Contraction options and optimal multiple-stopping
in spectrally negative Lévy models
This paper studies the optimal multiple-stopping problem arising in the context of the timing option to withdraw from a project in stages. The profits are driven by a general spectrally negative Lévy process. This allows the model to incorporate sudden declines of the project values, generalizing greatly the classical geometric Brownian motion model. We solve the one-stage case as well as the extension to the multiple-stage case. The optimal stopping times are of threshold-type and the value function admits an expression in terms of the scale function. A series of numerical experiments are conducted to verify the optimality and to evaluate the efficiency of the algorithm.
Optimal multiple-stopping; Spectrally negative Lévy processes; Real options
Mathematics Subject Classification (2010): Primary 60G40, Secondary 60J75
Consider a firm facing a decision of when to abandon or contract a project so as to maximize the total expected future cash flows. This problem is often referred to as the abandonment option or the contraction option. A typical formulation reduces to a standard optimal stopping problem, where the uncertainty of the future cash flow is driven by a stochastic process and the objective is to find a stopping time that maximizes the total expected cash flows realized until then. A more realistic extension is its multiple-stage version where the firm can withdraw from a project in stages.
In a standard formulation, given a discount rate and for a standard Brownian motion , and , one wants to obtain a stopping time of that maximizes the expectation
The profit collected continuously is modeled as the geometric Brownian motion less the constant operating expense . The value corresponds to the lump-sum benefits attained (or the costs incurred) at the time of abandonment. Here a technical assumption is commonly imposed so that the expectation is finite and the problem is non-trivial. The problem is rather simple mathematically; it reduces to the well-known perpetual American option (or the McKean optimal stopping problem). An explicit solution can be attained even when is generalized to a Lévy process (see, e.g., Mordecki ).
In this paper, we generalize the classical model by extending from Brownian motion to a general Lévy process with negative jumps (spectrally negative Lévy process), and consider the optimal stopping problem of the form:
We obtain the optimal stopping time as well as the value function for the case is increasing and admits the form for some positive constants , and . We also show the optimality among all stopping times of threshold type (see (2.4) below) when is relaxed to be a general decreasing and concave function. The decreasing property of reflects the fact that the cost of abandoning a project is higher when the project is large.
We further extend it to the multiple-stage case where one wants to obtain a set of stopping times such that a.s. and achieve
when and (with ), for each , satisfy the same assumptions as in the one-stage case. The multiple-stopping problem arises frequently in real options (see e.g. ) and is well-studied particularly for the case is driven by Brownian motion. In mathematical finance, similar problems are dealt in the valuation of swing options [11, 12] with refraction times between any consecutive stoppings.
Although the use of Brownian motion is fairly common in real options, empirical evidence suggests that the real world is not Gaussian, but with significant skewness and kurtosis (see, e.g., [9, 14, 35]). Dixit and Pindyck  considered the case with jumps of a fixed size with Poisson arrivals. Boyarchenko and Levendorskiĭ  considered the EPV approach for a general Lévy process satisfying the (ACP)-condition (with a focus on exponential-type jumps for illustration); they solved a related multiple-stage problem with being constant. The Lévy model is in general less tractable than the continuous diffusion counterpart, especially when the lump-sum reward function is not a constant. When jumps are involved, the process can potentially jump over a threshold level, requiring one to compute the overshoot distributions that depend significantly on the form of the Lévy measure. Technical details are further required when it has jumps of infinite activity or infinite variation. For related literature, we refer the reader to, among others, [1, 4, 18, 25, 28] for optimal stopping problems and [6, 7, 16, 21] for optimal stopping games of spectrally negative Lévy processes. For a general reference on optimal stopping problems, see, e.g., .
In this paper, we take advantage of the recent advances in the theory of the spectrally negative Lévy process (see, e.g., [8, 24]). In particular, we use the results by Egami and Yamazaki , where we obtained and showed the equivalence of the continuous/smooth fit condition and the first-order condition in a general optimal stopping problem. Unlike the two-sided jump case, the identification of the candidate optimal stopping time can be conducted efficiently without intricate computation. The resulting value function can be written in terms of the scale function, which also can be computed efficiently by using, e.g., [19, 33]. The extension to the multiple-stage case can be carried out without losing generality. The resulting optimal stopping times are of threshold type with possibly simultaneous stoppings, and the value function again admits the form in terms of the scale function. We also conduct a series of numerical experiments using the spectrally negative Lévy process with phase-type jumps so as to verify the optimality of the proposed strategies as well as the efficiency of the proposed algorithm.
The rest of the paper is organized as follows. In Section 2, we review the spectrally negative Lévy process and the scale function and then solve the one-stage problem. In Section 3, we extend it to the multiple-stage problem. In Section 4, we verify the optimality and efficiency of the algorithm through a series of numerical experiments. Section 5 concludes the paper.
2. One-stage Problem
Let be a probability space hosting a spectrally negative Lévy process characterized uniquely by the Laplace exponent
where , and is a Lévy measure concentrated on such that
Here and throughout the paper is the conditional probability where and is its expectation. We exclude the case when is a negative of a subordinator (i.e. it has monotone paths a.s.) and we shall further assume that the Lévy measure is atomless:
We assume that does not have atoms.
In addition, we assume the following regarding the tail of the Lévy measure.
We assume that there exists some such that
In particular, this guarantees that .
This section considers the one-stage optimal stopping problem of the form (1.2) where the supremum is taken over the set (or a subset) of stopping times with respect to the filtration generated by . We assume the running payoff function to be increasing. The stochastic process models the state of the project and the monotonicity of means that it yields higher rewards when is high. Typically one assumes as in (1.1) and this is clearly a special case of our model. Regarding the terminal reward function , we consider two cases: (i) when is a sum of linear and exponential functions (Assumption 2.3 below) and (ii) when is a general decreasing and concave function (Assumption 2.4 below).
The results discussed in this section are applications of Egami and Yamazaki  and will be extended to the multiple-stage problem in the next section. Fix . Let be the set of all -valued -stopping times and define for any ,
We then obtain under Assumption 2.4 below a weaker version of optimality:
over the set of all first down-crossing times,
with by convention. This form of optimality is often used in real options and also in the field of corporate finance and credit risk as exemplified by Leland’s endogenous default model [26, 27]. In practice, a strategy must be simple enough to implement and it is in many cases a reasonable assumption to focus on the set of stopping times of threshold type as in (2.4). Because , it is clear that . For the rest of the paper, let , , for any measurable function .
2.1. Review of scale functions and Egami and Yamazaki 
For any spectrally negative Lévy process, there exists a function called the (r-)scale function
which is zero on , continuous and strictly increasing on , and is characterized by the Laplace transform:
Here, the Laplace exponent in (2.1) is known to be zero at the origin and convex on ; is well-defined and is strictly positive whenever . We also define the second scale function:
As in Lemmas 8.3 and 8.5 of Kyprianou , for each , the functions and can be analytically extended to . Fix and define , as the Laplace exponent of under with the change of measure , ; as in page 213 of , for all ,
If and are the scale functions associated with under (or equivalently with ). Then, by Lemma 8.4 of ,
In particular, by setting (or equivalently ), we can define
The smoothness and asymptotic behaviors around zero of the scale function are particularly important in our analysis. We summarize these in the remark given immediately below.
which equals for . By combining the compensation formula for Lévy processes and the resolvent measure written in terms of the scale function, this can be written in a semi-explicit form. Let
These integrals are well-defined if
If these are satisfied, we can write as in (2.9) for as the sum of the following three terms:
Egami and Yamazaki  obtained the first-order condition that makes vanish and showed that it is equivalent to the continuous fit condition when is of bounded variation and to the smooth fit condition when . Recall that is of bounded variation if and only if and
It has been shown that
In view of Remark 2.1(2), for the unbounded variation case, continuous fit holds whatever the choice of is, while, for the bounded variation case, it holds if and only if .
Furthermore, it has been shown by , on condition that there exists some satisfying
where is defined in (2.8). It is known that is increasing and hence, if is monotonically increasing, the down-crossing time for such with becomes a natural candidate for the optimal stopping time.
2.2. Exponential/Linear Case
We first consider the case where admits the form:
for some constants , and , , . We assume without loss of generality that for . The conditions (2.14) and (2.19) are satisfied by Assumption 2.2. For , we need a technical condition so that (2.13) is guaranteed. We summarize the conditions in the Assumption given below.
exists, where is the derivative of with respect to .
Moreover, this is finite. Indeed, by Assumption 2.3(1) and because is zero on the negative half line. On the other hand, because is increasing, .
By the convexity of , for any . The proof of the following lemma is given in Appendix A.1.
For every , we have
In view of (2.24) above, the function is clearly continuous and increasing. Therefore, if , there exists a unique root such that . Otherwise, let if and let if .
Except for the case is a constant, because increases to , we have .
This together with (2.5) gives, for any ,
The verification of optimality requires the following smoothness properties, whose proofs are given in Appendix A.2.
is on .
In particular, when is of unbounded variation, is on .
Herein, we add a remark concerning continuous/smooth fit. The following remark confirms the results in  and further verifies that smooth fit holds whenever is of unbounded variation even when . This observation only requires the asymptotic behavior of the scale function near zero as in Remark 2.1(2).
Remark 2.4 (continuous/smooth fit).
We now state the main results of this subsection. The proof is given in Appendix A.3.
2.3. For a general concave and decreasing
We now relax the assumption on and consider a general concave and decreasing function . We also drop the continuity assumption on .
Under this assumption, we see that as in (2.18) is continuous and increasing. Indeed, we have
Moreover, is increasing by the concavity of . On the other hand, is increasing because is. Therefore, we again define in the same way as the unique root of (if it exists). The proof of the following result is given in Appendix A.4.
3. Multiple-stage problem
In this section, we extend to the scenario the firm can decrease its involvement in the project in multiple stages as defined in (1.3). As in the one-stage case, we consider two modes of optimality:
for all where we define for notational brevity and the supremum is, respectively, over the set of increasing sequences of stopping times,
and over the set of increasing sequences of down-crossing times,
Clearly, and hence .
We first consider the case admits the form
for some constants , , , , and show the optimality in the sense of (3.1) as an extension of Proposition 2.1. We then consider a more general case where is twice-differentiable, concave and monotonically decreasing and show the optimality over as an extension of Proposition 2.2. Regarding the running reward function , define the differences:
For each , we assume that and satisfy Assumption 2.3.
For each , we assume that and satisfy Assumption 2.4.
As is clear from the problem structure, simultaneous stoppings (i.e. a.s. for some and ) may be optimal in case it is not advantageous to stay in some intermediate stages. For this reason, define, for any subset ,
and consider an auxiliary one-stage problem (1.2) with and . Notice that Assumption 3.1 (resp. Assumption 3.2) guarantees that and also satisfy Assumption 2.3 (resp. Assumption 2.4) for any . Hence Propositions 2.1 and 2.2 apply.
Now let be the root of if it exists. If , we set ; if , we set . For simplicity, let for any . Also define
Fix any and , and consider the problems:
Suppose Assumption 3.1.
If , then
and the stopping time is optimal.
If , for any with the optimal stopping time a.s.
If , it is never optimal to stop, and the value function is given by