The problem of European-style option pricing in time-changed Lévy models in the presence of compound Poisson jumps is considered. These jumps relate to sudden large drops in stock prices induced by political or economical hits. As the time-changed Lévy models, the variance-gamma and the normal-inverse Gaussian models are discussed. Exact formulas are given for the price of digital asset-or-nothing call option on extra asset in foreign currency. The prices of simpler options can be derived as corollaries of our results and examples are presented. Various types of dependencies between stock prices are mentioned.

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 [

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 [

The normal-inverse Gaussian distribution was introduced in Barndorff-Nielsen [

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 [

We suggest that the risky asset log-returns

It is easy to observe that the problem of pricing the options with payoffs (

Next, it is suggested in our model that the stock prices satisfy the inequality

Since our model is not the classical two-asset financial market model (see for example the book by Shiryaev [

Similarly to (

Next, we introduce some necessary notations. We denote as

The gamma process

Throughout this section, we assume that the subordinators in (

To model dependencies in the subordinators, let us assume that in (

Since for a gamma distribution

The following example illustrates how Theorem

Let

Theorem

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

Example

Assume that

Next, let

Combining together (

Now we will consider the case when the indicator stock

One could notice that the result symmetric to Theorem

Let

Similarly to (

The next theorem suggests the conditions of dependence in (

The following example applies the result of Theorem

Assume that

The next two theorems are analogues of Theorem

The example below gives us the price of the standard asset-or-nothing digital option computed in Ivanov and Temnov [

Let

Theorem

The paper suggests a foundation for computing of European-style options in the variance-gamma and normal inverse-Gaussian models with extra compound Poisson negative jumps. It is intended to calculate the option prices basing on the knowledge of the price of the digital asset-or-nothing call option in foreign currency. The payoffs of the discussed option build on the values of three risky assets which are assumed to be dependent on each other. Various types of the dependencies between the risky asset prices are considered. The price of the option exploits the values of some special mathematical functions including the hypergeometric ones. A future investigation can relate to discussion of specific types of the compound Poisson process or possibility of the jump in the linear drift, see Ivanov [

We have that the conditional expectation

Let

Set

Next, we pass to the computing of the conditional expectation

Let us notice that the condition (

Set

The result of Theorem

Since

Under the conditions of Theorem

Because

The condition (

Keeping in mind the conditions of Theorem

The condition (

We have similarly to (

Since

If

When

If

The condition (

Since

We have from (

The condition (