An Optimal Stopping Problem with a Reward

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Specifically, Osprey's debt indentures contained “Note Trigger Events” de- ...... constrained by the protective covenant and bear losses (transfer value) in the event .... [21] Peskir, G., Shiryaev, A.N. Optimal stopping and free-boundary problems, ...
An Optimal Stopping Problem with a Reward Constraint ∗ J´erˆome Detemple†

Weidong Tian ‡

Jie Xiong §

Abstract: This article studies an optimal stopping problem with an endogenous constraint on the set of admissible stopping times. The constraint stipulates that continuation is permitted, at any given date t, only if the endogenous reward achieved exceeds a prespecified threshold. Characterizations of the value function and the optimal stopping time are presented. An application to the pricing of corporate claims, when the capital structure of the firm includes equity-trigger debt, is carried out. JEL Classification: G11. Keywords: Optimal Stopping, Endogenous Constraint, Protective Covenant, Equity-trigger Protection AMS 2000 Subject Classification: Primary: 60G40, 93E20; Secondary: 91A10, 91A11. ∗

We would like to thank the editor, the associate editor, and two anonymous referees for their constructive comments and insightful suggestions that improved the paper. Jie Xiong gratefully acknowledges research support from National Science Foundation grant DMS-0906907. Address correspondence to Weidong Tian, Belk College of Business, University of North Carolina at Charlotte, NC, 28223, USA. Email: [email protected]. † School of Management, Boston University, Boston, MA, 02215, USA. Email: [email protected]. ‡ Belk College of Business, University of North Carolina at Charlotte, NC, 28223, USA. Email: [email protected]. § Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA. Email: [email protected].

1

1 Introduction This article studies an optimal stopping problem with a reward constraint. To set the stage, let (Ω, F, P, {Ft : t ∈ T }) be a filtered probability space with time set T and satisfying the usual conditions. Let X ≡ {Xt : t ∈ T } be a progressively measurable non-negative stochastic process defined on (Ω, F, P, {Ft : t ∈ T }). Let F be a positive real number and define, for all t ∈ T , the subclass of stopping times © £ ¤ ª D(t; F ) ≡ τ ∈ St : ∀s ∈ [t, τ ) ∩ S, Es e−β(τ −s) Xτ > F, a.s. where S is the class of stopping times with values in T of the filtration {Ft : t ∈ T }, St = {τ ∈ S : τ ≥ t}, β ≥ 0 is a constant discount rate, and Es [·] is the conditional expectation given Fs . D(t; F ) is the set of admissible stopping times in the following constrained optimization problem £ ¤ Zt ≡ ess sup Et e−β(τ −t) Xτ

(1.1)

τ ∈D(t;F )

at time t ∈ T . The random variable Zt is the maximum expected reward at time t and Z ≡ {Zt : t ∈ T } is the maximum expected¤ reward process (or value process) associated with the problem £ −β(τ −s) (1.1). Note that Es e Xτ is the discounted time s-value of Xτ , where discounting £ −β(τ −s) is¤ at the rate β. A stopping time τ belongs to D(t; F ) if the associated reward Es e Xτ is strictly greater than the prespecified floor F at any stopping time s strictly prior to τ . Conversely, τ does not belong to D(t; F ) if there exists a stopping time s, strictly prior to τ , at which the discounted time s-value of Xτ fails to exceed the floor F . The constrained stopping problem is to solve (1.1) at time 0. The optimal stopping problem Z0 under the reward constraint D(0; F ) is motivated by a pricing problem in finance. Constraints of this form emerge when a firm uses bonds with certain types of protective covenants in order to finance its activities. The classic debt contract specifies a bullet payment (face value) at a given maturity date. Default occurs if the firm, at the maturity date, is unable to fulfill the payment promised to debtholders. This is the standard situation studied in the early literature dealing with structured credit models (Black and Scholes (1973), Merton (1974)). When the debt contract has a safety covenant with exogenous trigger, default occurs when the safety provision is activated (Black and Cox (1976), Leland (1994)). Recently, safety clauses involving various types of endogenous trigger events have appeared in corporate bond issues. An equity-trigger safety covenant, for instance, stipulates that debtholders have the right to demand immediate repayment of the debt’s face value if the stock price falls below a prespecified threshold. Given that immediate repayment is often optimal for debtholders, equityholders effectively face a constrained stopping time problem of the type described above. The reason is as follows. The equity payoff at the liquidation time τ is Xτ = (Vτ − K)+ where K is the debt’s face value and V £is the firm’s¤ asset value process. The equity price at time t, given the liquidation time τ , is Et e−r(τ −t) Xτ . As equityholders can choose the time at which to repay the debt, as long as the endogenous equity price exceeds the threshold F , their ability 2

to postpone repayment is restricted by the covenant constraint D (t; F ). The equity value at time t is the value Zt in problem (1.1). A concrete example is drawn from the history of Enron. In 1998 Enron created Osprey, a special purpose vehicle that raised $100 million in equity and another $1.4 billion in debt. In order to find investors willing to hold Osprey’s debt, Enron issued a guarantee backing the debt with Enron’s equity. Specifically, Osprey’s debt indentures contained “Note Trigger Events” designed to protect the debtholders. A specific clause stipulated that (see Enron 10-Q, 2001-Q3) Enron had to repay certain debt notes in the event of “an Enron senior unsecured debt rating below investment grade by any of the three major credit rating agencies concurrent with an Enron stock closing price of $59.78 per share or below.” This safety covenant shows that Osprey’s debt was effectively an obligation of the parent company, partly triggered by an event determined by Enron’s stock (see Powers (2002) for details about Enron’s special purpose vehicle and the structure of debt with an equity-trigger covenant). Problem (1.1) captures essential elements involved in corporate structures of this type. Its solution is important for a sound analysis of the credit worthiness of the firm. The article is divided into two parts. The first part solves the optimal stopping problem (1.1) under the assumption Assumption A: The discounted payoff process {e−βt Xt : t ∈ T } is a submartingale. The second part uses the solution to study the pricing of corporate claims when the firm is partly financed by debt with equity-trigger safety covenants. Assumption A is maintained throughout the paper, unless explicitly noted. At first sight, Assumption A might seem rather restrictive. Under it, the classical optimal stopping problem is indeed trivial as delaying exercise is always beneficial. The solution of this problem goes back to Snell (1952) and there is a vast literature on it (see Peskir and Shiryaev (2006) for recent developments in general settings). It turns out, however, that the optimal stopping problem under constraint (1.1) is fairly nontrivial. From a mathematical point of view it would obviously be interesting to study (1.1) without this assumption. But, there are also several reasons to impose it. First, the assumption implies a time-consistency property of the optimal stopping problem. This property plays a key role in the resolution. Second, from an application perspective, Assumption A is satisfied in standard credit models such as in Merton (1974). In these models the discounted payoff to the equity issued by the firm is a submartingale. Optimal stopping problems arise in a range of applications in finance. It is well known, for instance, that the classic optimal stopping problem is equivalent to the American-style contingent claim pricing problem (see Bensoussan (1984) and Karatzas (1988)). Optimal stopping methods have also been used extensively in the real options literature (see Brennan and Schwartz (1985), McDonald and Siegel (1986) and Villeneuve (2007)). Applications in corporate finance abound (see Leland (1994) and Dorobantu (2009)). In capital structure models with a convertible-callable debt, the proper valuation of corporate securities requires elements of stochastic game theory. Financing structures of this type effectively give rise to complex stopping time problems in which the timing decisions of stakeholders with different incentives 3

affect the payoffs of all the claimants (e.g., Kifer (2000), Shreve and Sirbu (2006)). The current paper contributes to the finance literature by dealing with the pricing of corporate securities when the capital structure includes a debt contract with an equity-trigger safety covenant. The analysis focuses on situations where the optimal decision of debtholders is to request immediate payment when the protection is activated. The layout of the paper is as follow. General results concerning the solution to the optimal stopping problem (1.1) are presented in Section 2. Section 3 establishes stronger results in the discrete-time case. Section 4 discusses the application to the asset pricing problem described above. A closed form solution is provided in Section 5, when the firm is financed by an equitytrigger consol. Conclusions are formulated in Section 6.

2 Optimal stopping problem with reward constraints This section studies the optimal stopping time problem subject to reward constraints. A characterization of the solution is provided in Theorem 2.5, which is the central result of the section. The uncertainty setting is as described in the introduction. The time set T can be a discrete time set, such as {0, 1, · · · , N }, or {0, 1, · · · , ∞}, or a continuous time set, such as [0, T ], or [0, ∞] (See Karatzas and Shreve S(1998), Appendix D). Following standard conventions, if T = {0, 1, · · · , ∞}, then F∞ ≡ σ( 0≤n ν} ⊆ Eν e−β(ρ−ν) Xρ > F , a.s. where Sτ = {ρ ∈ S : ρ ≥ τ }. The set D(τ ; F ) extends the definition of D(t; F ) to situations where starting dates are stopping times. The set D(τ ; F ) is understood to include τ (immediate exercise is an admissible strategy). The notion of time-consistency plays an important role in the analysis. Definition (TC): The set {D(τ ; F ) : τ ∈ S} is time consistent if for τ ∈ D(ν; F ), the inclusion D(τ ; F ) ⊆ D(ν; F ) holds. 4

Time-consistency is a natural assumption in the context of dynamic decision problems. In essence, the property states that stopping times that are admissible at a future admissible time are also admissible from the standpoint of the current time. Under this assumption the choice set is not altered as time elapses. Such a property is at the heart of the recursive structure of a dynamic decision problem. It effectively ensures that Bellman’s optimality principle applies. Dynamic consistency plays a key role in a variety of economic contexts. For recent applications of the property, see Artzner, Delbaen, Eber, Heath and Ku (2007), Detlefsen and Scandolo (2005) and Riedel (2009). The next lemma demonstrates that time-consistency holds under certain conditions, including Assumption A. Lemma 2.1 Time consistency (TC) holds under the following alternative conditions: © ª (i) the discounted payoff process e−βt Xt : t ∈ T satisfies Assumption A, (ii) the discounted payoff process fails to satisfy Assumption A, but β = 0 and T is bounded. Proof: (i). Suppose that Assumption A holds and let τ ∈ D(ν; F ). It is shown next that ρ ∈ D(τ ; F ) implies ρ ∈ D(ν; F ). Assume first that T is bounded. Doob’s optional sampling theorem applies because of the uniform integrability condition (2.2) (see Karatzas and Shreve 3.22). Then by £ (1988), Theorem ¤ Assumption A and Doob’s optional sampling theorem, Eτ e−β(ρ−τ ) Xρ ≥ Xτ . Therefore, for any α ≥ ν, on the event {τ > α}, £ ¤ £ ¤ £ £ ¤ ¤ Eα e−β(ρ−α) Xρ = Eα e−β(ρ−τ ) Xρ e−β(τ −α) = Eα Eτ e−β(ρ−τ ) Xρ e−β(τ −α) £ ¤ ≥ Eα e−β(τ −α) Xτ . Because τ ∈ D(ν; F ), it follows that © £ ¤ ª {τ > α} ⊆ Eα e−β(ρ−α) Xρ > F , a.s. As τ ∈ D(ν; F ) and ρ ∈ D(τ ; F ), this implies © £ ¤ ª {ρ > α} = {ρ > τ > α} ∪ {ρ > α ≥ τ } ⊆ Eα e−β(ρ−α) Xρ > F , a.s. This proves ρ ∈ D(ν; F ). Suppose next that T is unbounded. By Assumption A and the uniform integrality condition (2.2), the submartingale convergence theorem (see Karatzas and Shreve (1988), Chapter 1, Theorem 3.15) ensures that limt→∞ e−βt Xt exists. Hence the optional sampling theorem can be applied and the rest of the proof is the same as above. (ii). The proof for this case parallels the proof for bounded time set T when Assumption A holds. 2 The next example shows that time consistency can fail if Assumption A is not satisfied. 5

Example 2.1 Consider a standard three-period Binomial model on the space Ω ≡ {ω = ω 1 ω 2 ω 3 : ω i ∈ {H, T }, i = 1, 2, 3} . The risky asset price process S is given by S0 = 2, u = 1.1, d = 1/u where u (resp. d) is the gross return if the price moves up (resp. down). The interest rate r is 6%. The discount rate β = log(1 + r) in the optimal stopping problem. The risk neutral probability p of an up-move is determined by pu + (1 − p)d =

1 . 1+r

n

1 S (1+r)t t

o

Under the probability measure p, the discounted price process : t = 0, 1, 2, 3 is a martingale (see Shreve (2000)). Consider a capped option with cap L = 2 and strike K = 0.5. Its payoff process is Xt = (min(St , L) − K)+ , t = 0, 1, 2, 3. It is easy to verify that the discounted payoff fails to satisfy Assumption A. It can also be shown that time consistency is violated in this example. o n 1 X : t = 0, 1, 2, 3 Proof: It is straightforward to verify that the discounted payoff process (1+r) t t is not a submartingale. Hence Assumption A fails. Select F = 0.82. Note that 3 ∈ D(1; F ) and 1 ∈ D(0; F ). But 3 does not belong to D(0; F ) because · ¸ 1 E X3 = 0.8174 < F. (1 + r)3 Time consistency fails.

2

Before proving the main result of the section, Theorem 2.5, several lemmas are in order. Lemma 2.2 The set D(t; F ) has the following properties. (1) If τ , ρ ∈ D(t; F ), then τ ∨ ρ ∈ D(t; F ). (2) D(t; F ) is left closed: if τ n ∈ D(t; F ), τ n ↑ τ , a.s., then τ ∈ D(t; F ). Proof: (1). By assumption

¤ £ Eτ e−β(τ ∨ρ) Xτ ∨ρ ≥ e−βτ Xτ .

Then, for each ν ∈ St , ¤ ¤ £ £ ¤ £ 1τ >ν Eν e−β(τ ∨ρ−ν) Xτ ∨ρ = Eν Eτ e−β(τ ∨ρ−ν) Xτ ∨ρ 1τ >ν ≥ Eν [e−β(τ −ν) Xτ 1τ >ν ] 6

so that

¤ ª © £ {τ > ν} ⊆ Eν e−β(τ ∨ρ−ν) Xτ ∨ρ > F .

Similarly,

© £ ¤ ª {ρ > ν} ⊆ Eν e−β(τ ∨ρ−ν) Xτ ∨ρ > F .

Hence, © £ ¤ ª {τ ∨ ρ > ν} ⊆ Eν e−β(τ ∨ρ−ν) Xτ ∨ρ > F so that, τ ∨ ρ ∈ D(t; F ). (2). Consider a sequence {τ n } such that τ n ∈ D(t; F ) for each n and τ n ↑ τ , a.s.. For any ν ∈ St , on the event {τ n > ν}, it holds that £ ¤ £ £ ¤ ¤ £ ¤ Eν e−β(τ −ν) Xτ = Eν Eτ n e−βτ Xτ eβν ≥ Eν e−β(τ n −ν) Xτ n . (2.4) Hence, for every ν ∈ St , {τ > ν} ⊆

[

{τ n > ν} ⊆

n



ª £ ¤ 1τ n >ν Eν e−β(τ n −ν) Xτ n > F

n

© £ ¤ ª ⊆ Eν e−β(τ −ν) Xτ > F . So, τ ∈ D(t; F ), i.e., D(t; F ) is left closed.

2

Lemma 2.3 The value process {Zt : t ∈ T } satisfies £ ¤ £ ¤ Ev e−βτ Zτ = ess sup Ev e−βρ Xρ . ρ∈D(τ ;F )

Proof: For any ρ ∈ D(τ ; F ), £ ¤ £ £ ¤ ¤ £ ¤ Ev e−βρ Xρ = Ev Eτ e−β(ρ−τ ) Xρ e−βτ ≤ Ev Zτ e−βτ . Hence,

£ ¤ £ ¤ ess sup Ev e−βρ Xρ ≤ Ev e−βτ Zτ . ρ∈D(τ ;F )

Moreover, by Lemma 2.2, D(τ ; F ) is closed under© pairwise Then there exists a ¤ª £ −β(ρmaximization. −τ ) n Xρn is nondecreasing and sequence {ρn } ⊆ D(τ ; F ) such that the sequence Eτ e ¤ £ (2.5) Zτ = lim Eτ e−β(ρn −τ ) Xρn . n→∞

Hence ¤ £ Ev e−βτ Zτ =

¤ £ ¤ £ lim Ev e−βρn Xρn ≤ ess sup Ev e−βρ Xρ .

n→∞

ρ∈D(τ ;F )

2 7

For a bounded time set T with maximal element T , let D∗ (τ ; F ) ≡ {ρ ∈ D(τ ; F ) : ρ > τ , a.s., over {τ < T }} be the set of admissible stopping times that are strictly posterior to τ on the event {τ < T }. The optimal stopping problem restricted to this subset is £ ¤ Zτ∗ ≡ ess sup Eτ e−β(ρ−τ ) Xρ , ∀τ ∈ S (2.6) ρ∈D∗ (τ ;F )

and Zτ∗ = 0 if D∗ (τ ; F ) is empty. The following result is adapted from Karatzas and Shreve (1998), Appendix D. Lemma 2.4 For each τ ∈ S, Zτ = max {Xτ , Zτ∗ }. Theorem 2.5 The following properties hold. © ª (1) The discounted value process e−βt Zt : t ∈ T is a submartingale. (2) Let ρ∗ = ess sup {ρ : ρ ∈ D(0; F )}. Then ρ∗ is an admissible time for the optimal stopping problem Z0 . (3) © The stopping time ρ∗ isªoptimal for Z0 . Moreover, Zρ∗ = Xρ∗ and the stopped process ∗ e−β(t∧ρ ) Zt∧ρ∗ : t ∈ T is a martingale. Proof: (1). It is first shown that e−βs Et [Zs ] ≥ e−βt Zt , for all t ≤ s, s ∈ T . It suffices to prove £ ¤ Et e−βρ Xρ ≤ e−βs Et [Zs ] , ∀ρ ∈ D(t; F ). Let ρ ∈ D(0; F ) and take ρ1 = ρ ∨ s. Then, for any ν ∈ Ss , © £ ¤ ª {ρ1 > ν} = {ρ > ν} ⊆ Eν e−β(ρ−ν) Xρ > F so that ρ1 ∈ D(s; F ). Thus, ¤ £ ¤ £ ¤ £ Et e−βρ Xρ = Et e−βρ Xρ 1ρ≤s + Et e−βρ Xρ 1ρ>s ¤ £ ¤ ¢ £¡ ≤ Et Eρ e−βs Xs 1ρ≤s + Et e−βρ Xρ 1ρ>s ¤ £ ¤ £ ¤¤ £ £ = Et Eρ e−βρ1 Xρ1 = Et e−βρ1 Xρ1 ≤ Et e−βs Zs where the first inequality follows from the submartingale property of the discounted payoff process. Optimizing over ρ ∈ D(t; F ) on the left hand side shows that e−βt Zt ≤ e−βs Et [Zs ]. 8

This holds for any time s finite. The property for s = ∞, follows from condition (2.3), which ensures that the submartingale {e−βt Zt } converges as t goes to infinity. (2). By Lemma 2.2, D (t; F ) is closed under pairwise maximization. Then there exists a nondecreasing sequence {ρn } of stopping times in D(0; F ) converging to ρ∗ (see Karatzas and Shreve (1998), Appendix A). Hence, invoking Lemma 2.2 again, ρ∗ ∈ D(0; F ). £ ¤ (3). It is first shown that Zν = Eν e−β(τ −ν) Zτ if τ ∈ D(ν; F ). Consider first the case where T is bounded. By virtue of Doob’s optional sampling theorem and property (1), £ ¤ e−βν Zν ≤ Eν e−βτ Zτ , ∀ν ≤ τ . (2.7) As D (t; F ) is closed under pairwise maximization by Lemma 2.2, the opposite inequality follows from Lemma 2.3. By definition and property (2), it is seen that, ρ∗ ∈ D(t ∧ ρ∗ ; F ) for every t ∈ T . Then, for every t < s in T , £ ¤ ∗ ∗ e−β(t∧ρ ) Zt∧ρ∗ = Et∧ρ∗ e−βρ Zρ∗ £ £ ¤¤ ∗ = Et∧ρ∗ Es∧ρ∗ e−βρ Zρ∗ £ ¤ ∗ = Et∧ρ∗ e−β(s∧ρ ) Zs∧ρ∗ where the last© equality follows fromªthe fact that ρ∗ ∈ D(s ∧ ρ∗ ; F ) and the proof of the first ∗ part. Hence, e−β(t∧ρ ) Zt∧ρ∗ : t ∈ T is a martingale. In particular, £ ¤ ∗ Z0 = E e−βρ Zρ∗ .

(2.8)

It is next shown that Zρ∗ = Xρ∗ . Suppose that the set D∗ (ρ∗ ; F ) = ∅. Then, by Lemma 2.4, Zρ∗ = Xρ∗ . Otherwise, let τ ∈ D∗ (ρ∗ ; F ). As ρ∗ ∈ D(0; F ), by Lemma 2.1, τ ∈ D(0; F ). ∗ ∗ Thus, τ ≤ ρ∗ , a.s. by the T definition of ρ . That is, P ({τ ≤ ρ }) = 1. By the definition of ∗ ∗ ∗ D (ρ ; F ), P ({τ ≤ ρ } {τ < T }) = 0. Then, ³ ´ \ P {τ ≤ ρ∗ } {τ = T } = 1. (2.9) Thus P ({ρ∗ = T }) = 1. Hence D∗ (ρ∗ ; F ) is empty. Applying Lemma 2.4 completes the proof. If T is unbounded, condition (2.3), the martingale convergence theorem, and property (1) ensure that e−βt Zt has a limit when t → ∞. Hence Doob’s optional sampling theorem applies and the proof parallels the one above. 2 Theorem 2.5 characterizes the discounted value process {e−βt Zt : t ∈ T } under Assumption A. Counterexamples provided below will show that these properties do not hold if Assumption A is violated. The next theorem deals with the undiscounted case. This result is useful for the construction of counterexamples highlighting the role of Assumption A. 9

Theorem 2.6 Assume β = 0 and T is bounded. An admissible stopping time τ ∗ ∈ D(0; F ) is optimal for the optimal stopping problem at time t = 0 if and only if Zτ ∗ = Xτ ∗ , a.s. and E [Zτ ∗ ] = Z0 . Proof: If Zτ ∗ = Xτ ∗ and E [Zτ ∗ ] = Z0 , then Z0 = E [Zτ ∗ ] = E [Xτ ∗ ] .

(2.10)

Hence τ ∗ is optimal for Z0 . Conversely, assume that τ ∗ is optimal. By definition Z0 = E [Xτ ∗ ] = ess sup E [Xρ ] .

(2.11)

ρ∈D(0;F )

As τ ∗ ∈ D(0; F ), by Lemma 2.1, D(τ ∗ ; F ) ⊆ D(0; F ). Then by (2.11), Z0 ≥ ess

sup ρ∈D(τ ∗ ;F )

E [Xρ ] = E [Zτ ∗ ]

(2.12)

where the last equality follows from Lemma 2.3. As Z0 = E [Xτ ∗ ], this shows that E [Xτ ∗ ] ≥ E [Zτ ∗ ] .

(2.13)

This inequality, along with Zτ ∗ ≥ Xτ ∗ implies that Zτ ∗ = Xτ ∗ , a.s.. Moreover, using (2.11) again, Z0 = E [Zτ ∗ ] . 2

3 The discrete time model This section proves a stronger version of Theorem 2.5 for the discrete-time case. Constructive characterizations of the value process and the optimal stopping time are given in Theorems 3.3 and 3.4. Examples are then displayed to highlight the role of Assumption A. The next lemma is straightforward and its proof is omitted. Lemma 3.1 If T is unbounded, then ¤ ª ª © © £ D(n; F ) = τ ∈ Sn : ∀k ≥ n, {τ > k} ⊆ Ek e−β(τ −k) Xτ > F , a.s. . If T = {0, 1, . . . , N }, then ¤ ª ª © © £ D(n; F ) = τ ∈ Sn,N : ∀k ≥ n, {τ > k} ⊆ Ek e−β(τ −k) Xτ > F , a.s. where Sn and Sn,N are the sets of stopping times for unbounded and finite time set, respectively. 10

Lemma 3.2 Consider a nondecreasing sequence of F-measurable non-negative random variables xn with limit x∗ = limn→∞ xn < ∞, a.s., let B = {x∗ > F } , F ≥ 0 and suppose P (B) > 0. Then, for every ² > 0, there exist positive integers N1 (²) and N2 (²) such that ∀m ≥ N2 (²), µ \½ ¾¶ 1 1 ∗ xm (ω) > x (ω) − P B , xm (ω) > F + ≥ P (B) − 2². 2N1 (²) 2N1 (²) Moreover, N1 (²) can be chosen so that N1 (²) → ∞ when ² → 0. Proof: First, note that

¾ \½ 1 ∗ , a.s. B= B x >F+ n n=1 ¡ T© ∗ ª¢ Then, limn→∞ P B x > F + n1 = P (B). Hence there exists a positive integer N1 (²) such that ¾¶ µ \½ 1 ∗ P B x >F+ > P (B) − ², ∀n ≥ N1 (²) . n T Moreover, N1 (²) can be chosen so that N1 (²) → ∞ as ² → 0 (because P (B {x∗ > F + y}) is a decreasing function of y). ∞ [

Next, note also that, B=

∞ [

\½ B xm > x ∗ −

m=1

It follows that

1 2N1 (²)

¾ .

µ \½ lim P B xm > x∗ −

¾¶ 1 = P (B) . m→∞ 2N1 (²) Hence, there exists a positive integer N2 (²) such that µ \½ ¾¶ 1 ∗ P B xm > x − > P (B) − ², ∀m ≥ N2 (²) . 2N1 (²) Therefore, for every m ≥ N2 (²) µ \½ ¾¶ 1 1 ∗ ∗ P (B) − P B x >F+ , xm > x − N1 (²) 2N1 (²) µ \½ ¾¶ µ \½ ¾¶ 1 1 ∗ ∗ ≤ P B x ≤F+ +P B xm ≤ x − N1 (²) 2N1 (²) ≤ 2². As,

½

¾ 1 1 ∗ x >F+ , xm > x − N1 (²) 2N1 (²) ½ ¾ 1 1 ∗ , xm > x − ⊆ xm > F + 2N1 (²) 2N1 (²) ∗

11

it follows that µ \½ P B xm > F +

1 1 , x m > x∗ − 2N1 (²) 2N1 (²)

¾¶ ≥ P (B) − 2².

This completes the proof of the lemma.

2

The next theorem characterizes the value process when Assumption A is not imposed. Theorem 3.3 Let {Xn } be a nonnegative adapted process, which may or may not satisfy Assumption A. Let Zn and ZnN be the value functions at date n associated with the discrete time sets T = {0, 1, . . . , ∞} and T = {0, 1, . . . , N }. Then ¡ ¢ Zn = H Xn , e−β En [Zn+1 ] (3.14) where

½ H(x, y) =

x, max {x, y} ,

y≤F . y>F

© ª When T is finite, ZnN : 0 ≤ n ≤ N is determined recursively by ¡ ¢ N ZnN = H Xn , e−β En [Zn+1 ] ,n < N

(3.15)

subject to the boundary condition ZNN = XN . Moreover, Zn = lim ZnN . N →∞

(3.16)

Proof: The proof for ZnN is similar to the proof for Zn . The demonstration focuses on the latter. The proof is divided into several steps. © ª Step 1. It is first shown that, Zn ≤ max Xn , e−β En [Zn+1 ] . For τ ∈ D(n; F ), note that τˆ ≡ max {n + 1, τ } ∈ D(n + 1; F ). Hence £ ¤ ¤ ¤ £ £ En e−β(τ −n) Xτ = En e−β(τ −n) Xτ 1{τ =n} + En e−β(ˆτ −n) Xτˆ 1{τ >n} ¤ ¤ £ £ £ ¤ = En e−β(τ −n) Xτ 1{τ =n} + En e−β En+1 e−β(ˆτ −n−1) Xτˆ 1{τ >n} ≤ Xn 1{τ =n} + e−β En [Zn+1 ] 1{τ >n} © ª ≤ max Xn , e−β En [Zn+1 ] as announced. Step 2. Define the set

© ª B ≡ e−β En [Zn+1 ] > F ∈ Fn . 12

It is now shown that Zn 1B c = Xn 1B c .

(3.17)

In fact, as τˆ ∈ D(n + 1; F ) and τ ∈ D(n; F ), it follows that © £ ¤ ª {τ > n} ⊆ En e−β(τ −n) Xτ > F, τ > n © £ £ ¤¤ ª ⊆ e−β En En+1 e−β(ˆτ −n−1) Xτˆ > F © ª ⊆ e−β En [Zn+1 ] > F = B. Hence, B c ⊆ {τ = n} for every τ ∈ D(n; F ). This proves that Zn 1B c = Xn 1B c . Step 3. This step establishes Zn 1B ≥ e−β En [Zn+1 ] 1B

(3.18)

where B is defined in Step 2. As D(n + 1; F ) is closed under pairwise£ maximization, there ¤ exists a sequence of stopping −β(τ m −n−1) times τ m ∈ D(n + 1; F ) such that En+1 e Xτ m increases to Zn+1 as m → ∞ (see Karatzas£ and Shreve (1998), Appendix A). By the monotone convergence theorem, the ¤ −β(τ m −n) −β sequence En e Xτ m increases to e En [Zn+1 ]. Using Lemma 3.2, it then follows that, for every ² > 0, there exists N1 (²) , N2 (²) such that ∀ m ≥ N2 (²), P (B1 ) ≥ P (B) − 2², where B1 = B ∩ B11 ∩ B12 , with, ½ £ ¤ B11 = En e−β(τ m −n) Xτ m > F + B12

1 2N1 (²)

½ £ ¤ = En e−β(τ m −n) Xτ m > e−β En [Zn+1 ] −

Define,

½ τ˜m =

¾

1 2N1 (²)

¾ .

£ ¤ n if En £e−β(τ m −n) Xτ m ¤ ≤ F τ m if En e−β(τ m −n) Xτ m > F.

It is easy to verify that τ˜m ∈ D(n; F ). Then, £ ¤ Zn 1B ≥ En e−β(˜τ m −n) Xτ˜m 1B ¤ £ = En e−β(τ m −n) Xτ m 1{˜τ m >n}∩B + Xn 1{˜τ m =n}∩B . As B1 ⊆ {˜ τ m > n} , B then,

\

{˜ τ m = n} ⊆ B \ B1

³ \ ´ P B {˜ τ m = n} ≤ P (B \ B1 ) ≤ 2². 13

(3.19)

It follows that, £ ¤ Zn 1B ≥ En e−β(τ m −n) Xτ m 1B1 + Xn 1B T{˜τ m =n} £ ¤ +En e−β(τ m −n) Xτ m 1(B T{˜τ m >n})\B1 µ ¶ 1 −β ≥ e En [Zn+1 ] − 1B1 + Xn 1B T{˜τ m =n} 2N1 (²) £ ¤ +En e−β(τ m −n) Xτ m 1(B T{˜τ m >n})\B1 1 ≥ e−β En [Zn+1 ] 1B − e−β En [Zn+1 ] 1B\B1 − 1B 2N1 (²) 1 £ ¤ +En e−β(τ m −n) Xτ m 1(B T{˜τ m >n})\B1 + Xn 1B T{˜τ m =n} . By the dominated convergence theorem, letting ² → 0, then gives Zn 1B ≥ e−β En [Zn+1 ] 1B . Step 4. Given that Zn ≥ Xn , by Step 3, ¡ ¢ Zn 1B ≥ H Xn , e−β En [Zn+1 ] 1B . Clearly

¡ ¢ Zn 1B c ≥ H Xn , e−β En [Zn+1 ] 1B c .

Hence

¡ ¢ Zn ≥ H Xn , e−β En [Zn+1 ] .

Conversely, by Step 1,

¡ ¢ Zn 1B ≤ H Xn , e−β En [Zn+1 ] 1B

and, by Step 2,

¡ ¢ Zn 1B c ≤ H Xn , e−β En [Zn+1 ] 1B c . ¡ ¢ ¡ ¢ Thus, Zn ≤ H Xn , e−β En [Zn+1 ] . Hence Zn = H Xn , e−β En [Zn+1 ] is proved. Step 5. To demonstrate the last part, note that ZnN ≤ ZnN +1 . Let Zn∞ = limN ↑∞ ZnN . Then £ ¤ Zn∞ = ess sup τ ∈D(∞;F ) En e−β(τ −n) Xτ (3.20) where D(∞; F ) denotes the set of all bounded stopping times in D(n; F ). Thus Zn∞ ≤ Zn .

(3.21)

On the other hand, for every τ ∈ D(n; F ), by Fatou’s lemma, ¤ £ ¤ £ En e−β(τ −n) Xτ ≤ lim inf En e−β(τ ∧k−n) Xτ ∧k . k→∞

Thus Zn ≤ Zn∞ . 14

(3.22)

Combining the two inequalities (3.21) and (3.22) gives Zn = Zn∞ = lim ZnN . N →∞

This completes the proof of the theorem.

2

Theorem 3.3 provides a backward equation for the value process Z for an arbitrary nonnegative process X. It effectively shows that it is possible to construct Z by a recursive algorithm in the discrete-time model. According to Theorem 2.5, if the discounted payoff process is a submartingale, the discounted value process is a martingale up to the optimal stopping time. The next theorem describes the optimal stopping time for the discrete case when this condition holds. Theorem 3.4 The following properties hold. © ª (1) The discounted value process e−βn Zn : n ∈ T is a submartingale. © ª ∗ (2) Let τ ∗ ≡ inf {n < ∞ : Zn ≤ F } or infinity if no such n exists. Then e−β(n∧τ ) Zn∧τ ∗ : n ∈ T is a martingale. (3) τ ∗ is the optimal stopping time in D(0; F ), and £ ¤ £ ¤ ∗ ∗ Z0 = E e−βτ Zτ ∗ = E e−βτ Xτ ∗ . © ª (4) Let τ ∗N ≡ inf n ≤ N : ZnN ≤ F ∧ N . Then τ ∗N is the optimal stopping time in D(0; F ) with T = {0, 1, . . . , N }. Proof: (1) This follows from Theorem 2.5. (2) By Doob’s optional © −β(n∧τ ª sampling theorem, for every stopping time τ , the stopped process ) e Zn∧τ : n ∈ T is a submartingale. It remains to be shown that £ ¤ ∗ ∗ En e−β((n+1)∧τ ) Z(n+1)∧τ ∗ ≤ e−β(n∧τ ) Zn∧τ ∗ .

(3.23)



On {τ ∗ ≤ n}, both sides of (3.23) equal e−βτ Zτ ∗ . On {τ ∗ > n}, it must be that e−β En [Zn+1 ] > F ; otherwise, by the submartingale property Zn ≤ e−β En [Zn+1 ] ≤ F which implies τ ∗ ≤ n, a contradiction. Therefore, e−β En [Zn+1 ] > F . By Theorem 3.3, © ª Zn = max Xn , e−β En [Zn+1 ] ≥ e−β En [Zn+1 ] (3.24) which establishes inequality (3.23). 15

(3) By the martingale property in Theorem 3.4 (2), ¤ £ ¤ £ ∗ ∗ Z0 = E e−βτ Zτ ∗ = E e−βτ Xτ ∗ where the second equality follows from © ª {n : Zn ≤ F } ⊆ Zn ≤ F, e−β En [Zn+1 ] ≤ F ⊆ {n : Zn = Xn } .

(3.25)

(3.26)

Finally, it is shown that τ ∗ ∈ D(0; F ). For any s ≥ 0 ¤and any£ path in {τ¤∗ > s}, the inequality £ −βτ ∗ ∗ ∗) −β(s∧τ Zτ ∗ = Es e−βτ Xτ ∗ . Thus Zs∧τ ∗ = Es e Zs∧τ ∗ > F holds. But, e £ ¤ ∗ Es e−β(τ −s) Xτ ∗ = Zs∧τ ∗ > F over {τ ∗ > s}. That is τ ∗ ∈ D(0; F ). The proof of Theorem 3.4 (4) parallels that of Theorem 3.4 (3).

2

Consider now the stopping time TF = inf{t : Xt < F }. The next example shows that TF might be not an admissible stopping time even though Assumption A is satisfied. Example 3.1 Consider a standard Binomial model on the space Ω ≡ {ω = ω 1 ω 2 ω 3 ω 4 : ω i ∈ {H, T }, i = 1, 2, 3, 4}. Assume that S0 = 2, u = 1.2, d = 1/u, β = 0. Let X = (S − 1.5)+ and F = 0.5. Then TF ∈ / D(0; F ). Proof: In the risk neutral world, the probability p of an up-move is characterized by pu + (1 − p)d = 1. Under the risk neutral probability, the process X is a submartingale and Assumption A is satisfied. A simple calculation shows that E[XTF ] = 0.5, implying TF ∈ / D(0; F ). 2 Instead of TF consider the stopping time τ F ≡ inf{t : Xt ≤ F } and suppose that Assumption A holds. It is then easy to show that τ F is an admissible stopping time in D(0; F ). But it can also be shown that τ F is not always the optimal stopping time in D(0; F ). The following examples show that Theorem 3.4 (1)-(4) and Theorem 2.5 (1) - (3) are not always true when Assumption A is not satisfied. Note that time consistency holds in these examples, by Lemma 2.1. Time consistency is therefore not sufficient for Theorems 2.5 and 3.4. Example 3.2 Consider a standard Binomial model on the space Ω ≡ {ω = ω 1 ω 2 ω 3 : ω i ∈ {H, T }, i = 1, 2, 3}. 16

Assume that S0 = 2, u = 2, d = 1/2, β = 0 and that the probability p of an up-move is 1/3. Let X = (S − 1)+ ∧ 3. Then {Xn : n = 0, 1, 2, 3} is not a submartingale. The value process Z depends on the parameter F as follows: If F ≥ 1/3, then Zt = Xt , ∀t; if F < 1/3, then Zt = Zta , the value of the optimal stopping problem without constraint, ∀t. Moreover, for F = 1/4, the optimal stopping time τ ∗ for the constrained problem is not the same as inf{n : Zn ≤ F }. Proof: It is easy to see that {Xn : n = 0, 1, 2, 3} is not a submartingale. To see that the value process Z depends on the parameter F , first note that Z2 (ω) = X2 (ω), ∀ω = ω 1 ω 2 ω 3 . Next, by Theorem 3.3, if F < 1/3, Z1 (T ω 2 ω 3 ) = 1/3; if F ≥ 1/3, then Z1 (T ω 2 ω 3 ) = 0, ∀ω 2 , ω 3 ∈ {H, T }. As claimed, the value process depends on the parameter F . To show the assertion in the last part of the example, let F = 1/4. Then Zt = Zta , the unconstrained value process. By Theorem 2.6, the optimal stopping time for the constrained problem (1.1) is given by τ ∗ (Hω 2 ω 3 ) = 1, τ ∗ (T ω 2 ω 3 ) = 2, ∀ω 2 , ω 3 ∈ {H, T }. Clearly, τ ∗ 6= inf{n : Zn ≤ F }.

2

Example 3.2 shows that Theorem 3.4 (1) and (4) do not always hold (for F ≥ 1/3 in Example 3.2) if the discounted payoff process is not a submartingale. The next example shows that the discounted value process is not always a martingale up to the optimal stopping time. Consequently, Theorem 3.4 (2) - (3) and Theorem 2.5 need not hold, if Assumption A is removed. Example 3.3 Example 3.2 is extended to a five-step binomial model with the same initial asset price S0 and same parameters u = 2, d = 1/2, β = r = 0, p = 1/3. The space of states is Ω ≡ {ω = ω 1 · · · ω 5 : ω i ∈ {H, T }, i = 1, 2, 3, 4, 5}. Let X = (S − 2)+ ∧ 5 and F = 1/4. Let τ ∗ be the optimal stopping time characterized in Theorem 2.6. Then {Zt∧τ ∗ : t = 0, 1, 2, 3, 4, 5} is not a martingale. Proof: It is easy to verify that {Xt } is not a submartingale. A straightforward calculation, combined with Theorem 3.3, establish that Z0 = 173/243. Moreover, the optimal stopping time τ ∗ , determined by Theorem 2.6, is given by, ∀ω 5 ∈ {H, T }, τ ∗ (HHHHω 5 ) = τ ∗ (HHT Hω 5 ) = τ ∗ (HT HHω 5 ) = τ ∗ (T HHHω 5 ) = 4 17

and τ ∗ = 5 in other states. With this definition of τ ∗ , the process {Zi∧τ ∗ : i = 1, 2, 3, 4, 5} is not a martingale. For instance, Z4∧τ ∗ (HT HH) = 5. But 2 1 E4∧τ ∗ [Z5∧τ ∗ ] (HT HH) = 5 + 2 = 3. 3 3

(3.27)

Clearly E4∧τ ∗ [Z5∧τ ∗ ] 6= Z4∧τ ∗ . 2 These examples illustrate the importance of Assumption A for the resolution of the optimal stopping problem (1.1). In applications, the payoff process X is often derived from a homogeneous Markov process and a payoff function g(·). In this case Assumption A is closely related to a convexity property of the payoff function g(·). To see this point, consider a homogeneous Markov process {Vn : n = 0, 1, · · · } and a continuous payoff function g(·). Let Xn = g(Vn ). For each n ≥ 1 define Ten f (x) ≡ Ex f (Vn ) and Te = Te1 . It is well known that {e−βt g(Vt )} is a submartingale if and only if g (x) ≤ e−β Teg (x) . (3.28) £ −βτ ¤ Recall that g (x) is β-excessive in the sense of Dynkin (1963) if g (x) ≥ Ex e g (Vτ ) . It is also well known that every excessive function, for a one-dimensional Brownian motion, is concave and vice-versa (see Dynkin and Yushkevich (1969) and Dayanik and (2003)). © Karatzas ª −βt Therefore, the submartingale assumption on the discounted payoff process e g (Vt ) corresponds to a “convexity” property of the payoff function g (x). In many applications in finance, the payoff function g (x) is convex and the underlying process V has properties ensuring that the discounted payoff is a submartingale (see Black and Scholes (1973) and Merton (1973)). Assumption A covers these standard cases.

4 Equity and Debt Valuation with Equity-Trigger Covenants This section studies the valuation of corporate claims when the firm is financed by equity and by debt with an equity-trigger safety covenant. The time period is assumed to be T = [0, T ] with T < ∞. Suppose that the value of the firm, V follows a geometric Brownian motion under the risk neutral measure dVt = rdt + σdWt . Vt

(4.1)

In equation (4.1), W is a Brownian motion process, r is the interest rate and σ is the volatility of the return on the assets of the firm. The interest rate is assumed to be positive. The firm is financed by a combination of equity and debt. Debt takes the form of a zero-coupon bond with face value K, maturity date T and carries a protective covenant. The safety clause stipulates 18

that debtholders have the right to request immediate repayment of K if the equity price falls below a given trigger level F . Upon liquidation, the debtholders receive min{V, K} and the equityholders receive the residual value max{V − K, 0}. In the event of liquidation, the payoff function for equityholders is g (x) ≡ max {x − K, 0}. In the complementary event, equityholders own the assets of the firm, but have no contractual payment obligation to debtholders before the maturity date. Let Xt = g(Vt ). If τ is an admissible liquidation time, then the equity price at time t under this liquidation policy is Et [e−r(τ −t) Xτ ] (see Merton (1974) for details). Let us first characterize the set of admissible liquidation times and the debtholders’ optimal decision. Under the model’s assumptions, the discounted payoff process © −rt ª e (Vt − K)+ : t ∈ [0, T ] is a submartingale. Assumption A thus holds. By virtue of Doob’s optional sampling theorem, for every τ 1 ≤ τ 2 , £ ¤ (Vτ 1 − K)+ ≤ Eτ 1 e−r(τ 2 −τ 1 ) (Vτ 2 − K)+ . (4.2) Equivalently,

£ ¤ min (Vτ 1 , K) ≥ Eτ 1 e−r(τ 2 −τ 1 ) min (Vτ 2 , K) .

(4.3)

It is therefore always optimal for debtholders to request immediate repayment when the equity price falls below F . From the perspective of equityholders, the set of admissible stopping times at time t is D(t; F ). The corresponding equity price is Zt . In essence, Assumption A states that the discounted equity payoff process is a submartingale in the presence of zero-coupon debt. In a general capital structure with positive-coupon equitytrigger debt, the discounted equity payoff process might be not a submartingale (Assumption A might not hold). Moreover, the equity price is not always the solution to the optimal stopping problem (1.1). The reason is simple. Because of positive coupon payments, the debtholders’ decision, when the safety clause is activated, depends on the immediate repayment as well as possible future coupon payments. Requesting immediate repayment may be suboptimal if future coupon payments are sufficiently large. The equity price is then the solution of a stochastic game between debtholders and equityholders and the representation (1.1) does not apply. Let us return to the example of equity-trigger zero-coupon debt. By the strong Markov property, the equity price at time t is £ ¤ Z (V, t; F ) = ess sup Et,V e−r(τ −t) Xτ τ ∈D(t;V,F )

where © © ª ª D(t; V, F ) = τ ∈ St,T : ∀ν ∈ St,T , {τ > ν} ⊆ Eν [e−r(τ −ν) Xτ |Vt = V ] > F , a.s. and Et,V [·] is the conditional expectation given Ft and Vt = V . The dependence on V is made explicit given its importance in the analysis below. Similarly, let £ ¤ Z ∗ (V, t; F ) = ess sup Et,V e−r(τ −t) Xτ τ ∈D∗ (t;V,F )

19

where D∗ (t; V, F ) = {τ ∈ D(t; V, F ) : τ > t, a.s.} and Z (V, T ; F ) = g(V ). By Proposition 2.4, Z (V, t; F ) = max {g (V ) , Z ∗ (V, t; F )} . (4.4) © −βt ª By Theorem 2.5, e Z (Vt , t; F ) : 0 ≤ t ≤ T ª is a submartingale. Moreover, the stopped © −β(t∧ρ ∗) ∗ process e Z (Vt∧ρ∗ , t ∧ ρ ; F ) : 0 ≤ t ≤ T is a martingale, where ρ∗ is the optimal stopping time. The next theorem establishes the main properties of the function Z(V, t; F ) in this context. Theorem 4.1 (1) There exists a function {B(t) : 0 ≤ t ≤ T } such that Z (V, t; F ) > F if and only if V > B (t). The set G ≡ {(V, t) : V > B (t) , 0 ≤ t ≤ T } is the region where the endogenous debt covenant constraint is inactive. (2) In the inactive region, the function Z (V, t; F ) satisfies the partial differential equation LZ (V, t; F ) = 0 where

∂f (V, t) 1 ∂ 2 f (V, t) ∂f (V, t) rV + σ 2 V 2 + − rf (V, t) . ∂V 2 ∂V 2 ∂t In particular, the partial derivatives Z, ZV V , ZV , and Zt exist and are continuous in G. Lf ≡

Theorem 4.1 (1) shows that the region in which the unknown value process satisfies the constraint can be expressed in terms of a region for the known underlying stochastic process. This considerably simplifies the analysis and transposes it into a more standard framework similar to that for American-style derivatives pricing. The boundary B(·) can be interpreted as an automatic liquidation boundary. Theorem 4.1 (2) effectively describes the behavior of the value function in the inactive region. In order to provide a complete characterization of the solution, boundary conditions on the first-order partial derivative of the function Z (V, t; F ) must be added. Lemma 4.2

(1) If x ≤ y, then D(t; x, F ) ⊆ D(t; y, F ).

(2) If x ≤ y, then Z (x, t; F ) ≤ Z (y, t; F ). Proof: (1) For every τ ∈ D(t; x, F ) and ν ∈ St,T the inclusion © ª {τ > ν} ⊆ Et,x,ν [e−r(τ −ν) g (Vτ )] > F

(4.5)

holds. Invoking the comparison theorem for solutions of SDE (see Karatzas and Shreve (1991), Proposition 2.18), gives (4.6) Vτt,x ≤ Vτt,y 20

where Vτt,x denotes Vτ started at Vt = x. Hence £ ¤ £ ¤ Et,x,ν e−r(τ −ν) g (Vτ ) ≤ Et,y,ν e−r(τ −ν) g (Vτ ) .

(4.7)

It follows that τ ∈ D(t; y, F ), and therefore D(t; x, F ) ⊆ D (t; y, F ). This proves (1). Property (2) follows from (1). 2 Define the boundary B (t) ≡ inf {V > 0 : Z (V, t; F ) > F } ,

∀t ∈ [0, T ] .

(4.8)

As Z(V, T ; F ) = max{V − K, 0}, the equity price Z(V, T ; F ) at time T is strictly greater than F if and only if V > F +K. Hence B (T ) = K +F . Then by Lemma 4.2, for every V > B (t), the value function satisfies Z (V, t; F ) > F . The region G ≡ {(V, t) ∈ R+ × [0, T ] : V > B (t)} is the inactive region. This proves Theorem 4.1 (1). To prove Theorem 4.1 (2), the following proposition is useful. Proposition 4.3 The following properties are satisfied. (1) Z (V, t; F ) is non-increasing with respect to time t. (2) {B (·)} is a nondecreasing curve. (3) {B (·)} is right continuous. Proof: (1). For any t < s, it can easily be shown that D∗ (s; V, F ) ⊆ D∗ (t; V, F ). Then Z ∗ (V, t; F ) ≥ Z ∗ (V, s; F ). Using Lemma 2.4, it then follows that Z (V, t; F ) ≥ Z (V, s; F ). This proves (1). (2). Consider a sequence Vn ↓ B (s). Then Z (Vn , s; F ) > F , and for t ≤ s, the inequality Z (Vn , t; F ) > F holds. Hence B (t) ≤ limn→∞ Vn = B (s). One concludes that the boundary curve {B (·)} is nondecreasing. (3). Assume that B(t) is not right continuous. Then there exists tn ↓ t, V > 0 such that B(tn ) ≥ V, ∀n; B(t) < V. Equivalently, Z(V, tn ; F ) ≤ F, ∀n; Z(V, t; F ) > F. Then there exists a stopping time τ ∈ D(t; F ) such that ¤ £ Et,V e−r(τ −t) Xτ > F. It must be shown that τ ∨ tn ∈ D(tn ; F ). In fact, for all tn ≤ s ≤ τ , ¤ £ ¤ £ Hsτ ∨tn = Es e−r(τ ∨tn −s) Xτ ∨tn = Es e−r(τ −s) Xτ > F. 21

(4.9)

(4.10)

Hence τ ∨ tn ∈ D(tn ; F ). Therefore, £ ¤ Etn ,V e−r(τ ∨tn −tn ) Xτ ∨tn ≤ Z(V, tn ; F ) ≤ F.

(4.11)

Taking n → ∞ and using the dominated convergence theorem, it holds that £ ¤ Et,V e−r(τ −t) Xτ ≤ F.

(4.12)

The last inequality contradicts formula (4.10).

2

Proof of Theorem 4.1 (2). The proof is similar to that in Jacka (1991) for the American put option. Take a point (V, t) in the inactive region, and choose a rectangle R = (V1 , V2 ) × (t1 , t2 ) which is contained in the inactive region (the existence of this rectangle follows from the right continuity property of B(t), proved by Proposition 4.3). Let ∂0 R = ∂R \ ((V1 , V2 ) × {t2 }) be the “parabolic boundary” of this rectangle, and consider the initial-boundary value problem ½ Lf = 0, (V, t) ∈ R f (V, t) = Z (V, t; F ) , (V, t) ∈ ∂0 R. The classical theory for parabolic partial differential equations guarantees the existence of a unique solution f with fV V , fV and ft continuous. It is now shown that f (V, t) and Z (V, t; F ) agree on R. Let (V0 , t0 ) ∈ R be given, and consider the stopping time τ representing the first hitting time of ∂0 R. Also define the process Nt = e−rt f (Vt , t). Then, by Itˆo’s rule, N.∧τ is a bounded martingale, and £ ¤ f (V0 , t0 ) = N0 = E0 [Nτ ] = E0 e−rτ Z (Vτ , τ ; F ) . (4.13) Note that (τ , Vτ ) belongs to the inactive region. In fact, τ is less than the optimal stopping time τ V . To prove the property τ ≤ τ V , it suffices to show that Z (Vτ V , τ V ; F ) = g (Vτ V ) ≤ F , (invoking Theorem 2.5). Because of the Markov property, it suffices to show that this holds at t = 0 in the event τ V = 0. The proof is by contradiction. Suppose g (V0 ) > F . As limt→0,V →V0 e−rt g (V ) = g (V0 ) > F , there exists an open neighborhood U of (0, V0 ) such that e−rt g (V ) > F for every (t, V ) ∈ U . Let τ 1 be the first hitting time of ∂U . Then τ 1 > 0 and τ 1 is an admissible stopping time by construction. This contradicts τ V = 0. Therefore τ ≤ τ V .Thus, by the optional sampling theorem and Theorem 2.5, it must be that £ ¤ (4.14) Z (V0 , t0 ; F ) = E0 e−rτ Z (Vτ , τ ; F ) . Thus f (V0 , t0 ) = Z (V0 , t0 ; F ), which completes the proof.

2

The results above show that a precise characterization of the equity price depends on the boundary B(·). In the next section, the boundary B(·) is identified by using the solution of the infinite horizon model.

22

5 Equity-Trigger Consol The firm is still assumed to be financed by debt and equity, but the debt takes the form of a consol (an infinite maturity debt instrument) with an equity-trigger safety covenant. There are no coupon payments. The protection is activated when the equity price reaches or falls below a threshold F . Without the equity-trigger protective covenant, the debt is worthless. The equity price is then the same as the firm value. The equity-trigger covenant gives debtholders the right to demand a bullet payment K at times at which the equity price falls below the threshold F . In the complementary event equityholders can choose whether to repay the debt or not. The maturity date of the contract is determined endogenously by the optimal actions of the two parties. Let the set of observed times be T = [0, ∞]. Given the infinite horizon nature of the model, the protection boundary B is a flat line. Moreover, as Zt ≤ Zta ≤ Vt , £ ¤ £ ¤ sup E e−rt Zt ≤ sup E e−rt Vt < ∞ 0≤t B. In the region {V ≤ B}, E(V ) = max{V − K, 0}. Let E (V ) = c1 V + c2 V −α , α =

2r σ2

for some real constants c1 , c2 . Using (V − K)+ ≤ c1 V + c2 V −α ≤ V and letting V → ∞, shows that c1 = 1. Hence E (V ) = V + c2 V −α . 23

(5.16)

On the other hand, limV ↓B E (V ) = F , so that B + c2 B −α = F implying c2 = (F − B) B α and

µ ¶−α V E (V ) = V − (B − F ) B

(5.17)

over the region {V > B}. To find the protection boundary B, let us now make use of Theorem 2.5 and Theorem 3.4. Let τ ∗ be the first hitting time of V across B or τ ∗ = ∞ if no such time exists. Assume that V0 > B. ªIt was shown that τ ∗ is the optimal stopping time for © −r(τ ∗ ∧t) equityholders, and e (Vτ ∗ ∧t ) is aªmartingale. As {e−rt Vt } is a martingale, ©E −r(τ © −r(τ ∗ ∧t) by Doob’s ª ∗ ∧t) optional sampling theorem, e Vτ ∗ ∧t is a martingale. Therefore, e D (Vτ ∗ ∧t ) is a martingale and by dominated convergence theorem, £ ¤ £ ∗ ∗¤ D (V0 ) = E e−rτ min {Vτ ∗ , K} = min {B, K} E e−rτ . (5.18) It is straightforward to show that £

−rτ ∗

E e

¤

µ =

V0 B

¶−α (5.19)

.

Hence µ D (V0 ) = min {B, K}

V0 B

¶−α .

(5.20)

On the other hand, by formula (5.17), µ D (V0 ) = (B − F )

V0 B

¶−α .

The last two equations imply min {B, K} = B − F , so that B = F + K.

(5.21) 2

There are two remarkable implications of the solution displayed in Theorem 5.1. First, the smooth fit condition does not hold. To see this, note that ½ K V 1 + α F +K ( K+F )−α−1 if V > F + K 0 E (V ) = (5.22) 1{V ≥K} if V ≤ F + K and lim E 0 (V ) = 1 + α

V ↓F +K

K > 1 = lim E 0 (V ). V ↑F +K K +F 24

(5.23)

Figure 1: Equity value with equity-triggered consol This figure displays the equity price in the presence of an equity-triggered consol. The equity price is an increasing function of the asset value. It is convex when the protective covenant is active and concave otherwise. Parameter values are r = 5%, σ = 30%. Equity value with consol debt 300

250

Equity price Asset value

Equity value

Protection Trigger 200

150

100

50

0 0

50

100

150

200

250

300

Asset value

From a technical point of view, this result means that the equity price function is not continuously differentiable (not C 1 ). Differentiability fails at the endogenous protection boundary (the smooth fit condition fails). Intuitively, this behavior follows from the fact that equityholders are constrained by the protective covenant and bear losses (transfer value) in the event that the protection becomes active. If they were able to select the debt repayment time in an unconstrained manner, they would postpone it indefinitely, thereby extracting all the asset value. This sets this type of contract and financial structure apart from the standard cases studied in the previous literature on American-style derivatives pricing. Figure 1 illustrates the equity price behavior with respect to the underlying asset value. The plot displays the behavior for a protection floor F = 50. The lack of smoothness at V = F + K = 150 is apparent. It is remarkable to note the concave structure of the equity price over part of the domain. When the protective covenant is inactive, equityholders bear the risk of unwanted liquidation in the event that the safety covenant is triggered in the future. The risk of future suboptimal liquidation is a source of concavity in this example (see Leland (1994) for other examples of concave equity price in the presence of protected debts). On the other hand, when V ≤ F + K, the equity price is convex, which reflects to the fact that equityholders lose their option to delay repayment when the protective covenant is triggered. Second, the equity price is discontinuous with respect to the protection floor F . When F = 0 the value of debt is null (as debtholders never receive the bullet payment) and the equity price is equal to the value of the assets in place V . However, taking the limit of the equity price E(V )

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as the floor F approaches zero gives lim E(V ) = V − K( F ↓0

V −α ) . K

(5.24)

This shows that the equity price is not a continuous function of F at the point F = 0. The V −α difference between the equity value V , when F = 0, and the limit equity value V − K( K ) , V −α is K( K ) . This amount is the present value of K in the event of future liquidation. We are now ready to solve the valuation problem for the case of finite maturity debt. In fact, it is shown next that the protection boundary is also a flat line in this situation. Theorem 5.2 For finite maturity debt with an equity-trigger safety covenant, the endogenous protection boundary is B(t) = F + K. Moreover, the equity price at time t is Z(V, t; F ) = Et [e−r(τ −t) (Vτ − K)+ 1{τ F . But, Z(V ; T, F ) ≤ Z(V ; ∞, F ) = E(V ). It follows that E(V ) > F . But, by Theorem 5.1, the protection boundary is F + K when the firm is financed by a consol. Thus, E(V ) ≤ F when V < F + K, a contradiction. It follows that B(0, T ) ≥ F + K. 2 A closed-form expression for Z(V ; T, F ) can be derived by using results in Leland and Toft (1996). As the boundary B = F + K is generated by the protection covenant, the smoothpasting condition does not hold for equity-trigger debt. In contrast, the constant bankruptcy boundary in Leland and Toft (1996) is determined by using a smooth-pasting condition for the equity value.

6 Conclusion This paper studies an optimal stopping problem when the set of admissible stopping times is subject to a constraint on the endogenous reward function. The problem considered differs from the classic optimal stopping problem by the nature of the constraint, which is only “left closed”. This type of restriction introduces substantial challenges for the resolution of the problem. 26

Bhanot and Mello (2006) study debt with a rating-trigger safety covenant where the trigger event is exogenously determined. This paper examines the pricing problem when the safety covenant is triggered endogenously, based on the stock price. The pricing methodology developed in this paper can be extended to the case where the rating depends on the stock price and debt carries an equity-trigger covenant and a rating-trigger covenant, as in the case of Enron. The method is also useful for the analysis of complex corporate securities with safety covenants based on endogenous values. The analysis of these securities in more general models of the firm is left for future study.

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