The First Passage Time Problem for Mixed-Exponential Jump Processes with Applications in Insurance and Finance
This paper studies the first passage times to constant boundaries for mixed-exponential jump diffusion processes. Explicit solutions of the Laplace transforms of the distribution of the first passage times, the joint distribution of the first passage times and undershoot (overshoot) are obtained. As applications, we present explicit expression of the Gerber-Shiu functions for surplus processes with two-sided jumps, present the analytical solutions for popular path-dependent options such as lookback and barrier options in terms of Laplace transforms, and give a closed-form expression on the price of the zero-coupon bond under a structural credit risk model with jumps.
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 the theory of 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 discounted penalty function or the Gerber-Shiu function as well as the expected total discounted dividends up to ruin. See, for example, Klüppelberg et al. , Mordecki , Xing et al. , Cai et al. , Zhang et al. , Chi , and Chi and Lin . In the setting of mathematical finance, the first passage time plays a crucial role for the pricing of many path-dependent options and American-type and Russian-type options; see, for example, Kou , Kou and Wang [9, 10], Asmussen et al. , Levendorskiǐ , Alili and Kyprianou , Cai et al. , and Cai and Kou , as well as certain credit risk models; see, for example, Hilberink and Rogers , Le Courtois and Quittard-Pinon , and Dong et al. . Many optimal stopping strategies also turn out to boil down to the first passage problem for jump diffusion processes; see, for example, Mordecki . In queueing theory one-sided and two-sided first-exit problems for the compound Poisson processes and jump diffusion processes with two-sided jumps have been playing a central role in a single-server queueing system with random workload removal; see, for example, Perry et al. . Usually, when we study the first passage problem, the models with two-sided jumps are more difficult to handle than those with one-sided jumps, because the undershoot and overshoot problem could not be avoided. Despite the maturity of this field of study, it is surprising to note that, until very recently, it can only be solved for certain kinds of jump distributions, such as the Kou’s double exponential jump diffusion model (see Kou  and Kou and Wang ). Recently, Cai and Kou  proposed a mixed-exponential jump diffusion process to model the asset return and found an expression for the joint distribution of the first passage time and the overshoot for a mixed-exponential jump diffusion process. In the most recent paper of Wen and Yin , two-sided first-exit problem for a jump process having jumps with rational Laplace transform was studied. However, determination of the coefficients in expressions of the above two papers still remains a mathematical and computational challenge. In this paper, we will further study the first passage problems in Cai and Kou  and give an explicit expression for the joint distribution of the first passage time and the overshoot for a mixed-exponential jump process with or without a diffusion. Moreover, we present several applications in insurance risk theory and in finance.
The rest of the paper is organized as follows. In Section 2, the model assumptions are formulated. In Section 3, we study the one-sided passage problem from below or above for compound Poisson process and jump diffusion process. In Section 4, we give explicit expression of the Gerber-Shiu function with two-sided jumps. In Section 5, we present the analytical solutions to the pricing problem of one barrier options and lookback options, and in the last section we derive a closed-form expression for the price of the zero-coupon bond.
2. Mathematical Model
A jump diffusion process is defined as where is the starting point of , is a standard Brownian motion with , is a Poisson process with rate , constants , represent the drift and the volatility of the diffusion part, respectively, and the jump sizes are independent and identically distributed random variables. We assume that are identically distributed as the canonical random variable with probability density function . Moreover, it is assumed that , , and are independent. When , the process (1) is the so-called compound Poisson process with positive and negative jumps and linear deterministic decrease or increase between jumps according to or . The processes cover many models appearing in the literature such as the compound Poisson risk models, the perturbed compound Poisson risk models, and their dual models. From now on, we will denote by the probabilities such that, under , with probability one. Moreover, will be the expectation operator associated to . For convenience, we will write and .
It is easy to see that is a special case of Lévy processes with two-sided jumps, whose infinitesimal generator of is given by for any twice continuously differentiable function . The moment generating function of is , , , where , called the exponent of the Lévy process , is defined as For more about the general Lévy processes, we refer to Bertoin , Kyprianou , and Doney .
3. First Passage Problems
We now turn to one-sided passage problems for the Lévy process (1). For two flat barriers and (), define the first downward passage time under and the first upward passage time over by with the convention that . In the next two subsections we will investigate the distributions of the following quantities: first upward passage time and overshoot ; first downward passage time and undershoot .
3.1. One-Sided Exit from above
In this subsection we assume that the downward jumps have an arbitrary distribution with density and Laplace transform , while the upward jumps are mixed-exponential; that is, where constants , , , , and .
The Lévy exponent of is given by
Using the same argument as in Cai and Kou  we have the following.
Lemma 1. (i) For sufficiently large , if or and , then the equation has exactly distinct positive roots satisfying
(ii) If and , then the equation has exactly distinct positive roots satisfying
Cai and Kou  found the joint distribution of the first passage time and in case under the additional assumption is also mixed-exponential. However, for a general in case the upward jumps are mixed-exponential (cf. Yin et al. ), for any sufficiently large , , and , we have where is a vector uniquely determined by the following system , where is an matrix is an diagonal matrix, and is an -dimensional vector
In this paper we will determine the coefficients ’s explicitly. Moreover, we also consider the cases , and , .
Theorem 2. For any sufficiently large , one has,(i)for and , (ii)for , , (iii)for , (iv)for , , (v)for , where are the positive roots of the equation , is the Dirac delta at , and
Proof. We prove the result for the case only; the rest of the cases can be proved similarly. To prove Theorem 2, the most difficult part is to find the inverse of matrix . For simplicity, we write
Note that can be written as , where is a diagonal matrix, is a Cauchy matrix of order which is invertible, and the inverse is given by , where Here, Then the inverse of is given by The determinant of is given by (see Calvetti and Reichel ) After some algebra, where is the Schur complement of the block in , which is a matrix of order 1. By Schur’s formula (see Zhang ), Moreover, by Banachiewicz inversion formula (see Zhang ), the inverse of is given by After some algebra, we have Now by solving we find that from which and from (9) we get (12).
By the fractional expansion, where the coefficients ’s are defined in the theorem. Substituting (30) into (12) and inverting it on immediately lead to (13). Equations (14)–(16) are direct consequence of (13). This ends the proof of Theorem 2.
Example 3. Let ; several expressions are obtained by Theorem 2. When or and , for , , and , we recover the following three formulae which are obtained by Kou and Wang : When and , then for , , and ,
3.2. One-Sided Exit from below
In this subsection we assume that the upward jumps have an arbitrary distribution with Laplace transform , while the downward jumps are mixed-exponential; that is, where constants , , , , and . By (3), the Lévy exponent of is given by
By replacing by in the previous section, we get the main finding in this section.
Theorem 4. For any sufficiently large , one has, (i)for , , (ii)for , , (iii)for , (iv)for , (v)for , where are the negative roots of the equation and
Example 6. Let in Theorem 4. When or and , for and , When and , then for and ,
4. Applications to Gerber-Shiu Functions
We consider an insurance risk model in which the insurer’s surplus process is defined as where is defined by (1) with jump density (33). The time of (ultimate) ruin is defined as , where if ruin does not occur in finite time. As applications, we obtain the following special case of the Gerber-Shiu functions for surplus processes with two-sided jumps: where is interpreted as the force of interest and is a nonnegative function defined on . Note that a more general form of Gerber-Shiu function was originally introduced in Gerber and Shiu  for the classical risk model.
From Theorem 4(ii) we get the following result.
Corollary 7. Suppose that drifts to ; then one has where ’s, ’s, and ’s are defined as in Theorem 4 and
Remark 8. We compare our results with the existing literature. In case and has a double exponential distribution, the result (45) was found by Cai et al. . For and , the result (45) was found by Albrecher et al. [29, ]. For , the result (45) was found by Albrecher et al. [29, ]. For and , the results (45)–(47) were found by Cheung (see Albrecher et al. [29, PP. 443-444]).
5. Applications to Pricing Path-Dependent Options
As applications of our model in finance, we will study the risk-neutral price of barrier and lookback options. These options have a fixed maturity and a payoff that depends on the maximum (or minimum) of the asset price on . The asset price process under a risk-neutral probability measure is assumed to be , where is given by (1), . We are going to derive pricing formulae for standard single barrier options and lookback options, based on the results obtained in Section 3.
5.1. Lookback Options
The value of a lookback option depends on the maximum or minimum of the stock price over the entire life span of the option. Let the risk-free interest rate be . Given a strike price and the maturity , it is well known that (see, e.g., Schoutens ) using risk-neutral valuation and after choosing an equivalent martingale measure the initial (i.e., ) price of a fixed-strike lookback put option is given by The initial price of a fixed-strike lookback call option is given by The initial price of a floating-strike lookback put option is given by The initial price of a floating-strike lookback call option is given by
In the standard Black-Scholes setting, closed-form solutions for lookback options have been derived by Merton  and Goldman et al. . For the double mixed-exponential jump diffusion model, Cai and Kou  derived the Laplace transforms of the lookback put option price with respect to the maturity ; however, the coefficients do not determinate explicitly.
We will only consider lookback put options because lookback call options can be obtained similarly. For jump diffusion process (1) with jump size density (5), the condition is imposed to ensure that the expectation of is well defined.
Theorem 9. For all sufficiently large , one has,(i)for , (ii)then where are the positive roots of the equation and
Proof. (i) We prove it along the same line as in Cai and Kou . Set ; then
It follows that
The result follows from Theorem 2 and (57).
(ii) Since it follows that The result follows from Theorem 2 and (59).
5.2. Barrier Options
The generic term barrier options refers to the class of options whose payoff depends on whether or not the underlying prices hit a prespecified barrier during the options’ lifetimes. There are eight types of (one dimensional, single) barrier options: up- (down) and-in (out) call (put) options. For more details, we refer the reader to Schoutens . Kou and Wang  obtain closed-form price of up-and-in call barrier option under a double exponential jump diffusion model; Cai and Kou  obtain closed-form expressions of the up-and-in call barrier option under a double mixed-exponential jump diffusion model. Here, we only illustrate how to deal with the down-and-out call barrier option because the other seven barrier options can be priced similarly. For jump diffusion process (1) with jump size density (33), given a strike price and a barrier level , under the risk-neutral probability measure , the price of down-and-out call option is defined as Let and . Then
Theorem 10. For any and , then where are the negative roots of the equation and
6. The Price of the Zero-Coupon Bond
In this section, we give a simple application on the price of the zero-coupon bond under a structural credit risk model with jumps. As in Dong et al. , we assume that the total market value of a firm under the pricing probability measure is given by where is positive constant and is defined as (1). For , define the default time as If we set , then Given and a short constant rate of interest , Dong et al.  have shown that the Laplace transform of the fair price of a defaultable zero-coupon bound at time 0 with maturity is given by where is a constant. When the jump size distribution is a double hyperexponential distribution, a closed-form expression is obtained, but the coefficients cannot be determined explicitly (except for ). Now applying the result in Section 3.2, we get the following result.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the anonymous referee’s careful reading and detailed helpful comments and constructive suggestions, which have led to a significant improvement of the paper. The research was supported by the National Natural Science Foundation of China (no. 11171179) and the Research Fund for the Doctoral Program of Higher Education of China (no. 20133705110002).
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