The first passage time problem for mixed-exponential jump processes with applications in insurance and finance

This paper stidies 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.


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 the theory of actuarial mathematics, the problem of first exit from a halfline 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 e.g. Klüppelberg Dong et al. (2011). Many optimal stopping strategies also turn out to boil down the first passage problem for jump diffusion processes, see e.g. Mordecki (2002). 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 e.g. Perry et al. (2002). 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 (2002), Kou and Wang (2003)). Recently, Cai and Kou (2011) 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. However, determination of the coefficients in their expression still remains a mathematical and computational challenge; See also  for the case of a hyper-exponential jump diffusion process. In this paper we will further study the first passage problems in Cai and Kou (2011) 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.

Mathematical model
process with rate λ, constants µ ∈ R, σ ≥ 0 represent the drift and the volatility of the diffusion part respectively, and the jump sizes {Y i ; i ≥ 1} are independent and identically distributed random variables whose probability density function is given by f Y (y). Moreover, it is assumed that {W t }, {N t } and {Y i } are independent. When σ = 0, the process (2.1) is the so called the compound Poisson process with positive and negative jumps and linear deterministic decrease or increase between jumps according to µ < 0 or µ > 0. 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 shall denoted by {P x : x ∈ R} probabilities such that under P x , X(0) = x with probability one. Moreover, E x will be the expectation operator associated to P x .
For convenience, we shall write P = P 0 and E = E 0 .
It is easy to see that X is a special case of Lévy processes with two-sided jumps, whose infinitesimal generator of X(t) is given by for any twice continuously differentiable function g. The moment generating function of X(t) is E(e zX(t) ) = e ψ(z)t , t ≥ 0, ℜ(z) = 0, where ψ(z), called the exponent of the Lévy process X, is defined as For more about general Lévy processes, we refer to Bertion (1996), Kyprianou (2006) and Doney (2007).

First passage problems
We now turn to one-sided passage problems for the Lévy process ( with the convention that inf ∅ = ∞. In the next two subsections we will investigate the distributions of the following quantities: first upward passage time τ + H and overshoot

One sided exit from above
In this subsection we assume that the downward jumps have an arbitrary distribution with density f − and Laplace transformf − , while the upward jumps are mixed-exponential, i.e.
The Lévy exponent of X is given by Using the same argument as in Cai and Kou (2011) we have the following (ii) If µ ≤ 0 and σ = 0, then the equation ψ 2 (z) = α has exactly m distinct positive roots Cai and Kou (2011) found the joint distribution of the first passage time τ + H and X(τ + H ) in the case σ > 0 under the additional assumption f − (y) is also mixed-exponential. However, for a general f − (y) as in the case of the upward jumps are mixed-exponential (cf. Yin, Shen and Wen (2013)), for any sufficiently large α > 0, θ < η 1 and x < H, we have where w := (w 1 , · · · , w m+1 ) ′ is a vector unique determined by the following system ABw = B is an (m + 1) × (m + 1) diagonal matrix and J is an (m + 1)-dimensional vector In this paper we will determined the coefficients w l 's explicitly. Moreover, we also consider the cases µ > 0, σ = 0 and µ ≤ 0, σ = 0.
Proof We prove the result for the case σ > 0 only, the rest cases can be proved similarly. To prove Theorem 3.1, the most difficult part is to find the inverse of matrix A. For simplicity, we write Note that A 22 can be written as ,j≤m is a Cauchy matrix of order m which is invertible and the inverse is given by . Here Then the inverse of A 22 is given by The determinant of C 1 is given by (see Calvetti and Reichel (1996)): .
After some algebra, is the Schur complement of the block A 22 in A, which is a matrix of order 1. By Schur's formula (see Zhang (2005)), Moreover, by Banachiewicz inversion formula (see Zhang (2005)), the inverse of A is given by .
After some algebra, we have . Now by solving ABw = J we find that from which and (3.2) we get (3.3).
By the fractional expansion, This ends the proof of Theorem 3.1.

One sided exit from below
In this subsection we assume that the upward jumps have an arbitrary distribution with Laplace transformf + , while the downward jumps are mixed-exponential, i.e.
Theorem 3.2. For any sufficiently large α > 0, we have

Applications to Gerber-Shiu functions
We consider an insurance risk model in which the insurer's surplus process is defined as where X(t) is defined by (2.1) with jump density (3.9). The time of (ultimate) ruin is defined as τ = inf{t ≥ 0 : U t ≤ 0}, where τ = ∞ 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 α > 0 is interpreted as the force of interest and w is a non-negative function   4) where B k 's, A kl 's and r k 's are defined as in Theorem 3.2, and A kl η l e −η l y e −r k u .

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 T and a payoff that depends on the maximum (or minimum) of the asset price on [0, T]. The asset price process {S(t) : t ≥ 0} under a risk-neutral probability measure P is assumed as S(t) = S(0)e X(t) , where X(t) is given by (2.1), S(0) = e X(0) . We are going to derive pricing formulae for standard single barrier options and lookback options, based on the results obtained in Section 3.

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 r > 0. Given a strike price K and the maturity T , it is well-known that (see e.g. Schoutens (2006)) using risk-neutral valuation and after choosing an equivalent martingale measure P the initial (i.e. t = 0) 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 (1973) and Goldman et al. (1979). For the double mixedexponential jump diffusion model, Cai and Kou (2011) derived the Laplace transforms of the lookback put option price with respect to the maturity T , however, the coefficients do not determinate explicitly.
We shall only consider lookback put options because lookback call options can be obtained similarly. For jump diffusion process (2.1) with jump size density (3.1), the condition η 1 > 1 is imposed to ensure that the expectation of e −rt S t well defined.
Proof (i). We prove it along the same line as in Cai and Kou (2011). Set k = ln K S 0 ≥ 0, then The result follows from Theorem 3.1 and (5.1). (ii). Since The result follows from Theorem 3.1 and (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 (2006). Kou  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 (2.1) with jump size density (3.9), given a strike price K and a barrier level U, under the risk-neutral probability measure P, the price of down-and-out call option is defined as Let h = ln U S 0 and k = − ln K. Then DOC(k, T ) : Theorem 5.2. For any 0 < φ < η 1 − 1 and r + ϕ > ψ 1 (φ + 1), then where where −R 1 , · · · , −R J are the negative roots of the equation ψ 2 (r) = r + ϕ, and J = m + 1, σ > 0, or σ = 0 and µ < 0, m, σ = 0 and µ ≥ 0, .
Proof Using the same argument as that of the proof of Theorem 5.  where R ∈ [0, 1] is a constant. When the jump size distribution is a double hyperexponential distribution, a closed-form expression is obtained, but the coefficients can not determined explicitly (except for n = 2). Now applying the result in Section 3.2, we get the following result: Corollary 6.1. If the process X(t) is defined as (2.1) has jump size density (3.9), we haveB where −ρ 1 , · · · , −ρ J are the negative roots of the equation ψ 2 (ρ) = γ + r, and J = m + 1, σ > 0, or σ = 0 and µ < 0, m, σ = 0 and µ ≥ 0, , j = 1, · · · , J.