Abstract and Applied Analysis
Volume 2016 (2016), Article ID 5914657, 8 pages
http://dx.doi.org/10.1155/2016/5914657

Research Article

A Note on First Passage Functionals for Lévy Processes with Jumps of Rational Laplace Transforms

Djilali Ait-Aoudia

Département de Sciences de la Décision, École des HEC, Montréal, QC, Canada

Received 22 July 2016; Revised 24 August 2016; Accepted 29 September 2016

Academic Editor: Lucas Jodar

Copyright © 2016 Djilali Ait-Aoudia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper investigates the two-sided first exit problem for a jump process having jumps with rational Laplace transform. The corresponding boundary value problem is solved to obtain an explicit formula for the first passage functional. Also, we derive the distribution of the first passage time to two-sided barriers and the value at the first passage time.

1. Introduction

One-sided and two-sided exit problems for the compound Poisson processes and jump-diffusion processes with two-sided jumps have been applied widely in a variety of fields. For example, in actuarial mathematics, the problem of first exit from a half-line is of fundamental interest with regard to the classical ruin problem and the expected discount penalty function or the Gerber-Shiu function as well as the expected total; see, for example, Mordecki [1], Xing et al. [2], Zhang et al. [3], Lewis and Mordecki [4], and Avram et al. [5]. In mathematical finance, the first passage time plays a crucial role for the pricing of many path-dependent options and American type options; see, for example, Geman and Yor [6], Bertoin [7], Kyprianou [8], Rogers [9], Avram et al. [10], and Alili and Kyprianou [11]. Recently, Cai [12] investigated the first passage time of hyperexponential jump-diffusion process. Cai and Kou [13] proposed a mixed-exponential jump-diffusion process to model the asset return and found an expression for the joint distribution of the first exit problem for a jump and overshoot for a mixed-exponential jump-diffusion process. Chen et al. [14] and Yin et al. [15] discussed the first passage functional for hyperexponential jump-diffusion process.

Motivated by works mentioned above, the main objective of this paper is to study the first exit time of the two-sided first exit problem for jump-diffusion process having jump with rational Laplace transform proposed by Lewis and Mordecki [4]; see also Kuznetsov [16]. This extends recent results obtained in Chen et al. [14, Theorem ] on the hyperexponential jump-diffusion process.

The rest of the paper is organized as follows. In Section 2, we introduce the jump-diffusion process having jumps with rational Laplace transform. Section 3 concentrates on deriving the joint distribution of first exit time and a nonnegative bounded measurable function of the process value at the first exit time to two flat barriers. Section 4 presents the analytical solution to the pricing problem of standard double-barrier options.

2. The Model

A Lévy jump-diffusion process is defined as

where and represent the drift and volatility of the diffusion part, respectively, is a (standard) Brownian motion, is a homogeneous Poisson process with rate , and are independent and identically distributed random variables supported in ; moreover, , and are mutually independent; finally, the probability density function (pdf) of is given by

where , such that , and and that and for all .

Another important tool to establish the key result of the article is the infinitesimal generator of . Note that is a Markovian process and its infinitesimal generator is given by

for any bounded and twice continuously differentiable function .

Throughout the rest of the paper, the law of such that is denoted by and the corresponding expectation by ; we write and when . The Lévy exponent of is given by

Accordingly, is a rational function on . The equation with , and yields zeros with and ; see Kuznetsov [16].

Let us denote the zeros of in the half-plane as . Also, we assume that all zeros of are simple.

3. Distribution of the First Passage Time to Two Flat Barriers

Define to be the first passage time of to two flat barriers and with ; that is,

And let

where and is nonnegative bounded measurable function.

Now, by Feynman-Kac formula (see, e.g., Theorem , Karatzas and Shreve [17]) we have that must satisfy

Our goal in this section is to solve the boundary problem (7) and find explicit formulae for . We first show that satisfies an integrodifferential equation and then derive an ordinary differential equation for . Based on the ODE, we show that can be written as a linear combination of known exponential functions. As a consequence, its explicit form is obtained, for instance, choosing to be ; it is easy to derive the joint distribution of the first passage time of to two flat barriers and the process value at the first passage time.

Now, let ; then is a polynomial whose zero coincides with those of . Also, denote by the differential operator such that its characteristic polynomial is .

The following lemma will be needed for our proof of Proposition 2.

Lemma 1. Let indicate the th derivative with respect to of any differentiable function and define

with being the generalized Leibniz operator such that

Then, for all ,

Proof. We proceed by induction on . For , we have

Moreover,

It follows inductively that

which is the desired result.

We may now state the following.

Proposition 2. Suppose a bounded solution defined on to the boundary value problem (7) exists. Then on , is infinitely differentiable and satisfies the ODE,

Hence, on , for some constants .

Proof. Applying the infinitesimal generator to the function , we obtain

Next, will be shown to satisfy an . Using Lemma 1, we get, for , ,

The same computation will yield, for , ,

Now, since and then, thanks to Proposition in the work of Chen et al. [18], is infinitely differentiable on and

Hence, (18) transforms the integrodifferential equation into an , where is high order differential operator.
To complete the proof, must be shown to coincide with . To show that the characteristic polynomials of and will suffice, write as the characteristic polynomial of . Then, by (18), is given by

This equation reveals that the characteristic polynomial of equals that, , of , which completes the proof.

Proposition 3. Suppose that is a bounded solution to the boundary value problem (7) and, on for some constants . Then the constant vector satisfies the equation

where is nonsingular matrix given by

and ,

Proof. Since and on , which entails

furthermore

Now, since ,

with being the incomplete gamma function (see Gradshteyn and Ryzhik [19, page 342]).
Consequently, by combining (26) and (27) and taking into account the fact that for all , we obtain

Comparing and yields (20), which entails the desired result.

Lemma 4. For a given value of the matrix given by (21) is invertible.

Proof. Assume that for some vector . Consider the function for and , otherwise, with to be the distinct zeros of the equation . Since and is a solution to the boundary value problem,

From the uniqueness of solutions to the boundary value problem (30), in . Now, since are linearly independent then and is invertible.

In the following, is written for the usual inner product of the vectors and in and for every real value , , where are the roots of the equation . Our main result is the following.

Theorem 5. For any and a nonnegative bounded measurable function on , the following assertions are equivalent:

(a), where .(b)The function solves the boundary problem (7).(c)The function is given by the formula

with and and are given by formulas (21) and (24), respectively.

Proof. The fact that implies is straightforward consequence of Proposition 3. Conversely, consider the function for and otherwise, where is a bounded function on and ’s are given constants. Then the same reasoning as in Proposition 3 shows that

Thanks to (20), we conclude that implies .
Let us finally assume that holds. Then by Feynman-Kac formula, the function solves the boundary problem (7); hence holds. Conversely, thanks to Proposition in the work of Chen et al. [18], the boundary problem (7) has a unique solution; consequently implies . The proof is complete.

As an illustration of the main result of Theorem 5, we can obtain closed-form expressions for the expectations of a variety of functions with respect to and . For instance, choosing in the above theorem, we can derive the joint Laplace transform of , which is presented in the following corollary.

Corollary 6. For any , ,

where , is given by formula (21), and is given by

As another consequence of Theorem 5 and Lebesgue’s dominated convergence theorem, we get the following for the asymptotic case when and , respectively.

Corollary 7. For two flat barriers and , define the first downwards passage time under and the first upwards passage time over by

Then for , one has the following:

with

4. Pricing Double-Barrier Options

We now show how our theoretical results can be easily applied to derive pricing formulae for standard double-barrier options. We assume the asset price process under the risk-neutral probability measure is defined as . The log-return process is given by (1) where and (i.e., ), where is the risk-free rate. More recently, Cai et al. [20] presented the following.

The payoff of a standard double-barrier option is activated (knocked in) or extinguished (knocked out) when the price of the underlying asset crosses barriers. For example, a knock-out call option will not give the holder the payoff of a European call option unless the underlying price remains within a prespecified range before the option matures. More precisely, consider an interval and the initial asset price is in it. The holder will receive at maturity , where

Under the risk-neutral probability measure, the price of such option with maturity and strike is given by

Make a change variable . Then, the expectation can be represented as

Define and as the double Laplace transforms of the price in (40) and the delta with respect to and , respectively; that is,

Theorem 8. For any and

where and associated with is defined as in Theorem 5 and is given by formula (34).

Proof. Equation (43) is an easy consequence of (42). To prove (43), using an idea of Kou et al. [21] (see also Cai et al. [20]) along with a change of the order of integration and the integral with respect to , we obtain

Now, we suppose that and applying Itô’s formula to the process , we obtain

is a local martingale starting from . Since , it follows from Fubini’s theorem that

It follows from the dominated convergence theorem that is actually a martingale. In particular

Combining (44) and (47) and applying Corollary 6 we can therefore conclude that

where and associated with is defined as in Theorem 5 and is given by formula (34), which ends the proof.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

E. Mordecki, “Ruin probabilities for Lévy processes with mixed-exponential negative jumps,” Theory of Probability & Its Applications, vol. 48, no. 1, pp. 170–176, 2003.

X. Xing, W. Zhang, and Y. Jiang, “On the time to ruin and the deficit at ruin in a risk model with double-sided jumps,” Statistics and Probability Letters, vol. 78, no. 16, pp. 2692–2699, 2008.

Z. M. Zhang, Y. Yang, and S. M. Li, “The perturbed compound poisson risk model with two-sided jumps,” Journal of Computational and Applied Mathematics, vol. 233, pp. 1773–1784, 2010.

A. L. Lewis and E. Mordecki, “WienerHopf factorization for Lévy processes having positive jumps with rational transforms,” Journal of Applied Probability, vol. 45, no. 1, pp. 118–134, 2008.

F. Avram, Z. Palmowski, and M. R. Pistorius, “On Gerber-Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function,” The Annals of Applied Probability, vol. 25, no. 4, pp. 1868–1935, 2015.

H. Geman and M. Yor, “Pricing and hedging double-barrier options: a probabilistic approach,” Mathematical Finance, vol. 6, no. 4, pp. 365–378, 1996.

J. Bertoin, Lévy Processes, Cambridge University Press, 1998.

A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer, Berlin, Germany, 2nd edition, 2014.

L. C. Rogers, “Evaluating first-passage probabilities for spectrally one-sided Lévy processes,” Journal of Applied Probability, vol. 37, no. 4, pp. 1173–1180, 2000.

F. Avram, A. E. Kyprianou, and M. R. Pistorius, “Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options,” The Annals of Applied Probability, vol. 14, no. 1, pp. 215–238, 2004.

L. Alili and A. E. Kyprianou, “Some remarks on first passage of Lévy processes, the American put and pasting principles,” The Annals of Applied Probability, vol. 15, no. 3, pp. 2062–2080, 2005.

N. Cai, “On first passage times of a hyper-exponential jump diffusion process,” Operations Research Letters, vol. 37, no. 2, pp. 127–134, 2009.

N. Cai and S. G. Kou, “Option pricing under a mixed-exponential jump diffusion model,” Management Science, vol. 57, no. 11, pp. 2067–2081, 2011.

Y. T. Chen, Y. C. Sheu, and M. C. Chang, “A note on first passage functionals for Hyper-Exponential jump-diffusion process,” Electronic Communications in Probability, vol. 18, pp. 1–8, 2013.

C. Yin, Y. Wen, Z. Zong, and Y. Shen, “The first passage time problem for mixed-exponential jump processes with applications in insurance and finance,” Abstract and Applied Analysis, vol. 2014, Article ID 571724, 9 pages, 2014.

A. Kuznetsov, “On the distribution of exponential functionals for Lévy processes with jumps of rational transform,” Stochastic Processes and their Applications, vol. 122, no. 2, pp. 654–663, 2012.

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 2nd edition, 1991.

Y.-T. Chen, C.-F. Lee, and Y.-C. Sheu, “An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model,” Finance and Stochastics, vol. 11, no. 3, pp. 323–355, 2007.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, San Diego, Calif, USA, 6th edition, 2000.

N. Cai, N. Chen, and X. Wan, “Pricing double-barrier options under a flexible jump diffusion model,” Operations Research Letters, vol. 37, no. 3, pp. 163–167, 2009.

S. G. Kou, H. Petrella, and H. Wang, “Pricing path-dependent options with jump risk via Laplace transforms,” The Kyoto Economic Review, vol. 74, pp. 1–23, 2005.