Contraction options in spectrally negative Lévy models

# Contraction options and optimal multiple-stopping in spectrally negative Lévy models

Kazutoshi Yamazaki
###### Abstract.

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.

This version: August 6, 2019. The author thanks the anonymous referee for his/her thorough reviews and insightful comments that help improve the presentation of the paper. K. Yamazaki is in part supported by MEXT KAKENHI grant numbers 22710143 and 26800092, JSPS KAKENHI grant number 23310103, the Inamori foundation research grant, and the Kansai University subsidy for supporting young scholars 2014.
Department of Mathematics, Faculty of Engineering Science, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka 564-8680, Japan. Email: kyamazak@kansai-u.ac.jp. Tel: +81-6-6368-1527.

Key words: Optimal multiple-stopping; Spectrally negative Lévy processes; Real options
Mathematics Subject Classification (2010): Primary 60G40, Secondary 60J75

## 1. Introduction

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

 E[∫τ0e−rt(eXt−δ)dt+e−rτK1{τ<∞}]. (1.1)

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 [29]).

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:

 supτE[∫τ0e−rtf(Xt)dt+e−rτg(Xτ)1{τ<∞}]. (1.2)

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

 supτ(1)≤⋯≤τ(M)M∑m=1E[∫τ(m)τ(m−1)e−rtFm(Xt)dt+e−rτ(m)gm(Xτ(m))1{τ(m)<∞}] (1.3)

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. [15]) 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 [15] considered the case with jumps of a fixed size with Poisson arrivals. Boyarchenko and Levendorskiĭ [10] 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., [31].

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 [17], 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

 ψ(β):=logE0[eβX1]=cβ+12σ2β2+∫(0,∞)(e−βz−1+βz1{0

where , and is a Lévy measure concentrated on such that

 ∫(0,∞)(1∧z2)Π(dz)<∞. (2.2)

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:

###### Assumption 2.1.

We assume that does not have atoms.

In addition, we assume the following regarding the tail of the Lévy measure.

###### Assumption 2.2.

We assume that there exists some such that

 ∫[1,∞)eϵuΠ(du)<∞.

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 [17] 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 ,

 u(x,τ)≡u(x,τ;f,g):=Ex[∫τ0e−rtf(Xt)dt+e−rτg(Xτ)1{τ<∞}],x∈R. (2.3)

After a brief review on the scale function and the results of [17], we shall solve, under Assumption 2.3 below, the problem:

 u(x):=supτ∈Su(x,τ).

We then obtain under Assumption 2.4 below a weaker version of optimality:

 ˜u(x):=supτ∈˜Su(x,τ),

over the set of all first down-crossing times,

 ˜S:={τA:A∈R},

where

 τA:=inf{t>0:Xt≤A},A∈R, (2.4)

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 [17]

For any spectrally negative Lévy process, there exists a function called the (r-)scale function

 W(r):R→[0,∞),

which is zero on , continuous and strictly increasing on , and is characterized by the Laplace transform:

 ∫∞0e−sxW(r)(x)dx=1ψ(s)−r,s>Φr,

where

 Φr:=sup{λ≥0:ψ(λ)=r},r≥0.

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:

 Z(r)(x):=1+r∫x0W(r)(y)dy,x∈R.

As we shall see below, the pair of scale functions and play significant roles in our problems; for a comprehensive account of the scale function, we refer the reader to, e.g., [8, 22, 24].

Recall (2.4) and define the first up-crossing times of by . Then, for any and , as summarized in Theorem 8.1 of [24],

 Ex[e−rτ+b1{τ+b<τ0}]=W(r)(x)W(r)(b),Ex[e−rτ01{τ+b>τ0}]=Z(r)(x)−Z(r)(b)W(r)(x)W(r)(b),Ex[e−rτ0]=Z(r)(x)−rΦrW(r)(x). (2.5)

As in Lemmas 8.3 and 8.5 of Kyprianou [24], 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 [24], for all ,

 ψa(β):=(aσ2+c−∫(0,1)u(e−au−1)Π(du))β+12σ2β2+∫(0,∞)(e−βu−1+βu1{0

If and are the scale functions associated with under (or equivalently with ). Then, by Lemma 8.4 of [24],

 W(r−ψ(a))a(x)=e−axW(r)(x),x≥0. (2.7)

In particular, by setting (or equivalently ), we can define

 WΦr(x):=W(0)Φr(x)=e−ΦrxW(r)(x),x∈R (2.8)

which satisfies

 ∫∞0e−βxWΦr(x)dx =1ψ(β+Φr)−r,β>0.

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.

###### Remark 2.1.
1. Assumption 2.1 guarantees that is on . In particular, when , then is on . Fore more details on the smoothness of the scale function, see Chan et al. [13].

2. As in Lemmas 4.3 and 4.4 of [25],

 W(r)(0) W(r)′(0+)

where , which is finite when is of bounded variation.

In [17], we have shown that a candidate optimal stopping time can be efficiently identified using the scale function. Define the expected payoff corresponding to the down-crossing time (2.4) by

 uA(x) :=u(x,τA),x,A∈R, (2.9)

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

 Ψf(A) (2.10) Θf(x;A) :=∫xAW(r)(x−y)f(y)dy,x,A∈R, (2.11)

and

 (2.12)

These integrals are well-defined if

 ∫∞0e−Φry|f(y+A)|dy<∞,A∈R, (2.13) g∈C2,and∫[1,∞)Π(du)maxA−u≤ζ≤A|g(ζ)−g(A)|<∞,A∈R. (2.14)

If these are satisfied, we can write as in (2.9) for as the sum of the following three terms:

 Γ1(x;A):=g(A)[Z(r)(x−A)−rΦrW(r)(x−A)],Γ2(x;A):=W(r)(x−A)ρ(r)g,A−φ(r)g,A(x),Γ3(x;A):=W(r)(x−A)Ψf(A)−Θf(x;A). (2.15)

Egami and Yamazaki [17] 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

 ∫(0,∞)(1∧z)Π(dz)<∞. (2.16)

It has been shown that

 uA(A+)=g(A)+W(r)(0)Λ(A),A∈R, (2.17)

where

 Λ(A)≡Λ(A;f,g):=−rΦrg(A)−σ22g′(A)+ρ(r)g,A+Ψf(A),A∈R. (2.18)

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 [17], on condition that there exists some satisfying

 ∫[1,∞)Π(du)sup0≤ξ≤δ|g(A+ξ)−g(A+ξ−u)|<∞, (2.19)

we have

 ∂∂AuA(x)=−eΦr(x−A)W′Φr(x−A)Λ(A),x>A, (2.20)

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:

 g(x)=K−bx−N∑i=1cieaix,x∈R, (2.21)

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.

###### Assumption 2.3.

We assume the following.

1. is continuous, piecewise differentiable, and increasing. In addition, the growth of as is at most polynomial and , ; these guarantee (2.13).

2. admits the form (2.21) for some , , and strictly positive constants and , , such that for any ,

###### Remark 2.2.

Assumptions 2.2 and 2.3(1) guarantee that for all ; for its proof, see, e.g., [34]. By this and Corollary 8.9 of [24],

 E[∫∞0e−rtf(Xt)dt]=∫∞−∞(Φ′re−Φr(y−x)−W(r)(x−y))f(y)dy (2.22)

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, .

With Assumption 2.3, we simplify (2.18) using

 (2.23)

By the convexity of , for any . The proof of the following lemma is given in Appendix A.1.

###### Lemma 2.1.

For every , we have

 Λ(A)=−rΦrK+b(rΦ2r+rA−ψ′(0+)Φr)+N∑i=1cieaiAϖr(ai)+Ψf(A). (2.24)

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 .

###### Remark 2.3.

Except for the case is a constant, because increases to , we have .

With our assumption on the form of , the value function can be written succinctly. By Proposition 2 of Avram et al. [5] and because by Assumption 2.2,

 Ex[e−rτ0Xτ0]=¯¯¯¯Z(r)(x)−ψ′(0+)Z(r)(x)−1r−r−ψ′(0+)ΦrΦ2rW(r)(x),x∈R,

where

 ¯¯¯¯Z(r)(x):=∫x0Z(r)(y)dy,x∈R.

This together with (2.5) gives, for any ,

 Ex[e−rτAXτA]=¯¯¯¯Z(r)(x−A)+(A−ψ′(0+)r)Z(r)(x−A)+ψ′(0+)r−r−ψ′(0+)Φr+rAΦrΦ2rW(r)(x−A).

With the help of Exercise 8.7(ii) of [24], the expression (2.9) becomes

 (2.25)

Moreover, if , by how is chosen as in (2.24) and by (2.7), it can be simplified to

 (2.26)

The verification of optimality requires the following smoothness properties, whose proofs are given in Appendix A.2.

###### Lemma 2.2.

Suppose .

1. is on .

2. 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 [17] 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).

Suppose .

1. Continuous fit holds (i.e. ) because, by (2.26), and .

2. In particular, when is of unbounded variation, smooth fit holds (i.e. ) because

 u′A∗(x) =KrW(r)(x−A∗)−b[Z(r)(x−A∗)+r(A∗−ψ′(0+)r)W(r)(x−A∗)] −N∑i=1cieaix(r−ψ(ai))W(r−ψ(ai))ai(x−A∗)−N∑i=1aicieaixZ(r−ψ(ai))ai(x−A∗)−Θ′f(x;A∗) x↓A∗−−−→−b−N∑i=1aicieaiA∗=g′(A∗),

thanks to , and ; see also the proof of Lemma 2.2.

We now state the main results of this subsection. The proof is given in Appendix A.3.

###### Proposition 2.1.
1. If , the stopping time is optimal over and the value function is as in (2.26) for all .

2. If , immediate stopping is always optimal and for any .

3. If , it is never optimal to stop (i.e.  a.s. is optimal), and the value function is that is given in (2.22).

### 2.3. For a general concave and decreasing g

We now relax the assumption on and consider a general concave and decreasing function . We also drop the continuity assumption on .

###### Assumption 2.4.

We assume the following.

1. is increasing. In addition, the growth of as is at most polynomial and , .

2. is twice-differentiable, concave and monotonically decreasing such that (2.14) and (2.19) hold.

Under this assumption, we see that as in (2.18) is continuous and increasing. Indeed, we have

 ∂∂A[−rΦrg(A)−σ22g′(A)]=−rΦrg′(A)−σ22g′′(A)≥0.

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.

###### Proposition 2.2.

Suppose Assumption 2.4.

1. When , then is optimal over and the value function is given by

 ˜u(x)=uA∗(x)=g(A∗)Z(r)(x−A∗)+W(r)(x−A∗)σ22g′(A∗)−φ(r)g,A∗(x)−Θf(x;A∗),x>A∗. (2.27)

For , we have .

2. If , immediate stopping is always optimal and for any .

3. If , then a.s. is optimal over and the value function is that is given in (2.22).

## 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:

 U(M)(x) :=sup(τ(1),…,τ(M))∈SMM∑m=1Ex[∫τ(m)τ(m−1)e−rtFm(Xt)dt+e−rτ(m)gm(Xτ(m))1{τ(m)<∞}], (3.1) ˜U(M)(x) :=sup(τ(1),…,τ(M))∈˜SMM∑m=1Ex[∫τ(m)τ(m−1)e−rtFm(Xt)dt+e−rτ(m)gm(Xτ(m))1{τ(m)<∞}], (3.2)

for all where we define for notational brevity and the supremum is, respectively, over the set of increasing sequences of stopping times,

 SM:={τ(m)∈S,1≤m≤M:τ(1)≤⋯≤τ(M)},

and over the set of increasing sequences of down-crossing times,

 ˜SM:={τ(m)=τAm∈˜S,1≤m≤M:A1≥A2⋯≥AM}.

Clearly, and hence .

We first consider the case admits the form

 gm(x):=Km−bmx−Nm∑i=1cmieamix1≤m≤M, (3.3)

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:

 fm:=Fm−Fm+1,1≤m≤M,

with . As is also assumed in [10], we consider the case is increasing for each . Using the notation as in (2.3), we can then write for all

 U(M)(x)=sup(τ(1),…,τ(M))∈SMM∑m=1u(x,τ(m);fm,gm), (3.4) ˜U(M)(x)=sup(τ(1),…,τ(M))∈˜SMM∑m=1u(x,τ(m);fm,gm). (3.5)

In summary, we assume Assumptions 3.1 and 3.2 below for (3.1) and (3.2), respectively.

###### Assumption 3.1.

For each , we assume that and satisfy Assumption 2.3.

###### Assumption 3.2.

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 ,

 gI:=∑i∈IgiandfI:=FminI−FmaxI+1, (3.6)

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.

Let

 Λm(A):=Λ(A;fm,gm),A∈R,1≤m≤M, (3.7)

as the function (2.18) for . Because and for any measurable functions and in view of (2.10) and (2.12), we see that

 ΛI(A):=Λ(A;fI,gI)=Λ(A;∑m∈Ifm,∑m∈Igm)=∑m∈IΛm(A) (3.8)

is increasing and corresponds to the function (2.18) for . In particular, under Assumption 2.3, this reduces by Lemma 2.1 to

 ΛI(A)

Now let be the root of if it exists. If , we set ; if , we set . For simplicity, let for any . Also define

 uIA(x) :=u(x,τA;fI,gI),x,A∈R.

With these notations, the following is immediate by Propositions 2.1 and 2.2.

###### Corollary 3.1.

Fix any and , and consider the problems:

Suppose Assumption 3.1.

1. If , then

 uI(x)=uIA∗I(x)=∑m∈I(KmZ(r)(x−A∗I)−bm[¯¯¯¯Z(r)(x−A∗I)+(A∗I−ψ′(0+)r)Z(r)(x−A∗I)+ψ′(0+)r] −Nm∑i=1cmieamixZ(r−ψ(ami))ami(x−A∗I))−ΘfI(x;A∗I),

and the stopping time is optimal.

2. If , for any with the optimal stopping time a.s.

3. If , it is never optimal to stop, and the value function is given by

 uI(x)=∫∞−∞(Φ′re−Φr(y−x)−W(r)(x−y))fI(y)dy. (3.9)

Suppose Assumption