb−u)at. For more properties of this process, see the paper by Yor [45] or the monograph by Applebaum [2].

Throughout this section, we assume that the subordinators in (1) and (4) are the gamma processes with unit mean rate, i.e. (8) ϑtj=γtj(aj)=γtj(aj,aj). Then the processes Xtj in (4) become the variance-gamma processes, see Madan et al. [33] or Seneta [40] for more details.

To model dependencies in the subordinators, let us assume that in (8) the subordinators (9) γtj=κjγt(a)+κj1γt1+κ˜jγ˜tj(a˜j),j=2,3, where all the gamma processes with unit mean rate γt,γt1,γ˜t2,γ˜t3 are mutually independent, κj,κj1,κ˜j≥0 and κj+κj1+κ˜j=1 , j=2,3 .

Since for a gamma distribution γ the identity uγ(u1,u2)=Lawγ(u1,u2u) is satisfied, we have from (9) that aj=aκj if κj≠0 , aj=a1κj1 if κj1≠0 , aj=a˜jκ˜j if κ˜j≠0 and hence the equality (10) (aj−aκj)I{κj>0}=(aj−a1κj1)I{κj1>0}=(aj−a˜jκ˜j)I{κ˜j>0}=0 0\}}}=\bigg({a_{j}}-\frac{{a_{1}}}{{\kappa _{j1}}}\bigg){I_{\{{\kappa _{j1}}>0\}}}=\bigg({a_{j}}-\frac{{\tilde{a}_{j}}}{{\tilde{\kappa }_{j}}}\bigg){I_{\{{\tilde{\kappa }_{j}}>0\}}}=0\]]]> holds. Next, because the identity γ(u1,u)+γ˜(u2,u)=Lawγ(u1+u2,u) holds for arbitrary independent gamma distributions γ and γ˜ , one could observe from (9), as γt,γt1,γ˜tj are mutually independent, that (11) aj=aI{κj>0}+a1I{κj1>0}+a˜jI{κ˜j>0} 0\}}}+{a_{1}}{I_{\{{\kappa _{j1}}>0\}}}+{\tilde{a}_{j}}{I_{\{{\tilde{\kappa }_{j}}>0\}}}\]]]> in our model, j=2,3 . Alternatively, the identities (10) and (11) can be seen from the equality for characteristic functions of (9) (ajaj−iu)ajt=(a/κja/κj−iu)at(a1/κj1a1/κj1−iu)a1t(a˜j/κ˜ja˜j/κ˜j−iu)a˜jt if all κj>0 0$]]>, κj1>0 0$]]>, κ˜j>0 0$]]>. The theorem below gives us the price (6) in the case of the independent Brownian motions in (1).

The following example illustrates how Theorem 1 works when Ztj are standard Poisson processes.

Theorem 2 computes us the price (6) in the case when the exchange rate St3 and the underlying asset St2 are strongly dependent but the indicator stock St1 is weakly dependent on them.

The next theorem considers the case when all risky assets are strongly dependent.

Example 2 shows how Theorem 3 can be applied to the problem of pricing of the standard European call option in foreign currency which has the payoffs at expiry ST2(ST1−K)+ .

Now we will consider the case when the indicator stock St1 and the exchange rate St3 are strongly dependent but the underlying asset St2 is weakly dependent on them. Theorem 4.

Assume that in (1) ρ23=ρ12=0 , the subordinators are gamma distributed, satisfy (8)–(9), and γt3=γt1 , γt2=κ21γt1+κ˜2γ˜t2 . Let b=β3+σ322+κ21(β2+σ222). Then (12) is satisfied with DC(n1,n2,n3)=e(μ2+μ3−r)Ta1a1T(a1−b)a1TΓ(a1T)2πE(e−∑l=0n2ξ2l)E(e−∑l=0n3ξ3l)×(a˜2a˜2−κ˜2(β2+σ222))a˜2T(ΛP( ∑l=0n1ξ1l=μ1T−K)+E(Ξ( ∑l=0n1ξ1l)I{∑l=0n1ξ1l≠μ1T−K})), where Λ=Γ(a1T+12)(B(12,a1T)2+β1+σ1σ3σ1a1−bG(a1T+12,12,32;−(β1+σ1σ3)22(a1−b)σ12)) and Ξ(x) is defined by (14) with q=β1+σ1σ3(β1+σ1σ3)2+2(a1−b)σ12 and s=s(x)=(μ1T−K−x)(β1+σ1σ3)2+2(a1−b)σ12σ1.