The Leland-Toft optimal capital structure model under Poisson observations
Zbigniew Palmowski, José Luis Pérez, Budhi Arta Surya, Kazutoshi Yamazaki
TTHE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSONOBSERVATIONS
ZBIGNIEW PALMOWSKI ∗ , JOS ´E LUIS P ´EREZ ∗∗ , BUDHI ARTA SURYA † , AND KAZUTOSHI YAMAZAKI ‡ A BSTRACT . This paper revisits the optimal capital structure model with endogenous bankruptcy, first stud-ied by Leland [39] and Leland and Toft [40]. Unlike in the standard case, where shareholders continuouslyobserve the asset value and bankruptcy is executed instantaneously and without delay, the information ofthe asset value is assumed to be updated only at intervals, modeled by the jump times of an independentPoisson process. Under the spectrally negative L´evy model, we obtain the optimal bankruptcy strategyand the corresponding capital structure. A series of numerical studies enable analysis of the sensitivity ofobservation frequency in the optimal solutions, optimal leverage and credit spreads.
Keywords:
Credit risk, optimal capital structure, spectrally negative L´evy processes, scale functions
JEL Classification:
D92, G32, G33
Mathematics Subject Classification (2010):
NTRODUCTION
The study of capital structures dates back to the seminal work by Modigliani and Miller [47], which shows that, in a frictionless economy, the value of a firm is invariant to the choice of capital structures. While the Modigliani-Miller (MM) theory is regarded as an effective starting point for research on capital structures and has providedvaluable insights in the field, it is not directly applicable to businesses. In reality, selection of capital structuresis not perfectly random. Instead, it depends significantly on factors such as industry type, county and corporatelaw. In the field of corporate finance, various approaches have been taken to explain how much debt a firm shouldissue. A reasonable conclusion can be obtained only after challenging some of the assumptions of the classicalMM theory.The trade-off theory is one well-known approach for the study of capital structures. While various frictions mayaffect a firm’s decisions, (1) bankruptcy costs and (2) tax benefits are believed to be the most important factors. Byissuing debt, bankruptcy costs increase, while at the same time the firm can enjoy tax shields for coupon paymentsto the bondholders. The trade-off theory states that firms issue the appropriate debt to solve the trade-off between ∗ Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspia´nskiego 27,50-370 Wrocław, Poland. Email: [email protected] . ∗∗ Department of Probability and Statistics, Centro de Investigaci´on en Matem´aticas, A.C. Calle Jalisco S/N C.P. 36240,Guanajuato, Mexico. Email: [email protected] . † School of Mathematics and Statistics, Victoria University of Wellington, Gate 6, Kelburn PDE, Wellington 6140, NewZealand. Email: [email protected] . ‡ Department of Mathematics, Faculty of Engineering Science, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka564-8680, Japan. Email: [email protected] . a r X i v : . [ q -f i n . P R ] M a r Z. PALMOWSKI, J.L. P ´EREZ, B. A. SURYA, AND K. YAMAZAKI minimizing bankruptcy costs and maximizing tax benefits. To formulate this optimization problem, one needs anefficient and realistic way of modeling not only bankruptcy but also tax benefits, which depend heavily on thedynamics of the firm’s asset value. For more details on the trade-off theory and its review, see, e.g., [33, 28, 31].Classically, there are two models of bankruptcy in credit risk: the structural approach and the reduced-formapproach (see [13]). The former, first proposed by Black and Cox [14], models bankruptcy time as the first timethe asset value goes below a fixed barrier. The latter models it as the first jump epoch of a doubly stochastic process(known hereafter as the Cox process) where the jump rate is driven by another stochastic process. Both approacheswere developed extensively in the 2000s and are now commonly used throughout the asset pricing and credit riskliterature. An extension of the structural approach, which we call the excursion (Parisian) approach, models itas the first instance in which the amount of time the asset price stays continuously below a threshold exceeds agiven grace period . Motivated by the Parisian option, this is sometimes called the
Parisian ruin (see [21]). Inthe corporate finance literature, the approach has been used to model the reorganization process (Chapter 11), asin [27, 17]. Here, reorganization is undertaken whenever the asset value is below a threshold; although there isa chance of recovering to reach above the threshold, if reorganization time exceeds the grace period, the firm isliquidated. For more information, see the literature review in Section 1.3.1.1.
A new model of bankruptcy.
This paper considers the scenario where asset value information is updatedonly at epochs ( T λn ) n ≥ , given by the jump times of a Poisson process ( N λt ) t ≥ with fixed rate λ . Given a bank-ruptcy barrier V B , chosen by the equity holders, bankruptcy is triggered at the first update time where the assetprocess ( V t ) t ≥ is below V B : inf { T λi : V T λi < V B } . (1.1)This is also written as the classical bankruptcy time inf { t > V λt < V B } , (1.2)of the asset value if it is only updated at ( T λn ; n ≥ : V λt := V T λNλt , t ≥ . Here T λN λt is the most recent update time before t . In Figure 1, we plot sample paths of ( V t ) t ≥ , ( V λt ) t ≥ , ( T λn ) n ≥ and the corresponding bankruptcy time.The bankruptcy model (1.1) is closely related to the reduced-form and excursion approaches reviewed above.(1) The bankruptcy time (1.1) is equivalent to the Parisian ruin with the (constant) grace period replaced withan exponential time clock, the first epoch being the time spent continuously below V B for more than anindependent exponential time. For more details see Appendix A.(2) It is also equivalent to the bankruptcy time in the reduced-form credit risk model, where the bankruptcytime is the first jump time of the Cox process with hazard rate given by ( h t := λ { V t 1. Sample paths of the asset value ( V t ) t ≥ (black lines) and ( V λt ) t ≥ (horizontalblue lines) along with the Poisson arrival times ( T λn ) n ≥ (indicated by dotted verticallines). The red zone (0 , V B ) is given by the rectangle colored in red. The asset values atbankruptcy and other observation times are indicated by the red circle and blue triangles,respectively. Here, the bankruptcy time corresponds to T λ , but the asset value has crossed V B before and then recovered back before T λ . Note that ( V λt ) t ≥ has a positive jump at T λ . First, in reality, it is not possible to continuously observe the accurate status of a firm and make bankruptcydecisions instantaneously. In addition, unlike in the case of American options pricing, for which computer pro-grams can be set up to exercise automatically, in our case, information is acquired by humans. As observed in theliterature of rational inattention [54], the amount of information a decision maker can capture and handle is lim-ited, and instead they rationally decide to stay with imperfect information. Taking a bankruptcy decision requirescomplex information and it is more realistic to assume that the information for the decision makers is updated onlyat random discrete times. While they are expected to respond promptly, delays are inevitable and possibly have asignificant impact on bankruptcy costs.Second, the majority of the existing literature assumes continuous observation using a continuous asset valueprocess – in this case, the asset value at bankruptcy is, in any event, precisely V B . Unfortunately, it is unreasonableto assume that one can precisely predict the asset value at bankruptcy, which is in reality random. The randomnesscan be realized by adding negative jumps to the process. We underline that in our model this randomness can alsobe achieved by any choice (continuous or c´adl´ag) of the underlying process. See Figure 6 in Section 6.Third, this model generalizes the classical model and allows more flexibility by having one more parameter λ .The classical structural model (with instantaneous liquidation upon downcrossing the barrier) corresponds to thecase λ = ∞ and the no-bankruptcy model corresponds to the case λ = 0 . With careful calibration of λ , the model Z. PALMOWSKI, J.L. P ´EREZ, B. A. SURYA, AND K. YAMAZAKI can potentially estimate the bankruptcy costs and tax benefits more precisely. Typically, for calibration, creditspread data is used. As shown in the numerical results (see Figure 8), a variety of term structures can be achievedby choosing the value of λ .Finally, thanks to the equivalence of our bankruptcy time with the classical bankruptcy time (1.2) of the pro-cess ( V λt ) t ≥ , this research can be considered a contribution to the classical structural approach. Existing resultsfeaturing asset value processes with two sided jumps are rather limited. However, we provide a new analyticallytractable case for ( V λt ) t ≥ , containing two-sided jumps even when ( V t ) t ≥ does not have positive jumps (see Fig-ure 1). By appropriately selecting the driving process ( V t ) t ≥ as well as λ , it is possible to construct a wide rangeof stochastic processes with two-sided jumps.1.2. Contributions of the paper. This model is built based on the seminal paper by Leland and Toft [40], witha feature of endogenous default. While Leland [39]’s framework is more frequently used and is certainly moremathematically tractable, its extension [40] more accurately captures the flow of debt financing by successfullyavoiding the use of perpetual bonds assumed in [39].In addition, while the majority of papers in financial economics assume a geometric Brownian motion for theasset price ( V t ) t ≥ , we follow the works of Hilberink and Rogers [29], Kyprianou and Surya [38] and Surya andYamazaki [56] and consider an exponential L´evy process with arbitrary negative jumps (spectrally negative L´evyprocesses). Although it is more desirable to also allow positive jumps as in Chen and Kou [20], as discussed in[29], negative jumps occur more frequently and effectively model the downward risks. With the spectrally negativeassumption, semi-explicit expressions of the equity value as well as the optimal bankruptcy threshold are elicited,without focusing on a particular set of jump measures. Again, see the discussion above on how our model iscapable of modeling the two-sided jump case in the classical structural approach, even when a spectrally negativeL´evy process is used for ( V t ) t ≥ . For a more general study of financial models using L´evy processes, the readershould refer to Cont and Tankov [22].To solve the problem, recent developments of the fluctuation theory of L´evy processes are utilized. First, thefirm/debt/equity values are expressed in terms of the so-called scale functions , which exist for a general spectrallynegative L´evy process. These permit direct computation of the optimal bankruptcy barrier and the correspondingfirm/debt/equity values.With these analytical results, a sequence of numerical experiments can be conducted. Here, to easily com-prehend the impacts of the parameters describing the problem, we use a (spectrally negative) hyperexponentialjump diffusion (a mixture of Brownian motion and i.i.d. hyperexponentially distributed jumps), for which the scalefunction can be written as a sum of exponential functions. The equity/debt/firm values can be written explicitlyand the optimal bankruptcy barrier can be computed instantaneously by a classical bisection method. The optimalcapital structure is obtained by solving the two-stage optimization problem as proposed in [40]. In addition, withnumerical Laplace inversion, we also obtain the term structures of credit spreads and the density/distribution ofthe bankruptcy time and the corresponding asset value. Because various numerical experiments have already beenconducted in other papers, here we focus on analyzing the impacts of the frequency of observation λ . We verify theconvergence to the classical case of [29, 38], and also observe monotonicity, with respect to λ , of the bankruptcybarrier, firm value under the optimal capital structure, the optimal leverage, and the credit spread. HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 5 Related literature. Before concluding this section, we review several relevant papers motivating our prob-lem.The most relevant paper, to our best knowledge, is Francois and Mollerec [27], in which the authors modeled thereorganization process (Chapter 11) using the excursion approach with a deterministic grace period as describedabove. Broadie et al. [17] considered a similar model with an additional barrier for immediate liquidation uponcrossing, whereas Moraux [48] considered a variant of [27] using the occupation time approach , in which distresslevel accumulates without being reset each time the asset process recovers to a healthy state. These papers are basedon Leland [39], with perpetual bonds and asset values driven by geometric Brownian motions for mathematicaltractability. However, it is significantly more challenging than the classical structural approach and hence most ofthem rely on numerical approaches. In this paper, on the other hand, semi-analytical solutions for a more generalasset value process with jumps are obtained as a result of the use of Poisson arrival times for the update times.This paper is also motivated by Duffie and Lando [23], in which they modeled the asymmetry of informationbetween firms and bond investors. The authors assumed that bond investors cannot observe the firm’s assetsdirectly and that instead, they receive only periodic and imperfect accounting reports on the firm’s status. Underthese assumptions, the authors successfully explained the non-zero credit spread limit.Regarding the study of L´evy processes observed at Poisson arrival times, there has been substantial progress inthe last few years. Recently, Albrecher and Ivanovs [2] investigated close links between L´evy processes observedcontinuously and periodically. In results similar to those for the classical hitting time at a barrier, they foundthat the exit identities under periodic observation can be obtained, if the Wiener-Hopf factorization is known. Inparticular, when focusing on the spectrally one-sided case, these can be written in terms of the scale function. Forthe results of our paper, we use the joint Laplace transform of the bankruptcy time (1.1) and the asset value in thatinstance, which is obtained in [1, 2]. In addition, we obtain the resolvent measure killed at the first Poissoniandownward passage time (1.1) for the computation of tax benefits.Regarding the optimal stopping problems under Poisson observations, perpetual American options have beenstudied by Dupuis and Wang [24] for the geometric Brownian motion case. This has recently been generalizedto the L´evy case by P´erez and Yamazaki [51]. Several key studies have been performed on the application ofscale functions in optimal stopping in the continuous observation setting (e.g., [3, 7, 44, 53, 55]). The periodicobservation model is more frequently used in the insurance community, in particular in the optimal dividendproblem (see [6, 5, 49]).To the best of our knowledge, this is the first attempt to introduce Poisson observations in the problem of capitalstructures. We believe the techniques used in this paper can be used similarly in related problems described abovewhen the Poisson observation is introduced.1.4. Organization of the paper. The organization of this paper is as follows. In Section 2 we present formally themain problem that we work on in this article. In Section 3, we compute the equity value using the scale function,and, in Section 4, we identify the optimal barrier. Section 5 considers the two-stage problem to obtain the optimalcapital structure. Section 6 deals with numerical examples confirming theoretical results. Section 7 concludes thepaper. Long proofs are deferred to the Appendix. Z. PALMOWSKI, J.L. P ´EREZ, B. A. SURYA, AND K. YAMAZAKI 2. P ROBLEM F ORMULATION Let (Ω , F , P ) be a complete probability space hosting a L´evy process X = ( X t ) t ≥ . The value of the firm’sasset is assumed to evolve according to an exponential L´evy process given by, for the initial value V > , V t := V e X t , t ≥ . Let r > be the positive risk-free interest rate and ≤ δ < r the total payout rate to the firm’s investors. Weassume that the market is complete and this requires ( e − ( r − δ ) t V t ) t ≥ to be a P -martingale.The firm is partly financed by debt with a constant debt profile: it issues, for some given constants p, m > , newdebt at a constant rate p with maturity profile ϕ ( s ) := me − ms . In other words, the face value of the debt issued inthe small time interval ( t, t + d t ) that matures in the small time interval ( t + s, t + s + d s ) is approximately givenby pϕ ( s )d t d s . Assuming the infinite past, the face value of debt held at time that matures in ( s, s + d s ) becomes (cid:20)(cid:90) −∞ pϕ ( s − u )d u (cid:21) d s = pe − ms d s, (2.1)and the face value of all debt is a constant value, P := (cid:90) ∞ pe − ms d s = pm . For more details, see [29, 38].Let ( N λt ) t ≥ be an independent Poisson process with rate λ > and T := ( T λn ) n ≥ be its jump times. Supposethe bankruptcy is triggered at the first time of T the asset value process ( V t ) t ≥ goes below a given level V B > : T − V B := inf { S ∈ T : V S < V B } (2.2)with the convention inf ∅ = ∞ . In our model, it is more natural to assume that the bankruptcy decision can bemade at time zero. Hence, we modify the above and consider the random time T − V B := inf { S ∈ T ∪ { } : V S < V B } = T − V B { V ≥ V B } . (2.3)(i) Suppose V ≥ V B so that T − V B = T − V B .The debt pays a constant coupon flow at a fixed rate ρ > and a constant fraction < α < of the asset valueis lost at the bankruptcy time T − V B . In this setting, the value of the debt with a unit face value and maturity t > becomes d ( V ; V B , t ) := E (cid:34)(cid:90) t ∧ T − VB e − rs ρ d s (cid:35) + E (cid:20) e − rt { t Regarding the value of the firm , it is assumed that there is a corporate tax rate κ > and its (full) rebate oncoupon payments is gained if and only if V t ≥ V T for some given cut-off level V T ≥ (for the case V T = 0 , itenjoys the benefit at all times). Based on the trade-off theory (see e.g. [15]), the firm value becomes the sum of theasset value and total value of tax benefits less the value of loss at bankruptcy, given by V ( V ; V B ) := V + E (cid:34)(cid:90) T − VB e − rt { V t ≥ V T } P κρ d t (cid:35) − α E (cid:20) e − rT − VB V T − VB { T − VB < ∞} (cid:21) . (2.5)(ii) Suppose V < V B so that T − V B = 0 a.s. Then, D ( V ; V B ) = V ( V ; V B ) = (1 − α ) V. (2.6)The problem is to pursue an optimal bankruptcy level V B ≥ that maximizes the equity value , E ( V ; V B ) := V ( V ; V B ) − D ( V ; V B ) , (2.7)subject to the limited liability constraint , E ( V ; V B ) ≥ , V ≥ V B , (2.8)if such a level exists. Here, V B = 0 means that it is never optimal to go bankrupt with the limited liability constraintsatisfied for all V > . Note that when V < V B then (2.6) gives E ( V ; V B ) = 0 .3. C OMPUTATION OF THE EQUITY VALUE Suppose from now on that ( X t ) t ≥ is a spectrally negative L´evy process, that is a L´evy process without positivejumps. We denote by ψ ( θ ) := log E (cid:2) e θX (cid:3) , θ ≥ (3.1)its Laplace exponent with the right-inverse Φ( q ) := sup { s ≥ ψ ( s ) = q } , q ≥ . (3.2)3.1. Scale functions. The starting point of whole analysis is introducing the so-called q -scale function W ( q ) ( x ) ,with q ≥ and x ∈ R . It features invariably in almost all known fluctuation identities of spectrally negative L´evyprocesses; see Zolotarev [58] and Tak´acs [57] for the origin of this function. See also [38, 35] for a detailed review.Fix q ≥ . The q -scale function W ( q ) is the mapping from R to [0 , ∞ ) that takes value zero on the negative half-line, while on the positive half-line it is a continuous and strictly increasing function with the Laplace transform: (cid:90) ∞ e − θx W ( q ) ( x )d x = 1 ψ ( θ ) − q , θ > Φ( q ) . (3.3)Define also the second scale function: Z ( q ) ( x ; θ ) := e θx (cid:18) q − ψ ( θ )) (cid:90) x e − θz W ( q ) ( z )d z (cid:19) , x ∈ R , θ ≥ . In particular, for x ∈ R , we let Z ( q ) ( x ) := Z ( q ) ( x ; 0) and, for λ > , Z ( q ) ( x ; Φ( q + λ )) = e Φ( q + λ ) x (cid:18) − λ (cid:90) x e − Φ( q + λ ) z W ( q ) ( z )d z (cid:19) . In the next section, we see that the equity value (2.7) can be written in terms of the scale functions W ( q ) and Z ( q ) . Z. PALMOWSKI, J.L. P ´EREZ, B. A. SURYA, AND K. YAMAZAKI Related fluctuation identities. For y ∈ R , let P y be the conditional probability under which the initial valueof the spectrally negative L´evy process is X = y .Following equation (4.5) in [38] (see also Emery [26] and [8, eq. (3.19)]), the joint Laplace transform of thefirst passage time(3.4) τ − := inf { t ≥ X t < } and X τ − is given by the following identity H ( q ) ( y ; θ ) := E y (cid:20) e − qτ − + θX τ − { τ − < ∞} (cid:21) = Z ( q ) ( y ; θ ) − ψ ( θ ) − qθ − Φ( q ) W ( q ) ( y ) , (3.5)where y ∈ R , θ ≥ , and q ≥ . Similar results have been obtained for the Poisson observation case. Recall that T := ( T λn ; n ≥ is the set of jump times of an independent Poisson process. We define ˜ T − z := inf { S ∈ T : X S < z } , z ∈ R . (3.6)By equation (14) of Theorem 3.1 in [1], for θ ≥ and y ∈ R , J ( q,λ ) ( y ; θ ) := E y (cid:20) e − q ˜ T − + θX ˜ T − { ˜ T − < ∞} (cid:21) = λλ + q − ψ ( θ ) (cid:18) Z ( q ) ( y ; θ ) − Z ( q ) ( y ; Φ( q + λ )) ψ ( θ ) − qλ Φ( q + λ ) − Φ( q ) θ − Φ( q ) (cid:19) = (cid:104) − ( ψ ( θ ) − q )( θ − Φ( q )) (Φ( λ + q ) − θ )( λ + q − ψ ( θ )) (cid:105) Z ( q ) ( y ; θ ) − ( ψ ( θ ) − q )( θ − Φ( q )) (Φ( λ + q ) − Φ( q ))( λ + q − ψ ( θ )) (cid:0) Z ( q ) ( y ; Φ( λ + q )) − Z ( q ) ( y ; θ ) (cid:1) . (3.7) Remark 3.1. (1) We have J ( q,λ ) (0; 1) = λλ + q − ψ (1) − ψ (1) − qλ + q − ψ (1) Φ( q + λ ) − Φ( q )1 − Φ( q ) = 1 − ψ (1) − qλ + q − ψ (1) Φ( q + λ ) − − Φ( q ) > ,J ( q,λ ) (0; 0) = λλ + q − qλ + q Φ( q + λ ) − Φ( q )Φ( q ) = 1 − qλ + q Φ( q + λ )Φ( q ) > , where the positivity holds by the probabilistic expression of J ( q,λ ) as in (3.7) .(2) We have J ( q,λ ) ( y ; θ ) < , q > , θ ≥ , y ∈ R . (3.8) To see this, by the memoryless property of the exponential random variable, we can write, for some independentexponential random variable e λ , the first observation time at which X is below zero is τ − + e λ and hence ˜ T − isbounded from below by an exponential random variable. In addition, we must have X ˜ T − ≤ P y -a.s. and hencewe have (3.8) . In order to write the equity value, we obtain an expression for Λ ( r,λ ) ( y, z ) := E y (cid:34)(cid:90) ˜ T − z e − rt { X t ≥ log V T } d t (cid:35) , y, z ∈ R . (3.9)In Appendix B, we obtain the resolvent measure killed at ˜ T − z and the following result as a corollary. HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 9 Proposition 3.1. Fix y, z ∈ R . For V T > , we have Λ ( r,λ ) ( y, z ) = Z ( r ) ( y − z ; Φ( r + λ )) Φ( r + λ ) − Φ( r ) λ × (cid:16) r ) Z ( r + λ ) ( z − log V T ; Φ( r )) − λ Φ( r ) W ( r + λ ) ( z − log V T ) (cid:17) − W ( r + λ ) ( y − log V T ) { z> log V T } − W ( r ) ( y − log V T ) { z ≤ log V T } + λ { z> log V T } (cid:90) y − z W ( r ) ( y − z − u ) W ( r + λ ) ( u + z − log V T )d u, where W ( q ) ( y ) := (cid:82) y W ( q ) ( u )d u for all q > and y ∈ R .For V T = 0 , we have Λ ( r,λ ) ( y, z ) = (1 − J ( r,λ ) ( y − z ; 0)) /r. Expression for the equity value in terms of the scale function. Using the identities in Section 3.2, theequity value (2.7) can be written as follows. Here, we focus on the case V B > . The case V B = 0 (for which, aswe will see, only the case V T = 0 needs to be considered) is given later in (4.4).First by (3.7), we have, for q = r and q = r + m , E (cid:20) e − qT − VB V T − VB { T − VB < ∞} (cid:21) = V B J ( q,λ ) (cid:16) log VV B ; 1 (cid:17) and E (cid:20) e − qT − VB { T − VB < ∞} (cid:21) = J ( q,λ ) (cid:16) log VV B ; 0 (cid:17) . In addition, by (3.9), E (cid:34)(cid:90) T − VB e − rt { V t ≥ V T } d t (cid:35) = Λ ( r,λ ) (log V, log V B ) . Hence, we can write D ( V ; V B ) = P ρ + pr + m (cid:16) − J ( r + m,λ ) (cid:16) log VV B ; 0 (cid:17)(cid:17) + (1 − α ) V B J ( r + m,λ ) (cid:16) log VV B ; 1 (cid:17) , V ( V ; V B ) = V + P κρ Λ ( r,λ ) (log V, log V B ) − αV B J ( r,λ ) (cid:16) log VV B ; 1 (cid:17) , (3.10)and therefore, by taking their difference, the equity value is E ( V ; V B ) = V + P κρ Λ ( r,λ ) (log V, log V B ) − αV B J ( r,λ ) (cid:16) log VV B ; 1 (cid:17) − P ρ + pr + m (cid:16) − J ( r + m,λ ) (cid:16) log VV B ; 0 (cid:17)(cid:17) − (1 − α ) V B J ( r + m,λ ) (cid:16) log VV B ; 1 (cid:17) . (3.11) 4. O PTIMAL BARRIER Having the equity value E ( V ; V B ) given in (3.11) identified using equation (3.7) and Proposition 3.1, we areready to find the optimal barrier V ∗ B maximizing it. Our objective in this paper is to show that the optimal barrieris V ∗ B such that E ( V ∗ B ; V ∗ B ) = 0 , (4.1) if it exists, where, by (3.11) and Remark 3.1(1), for V B > , E ( V B ; V B ) = V B + P κρ Λ ( r,λ ) (log V B , log V B ) − αV B J ( r,λ ) (0; 1) − P ρ + pr + m (1 − J ( r + m,λ ) (0; 0)) − (1 − α ) V B J ( r + m,λ ) (0; 1)= V B [1 − αJ ( r,λ ) (0; 1) − (1 − α ) J ( r + m,λ ) (0; 1)] + P κρ Λ ( r,λ ) (log V B , log V B ) − P ρ + pλ + r + m Φ( r + m + λ )Φ( r + m ) . (4.2)4.1. Existence. We first show the condition for the existence of V ∗ B satisfying (4.1). To this end, we show thefollowing result; the proof is given in Appendix C.1. Lemma 4.1. The mapping z (cid:55)→ Λ ( r,λ ) ( z, z ) is non-decreasing on R with the limit lim z ↓−∞ Λ ( r,λ ) ( z, z ) = if V T > , λ + r Φ( r + λ )Φ( r ) if V T = 0 . F IGURE 2. Plots of V B (cid:55)→ E ( V B ; V B ) for V T = 0 , , , . . . , . Solid lines show forthe case V T = 0 and dotted lines for the other cases. The points at V ∗ B are indicated bycircles. The left plot is based on the parameter set in Case B in Section 6 (except V T ), andachieves V ∗ B > for all cases. The right plot is based on the same parameters except thatwe set κ = 0 . , m = 10 , ρ = 0 . and λ = 0 . to achieve V ∗ B = 0 when V T = 0 . This lemma leads to the following proposition. For numerical illustration, see Figure 2. Proposition 4.1. The mapping V B (cid:55)→ E ( V B ; V B ) is strictly increasing on (0 , ∞ ) with the limit: lim V B ↓ E ( V B ; V B ) = − P ρ + pλ + r + m Φ( r + m + λ )Φ( r + m ) if V T > , P κρλ + r Φ( r + λ )Φ( r ) − P ρ + pλ + r + m Φ( r + m + λ )Φ( r + m ) if V T = 0 , lim V B ↑∞ E ( V B ; V B ) = ∞ . HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 11 Proof. From Remark 3.1(2), we have − αJ ( r,λ ) (0; 1) − (1 − α ) J ( r + m,λ ) (0; 1) > . By this, Lemma 4.1 andbecause z (cid:55)→ Λ ( r,λ ) ( z, z ) is non-decreasing and bounded, the claim is immediate in view of the second equality of(4.2). (cid:3) (cid:3) Now by Proposition 4.1, we define the candidate optimal threshold V ∗ B formally, as follows.(1) For the case V T > and the case V T = 0 with P κρ λ + r Φ( r + λ )Φ( r ) − P ρ + pλ + r + m Φ( r + m + λ )Φ( r + m ) < , we set V ∗ B > such that E ( V ∗ B ; V ∗ B ) = 0 , whose existence and uniqueness hold by Proposition 4.1.(2) For the case V T = 0 with(4.3) P κρλ + r Φ( r + λ )Φ( r ) − P ρ + pλ + r + m Φ( r + m + λ )Φ( r + m ) ≥ , we set V ∗ B = 0 .The debt/firm/equity values for the case V ∗ B > can be computed by (3.10) and (3.11). For the case V ∗ B = 0 ,where necessarily V T = 0 , we have, for all V > , D ( V ; 0) = E (cid:20)(cid:90) ∞ e − ( r + m ) t ( P ρ + p ) d t (cid:21) = P ρ + pr + m , V ( V ; 0) = V + E (cid:20)(cid:90) ∞ e − rt P κρ d t (cid:21) = V + P κρr , and therefore E ( V ; 0) = V + P κρr − P ρ + pr + m . (4.4)4.2. Optimality. For the rest of this section, we show the following one of our main results. Theorem 4.1. The barrier V ∗ B is optimal for the problem of maximizing (2.7) subject to (2.8) . To prove the optimality, it is sufficient to show the following:(1) If V ∗ B > , every threshold V B < V ∗ B violates the limited liability constraint (2.8).(2) V ∗ B attains a higher equity value than any V B > V ∗ B does.(3) V ∗ B is feasible. Proposition 4.2. Suppose V ∗ B > . For V B < V ∗ B , the limited liability constraint (2.8) is not satisfied.Proof. By the (strict) monotonicity as in Proposition 4.1 and because E ( V ∗ B ; V ∗ B ) = 0 (given that V ∗ B > ), wehave E ( V B ; V B ) < for V B < V ∗ B . (cid:3) (cid:3) The proof of the following is given in Appendix C.2. Proposition 4.3. For V > V B > , we have ∂∂V B E ( V ; V B ) = − (Φ( r + m + λ ) − Φ( r + m )) H ( r + m ) (cid:16) log VV B ; Φ( r + m + λ ) (cid:17) L (log V, log V B ) V B (4.5) where H ( r + m ) is as in (3.5) and, for x, z ∈ R , L ( x, z ) := H ( r ) ( x − z ; Φ( r + λ )) H ( r + m ) ( x − z ; Φ( r + m + λ )) Φ( r + λ ) − Φ( r )Φ( r + m + λ ) − Φ( r + m ) × (cid:32) α (1 − J ( r,λ ) (0; 1)) e z + P κρ Φ( r + λ ) − Φ( r ) λ (cid:90) z − log V T −∞ H ( r + λ ) ( y ; Φ( r ))d y (cid:33) + (1 − α )(1 − J ( r + m,λ ) (0; 1)) e z − P ρ + pr + m (1 − J ( r + m,λ ) (0; 0)) . The proof of the following results are given in Appendices C.3 and C.4. Proposition 4.4. Suppose V B > V ∗ B ≥ . We have ∂∂V B E ( V ; V B ) < for V > V B . Hence, E ( V ; V B ) < E ( V ; V ∗ B ) for all V > V B . Proposition 4.5. For V > V B > , we have ∂∂V E ( V ; V B ) = 1 − V B V (cid:104) ∂∂V B E ( V ; V B ) + αJ ( r,λ ) (cid:16) log VV B ; 1 (cid:17) + (1 − α ) J ( r + m,λ ) (cid:16) log VV B ; 1 (cid:17)(cid:105) + P κρV R ( r,λ ) (cid:16) log VV B , log V T V B (cid:17) , where R ( r,λ ) is the resolvent density given in (B.3) . Proposition 4.6. We have E ( V ; V ∗ B ) ≥ for all V ≥ V ∗ B when V ∗ B > and for all V > when V ∗ B = 0 . In otherwords, V ∗ B is feasible.Proof. (i) Suppose V ∗ B > . Because R ( r,λ ) is the resolvent density, it is nonnegative. By this together withPropositions 4.4 and 4.5, for V > V ∗ B > , ∂∂V E ( V ; V ∗ B ) = 1 − V ∗ B V ∂∂V B E ( V ; V ∗ B ) − α V ∗ B V J ( r,λ ) (cid:16) log VV ∗ B ; 1 (cid:17) − (1 − α ) V ∗ B V J ( r + m,λ ) (cid:16) log VV ∗ B ; 1 (cid:17) + P κρV R ( r,λ ) (cid:16) log VV ∗ B , log V T V ∗ B (cid:17) ≥ − V ∗ B V (cid:104) αJ ( r,λ ) (cid:16) log VV ∗ B ; 1 (cid:17) + (1 − α ) J ( r + m,λ ) (cid:16) log VV ∗ B ; 1 (cid:17)(cid:105) ≥ − V ∗ B V ≥ , where the second inequality holds by Remark 3.1(2). Applying this and the fact that E ( V ∗ B ; V ∗ B ) = 0 when V ∗ B > ,the claim is immediate.(ii) For the case V ∗ B = 0 recall that necessarily V T = 0 , and hence by (4.4) we obtain that(4.6) ∂∂V E ( V ; 0) = 1 > . Moreover, by Remark 3.1(1) and because q (cid:55)→ J ( q,λ ) (0; 0) is non-increasing in view of its probabilistic expression,for m > , rλ + r Φ( r + λ )Φ( r ) = 1 − J ( r,λ ) (0; 0) ≤ − J ( r + m,λ ) (0; 0) = r + mλ + r + m Φ( r + λ + m )Φ( r + m ) . HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 13 By this and recalling inequality (4.3), we have that (cid:18) P κρr − P ρ + pr + m (cid:19) r + mλ + r + m Φ( r + λ + m )Φ( r + m ) ≥ P κρλ + r Φ( r + λ )Φ( r ) − P ρ + pλ + r + m Φ( r + m + λ )Φ( r + m ) ≥ . Hence, P κρr − P ρ + pr + m ≥ and therefore lim V ↓ E ( V ; 0) = P κρr − P ρ + pr + m ≥ . (4.7)Using (4.6) together with (4.7) completes the proof. (cid:3) (cid:3) Proof of Theorem 4.1 Now, by Propositions 4.2, 4.4 and 4.6, the proof of Theorem 4.1 is complete. (cid:3) Remark 4.1. Intuitively, as λ → ∞ , the optimal barrier is expected to converge to that in the classical case as in [38] . In order to confirm this assertion, we provide the following result; its proof is deferred to Appendix C.5. Lemma 4.2. Suppose V T > and let V B ≥ be fixed. We have lim λ →∞ λ + r + m Φ( λ + r + m ) E ( V B ; V B )= V B (cid:20) α ψ (1) − r − Φ( r ) + (1 − α ) ψ (1) − ( r + m )1 − Φ( r + m ) (cid:21) + P κρ Φ( r ) (cid:34)(cid:18) V B V T (cid:19) Φ( r ) ∧ (cid:35) − P ρ + p Φ( r + m ) . (4.8) This is consistent with identity (3.26) in [38] , where the optimal bankruptcy level is such that the right-hand sideof (4.8) vanishes. 5. T WO - STAGE PROBLEM We now obtain the optimal leverage by solving the two-stage problem as studied by [20, 39, 40] where the finalgoal is to choose P that maximizes the firm’s value V . For fixed V > , the problem is formulated as max P V ( V ; V ∗ B ( P ) , P ) (5.1)where we emphasize the dependency of V and V ∗ B on P .In this two-stage problem, it is worth investigating the shape of V ( V ; V ∗ B ( P ) , P ) with respect to P to confirmwhether it has a unique maximizer. Chen and Kou [20] verified the concavity in the continuous observation casewith a double jump diffusion as the underlying model and the assumption that V T = 0 .In this section, we show, in the periodic observation setting, the concavity for the case when V T = 0 and thefollowing assumption is satisfied. Assumption 5.1. The L´evy measure Π of the dual process − X has a completely monotone density, i.e. Π has adensity π whose n th derivative π ( n ) exists for all n ≥ and satisfies ( − n π ( n ) ( x ) ≥ , x > . Important examples satisfying Assumption 5.1 include (the spectrally negative versions of) hyperexponentialjump diffusion (as a generalization of [20]), variance gamma process [45], CGMY process [19], as well as mero-morphic L´evy process [34].To show this claim, first we show the following property. Lemma 5.1. Under Assumption 5.1, the mapping x (cid:55)→ H ( r ) ( x ; Φ( r + λ )) is decreasing.Proof. For the completely monotone case, it is known as in Theorem 2 of [42] that the scale function admits theform W ( r ) ( x ) = Φ (cid:48) ( r ) e Φ( r ) x − (cid:90) ∞ e − xt µ ( r ) (d t ) , x ≥ , for some finite measure µ ( r ) . Substituting this and using Fubini’s theorem, Z ( r ) ( x ; Φ( r + λ )) = e Φ( r + λ ) x (cid:18) − λ (cid:90) x e − Φ( r + λ ) z (cid:104) Φ (cid:48) ( r ) e Φ( r ) z − (cid:90) ∞ e − zt µ ( r ) (d t ) (cid:105) d z (cid:19) = e Φ( r + λ ) x (cid:32) − λ (cid:104) Φ (cid:48) ( r ) 1 − e − (Φ( r + λ ) − Φ( r )) x Φ( r + λ ) − Φ( r ) − (cid:90) ∞ (cid:90) x e − z ( t +Φ( r + λ )) d zµ ( r ) (d t ) (cid:105)(cid:33) = e Φ( r + λ ) x − λ (cid:104) Φ (cid:48) ( r ) e Φ( r + λ ) x − e Φ( r ) x Φ( r + λ ) − Φ( r ) − (cid:90) ∞ e Φ( r + λ ) x − e − tx t + Φ( r + λ ) µ ( r ) (d t ) (cid:105) . Now, substituting the above expressions in (3.5), we have H ( r ) ( x ; Φ( r + λ )) = Z ( r ) ( x ; Φ( r + λ )) − λ Φ( r + λ ) − Φ( r ) W ( r ) ( x )= e Φ( r + λ ) x − λ (cid:104) Φ (cid:48) ( r ) e Φ( r + λ ) x − e Φ( r ) x Φ( r + λ ) − Φ( r ) − (cid:90) ∞ e Φ( r + λ ) x − e − xt t + Φ( r + λ ) µ ( r ) (d t ) (cid:105) − λ Φ( r + λ ) − Φ( r ) (cid:104) Φ (cid:48) ( r ) e Φ( r ) x − (cid:90) ∞ e − tx µ ( r ) (d t ) (cid:105) = e Φ( r + λ ) x A + B ( x ) where A := 1 − λ Φ (cid:48) ( r )Φ( r + λ ) − Φ( r ) + (cid:90) ∞ λt + Φ( r + λ ) µ ( r ) (d t ) ,B ( x ) := λ (cid:90) ∞ e − xt (cid:20) r + λ ) − Φ( r ) − t + Φ( r + λ ) (cid:21) µ ( r ) (d t ) . Because lim x →∞ B ( x ) = 0 by monotone convergence and lim x →∞ H ( r ) ( x ; Φ( r + λ )) = 0 in view of the proba-bilistic expression (3.5), we must have that A = 0 . Hence, H ( r ) ( x ; Φ( r + λ )) = B ( x ) and its derivative becomes ∂∂x H ( r ) ( x ; Φ( r + λ )) = B (cid:48) ( x ) = − λ (cid:90) ∞ te − xt (cid:20) r + λ ) − Φ( r ) − t + Φ( r + λ ) (cid:21) µ ( r ) (d t ) < , where the negativity holds because Φ( r + λ ) > Φ( r ) > and hence the integrand is always positive. This showsthe claim. (cid:3) Now suppose V T = 0 so that V ( V ; V B , P ) = V + E (cid:34)(cid:90) T − VB e − rt P κρdt (cid:35) − α E (cid:20) e − rT − VB V T − VB { T − VB < ∞} (cid:21) . HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 15 By Proposition 3.1 and identity (4.2), the optimal barrier V ∗ B ( P ) is given by the root of E ( V B ; V B , P ) = 0 for thecase V ∗ B ( P ) > where E ( V B ; V B , P ) = V B + P κρr (1 − J ( r,λ ) (0; 0)) − αV B J ( r,λ ) (0; 1) − P ρ + pr + m (1 − J ( r + m,λ ) (0; 0)) − (1 − α ) V B J ( r + m,λ ) (0; 1) . (5.2)Recall that by (4.3) and p = mP , V ∗ B ( P ) = 0 ⇐⇒ lim V B ↓ E ( V B ; V B , P ) ≥ ⇐⇒ κρλ + r Φ( r + λ )Φ( r ) − ρ + mλ + r + m Φ( r + m + λ )Φ( r + m ) ≥ , which does not depend on the value of P . Hence, the criterion for V ∗ B ( P ) = 0 is irrelevant to the selection of P .(1) First consider the case κρ λ + r Φ( r + λ )Φ( r ) − ρ + mλ + r + m Φ( r + m + λ )Φ( r + m ) ≥ so that V ∗ B ( P ) = 0 for any choice of P > .In this case, V ( V ; V ∗ B ( P ) , P ) = V ( V ; 0 , P ) = V + P κρr , which is linear (and hence concave) in P .(2) Suppose κρ λ + r Φ( r + λ )Φ( r ) − ρ + mλ + r + m Φ( r + m + λ )Φ( r + m ) < so that V ∗ B ( P ) > is irrelevant to the selection of P .Because p = P m , by solving E ( V B ; V B , P ) = 0 with (5.2), V ∗ B ( P ) = − P κρr (1 − J ( r,λ ) (0; 0)) − P ρ + pr + m (1 − J ( r + m,λ ) (0; 0))1 − αJ ( r,λ ) (0; 1) − (1 − α ) J ( r + m,λ ) (0; 1) = εP, where ε := − κρr (1 − J ( r,λ ) (0; 0)) − ρ + mr + m (1 − J ( r + m,λ ) (0; 0))1 − αJ ( r,λ ) (0; 1) − (1 − α ) J ( r + m,λ ) (0; 1) > . Now, as in (3.10) and Proposition 3.1, the firm’s value is given by V ( V ; V ∗ B ( P ) , P ) = V + P κρr (cid:18) − J ( r,λ ) (cid:18) log VV ∗ B ( P ) ; 0 (cid:19)(cid:19) − αV ∗ B ( P ) J ( r,λ ) (cid:18) log VV ∗ B ( P ) ; 1 (cid:19) = V + P κρr (cid:18) − J ( r,λ ) (cid:18) log VεP ; 0 (cid:19)(cid:19) − αεP J ( r,λ ) (cid:18) log VεP ; 1 (cid:19) . Differentiating the above expression and using Lemmas C.1 and C.2 (in the appendix), we have ∂∂P V ( V ; V ∗ B ( P ) , P ) = κρr (cid:18) − J ( r,λ ) (cid:18) log VεP ; 0 (cid:19)(cid:19) − κρλ + r Φ( r + λ ) − Φ( r )Φ( r ) Φ( r + λ ) H ( r ) (cid:18) log VεP ; Φ( r + λ ) (cid:19) − αε ψ (1) − rλ + r − ψ (1) Φ( r + λ ) − Φ( r )1 − Φ( r ) (Φ( r + λ ) − H ( r ) (cid:18) log VεP ; Φ( r + λ ) (cid:19) . (5.3)Here by the convexity of ψ on [0 , ∞ ) , the coefficient ψ (1) − rλ + r − ψ (1) Φ( r + λ ) − Φ( r )1 − Φ( r ) (Φ( r + λ ) − is positive.First, the mapping x (cid:55)→ J ( r,λ ) ( x ; 0) = E x [ e − r ˜ T − { ˜ T − < ∞} ] = E [ e − r ˜ T −− x { ˜ T −− x < ∞} ] is decreasing, because ˜ T −− x is increasing in x . On the other hand, Lemma 5.1 shows that the mapping x (cid:55)→ H ( r ) ( x ; Φ( r + λ )) is decreasing aswell. Using these facts together with (5.3) we can conclude that ∂∂P V ( V ; V ∗ B ( P ) , P ) is decreasing in P , and thereforethat the firm’s value V ( V ; V ∗ B ( P ) , P ) is a concave function of P . In summary, we have the following. Theorem 5.1. Suppose V T = 0 and Assumption 5.1 is satisfied.(1) If κρ λ + r Φ( r + λ )Φ( r ) − ρ + mλ + r + m Φ( r + m + λ )Φ( r + m ) ≥ , then V ∗ B ( P ) = 0 for all P > and we have V ( V ; V ∗ B ( P ) , P ) = V + P κρr .(2) Otherwise, V ∗ B ( P ) = εP > for all P > and V ( V ; V ∗ B ( P ) , P ) is concave in P for any V > . 6. N UMERICAL E XAMPLES In this section, we confirm the analytical results obtained in the previous sections through a sequence of numer-ical examples. In addition, we study numerically the impact of the rate of observation λ on the optimal solutions,obtain the optimal leverage by considering the two-stage problem considered in (5.1), and analyze the behaviorsof credit spreads.Throughout this section, we use r = 7 . , δ = 7% , κ = 35% , α = 50% for the parameters of the problem asused in [29, 38, 39, 40]. Additionally, unless stated otherwise, we set ρ = 8 . and m = 0 . , which were usedin [20], P = 50 , and λ = 4 (on average four times per year). For the tax threshold, we set V T = P ρ/δ (6.1)as used in [38] and also suggested by [29, 40]. By the choice (6.1), necessarily V T > and hence V ∗ B > asdiscussed in Section 4.1.For the process ( X t ) t ≥ , we use a mixture of Brownian motion and a compound Poisson process with i.i.d.hyperexponential jumps: X t = µt + σB t − (cid:80) N t i =1 U i , t ≥ , where ( B t ) t ≥ is a standard Brownian motion, ( N t ) t ≥ is a Poisson process with intensity γ and ( U i ) i ≥ takes an exponential random variable with rate β i > with probability p i for ≤ i ≤ m , such that (cid:80) mi =1 p i = 1 . Note that this satisfies the completely monotonecondition given in Assumption 5.1. The corresponding Laplace exponent (3.1) then becomes ψ ( s ) = µs + 12 σ s + γ m (cid:88) i =1 p i (cid:18) β i β i + s − (cid:19) , s ≥ . This is a special case of the phase-type L´evy process [4] and its scale function has an explicit expression writtenas a sum of exponential functions; see e.g. [25, 35]. In particular, we consider the following two parameter sets: Case A (without jumps):: σ = 0 . , µ = − . , γ = 0 ; Case B (with jumps):: σ = 0 . , µ = 0 . , γ = 0 . , ( p , p ) = (0 . , . , and ( β , β ) = (9 , .Here, µ is chosen so that the martingale property ψ (1) = r − δ = 0 . is satisfied. In Case B , the jump size U models both small and large jumps (with parameters and ) that occur with probabilities . and . , respectively.6.1. Optimality. Under the parameter settings described above, we first confirm the optimality of the suggestedbarrier V ∗ B that satisfies E ( V ∗ B ; V ∗ B ) = 0 . Because the mapping V B (cid:55)→ E ( V B ; V B ) (given in (4.2)) is monotonicallyincreasing (see Proposition 4.1), the value of V ∗ B is computed by classical bisection methods. The correspondingcapital structure is then computed by (3.10) and (3.11).At the top of Figure 3, for Cases A and B , we plot V (cid:55)→ E ( V ; V ∗ B ) along with V (cid:55)→ E ( V ; V B ) for V B (cid:54) = V ∗ B .Here, we confirm Theorem 4.1: the level V ∗ B satisfies the limited liability constraint (2.8), and any level V B lower HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 17 than V ∗ B violates (2.8), while for V B larger than V ∗ B , E ( V ; V B ) is dominated by E ( V ; V ∗ B ) . The corresponding debtand firm values are also plotted in Figure 3.6.2. Sensitivity with respect to λ on the equity value. We now proceed to study the sensitivity of the optimalbankruptcy barrier and the equity value with respect to the rate of observation λ . On the left plot of Figure 4, weshow the equity value E ( · ; V ∗ B ) for various values of λ along with the classical (continuous-observation) case asobtained in [29, 38]. We see that the optimal barrier V ∗ B is decreasing in λ and converges to the optimal barrier, say ˜ V B , of the classical case. This confirms Remark 4.1.We also confirm the convergence of E ( V ; V ∗ B ) , to the classical case, say ˜ E ( V ; ˜ V B ) , for each starting value V .On the other hand, the monotonicity of E ( V ; V ∗ B ) with respect to λ fails. When V is small, the equity value tendsto be higher for small values of λ , but it is not necessarily so for higher values of V . In order to investigate this,we show in the bottom plots of Figure 4, the difference E ( V ; V ∗ B ) − ˜ E ( V ; ˜ V B ) . We observe also the differencesbetween Cases A and B – in Case A , a lower value of λ clearly achieves higher equity value when V is largewhereas this is not clear in Case B .6.3. Analysis of the bankruptcy time and the asset value at bankruptcy. While it was confirmed that thebarrier level V ∗ B is monotone in λ , it is not clear how the distributions of ( T − V ∗ B , V T − V ∗ B ) change in λ . Here, by takingadvantage of the joint Laplace transform ( q, θ ) (cid:55)→ J ( q,λ ) ( · ; θ ) as in (3.7), we compute numerically the densityand distribution of the random variables T − V ∗ B and V T − V ∗ B for each λ . We also obtain those in the classical case byinverting ( q, θ ) (cid:55)→ H ( q ) ( · ; θ ) as in (3.5).For Laplace inversion, we adopt the Gaver-Stehfest algorithm, which was suggested to use in Kou and Wang[32] (see also Kuznetsov [36] for its convergence results). The algorithm is easy to implement and only requiresreal values. While a major challenge is to handle the cases involving large numbers, our case can be handledwithout difficulty in the standard Matlab environment with double precision.In our case, the scale function W ( q ) is written in terms of a linear sum of e Φ( q ) x and e − ξ i,q x , ≤ i ≤ n ( n = 1 in Case A and n = 3 in Case B ), where Φ( q ) is as in (3.2) and − ξ i,q are the negative roots of ψ ( · ) = q . As inthe proof of Lemma 5.1, the terms for e Φ( q ) x all cancel out in the Laplace transforms J ( q,λ ) ( · ; θ ) and H ( q ) ( · ; θ ) .Hence, the algorithm runs without the need of handling large numbers even for high values of q . The same can besaid about the parameter θ .For the initial value V = 100 , we plot in Figure 5 the density and distribution functions of T − V ∗ B and in Figure6 those for V T − V ∗ B for the same parameter sets as used for Figure 4 (note that the value of V ∗ B depends on λ ). Forcomparison, those in the classical case (computed by inverting q, θ (cid:55)→ H ( q ) (log V ; θ ) ) are also plotted. It is notedthat in Figure 6, the distribution is not purely diffusive and instead the probability of the event V T − V ∗ B = V ∗ B isstrictly positive. In particular, for Case A , V T − V ∗ B = V ∗ B a.s. At least in our examples, the distribution functions for T − V ∗ B appear to be monotone in λ while they are not for V T − V ∗ B .6.4. Two-stage problem. Now we consider the two-stage problem (5.1). Recall, as confirmed in Theorem 5.1,that the firm value V ( V ; V ∗ B ( P ) , P ) is concave in P for the case V T = 0 . Here, in order to see if the concavityholds when V T > , we continue to use the tax cutoff level V T by (6.1) as a function of P . Case A : equity value V (cid:55)→ E ( V ; V B ) Case B : equity value V (cid:55)→ E ( V ; V B ) Case A : debt value V (cid:55)→ D ( V ; V B ) Case B : debt value V (cid:55)→ D ( V ; V B ) Case A : firm value V (cid:55)→ V ( V ; V B ) Case B : firm value V (cid:55)→ V ( V ; V B ) F IGURE The equity/debt/firm values as functions of V on ( V B , ∞ ) for V B = V ∗ B (solid)along with V B = V ∗ B exp( (cid:15) ) (dotted) for (cid:15) = − . , − . , . . . , − . , . , . , . . . , . . The valuesat V = V B are indicated by circles for V B = V ∗ B whereas those for V B < V ∗ B (resp. V B > V ∗ B ) areindicated by up (resp. down)-pointing triangles. HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 19 Case A: V (cid:55)→ E ( V ; V ∗ B ) Case B: V (cid:55)→ E ( V ; V ∗ B ) Case A: V (cid:55)→ E ( V ; V ∗ B ) − ˜ E ( V ; ˜ V B ) Case B: V (cid:55)→ E ( V ; V ∗ B ) − ˜ E ( V ; ˜ V B ) F IGURE (Top) The equity values E ( V ; V ∗ B ) (dotted) for λ = 1 , , , , , , along withthe classical case ˜ E ( V ; ˜ V B ) (solid). The corresponding values at V = V ∗ B are indicated by circles.(Bottom) The difference E ( V ; V ∗ B ) − ˜ E ( V ; ˜ V B ) for the same set of λ .For our numerical results, we set V = 100 and obtain V ∗ B for P running from to (leverage P/V runningfrom to ). The corresponding firm and debt values are computed for each P and V ∗ B = V ∗ B ( P ) , and is shown inFigure 7. For comparison, analogous results on the classical case are also plotted. Here, the concavity with respectto P is confirmed in all considered cases.Regarding the analysis with respect to λ , at least in these examples, we observe that the firm and debt valuesfor each P are monotone in λ and converge to those in the classical case. In addition, we see that the optimal facevalue P ∗ decreases in λ and converges to that in the classical case.6.5. The term structure of credit spreads. We now move onto the analysis of the credit spread. Let V B > be a fixed bankruptcy level. The credit spread is defined as the excess of the amount of coupon over the risk-freeinterest rate, required to induce the investor to lend one dollar to the firm until maturity time t . To be more precise,by finding the coupon rate ρ ∗ that makes the value of the debt d ( V ; V B , t ) defined in (2.4) of unit face value equal Case A: P ( T − V ∗ B ∈ d t ) / d t Case B: P ( T − V ∗ B ∈ d t ) / d t Case A: P ( T − V ∗ B ≤ t ) Case B: P ( T − V ∗ B ≤ t ) F IGURE 5. Density P ( T − V ∗ B ∈ d t ) / d t and distribution P ( T − V ∗ B ≤ t ) (indicated by dottedlines) for λ = 1 , , , , , , , the initial value V = 100 , and V ∗ B determined asin Figure 4. The classical cases are also shown by solid lines. These values are plottedagainst the logarithm of time. to one, the credit spread ρ ∗ − r is given after some rearrangement of (2.4) by CS λ ( t ) = rP E (cid:104)(cid:2) P − (1 − α ) V T V − B (cid:3) e − rT − VB { T − VB ≤ t } (cid:105) E (cid:2) − e − r ( t ∧ T − VB ) (cid:3) . (6.2)Before showing numerical results, we prove the following analytical limits. The proofs are deferred to Appen-dices C.6 and C.7. Proposition 6.1. For V (cid:54) = V B , we have lim t ↓ CS λ ( t ) = λP (cid:2) P − (1 − α ) V (cid:3) { V For V B > , V (cid:54) = V B , and t > , we have lim λ →∞ CS λ ( t ) = CS ( t ) . HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 21 Case A: P ( V T − V ∗ B ∈ d v ) / d v Case B: P ( V T − V ∗ B ∈ d v ) / d v Case A: P ( V T − V ∗ B ≤ v ) Case B: P ( V T − V ∗ B ≤ v ) F IGURE 6. Density P ( V T − V ∗ B ∈ d v ) / d v and distribution P ( V T − V ∗ B ≤ v ) (indicated by dottedlines) for λ = 1 , , , , , , , the initial value V = 100 , and V ∗ B determined as inFigure 4. The classical cases are also shown by solid lines, in which it has a positive massat the bankruptcy level. Remark 6.1. While theoretically the credit spread vanishes in the limit as in Proposition 6.1, we will see belowthat the rate of convergence can be controlled by the selection of X and λ and can be made very slow as shown inFigure 8. To compute credit spreads, we follow the procedures for Figure 6 (given in Appendix B) of [29].Fix V and m . The first step is to choose, for a selected leverage ≤ L ≤ , the face value of debt ˆ P ≡ ˆ P ( L ) and ˆ ρ = ˆ ρ ( L ) satisfying D ( V ; ˆ V ∗ B ) ≡ D ( V ; ˆ V ∗ B ; ˆ P , ˆ ρ ) = ˆ P and L = ˆ P / V ( V ; ˆ V B ) ≡ ˆ P / V ( V ; ˆ V B ; ˆ P , ˆ ρ ) where ˆ V ∗ B isthe optimal bankruptcy level when ρ = ˆ ρ and P = ˆ P . For this computation, at least in our numerical experiments,the mapping P (cid:55)→ P/ V ( V ; ˆ V B ; P, ρ ) , for fixed ρ , is monotonically increasing and hence the root ˆ P ( ρ ) solving L = ˆ P ( ρ ) / V ( V ; ˆ V B ; ˆ P ( ρ ) , ρ ) was obtained by classical bisection. In addition, ρ (cid:55)→ D ( V ; ˆ V ∗ B ; ˆ P ( ρ ) , ρ ) − ˆ P ( ρ ) was also monotone and hence the desired ˆ P and ˆ ρ were obtained by (nested) bisection methods. Case A: firm value Case B: firm value Case A: debt value Case B: debt value F IGURE The firm values (top) and debt values (bottom) as functions of the leverage P/V forthe two-stage problem for V = 100 . The periodic cases with λ = 1 , , , , , , (dotted)are indicated by dotted lines and the classical case corresponds to the solid lines. The points at P ∗ /V are indicated by the circles.For each leverage L , after ˆ P and ˆ ρ are computed, the second step is to obtain, for each maturity t > , the root ρ ∗ = ρ ∗ ( t ) such that d ( V ; ˆ V ∗ B , t ) ≡ d ( V ; ˆ V ∗ B , t ; ρ ∗ ) where d ( V ; ˆ V ∗ B , t ; ρ ) := E (cid:34)(cid:90) t ∧ T − ˆ V ∗ B e − rs ρ d s (cid:35) + E (cid:20) e − rt { t Case A with L = 50 Case B with L = 50 Case A with L = 75 Case B with L = 75 F IGURE 8. Term structure of credit spreads with respect to the logarithm of maturity for V = 100 . The periodic cases with λ = 1 , , , , , , (dotted) are indicated bydotted lines and the classical case corresponds to the solid lines. credit spread with respect to the log maturity for each λ . For comparison, we also plot those in the classical case.The spread appears to be monotone in λ and converges to those in the classical case for each maturity.Regarding the credit spread limit, while the convergence to zero has been confirmed in Proposition 6.1 for theperiodic case, the rate of convergence depends significantly on the selection of λ and the underlying asset priceprocess. In Case A (without negative jumps), it is clear that it vanishes quickly as in the classical case. On theother hand in Case B (where the credit spread limit in the classical case does not vanish), for large values of λ theconvergence is very slow. In view of these observations, with a selection of asset values with negative jumps andthe observation rate λ , it is capable of achieving realistic short-maturity credit spread behaviors.7. C ONCLUDING REMARKS We studied an extension of the Leland-Toft optimal capital structure model where the information on the assetvalue is updated only at the jump times of an independent Poisson process. In settings where the asset value follows λ ∞ ˆ P ˆ ρ ˆ V B Case A with L = 50 λ ∞ ˆ P ˆ ρ ˆ V B Case A with L = 75 λ ∞ ˆ P ˆ ρ ˆ V B Case B with L = 50 λ ∞ ˆ P ˆ ρ ˆ V B Case B with L = 75 T ABLE 1. Values of ˆ P , ˆ ρ and ˆ V B satisfying D ( V ; ˆ V ∗ B ) ≡ D ( V ; ˆ V ∗ B ; ˆ P , ˆ ρ ) = ˆ P and L =ˆ P / V ( V ; ˆ V B ) ≡ ˆ P / V ( V ; ˆ V B ; ˆ P , ˆ ρ ) for L = 50 , for each λ ( λ = ∞ corresponds to theclassical case). an exponential L´evy process with negative jumps, we obtained explicitly an optimal bankruptcy strategy and thecorresponding equity/debt/firm values. These analytical results enabled efficient conduct of numerical experimentsand further analysis of the impact of the observation rate on the optimal leverages and credit spreads.There are various venues for future research. First, it is a natural direction of research to consider the case inwhich the asset value process contains both positive and negative jumps. Because positive jumps do not have directinfluence on the model of the default, similar results are expected and, for example, the optimal barrier is likely tobe given by V B such that E ( V B ; V B ) = 0 . While the techniques using the scale function employed in this papercannot be directly applied to the two-sided jump cases, there are several potential alternative approaches. Oneapproach would be to add phase-type upward jumps to the spectrally negative L´evy process via fluid embeddingand construct a L´evy process with two-sided jumps in terms of a Markov additive process. To do this the phase-type jumps of the L´evy process can be substituted by linear stretches of unit slope. This procedure requires thoughadding a supplementary background Markov chain; see e.g. [30] for details. Another approach would be to focuson the L´evy process with two-sided phase-type distributed jumps and use them to approximate for a general case.This may be possible by combining the results of Asmussen et al. [4] and Albrecher et al. [1].Second, it is important to consider the constant grace period case described in (1) of Section 1.1. As discussed,this paper’s results, featuring exponential grace periods, may be used to approximate the constant case when the HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 25 grace period is short. However, an alternative approach is required when it is long. One potential approach wouldbe to use Carr’s randomization method [18] to approximate the constant period in terms of an Erlang randomvariable, or the sum of i.i.d. exponential random variables. As conducted in [41], a recursive algorithm may beconstructed to compute the required fluctuation identities. ACKNOWLEDGEMENTS The authors thank the anonymous referees and co-editor for careful reading of the paper and constructive com-ments and suggestions. They also thank Nan Chen, Sebastian Gryglewicz, and Tak-Yuen Wong for helpful com-ments and discussions. K. Yamazaki is supported by MEXT KAKENHI grant no. 17K05377. This paper wassupported by the National Science Centre under the grant 2016/23/B/HS4/00566 (2017-2020). Part of the workwas completed while Z. Palmowski was visiting Kansai University and Kyoto University at the invitation of K.Yamazaki. Z. Palmowski is very grateful for hospitality provided by Kazutoshi Yamazaki, Kouji Yano and TakashiKumagai. A PPENDIX A. R ELATION BETWEEN THE BAKRUPTCY MODEL (1.1) AND P ARISIAN RUIN .Let G denote the set of the starting points of the negative excursions of the shifted process ( V t − V B ) t ≥ , andconsider a set of mutually independent exponential random variables { e gλ : g ∈ G } , independent of ( V t ) t ≥ aswell, and g t := sup { s ≤ t : V s ≥ V B } be the last time before t the asset value was at or above V B (i.e., the startingpoint of the excursion). Then the Parisian ruin with exponential grace periods is defined as inf { t > V t < V B and t > g t + e g t λ } . (A.1)The equivalence to (1.1) can be easily verified. In each negative excursion with the starting time g for theshifted process ( V t − V B ) t ≥ between two Poissonian observation times, say T i ( g ) and T i ( g )+1 for some i ( g ) ≥ ,we consider the waiting time until the next observation T i ( g )+1 − g . Due to the lack of memory property of theexponential distribution and the strong Markov property, these waiting times are equal in distribution to a set ofmutually independent exponentially distributed random variables. Consequently, (1.1) can be written as (A.1) with e g t λ replaced by these independent exponential random variables. In fact, it has been shown in Remark 1.1 in [10]that the joint distribution of bankruptcy time (1.1) and the corresponding position of X is the same as that of (A.1)and the corresponding position of X (refer to [50, 9] for related literature).It is worth investing the impact of the randomness of the grace period. To this end, in Table 2, we comparethe expected discounted asset values at bankruptcy for the cases the grace periods are constant and exponentiallydistributed (with the common mean λ − ). When λ is low, the random (exponential) case tends to overestimatethe asset value, but as λ becomes larger (i.e. observation is more frequent), the differences become smaller. Thisimplies that when the observation is frequent, our model can approximate the constant grace period case reasonablywell. A PPENDIX B. P ROOF OF P ROPOSITION z T := z − log V T , z ∈ R . (B.1) λ constant exponential . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . λ constant exponential . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . Case A Case B T ABLE 2. The discounted asset values at bankruptcy E [ e − rτ − VB V τ − VB { τ − VB < ∞} ] when τ − V B is the bankruptcy time with constant and exponential grace periods with mean λ − . Theapproximated values via Monte Carlo simulation are displayed together with their 95%confidence intervals. We set r = 7 . and use the L´evy processes given in Cases A(without jumps) and B (with negative jumps) specified in Section 6 so that ( e − ( r − δ ) t V t ) t ≥ is a martingale for δ = 7% . The initial value of the process is and the bankruptcylevel V B is . We first obtain the q -resolvent measure of the spectrally negative L´evy process ( X t ) t ≥ killed at the stoppingtime (3.6) in terms of the function H ( q + λ ) ( · ; θ ) as in (3.5), and(B.2) I ( q,λ ) ( x, y ) := W ( q + λ ) ( x + y ) − λ (cid:90) x W ( q ) ( x − z ) W ( q + λ ) ( z + y )d z − Z ( q ) ( x ; Φ( q + λ )) W ( q + λ ) ( y ) , q > , x, y ∈ R . The proof of the following is given in Appendix D. Theorem B.1. For any bounded measurable function h : R → R with compact support, E x (cid:34)(cid:90) ˜ T − z e − qt h ( X t )d t (cid:35) = (cid:90) R h ( y + z ) R ( q,λ ) ( x − z, y )d y, x, z ∈ R , where R ( q,λ ) ( x, y ) := Z ( q ) ( x ; Φ( q + λ )) Φ( q + λ ) − Φ( q ) λ H ( q + λ ) ( − y ; Φ( q )) − I ( q,λ ) ( x, − y ) . (B.3)Using Theorem B.1, we show Proposition 3.1. The case V T = 0 is trivial and hence we assume V T > for therest. By integrating the density in Theorem B.1 and using (B.1), we can write (3.9) as Λ ( r,λ ) ( x, z ) = Z ( r ) ( x − z ; Φ( r + λ )) Φ( r + λ ) − Φ( r ) λ H ( z ) − I ( x, z ) , (B.4)where we define H ( z ) := (cid:90) z T −∞ H ( r + λ ) ( y ; Φ( r ))d y and I ( x, z ) := (cid:90) z T −∞ I ( r,λ ) ( x − z, y )d y, (B.5) HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 27 which are shown to be finite immediately below. The rest of the proof of Proposition 3.1 is devoted to the simpli-fication of the integrals H and I . Lemma B.1. For all y ∈ R , we have W ( r + λ ) ( y ) − λ (cid:82) y W ( r ) ( y − z ) W ( r + λ ) ( z )d z = W ( r ) ( y ) .Proof. We have ∂∂y (cid:16) W ( r + λ ) ( y ) − λ (cid:90) y W ( r ) ( y − z ) W ( r + λ ) ( z )d z (cid:17) = ∂∂y (cid:16) W ( r + λ ) ( y ) − λ (cid:90) y W ( r ) ( z ) W ( r + λ ) ( y − z )d z (cid:17) = W ( r + λ ) ( y ) − λ (cid:90) y W ( r ) ( z ) W ( r + λ ) ( y − z )d z = W ( r ) ( y ) , where the last equality holds by identity (6) of [43]. Integrating this and because W ( r + λ ) (0) = W ( r ) (0) = 0 , theproof is complete. (cid:3) Lemma B.2. We have, for x, z ∈ R , I ( x, z ) = W ( r + λ ) ( x − log V T ) { z T > } + W ( r ) ( x − log V T ) { z T ≤ } − λ (cid:90) x − z W ( r ) ( x − z − u ) W ( r + λ ) ( u + z T )d u { z T > } − Z ( r ) ( x − z ; Φ( r + λ )) W ( r + λ ) ( z T ) . Proof. For z T > , we have (cid:90) z T I ( r,λ ) ( x − z, y )d y = (cid:90) z T W ( r + λ ) ( x − z + y )d y − λ (cid:90) x − z W ( r ) ( x − z − u ) (cid:90) z T W ( r + λ ) ( u + y )d y d u − Z ( r ) ( x − z ; Φ( r + λ )) (cid:90) z T W ( r + λ ) ( y )d y = W ( r + λ ) ( x − log V T ) − W ( r + λ ) ( x − z ) − λ (cid:90) x − z W ( r ) ( x − z − u )( W ( r + λ ) ( u + z T ) − W ( r + λ ) ( u ))d u − Z ( r ) ( x − z ; Φ( r + λ )) W ( r + λ ) ( z T )= W ( r + λ ) ( x − log V T ) − W ( r ) ( x − z ) − λ (cid:90) x − z W ( r ) ( x − z − u ) W ( r + λ ) ( u + z T )d u − Z ( r ) ( x − z ; Φ( r + λ )) W ( r + λ ) ( z T ) , where we used x − z + z T = x − log V T (see (B.1)) in the second equality and Lemma B.1 in the last equality.On the other hand, because, as in Remark 4.3(ii) in [52], I ( r,λ ) ( x, y ) = W ( r ) ( x + y ) , y < , (B.6)we have (cid:90) ∧ z T −∞ I ( r,λ ) ( x − z, y )d y = (cid:90) ∧ z T −∞ W ( r ) ( x − z + y )d y = W ( r ) ( x − z + (0 ∧ z T )) . Now the result is immediate by summing up the two integrals and using (again see (B.1)) W ( r ) ( x − z + (0 ∧ z T )) = (cid:40) W ( r ) ( x − z ) if z T > ,W ( r ) ( x − log V T ) if z T ≤ . (cid:3) (cid:3) We note that (B.4) together with Lemma B.2 imply that(B.7) Λ ( r,λ ) ( z, z ) = Φ( r + λ ) − Φ( r ) λ (cid:90) z T −∞ H ( r + λ ) ( y ; Φ( r ))d y, z ∈ R . Lemma B.3. For z ∈ R , we have H ( z ) = 1Φ( r ) (cid:16) Z ( r + λ ) ( z T ; Φ( r )) − λ Φ( r + λ )Φ( r + λ ) − Φ( r ) W ( r + λ ) ( z T ) (cid:17) . (B.8) Proof. First, by (3.5), we have H ( r + λ ) ( y ; Φ( r )) = e Φ( r ) y (cid:18) λ (cid:90) y e − Φ( r ) u W ( r + λ ) ( u )d u (cid:19) − λ Φ( r + λ ) − Φ( r ) W ( r + λ ) ( y ) , y ∈ R , where, in particular, H ( r + λ ) ( y ; Φ( r )) = e Φ( r ) y for y < . For z T > , (cid:90) z T e Φ( r ) y (cid:90) y e − Φ( r ) u W ( r + λ ) ( u )d u d y = (cid:90) z T (cid:90) z T u e Φ( r ) y e − Φ( r ) u W ( r + λ ) ( u )d y d u = (cid:90) z T e Φ( r ) z T − e Φ( r ) u Φ( r ) e − Φ( r ) u W ( r + λ ) ( u )d u = 1Φ( r ) (cid:104) (cid:90) z T e Φ( r )( z T − u ) W ( r + λ ) ( u )d u − W ( r + λ ) ( z T ) (cid:105) , and hence (cid:90) z T H ( r + λ ) ( y ; Φ( r ))d y = (cid:90) z T (cid:20) e Φ( r ) y (cid:18) λ (cid:90) y e − Φ( r ) u W ( r + λ ) ( u )d u (cid:19) − λ Φ( r + λ ) − Φ( r ) W ( r + λ ) ( y ) (cid:21) d y = 1Φ( r ) (cid:104) e Φ( r ) z T − λ (cid:90) z T e Φ( r )( z T − u ) W ( r + λ ) ( u )d u − λ Φ( r + λ )Φ( r + λ ) − Φ( r ) W ( r + λ ) ( z T ) (cid:105) = 1Φ( r ) (cid:104) Z ( r + λ ) ( z T ; Φ( r )) − − λ Φ( r + λ )Φ( r + λ ) − Φ( r ) W ( r + λ ) ( z T ) (cid:105) . On the other hand, for z T ∈ R , (cid:82) z T ∧ −∞ H ( r + λ ) ( y ; Φ( r ))d y = (cid:82) z T ∧ −∞ e Φ( r ) y d y = e Φ( r )( z T ∧ / Φ( r ) . By summingup the integrals, we obtain (B.8). (cid:3) (cid:3) Now applying Lemmas B.2 and B.3 in (B.4), we get Proposition 3.1.A PPENDIX C. O THER PROOFS C.1. Proof of Lemma 4.1. For the case V T > , Λ ( r,λ ) ( z, z ) = E z (cid:2) (cid:90) ˜ T − z e − rt { X t ≥ log V T } d t (cid:3) = E (cid:2) (cid:90) ˜ T − e − rt { X t ≥ log V T − z } d t (cid:3) is clearly non-decreasing in z , and, by bounded convergence, lim z ↓−∞ Λ ( r,λ ) ( z, z ) = 0 . On the other hand, if V T = 0 , then, by Proposition 3.1 and Remark 3.1(1), Λ ( r,λ ) ( z, z ) = r (1 − J ( r,λ ) (0; 0)) = λ + r Φ( r + λ )Φ( r ) . HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 29 C.2. Proof of Proposition 4.3. We start from several key introductory identities. Fix q > . Because e θz Z ( q ) ( x − z ; θ ) = e θx (cid:18) q − ψ ( θ )) (cid:90) x − z e − θu W ( q ) ( u )d u (cid:19) , we have, for x (cid:54) = z , ∂∂z [ e θz Z ( q ) ( x − z ; θ )] = − e θz ( q − ψ ( θ )) W ( q ) ( x − z ) ,∂∂x Z ( q ) ( x − z ; θ ) = ∂∂x (cid:104) e θ ( x − z ) (cid:18) q − ψ ( θ )) (cid:90) x − z e − θu W ( q ) ( u )d u (cid:19) (cid:105) = θZ ( q ) ( x − z ; θ ) + ( q − ψ ( θ )) W ( q ) ( x − z ) . In particular, ∂∂x Z ( q ) ( x − z ; Φ( r + λ )) = Φ( r + λ ) Z ( q ) ( x − z ; Φ( r + λ )) − λW ( q ) ( x − z ) . (C.1)Moreover, we have, for x (cid:54) = z , ∂∂x J ( q,λ ) ( x − z ; θ )= λλ + q − ψ ( θ ) ∂∂x Z ( q ) ( x − z ; θ ) − ψ ( θ ) − qλ + q − ψ ( θ ) Φ( q + λ ) − Φ( q ) θ − Φ( q ) ∂∂x Z ( q ) ( x − z ; Φ( q + λ ))= λλ + q − ψ ( θ ) θZ ( q ) ( x − z ; θ ) − ψ ( θ ) − qλ + q − ψ ( θ ) Φ( q + λ ) − Φ( q ) θ − Φ( q ) Φ( q + λ ) Z ( q ) ( x − z ; Φ( q + λ ))+ ψ ( θ ) − qλ + q − ψ ( θ ) Φ( q + λ ) − θθ − Φ( q ) λW ( q ) ( x − z ) . (C.2)By setting θ = 0 , we obtain the following. Lemma C.1. We have, for x (cid:54) = z and q > , ∂∂z J ( q,λ ) ( x − z ; 0) = − ∂∂x J ( q,λ ) ( x − z ; 0) = qλ + q Φ( q + λ ) − Φ( q )Φ( q ) Φ( q + λ ) H ( q ) ( x − z ; Φ( q + λ )) . (C.3)Noting that ∂∂z [ e z J ( q,λ ) ( x − z ; 1)] = e z J ( q,λ ) ( x − z ; 1) − e z ∂∂x J ( q,λ ) ( x − z ; 1) , and using (C.2) with θ = 1 , wehave the following result. Lemma C.2. We have, for x (cid:54) = z and q > , ∂∂z [ e z J ( q,λ ) ( x − z ; 1)] = ψ (1) − qλ + q − ψ (1) Φ( q + λ ) − Φ( q )1 − Φ( q ) (Φ( q + λ ) − e z H ( q ) ( x − z ; Φ( q + λ )) . We will also need the following observation. Lemma C.3. We have, for z T (cid:54) = 0 and x > z , ∂∂z Λ ( r,λ ) ( x, z ) = − (Φ( r + λ ) − Φ( r )) λ H ( r ) ( x − z ; Φ( r + λ )) H ( z ) . (C.4) Proof. By differentiating the identity in Lemma B.2, for z T (cid:54) = 0 , by (C.1), ∂∂z I ( x, z ) = − λ ∂∂z (cid:90) x − z W ( r ) ( w ) W ( r + λ ) ( x − w − log V T )d w { z T > } + ∂∂x Z ( r ) ( x − z ; Φ( r + λ )) W ( r + λ ) ( z T ) − Z ( r ) ( x − z ; Φ( r + λ )) ∂∂z W ( r + λ ) ( z T )= λW ( r ) ( x − z ) W ( r + λ ) ( z T )+ [Φ( r + λ ) Z ( r ) ( x − z ; Φ( r + λ )) − λW ( r ) ( x − z )] W ( r + λ ) ( z T ) − Z ( r ) ( x − z ; Φ( r + λ )) W ( r + λ ) ( z T )= Z ( r ) ( x − z ; Φ( r + λ ))[Φ( r + λ ) W ( r + λ ) ( z T ) − W ( r + λ ) ( z T )] . (C.5)By (3.5) and (C.1), we can write(C.6) ∂∂x Z ( r ) ( x − z ; Φ( r + λ )) = (Φ( r + λ ) − Φ( r )) H ( r ) ( x − z ; Φ( r + λ )) + Φ( r ) Z ( r ) ( x − z ; Φ( r + λ )) . Using (B.5), we have, for x > z and z T (cid:54) = 0 ,(C.7) ∂∂z (cid:16) Z ( r ) ( x − z ; Φ( r + λ )) H ( z ) (cid:17) = − ∂∂x Z ( r ) ( x − z ; Φ( r + λ )) H ( z ) + Z ( r ) ( x − z ; Φ( r + λ )) H ( r + λ ) ( z T ; Φ( r )) . By (B.8) and (C.6), this equals − (Φ( r + λ ) − Φ( r )) H ( r ) ( x − z ; Φ( r + λ )) H ( z ) + Z ( r ) ( x − z ; Φ( r + λ )) (cid:16) H ( r + λ ) ( z T ; Φ( r )) − Φ( r ) H ( z ) (cid:17) . Furthermore, by (B.8), H ( r + λ ) ( z T ; Φ( r )) − Φ( r ) H ( z ) = H ( r + λ ) ( z T ; Φ( r )) − Z ( r + λ ) ( z T ; Φ( r )) + λ Φ( r + λ )Φ( r + λ ) − Φ( r ) W ( r + λ ) ( z T )= λ Φ( r + λ ) − Φ( r ) (Φ( r + λ ) W ( r + λ ) ( z T ) − W ( r + λ ) ( z T )) . In sum, we have ∂∂z (cid:16) Z ( r ) ( x − z ; Φ( r + λ )) H ( z ) (cid:17) = − (Φ( r + λ ) − Φ( r )) H ( r ) ( x − z ; Φ( r + λ )) H ( z )+ λ Φ( r + λ ) − Φ( r ) Z ( r ) ( x − z ; Φ( r + λ ))(Φ( r + λ ) W ( r + λ ) ( z T ) − W ( r + λ ) ( z T )) . (C.8)By applying (C.5) and (C.8) in (B.4), the proof is complete. (cid:3) (cid:3) HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 31 We now prove Proposition 4.3. Differentiating (3.11) and using Lemmas C.1, C.2 and C.3 give ∂∂V B E ( V ; V B ) = − V − B (cid:34) α ψ (1) − rλ + r − ψ (1) Φ( r + λ ) − Φ( r )1 − Φ( r ) (Φ( r + λ ) − V B + P κρ (Φ( r + λ ) − Φ( r )) λ (cid:90) log( V B /V T ) −∞ H ( r + λ ) ( y ; Φ( r ))d y (cid:35) H ( r ) (cid:16) log VV B ; Φ( r + λ ) (cid:17) − V − B (cid:34) (1 − α ) ψ (1) − r − mλ + r + m − ψ (1) Φ( r + m + λ ) − Φ( r + m )1 − Φ( r + m ) (Φ( r + m + λ ) − V B − P ρ + pr + m r + mλ + r + m Φ( r + m + λ ) − Φ( r + m )Φ( r + m ) Φ( r + m + λ ) (cid:35) H ( r + m ) (cid:16) log VV B ; Φ( r + m + λ ) (cid:17) , which reduces to (4.5) after simplification using Remark 3.1(1).C.3. Proof of Proposition 4.4. In view of the probabilistic expression (3.5), q (cid:55)→ H ( q ) ( x − z ; Φ( q + λ )) isnon-increasing for x, z ∈ R , and hence H ( r ) ( x − z ; Φ( r + λ )) H ( r + m ) ( x − z ; Φ( r + m + λ )) ≥ , for x, z ∈ R . On the other hand, because ψ is strictly convex and strictly increasing on [Φ(0) , ∞ ) , its right-inverse Φ is strictlyconcave, that is Φ (cid:48) ( r + λ + x ) − Φ (cid:48) ( r + x ) < for x, λ > . Therefore, Φ( r + λ ) − Φ( r )Φ( r + m + λ ) − Φ( r + m ) > for λ > . Combining these, Φ( r + λ ) − Φ( r )Φ( r + m + λ ) − Φ( r + m ) H ( r ) ( x − z ; Φ( r + λ )) H ( r + m ) ( x − z ; Φ( r + m + λ )) > . (C.9)By Remark 3.1(2) and because (3.5) implies that H ( r + λ ) is uniformly nonnegative, we have α (1 − J ( r,λ ) (0; 1)) e z + P κρ Φ( r + λ ) − Φ( r ) λ H ( z ) ≥ . Hence, by the previous inequality together with (B.7), (C.9), and (4.2), L (log V, log V B ) = H ( r ) (log V − log V B ; Φ( r + λ )) H ( r + m ) (log V − log V B ; Φ( r + m + λ )) Φ( r + λ ) − Φ( r )Φ( r + m + λ ) − Φ( r + m ) × (cid:32) α (1 − J ( r,λ ) (0; 1)) V B + P κρ Φ( r + λ ) − Φ( r ) λ (cid:90) log V B − log V T −∞ H ( r + λ ) ( y ; Φ( r ))d y (cid:33) + (1 − α )(1 − J ( r + m,λ ) (0; 1)) V B − P ρ + pr + m (1 − J ( r + m,λ ) (0; 0)) > α (1 − J ( r,λ ) (0; 1)) V B + P κρ Λ ( r,λ ) (log V B , log V B )+ (1 − α )(1 − J ( r + m,λ ) (0; 1)) V B − P ρ + pr + m (1 − J ( r + m,λ ) (0; 0))= E ( V B ; V B ) . In addition, because V B ≥ V ∗ B , by the monotonicity as in Proposition 4.1, we have E ( V B ; V B ) ≥ . Note when V ∗ B = 0 that E ( V B ; V B ) ≥ for all V B > by Proposition 4.1.Now, by Proposition 4.3 and recalling that H ( r + m ) is positive, ∂∂V B E ( V ; V B ) < − (Φ( r + m + λ ) − Φ( r + m )) H ( r + m ) (cid:16) log VV B ; Φ( r + m + λ ) (cid:17) E ( V B ; V B ) V B ≤ . (cid:3) C.4. Proof of Proposition 4.5. Using Lemma B.2 together with (C.1) and (C.5), for x (cid:54) = log V T and z ∈ R suchthat z T (cid:54) = 0 , ∂∂x I ( x, z ) = W ( r + λ ) ( x − log V T ) { z T > } + W ( r ) ( x − log V T ) { z T < } − ∂∂x λ (cid:90) x − z W ( r ) ( w ) W ( r + λ ) ( x − z − w + z T )d w { z T > } − ∂∂x Z ( r ) ( x − z ; Φ( r + λ )) W ( r + λ ) ( z T )= W ( r + λ ) ( x − log V T ) { z T > } + W ( r ) ( x − log V T ) { z T < } − λ (cid:90) x − z W ( r ) ( w ) W ( r + λ ) ( x − z − w + z T )d w { z T > } − λW ( r ) ( x − z ) W ( r + λ ) ( z T ) − (Φ( r + λ ) Z ( r ) ( x − z ; Φ( r + λ )) − λW ( r ) ( x − z )) W ( r + λ ) ( z T )= I ( r,λ ) ( x − z, z T ) − ∂∂z I ( x, z ) , (C.10)where we used (B.6) for the case z T < . Hence using (C.7) and (C.10) in (B.4), and by (B.3), ∂∂x Λ ( r,λ ) ( x, z ) = − ∂∂z Λ ( r,λ ) ( x, z ) + R ( r,λ ) ( x − z, − z T ) . (C.11)Now we write (3.11) as E ( V ; V B ) = A (log V, log V B ) + P κρ Λ ( r,λ ) (log V, log V B ) , (C.12)where A ( x, z ) := e x − αe z J ( r,λ ) ( x − z ; 1) − P ρ + pr + m (1 − J ( r + m,λ ) ( x − z ; 0)) − (1 − α ) e z J ( r + m,λ ) ( x − z ; 1) . Differentiating this with respect to x and z , we get ∂∂x A ( x, z ) = e x − αe z ∂∂x J ( r,λ ) ( x − z ; 1) + P ρ + pr + m ∂∂x J ( r + m,λ ) ( x − z ; 0) − (1 − α ) e z ∂∂x J ( r + m,λ ) ( x − z ; 1) ,∂∂z A ( x, z ) = − αe z J ( r,λ ) ( x − z ; 1) + αe z ∂∂x J ( r,λ ) ( x − z ; 1) − P ρ + pr + m ∂∂x J ( r + m,λ ) ( x − z ; 0) − (1 − α ) e z J ( r + m,λ ) ( x − z ; 1) + (1 − α ) e z ∂∂x J ( r + m,λ ) ( x − z ; 1) , and hence ∂∂x A ( x, z ) = e x − ∂∂z A ( x, z ) − αe z J ( r,λ ) ( x − z ; 1) − (1 − α ) e z J ( r + m,λ ) ( x − z ; 1) . (C.13) HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 33 Finally, using (C.11) and (C.13) in (C.12), we obtain that ∂∂V E ( V ; V B ) = 1 V ∂∂x (cid:104) A ( x, log V B ) + P κρ Λ ( r,λ ) ( x, log V B ) (cid:105)(cid:12)(cid:12)(cid:12) x =log V = 1 V (cid:104) V − ∂∂z A (log V, z ) | z =log V B − αV B J ( r,λ ) (log VV B ; 1) − (1 − α ) V B J ( r + m,λ ) (log VV B ; 1) − P κρ ∂∂z Λ ( r,λ ) (log V, z ) | z =log V B + P κρR ( r,λ ) (cid:16) log VV B , log V T V B (cid:17)(cid:105) which reduces to the desired expression by noting that (C.12) gives ∂∂V B E ( V ; V B ) = 1 V B (cid:104) ∂∂z A (log V, z ) | z =log V B + P κρ ∂∂z Λ ( r,λ ) (log V, z ) | z =log V B (cid:105) . C.5. Proof of Lemma 4.2. First we note by Theorem VII.4 in [11], that for q ≥ λ →∞ Φ( λ + r + m )Φ( λ + q ) = 1 . (C.14)On the other hand, identity (B.7) implies, for V B > , that λ Φ( r + λ ) − Φ( r ) Λ ( r,λ ) (log V B , log V B ) = r ) (cid:16) V B V T (cid:17) Φ( r ) if V B /V T < , r ) + (cid:82) log( V B /V T )0 H ( r + λ ) ( y ; Φ( r ))d y if V B /V T ≥ , where we used H ( r + λ ) ( y ; Φ( r )) = exp(Φ( r ) y ) for y ≤ . In addition, by the probabilistic expression of theprobabilistic expression of H ( r + λ ) given in (3.5) and using dominated convergence, we have lim λ →∞ (cid:90) log( V B /V T )0 H ( r + λ ) ( y ; Φ( r ))d y = 0 . This together with (C.14) gives lim λ →∞ λ + r + m Φ( λ + r + m ) Λ ( r,λ ) (log V B , log V B ) = 1Φ( r ) (cid:34)(cid:18) V B V T (cid:19) Φ( r ) ∧ (cid:35) . From Remark 3.1(1) and (C.14), we can conclude that, for q ≥ , λ + r + m Φ( λ + r + m ) (1 − J ( q,λ ) (0; 1)) = λ + r + m Φ( λ + r + m ) ψ (1) − qλ + q − ψ (1) Φ( q + λ ) − − Φ( q ) λ ↑∞ −−−→ ψ (1) − q − Φ( q ) . Combining these and (4.2), we obtain (4.8).C.6. Proof of Proposition 6.1. Fix t > . Let us define the event E := { N λt = 1 } = { T λ ≤ t, T λ > t } = { T λ ≤ t, S > t − T λ } where S := T λ − T λ has the exponential distribution with the parameter λ . Note that E ∩ { T − V B < t } = { T λ ≤ t, V T λ < V B , S > t − T λ } . (C.15)We start from analyzing the numerator of (6.2). We decompose it as follows: f ( t ) := E (cid:104)(cid:2) P − (1 − α ) V T − VB (cid:3) e − rT − VB { T − VB ≤ t } (cid:105) = f ( t ) + f ( t ) , (C.16) where f ( t ) := E (cid:104)(cid:2) P − (1 − α ) V T − VB (cid:3) e − rT − VB { T − VB ≤ t } E (cid:105) ,f ( t ) := E (cid:104)(cid:2) P − (1 − α ) V T − VB (cid:3) e − rT − VB { T − VB ≤ t } E c (cid:105) . Here, by (C.15) and because S is an independent exponential random variable with parameter λ , f ( t ) = E (cid:104)(cid:2) P − (1 − α ) V T λ (cid:3) e − rT λ { T λ ≤ t, V Tλ HE LELAND-TOFT OPTIMAL CAPITAL STRUCTURE MODEL UNDER POISSON OBSERVATIONS 35 C.7. Proof of Proposition 6.2. By (3.5) and (3.7), we can write, for any θ ≥ , and q ≥ , J ( q,λ ) ( y ; θ ) = λλ + q − ψ ( θ ) (cid:18) H ( q ) ( y ; θ ) − ψ ( θ ) − qλ Φ( q + λ ) − Φ( q ) θ − Φ( q ) H ( q ) ( y ; Φ( q + λ )) (cid:19) . For y (cid:54) = 0 , we now note the following:(1) For the cases (i) y < or (ii) y > and X does not have a diffusion component, we have that H ( q ) ( y ; Φ( q + λ )) λ ↑∞ −−−→ (because the process does not creep downward as in Exercise 7.6 of [37])and Φ( q + λ ) − Φ( q ) λ is bounded for λ > cut-off from zero (which can be verified by the convexity of ψ ).(2) For the case y > and X has a diffusion component, H ( q ) ( y ; Φ( q + λ )) λ ↑∞ −−−→ E y [ e − qτ − { X τ − =0 } ] and Φ( q + λ ) − Φ( q ) λ λ ↑∞ −−−→ (because ψ ( θ ) ∼ σ θ as θ → ∞ where σ is the diffusion coefficient of X ).Hence, the previous arguments imply that, for y (cid:54) = 0 , lim λ →∞ J ( q,λ ) ( y ; θ ) = H ( q ) ( y ; θ ) , θ ≥ .Given that J ( q,λ ) ( y ; θ ) is the Laplace transform of the random vector ( ˜ T − ( λ ) , X ˜ T − ( λ ) ) (where we put ( λ ) to spellout the dependency on λ ), by L´evy’s Continuity Theorem we have that ( ˜ T − ( λ ) , X ˜ T − ( λ )) converges in distributionto ( τ − , X τ − ) . Hence, using Skorohod’s Representation Theorem (see Theorem 6.7 in [12]) as well as dominatedconvergence, we obtain, for V (cid:54) = V B , lim λ →∞ CS λ ( t ) = rP lim λ →∞ E log( V/V B ) (cid:104)(cid:2) P − (1 − α ) V B e X ˜ T − λ ) (cid:3) e − r ˜ T − ( λ ) { ˜ T − ( λ ) ≤ t } (cid:105) E log( V/V B ) (cid:2) − e − r ( t ∧ ˜ T − ( λ )) (cid:3) = rP E log( V/V B ) (cid:104)(cid:2) P − (1 − α ) V B e X τ − (cid:3) e − rτ − { τ − ≤ t } (cid:105) E log( V/V B ) (cid:2) − e − r ( t ∧ τ − ) (cid:3) = CS ( t ) . A PPENDIX D. P ROOF OF T HEOREM B.1From Theorem 2.7 of [35] for any Borel set A on [0 , ∞ ) , on R , and ( −∞ , respectively, E x (cid:104) (cid:90) τ − e − qt { X t ∈ A } d t (cid:105) = (cid:90) A (cid:104) e − Φ( q ) y W ( q ) ( x ) − W ( q ) ( x − y ) (cid:105) d y, x ≥ , (D.1) E x (cid:20)(cid:90) ∞ e − ( q + λ ) t { X t ∈ A } d t (cid:21) = (cid:90) A (cid:34) e Φ( q + λ )( x − y ) ψ (cid:48) (Φ( q + λ )) − W ( q + λ ) ( x − y ) (cid:35) d y, x ∈ R , (D.2) E x (cid:104) (cid:90) τ +0 e − ( q + λ ) t { X t ∈ A } d t (cid:105) = (cid:90) A (cid:16) e Φ( q + λ ) x W ( q + λ ) ( − y ) − W ( q + λ ) ( x − y ) (cid:17) d y, x ≤ (D.3)where τ − is defined in (3.4) and τ +0 := inf { t ≥ X t > } .We will prove the result for z = 0 and compute g ( x ) := E x (cid:104) (cid:90) ˜ T − e − qt h ( X t )d t (cid:105) , x ∈ R . 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