abstract. arxiv:2111.02554v1 [q-fin.mf] 3 nov 2021

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CALLABLE CONVERTIBLE BONDS UNDER LIQUIDITYCONSTRAINTS

DAVID HOBSON, GECHUN LIANG, HAODONG SUN∗

Abstract. This paper provides a complete solution to the callable convertible bond studied in[Liang and Sun, Dynkin games with Poisson random intervention times, SIAM Journal on Control andOptimization, 57(4): 2962–2991, 2019], and corrects an error in Proposition 6.2 of that paper. Thecallable convertible bond is an example of a Dynkin game, but falls outside the standard paradigmsince the payoffs do not depend in an ordered way upon which agent stops the game. We show howto deal with this non-ordered situation by introducing a new technique which may of interest in itsown right, and then apply it to the bond problem.

Key words. Constrained Dynkin game, callable convertible bond, order condition, saddle point.

AMS subject classifications. 60G40, 91A05, 91G80, 93E20.

1. Introduction. The holder of a perpetual bond receives a coupon from thefirm indefinitely. If the bond is convertible, the bondholder has the additional op-portunity to exchange the bond for a fixed number of units of the firm’s stock at amoment of the bondholder’s choosing. If the convertible bond is callable then thefirm has the option to call the bond on payment of a fixed surrender price to thebondholder. The problem of pricing the callable convertible bond involves finding thevalue of the bond and the optimal stopping rules for conversion (by the bondholder)and call (by the firm).

The callable convertible bond is an example of a Dynkin game. A Dynkin game [3]is a game played by two agents, each of whom chooses a stopping time. The gameinvolves a payment from the second player to the first. If the first player is first tostop (at τ) then the payment is Lτ ; if the second player is first to stop (at σ) then thepayment is Uσ; under a tie (τ = σ) then the payment isMτ . Here, (Lt)t≥0, (Ut)t≥0 and(Mt)t≥0 are adapted stochastic processes. For stopping rules τ and σ, the expectedvalue of the payment is J = J(σ, τ) = JU,M,L(σ, τ) = E[Lτ1τ<σ+Uσ1σ<τ+Mτ1τ=σ].The objective of Player 1 is to maximise J whilst the objective of Player 2 is tominimise J and this leads to two problem values v = infσ supτ J(σ, τ) and v =supτ infσ J(τ, σ) (respectively the upper and lower value) depending on whether wetake the perspective of the first or second player. Trivially v ≤ v: of great importanceis whether v = v in which case the Dynkin game is said to have a value v = v = v. See[6] and [10] for general treatments using the backward stochastic differential equationapproach and the Markovian setup, respectively, and [13] for an extensive survey withapplications to financial game options. One of the main ideas is to find a saddle point,i.e. a pair of stopping times (σ∗, τ∗) such that (i) J(σ∗, τ) ≤ J(σ∗, τ∗) for all τ and(ii) J(σ∗, τ∗) ≤ J(σ, τ∗) for all σ; then it is straightforward to show that the gamehas a value and (σ∗, τ∗) are optimal for Player 2 and Player 1 respectively.

Historically, the relative order of the payoff processes L, U and M in a Dynkingame has been important in proving that the game has a value. When M ≡ L,[3] proved that a Dynkin game has a value under the Mokobodzki condition whichstates that there exist two supermartingales whose difference lies between L and U .Such a condition was later relaxed in [16], see Kifer [13, Theorem 1] for a statement

∗Department of Statistics, University of Warwick, Coventry, CV4 7AL, U.K. Email adress:[email protected]; [email protected]; [email protected]

1

http://arxiv.org/abs/2111.02554v1

2 David Hobson, Gechun Liang and Haodong Sun

of the result under the order condition L ≤ M ≤ U and an integrability condition.Subsequently, the order condition was further relaxed in [14, 19, 23] by extending theclass of stopping strategies to randomized stopping times. [14] shows the game has avalue under fairly general conditions on the various payoffs, provided that agents areallowed to use randomised strategies. In particular the players can use these strategiesto ‘hide’ their stopping time from the opposing player. One important result is thatthe game is always terminated (i.e. at least one player chooses to stop) at or beforethe first moment t that Lt > Ut.

In general, the information structure is a crucial element in Dynkin games. See,for example, [7, 8] for the treatment of asymmetric/incomplete information and [1]for a robust version of a Dynkin game in which the players are ambiguous about theirprobability model. Moving beyond the classical set-up in a different direction, Liangand Sun [18] introduced a constrained Dynkin game in which the players’ stoppingstrategies are constrained to be event times of an independent Poisson process. Theconstrained Dynkin game is an extension of the Dynkin game in the same way thatthe constrained optimal stopping problem of [9] is an extension of a classical optimalstopping problem. When the problem involves a time-homogeneous payoff, a time-homogeneous diffusion process and an infinite time-horizon, the use of a Poissonprocess to determine the set of possible stopping times maximises the tractability ofthe constrained problem. Then under the order condition L = M ≤ U on the payoffs[18] prove existence of a constrained game value.

The Dynkin game formulation of the perpetual callable convertible bond was firstintroduced in [20] (see also [21] for the finite horizon counterpart). They reduced theproblem from a Dynkin game to an optimal stopping problem, and discussed whencall precedes conversion and vice versa. A key element of the problem specificationin [20] is that the firm’s stock price is calculated endogenously. Extensions of thisconvertible bond model include, for example, [2] which considered the problem of thedecomposition of a convertible bond into a bond component and an option component,[5] which studied the convertible bond with call protection, and [4] which added thetax benefit and bankruptcy cost to the convertible bond.

In contrast to the abovementioned works on convertible bonds, [18] introduceda constraint on the players’ stopping strategies by assuming that both bondholderand firm may only stop at times which are event times of an independent Poissonprocess. The idea is that the existence of liquidity constraints may restrict the players’abilities to stop at arbitrary stopping times. [18] assumes that the firm’s share price isexogeneous, and then the analysis is an extension of [9], [11], [12], [15] and [17] whichstudy optimal stopping problems under a liquidity constraint and their applications.In the same way that the classical callable convertible problem is related to a Dynkingame, the problem with liquidity constraints can be related to a Dynkin game undera condition that the stopping times lie in a restricted set.

The callable convertible bond was studied in [18] but, unfortunately, one of thecalculations in Proposition 6.2 of [18] is incorrect. [18] first reduced the original prob-lem from the state space x ∈ (0,∞) to (0, xλ] with xλ to be endogenously determined,and argued that the order condition L ≤ U would hold in the domain x ∈ (0, xλ],whereas the game would stop at the earliest Poisson arrival time after the firm’s stockprice Xx exceeds xλ. However, the threshold xλ in Proposition 6.2 of [18] was incor-rectly calculated, making the remaining analysis for the optimal stopping strategiesof the convertible bond in [18] void. It turns out xλ cannot be endogenously deter-mined because the game will not automatically stop after Xx exceeds xλ. One way to

Callable convertible bonds under liquidity constraints 3

correct this error is to enforce this stopping condition by assuming xλ is exogenouslygiven (e.g. xλ := K/γ) and introducing a forced conversion condition: conversion isassumed to automatically occur at the earliest Poisson arrival time after the stockprice process Xx first exceeds xλ. See [22, Chapter 2.5] for further details. However,one significant drawback of the above modification is that there is a possibility thatthe stock price Xx drops below any small number ε > 0 at the moment TM when theforced conversion is taking place, that is Px(Xx

TM< ε) > 0. This is obviously against

the interest of both the firm and the bondholder as neither of them would have in-centive to stop in such a situation. Moreover, the new convertible bond (with forcedconversion) is different from the original convertible bond that we are interested in.

One of the difficulties in the callable convertible bond problem is that, although itcan be recast in the standard form of a Dynkin game, the upper payoff process U doesnot necessarily dominate the lower payoff process L. As a result the general existenceresults for Dynkin games (see [16] or [13]) and especially the existence theorem inthe Poisson (constrained) case ([18, Theorem 2.3]) do not apply. One possibility isto try to remove the order condition in the Poisson case as in [14, 19, 23], but atbest that would give us an existence result, and we would like an explicit solution.Instead therefore, we take a different approach. The idea is to replace the originalproblem with a modified problem for which the order condition is satisfied (in both thestandard case with unconstrained stopping times, and the illiquid case where stoppingtimes are constrained to be event times of a Poisson process) and to which the theoryapplies. We then use a saddle-point argument to find the (explicit) solution to thismodified problem, and a further (general) argument to show that this saddle-point isalso a saddle-point for the original (constrained) perpetual callable convertible bondproblem.

2. The extended saddle point result. Recall the classical formulation of aDynkin game: for L = (Lt)t≥0, U = (Ut)t≥0, M = (Mt)t≥0 and

(2.1) JU,M,L(σ, τ) = E[Lτ1τ<σ + Uσ1σ<τ +Mτ1τ=σ],

define the upper and lower values

v ≡ vU,M,L(T ) = infσ∈T

supτ∈T

JU,M,L(σ, τ), v = vU,M,L(T ) = supτ∈T

infσ∈T

JU,M,L(τ, σ),

where T is a set of stopping times. If v = v then we say the game has a valuev = vU,M,L(T ) where v = v = v.

The Dynkin game was analysed in [3] and [16] (amongst others, see [13] for acomprehensive survey). Often this analysis is under a condition that the payoffs areordered: L ≤ M ≤ U , and when this condition fails there are simple examples whichshow that the game may not have a value.

Example 2.1. Suppose that Lt = Mt = 1t>1 and Ut = 1t≤1. Then,JU,M,L(σ, τ) = 11<τ≤σ+1σ<τ,σ≤1. It is clear that v = infσ supτ J

U,M,L(σ, τ) = 1,as, given σ, Player 1 who maximizes J can choose a stopping strategy τ∗ = σ+1σ≤1

(or more generally any stopping time for which τ ∈ (σ,∞) on σ ≤ 1 and τ ∈ (1, σ]on σ > 1.) On the other hand, v = supτ infσ J

U,M,L(σ, τ) = 0, as Player 2 whominimizes J can choose a stopping strategy σ∗ with σ∗ ∈ (1, τ) on τ > 1 and σ∗ ≥ τon τ ≤ 1.

Notwithstanding the above example, in any given problem the pragmatic ap-proach to finding a solution is to find a saddle point, i.e. to find a pair (σ∗, τ∗) ∈ T ×T


such that JU,M,L(σ∗, τ) ≤ JU,M,L(σ∗, τ∗) ≤ JU,M,L(σ, τ∗) for all (σ, τ) ∈ T ×T . Then,

vU,M,L(T ) ≤ supτ∈T

JU,M,L(σ∗, τ) ≤ JU,M,L(σ∗, τ∗) ≤ infσ∈T

JU,M,L(σ, τ∗) ≤ vU,M,L(T ).

Since trivially vU,M,L(T ) ≤ vU,M,L(T ), we conclude that vU,M,L(T ) = vU,M,L(T ) andthe game has a value.

Note that the existence of a saddle-point gives a direct proof of the existence of agame value, but leaves us no nearer to finding the game value unless we can identifythe optimisers σ∗ and τ∗. Nonetheless our first key result is an extension of thisargument.

Proposition 2.2. Suppose U ≤ U ≤ U , M ≤ M ≤ M , L ≤ L ≤ L and let T bea set of stopping times.

Suppose that there exists a pair of stopping times (σ∗, τ∗) ∈ T × T such that

(i) JU,M,L(σ∗, τ∗) = JU,M,L(σ∗, τ∗) = JU,M,L(σ∗, τ∗);

(ii) JU,M,L(σ∗, τ) ≤ JU,M,L(σ∗, τ∗) for any τ ∈ T ;(iii) JU,M,L(σ∗, τ∗) ≤ JU,M,L(σ, τ∗) for any σ ∈ T .Then, vU,M,L(T ) = vU,M,L(T ) = JU,M,L(σ∗, τ∗) and the game for payoff triple

(U,M,L) and stopping time set T has a value. Moreover (σ∗, τ∗) is a saddle pointfor the game with payoffs (U,M,L).

Proof. Since T remains constant throughout, to economise on notation we omitthe label T . It follows from monotonicity of the payoffs and (i)-(iii) that

JU,M,L(σ∗, τ) ≤ JU,M,L(σ∗, τ)

≤ JU,M,L(σ∗, τ∗) = JU,M,L(σ∗, τ∗),

and

JU,M,L(σ, τ∗) ≥ JU,M,L(σ, τ∗)

≥ JU,M,L(σ∗, τ∗) = JU,M,L(σ∗, τ∗).

Hence, (σ∗, τ∗) is indeed a saddle-point for the game with payoffs (U,M,L).Remark 2.3. Under the hypotheses of Proposition 2.2, (σ∗, τ∗) is also a saddle-

point for the games with payoffs (U,M,L) and (U,M,L) and these games also havevalue JU,M,L(σ∗, τ∗). Indeed, We have

JU,M,L(σ, τ∗) ≥ JU,M,L(σ, τ∗)

≥ JU,M,L(σ∗, τ∗) = JU,M,L(σ∗, τ∗),

which, when combined with (ii) of Proposition 2.2 gives the desired result for the gamewith payoff triple (U,M,L).

Conversely, JU,M,L(σ∗, τ) ≤ JU,M,L(σ∗, τ) ≤ JU,M,L(σ∗, τ∗) = JU,M,L(σ∗, τ∗),and the corresponding results follow for (U,M,L).

For general triples (U,U, U), (M,M,M) and (L,L, L) we cannot expect to findstopping times satisfying the conditions of the proposition. But, the following specialcase will prove to be exactly what we need for the callable convertible bond. The keypoint is that it translates a problem where the ordering condition L ≤ U is violated toa problem where it is satisfied, and then the general theory of [18] can be applied. Inparticular, the existence theorem for constrained games ([18, Theorem 2.3]) is validfor problems where the ordering constraint is satisfied.


Recall U ∨ L is the process given by (U ∨ L)t = maxUt, Lt.Corollary 2.4. Suppose that there exists a pair of stopping times (σ∗, τ∗) ∈

T × T such that(i) JU∨L,L,L(σ∗, τ∗) = JU,L,L(σ∗, τ∗);(ii) JU∨L,L,L(σ∗, τ) ≤ JU∨L,L,L(σ∗, τ∗) for any τ ∈ T ;(iii) JU,L,L(σ∗, τ∗) ≤ JU,L,L(σ, τ∗) for any σ ∈ T .Then vU,L,L(T ) = vU,L,L(T ) = JU,L,L(σ∗, τ∗) and the game for payoff triple

(U,L, L) and (constrained) stopping time set T has a value vU,L,L = J(σ∗, τ∗).Proof. Take U = U ∨L, U = U , L = L = L = M = M = M in Proposition 2.2.Example 2.5. Suppose that Lt = Mt = 1t≥1 and Ut = 1t<1. (Note the

different treatment of t = 1 when compared with Example 2.1.) Then, JU,L,L(σ, τ) =11≤τ≤σ + 1σ<τ,σ<1. Although U ≥ L is not satisfied, it is still possible to find asaddlepoint in the sense of Corollary 2.4. Indeed, (σ∗, τ∗) = (any value in T , τ∗ =

inft : t ≥ 1, t ∈ T ) is a saddle point. To see this, for U = U ∨L = 1, JU,L,L(σ, τ) =11≤τ≤σ + 1σ<τ. It is clear that

JU,L,L(σ∗, τ) ≤ JU,L,L(σ∗, τ∗) = JU,L,L(σ∗, τ∗) = JU,L,L(σ, τ∗) = 1,

so the extended saddle point conditions in Corollary 2.4 are satisfied. Hence the gamehas a value, and that value is 1.

3. The callable convertible bond: problem specification. From the bond-holder’s viewpoint, the discounted payoff of a perpetual callable convertible bondissued by a firm has the form

(3.1) P (σ, τ) =

∫ σ∧τ

0

e−ruc du+ e−rτγXxτ 1τ≤σ + e−rσK1σ<τ.

Here Xx represents the stock price process starting from Xx0 = x. The firm issues

convertible bonds as perpetuities with a constant coupon rate c ≥ 0. The investorthen purchases a share of this convertible bond at initial time t = 0. By holding theconvertible bond, the investor will continuously receive the coupon rate c from thefirm until the contract is terminated: (i) if first, (i.e. σ < τ) the firm calls the bond atsome stopping time σ, the bondholder will receive a pre-specified surrender price Kat time σ; (ii) if first (i.e. τ < σ) the investor chooses to convert their bond at somestopping time τ , the bondholder will obtain γXx

τ at time τ from converting their bondto γ shares of firm’s stock with a pre-specified conversion rate γ. If bondholder andfirm choose to stop the contract simultaneously (i.e. σ = τ) then the bondholders acttakes priority and the bondholder will obtain γXx

τ .Note that P (σ, τ) can be rewritten using integration by parts as

P (σ, τ) = c

r+ e−rτ

(

γXxτ −

c

r

)

1τ≤σ + c

r+ e−rτ

(

K −c

r

)

1τ>σ.

This is exactly of the form in (2.1) with Lt = Mt = cr + e−rt

(

γXxt − c

r

)

and Ut =cr + e−rt

(

K − cr

)

. Note that we do not have L ≤ U ; in particular if Xxt > K

γ then

Lt > Ut. Then Jx(σ, τ) := Ex[P (σ, τ)] is given by(3.2)

Jx(σ, τ) = Ex[ c

r+ e−rτ

(

γXxτ −

c

r

)

1τ≤σ + c

r+ e−rτ

(

K −c

r

)

1σ<τ

]

.

We work under the following assumption on the stock price process Xx:


Standing Assumption 3.1. The price process Xx of the firm’s stock follows

Xxt = x+

∫ t

0

(r − q)Xxs ds+

∫ t

0

σXxs dWs,

where the constants r, q, σ (with σ > 0) represent the risk-free interest rate, the divi-dend rate and the stock’s volatility, and W is a Brownian motion.

Following [18] (and also [9] and others) the liquidity constraint is modelled asfollows. Instead of allowing σ and τ to be any stopping times we assume that σ, τ ∈T = R(λ), where

R(λ) = η: η is a G-stopping time such that η(ω) = TN (ω) for some N ≥ 1.

Herein, G is the filtration generated by the underlying Brownian motion W (with itsnatural filtration FW ) and an exogenous Poisson process (with its natural filtrationH and jump times TNN≥1), i.e. G = F

W ∨ H. Then, in summary, we work on afiltered probability space (Ω,F ,G,P) which supports a Brownian motion W drivingthe price process and a Poisson process, and T is the set of TNN≥1-valued stoppingtimes.

It turns out that it is useful to consider the (upper and lower) value of the callableconvertible bond as functions of the initial value of the stock price. Then, with thesuperscript λ denoting the rate of the Poisson process we define the upper and lowervalue of the convertible bond with liquidity constraint via

vλca(x) = infσ∈R(λ)

supτ∈R(λ)

Jx(σ, τ),(3.3)

vλca(x) = supτ∈R(λ)

infσ∈R(λ)

Jx(σ, τ),(3.4)

where Jx is as given in (3.2). In the case where there are no liquidity constraintson the stopping times, the superscript λ is omitted. Our goal is first to show thatvλca = vλca as functions of x and hence that the game value vλca is well defined, andsecond to give an explicit form for vλca.

A key element of the financial specification of the callable convertible bond is thatif the firm calls at the same time as the bondholder then the bondholder’s actions takepriority. In particular, M ≡ L. A second key element is that if the firm calls then thebondholder is given a final opportunity to preempt the call, and to convert. Effectivelythen, the payoff when the firm calls is given by U ∨L (and not U). This suggests thatwe should consider the problem with modified payoffs (U , M , L) given by (U , M , L) =(U ∨ L,L, L) =

(

cr + e−rt

(

K ∨ γXxt − c

r

)

, cr + e−rt

(

K − cr

)

, cr + e−rt

(

K − cr

))

, forwhich the ordering condition holds. We aim to find a saddle-point for this modifiedproblem, and then to use Corollary 2.4 to deduce that it provides a solution to theoriginal problem.

For future reference, let αλ > 1 and βλ < 0 be the two characteristic roots of thedifferential operator

(3.5) Lλf :=1

2σ2x2f ′′ + (r − q)xf ′ − (r + λ)f, f ∈ C2(R+).

That is, (αλ, βλ) are the roots of Qλ(z) = 0 with Qλ(z) :=12σ

2z2 + (r − q − 12σ

2)z −(r + λ). For simplicity, we also write α := α0, β := β0, L := L0 and Q := Q0.

Lemma 3.2. For r 6= q and λ ≥ 0 we have Qλ(λ+rr−q ) > 0. Then, if q < r,

λ+rr−q > 1 and αλ ∈ (1, λ+r

r−q ) (and α ∈ (1, rr−q )). Conversely, if q > r then λ+r

r−q < 0

and βλ ∈ (λ+rr−q , 0).


Further, we have αλ < α(λ+rr ) and hence rαλ

λ+r < α.

Proof. We have Qλ(λ+rr−q ) = 1

2σ2(

(λ+rr−q )

2 − (λ+rr−q )

)

. If q > r then this is imme-

diately positive; if q < r then it is positive since λ+rr−q > 1. The final result follows

similarly.

4. Two auxiliary optimal stopping problems. In this section, we solve theoptimal stopping problems of the bondholder and the firm separately. They will serveas the building blocks for the construction of the optimal stopping strategies for theconvertible bond in the next section.

4.1. Optimal stopping of the bondholder. Consider the following optimalstopping problem

(4.1) vλco(x) = supτ∈R(λ)

Ex

[∫ τ

0

e−ruc du+ e−rτγXxτ

]

.

This is the problem facing the bondholder in the absence of any callable feature forthe firm. We aim to find τ∗ ∈ R(λ) to maximize the above expected discounted payoffof the bondholder. For later use, we also introduce an equivalent formulation, underwhich the bondholder finds an optimal number of Poisson jumps, i.e.

(4.2) vλco(x) = supN∈N (λ)

Ex

[

∫ TN

0

e−ruc du+ e−rTNγXxTN

]

,

where

N (λ) = G-stopping time N for N ≥ 1, with G = GTnn≥1.

Define x⋆co,λ by

(4.3) x⋆co,λ =

c(λ+ q)(α(λ + r) − βλr)

γr(λ+ r)(α(λ + q)− λ− βλq).

It will turn out that x⋆co,λ is a critical threshold both in the problem under consid-

eration, and for the callable convertible bond. The subscripts co and λ are intendedto convey that the quantity arises in the convertible bond problem under a liquidityconstraint based on the Poisson process with rate λ.

Lemma 4.1. Assume that r > 0 and q > 0. Let x⋆co,λ be given as in (4.3). Then

x⋆co,λ > max c

γr ,cγq

λ+qλ+r and x⋆

co,λ < 1γ

cr

αα−1 .

Proof. Given the formula (4.3), x⋆co,λ > c

γr is equivalent to βλ(r − q) < λ+ r. Ifr ≥ q this is immediate, and if r < q it follows from Lemma 3.2.

Moreover, x⋆co,λ > c

γqλ+qλ+r is equivalent to r > α(r− q). If r ≤ q this is immediate.

Otherwise, it follows from Lemma 3.2.

Finally, after some algebra it can be shown that x⋆co,λ < 1

γcr

αα−1 is equivalent to

βλ ((λ+ q)r + αλ(q − r)) < α(λ+ r)q.

Given the signs of βλ and α, it is sufficient to argue that (λ+q)r+αλ(q−r) > 0. But ifq ≥ r this is immediate, and if q < r then since α < r

r−q , we have (λ+q)r+αλ(q−r) =


λ(r + α(q − r)) + qr > qr > 0.Lemma 4.2. Suppose that r > 0 and q > 0. Then, the value function vλco(x) has

the explicit expression

(4.4) vλco(x) =

cr +

[

γx⋆co,λ − c

r

]

xα

(x⋆co,λ)

α , x < x⋆co,λ;

cλ+r + λγx

λ+q +[

qγx⋆co,λ

λ+q − cλ+r

]

xβλ

(x⋆co,λ

)βλ, x ≥ x⋆

co,λ.

The optimal stopping time for (4.1) is given by τ∗ = TNx⋆co,λ

with Ny defined as

(4.5) Ny = infn ≥ 1 : XxTn

≥ y.

Proof. Following along similar arguments in Theorem 1 of [9], it can be shownthat the value function vλco(x) satisfies the recursive equation

vλco(x) = Ex

[

∫ T1

0

e−ruc du+ e−rT1 maxvλco(XxT1), γXx

T1

]

= Ex

[∫ ∞

0

e−(r+λ)u(c+ λmaxvλco(Xxs ), γX

xs )du

]

,(4.6)

which, in turn, implies that vλco is a solution of the HJB equation

LλV (x) + c+ λmaxV (x), γx = 0.

(Moreover, vλco is the solution which is bounded at zero, and of linear growth atinfinity).

From the structure of the stopping problem, we expect that there is a continuationregion (0, z) with V > γx and a stopping region [z,∞) with V ≤ γx. On (0, z), vλcosolves LλV + c + λV ≡ LV + c = 0 and on [z,∞), it solves LλV + (c + γλx) = 0,together with boundary conditions that V (0) = c/r and V (x) is of linear growth. Inaddition we expect value matching and smooth fit at z (and that V (z) = γz).

We find that with z replaced by the optimal threshold x⋆co,λ, on (0, x⋆

co,λ),

(4.7) vλco(x) =c

r

[

1−xα

(x⋆co,λ)

α

]

+ γx⋆co,λ

xα

(x⋆co,λ)

α=

c

r+[

γx⋆co,λ −

c

r

] xα

(x⋆co,λ)

α,

and on [x⋆co,λ,∞)

(4.8) vλco(x) =c

λ+ r+

λγx

λ+ q+

[

qγx⋆co,λ

λ+ q−

c

λ+ r

]

xβλ

(x⋆co,λ)

βλ.

Furthermore, first order smooth fit gives that

α

x⋆co,λ

[

γx⋆co,λ −

c

r

]

=λγ

λ+ q+

βλ

x⋆co,λ

[

qγx⋆co,λ

λ+ q−

c

λ+ r

]

.

This can be solved to give

x⋆co,λ =

c(λ+ q)(α(λ + r) − βλr)

γr(λ+ r)(α(λ + q)− λ− βλq).


Given the lower bounds on x⋆co,λ of Lemma 4.1, it is clear from (4.7), (4.8)

and the smooth fit at x⋆co,λ that vλco(x) is increasing and convex in x. Then, since

limx↓0 vλco(x) = c

r > 0, and limx↑∞vλco(x)x = λγ

λ+q < γ it follows that vλco crosses the

line γx exactly once on (0,∞). In particular, vλco(x) > γx on x ∈ (0, x⋆co,λ) and

vλco(x) ≤ γx on x ∈ [x⋆co,λ,∞). Hence, Nx⋆

co,λ= n ≥ 1 : vλco(X

xTn

) ≤ γXxTn.

Finally, consider Y = (Yn)n≥1 given by

Yn =

∫ Tn

0

e−rucdu+ e−rTn maxvλco(XxTn

), γXxTn.

Using arguments similar to those in Theorem 1 of [9] it can be shown that Y is aG-supermartingale, and a G-martingale when n is replaced by n∧Nx⋆

co,λ. In turn, for

any N ∈ N (λ),

maxvλco(XxT1), γXx

T1 ≥ E

XxT1

[

∫ TN

T1

e−r(u−T1)c du+ e−r(TN−T1) maxvλco(XxTN

), γXxTN

]

≥ EXx

T1

[

∫ TN

T1

e−r(u−T1)c du+ e−r(TN−T1)γXxTN

]

.

Substituting the above inequality into the recursive equation (4.6) yields that

vλco(x) ≥ Ex

[

∫ TN

0

e−ruc du+ e−rTNγXxTN

]

.

The above inequality will become an equality with N = Nx⋆co,λ

. This proves theoptimality of TNx⋆

co,λ.

Lemma 4.2 implies that the whole region (0,∞) can be divided into the contin-uation region (0, x⋆

co) and the stopping region [x⋆co,∞). Furthermore, we have the

following properties of the value function vλco(x) and the optimal stopping time τ∗.Corollary 4.3. Suppose that r > 0 and q > 0 and that the surrender price

satisfies K ≥ γx⋆co,λ. Then, the value function satisfies vλco(x) ≤ K for x ≤ x⋆

co,λ.

Proof. Since vλco(x) is increasing in x, we have vλco(x) ≤ vλco(x⋆co,λ) for x ≤ x⋆

co,λ.

On the other hand, by the value matching at x⋆co,λ, we have vλco(x

⋆co,λ) = γx⋆

co,λ. Theconclusion then follows from the assumption γx⋆

co,λ ≤ K.Proposition 4.4. The value function satisfies the dynamic programming equa-

tion

(4.9) vλco(x) = Ex

[∫ η

0

e−ruc du+ e−rηvλco(Xxη )1η<τ∗ + e−rτ∗

γXxτ∗1τ∗=η

]

for any η ∈ R(λ) with η ≤ τ∗.Proof. First, integration by parts yields

vλco(x) =c

r+ E

x[

e−rτ∗

(

γXxτ∗ −

c

r

)]

.

Next, for any stopping time η ∈ R(λ) with η ≤ τ∗, by conditioning on Xxη , we have

vλco(x) =c

r+ E

x[

e−rηEXx

η

[

e−r(τ∗−η)(

γXxτ∗ −

c

r

)]]

=c

r+ E

x[

e−rη

1η<τ∗EXx

η

[

e−r(τ∗−η)(

γXxτ∗ −

c

r

)]

+ 1η=τ∗

(

γXxτ∗ −

c

r

)]

=c

r+ E

x[

e−rη

1η<τ∗

(

vλco(Xxη )−

c

r

)

+ 1η=τ∗

(

γXxτ∗ −

c

r

)]

.


Then, the dynamic programming equation (4.9) follows since∫ η

0e−rucdu = c

r (1 −e−rη).

4.2. Optimal stopping of the firm. For K ∈ ( cr , γx⋆co,λ), consider the follow-

ing optimal stopping problem

(4.10) vλf (x) = infσ∈R(λ),σ≤TNK/γ

Ex

[∫ σ

0

e−ruc du+ e−rσ maxγXxσ ,K

]

,

where Ny is as defined in (4.5). Taking the perspective of the firm we aim to findσ to minimize the modified expected discounted payoff of the bondholder, wherethe bondholder is not allowed to make any stopping decisions, except that at σ thebondholder may choose to preempt and to receive shares in preference to a cashpayment. In line with (4.2), the optimal stopping problem (4.10) also has an equivalentformulation(4.11)

vλf (x) = infN∈N (λ),N≤NK/γ

Ex

[

∫ TN

0

e−ruc du+ e−rTN (K1N<NK/γ + γXxTNK/γ

1N=NK/γ)

]

,

where we used the stopping condition that the bond is forced to stop at TNK/γ, and

that γXxTN

< K if N < NK/γ .

Lemma 4.5. Suppose that r > 0, q > 0 and K ∈ ( cr , γx⋆co,λ). Then, the value

function vλf (x) has the explicit expression

(4.12) vλf (x) =

cr +

(

K − cr

)

xα

(x⋆ca,λ)

α , x < x⋆ca,λ;

c+λKλ+r +A xαλ

(x⋆ca,λ)

αλ+B xβλ

(x⋆ca,λ)

βλ, x⋆

ca,λ ≤ x ≤ K/γ;

cλ+r + λγx

λ+q + C γβλxβλ

Kβλ, K/γ < x,

where the unknowns A, B, C and the optimal threshold x⋆ca,λ are given by

A =K − c

r

αλ − βλ

(

α−rβλ

λ+ r

)

> 0;(4.13)

B =K − c

r

αλ − βλ

(

rαλ

λ+ r− α

)

< 0;(4.14)

x⋆ca,λ =

K

γ

(

1

(K − cr )

λ+ r

(α(λ + r)− rβλ)

λK

λ+ q

(

1−(r − q)βλ

λ+ r

))−1/αλ

;(4.15)

C =Aαλ

βλ

(

K

γx⋆ca,λ

)αλ

+B

(

K

γx⋆ca,λ

)βλ

−λK

λ+ q

1

βλ.(4.16)

The optimal stopping time for (4.10) is given by σ∗ = TNx⋆ca,λ

with Ny defined in

(4.5).

The optimal threshold x⋆ca,λ for this problem will turn out to be an optimal

threshold for the callable convertible bond with liquidity constraint (for moderatevalues of surrender value K) which justifies the subscripts ca and λ.

Proof. Similar to Lemma 4.2, it can be shown that the value function vλf (x)


satisfies the recursive equation

vλf (x) = Ex

[

∫ T1

0

e−ruc du+ e−rT1(minvλf (XxT1),K1XT1

<K/γ + γXxT11XT1

≥K/γ)

]

= Ex

[∫ ∞

0

e−(r+λ)u(c+ λ(minvλf (Xss ),K1Xs<K/γ + γXx

s 1Xs≥K/γ))du

]

,(4.17)

which in turn implies that vλf (x) is a solution of the HJB equation

LλV + c+ λ(minV (x),K1x<K/γ + γx1x≥K/γ) = 0.

Moreover, vλf is bounded at zero and of linear growth.To solve the above equation, we first consider x < K/γ. For small x, we expect

that the firm will continue, so vλf < K and vλf solves LV + c = 0 subject to V (0) = cr .

For moderate x, we expect that the firm will choose to call, so vλf > K and vλf solvesLλV + (c+ λK) = 0. Next, we consider x ≥ K/γ. The firm will be forced to stop inthis situation, so vλf solves LλV + (c+ λγx) = 0 subject to V being of linear growth.

The boundary between small and moderate x occurs at x where V (x) = K. Theboundary between moderate and large x occurs at x where γx = K. We expect valuematching and first order smooth fit at both boundaries, which lead to the expression(4.12) and (writing θ = K

γx⋆ca,λ

), the four algebraic equations

K =c+ λK

λ+ r+A+B;(4.18)

(

K −c

r

) α

x⋆ca,λ

= Aαλ

x⋆ca,λ

+Bβλ

x⋆ca,λ

;(4.19)

c+ λK

λ+ r+Aθαλ +Bθβλ =

c

λ+ r+

λK

λ+ q+ C;(4.20)

γ

KAαλθ

αλ +γ

KBβλθ

βλ =λγ

λ+ q+ Cβλ

γ

K.(4.21)

We next solve (4.18)-(4.21). Multiplying (4.19) by x⋆ca,λ eliminates x⋆

ca,λ, and then(4.18) and the modified (4.19) can be solved to give expressions for A and B in (4.13)and (4.14), respectively. Note that since K > c

r we have A > 0 and B < 0 byLemma 3.2. (4.20) and (4.21) can then be solved for C and θ. Indeed, we have

C =λK(q − r)

(λ+ r)(λ + q)+Aθαλ +Bθβλ =

Aαλθαλ

βλ+Bθβλ −

λK

λ+ q

1

βλ.

We can use the last equality to find x⋆ca,λ given in (4.15), and then this gives C in

(4.16).We are left to show x⋆

ca,λ < Kγ , v

λf (x) < K on x ∈ (0, x⋆

ca,λ) and vλf (x) > K on

x ∈ (x⋆ca,λ,∞). The first statement will be proved in Proposition 4.6 below. The

second and third statements follow from the increasing property of vλf (x) in x and

the value matching condition vλf (x⋆ca,λ) = K. (In particular, it is easily checked that

ddxv

λf > 0 on (0, x⋆

ca,λ) ∪ (x⋆ca,λ,K/γ), where for the second statement we use that

Aαλ > 0 and Bβλ > 0. Finally, on (K/γ,∞) we have that, irrespective of the sign C,ddxv

λf is monotonic in x, since the values of the derivative are positive at both ends of

the interval, it must be positive throughout.) The three statements then imply that

Nx⋆ca,λ

= n ≥ 1 : vλf (XxTn

) ≥ K ≤ NK/γ .


Finally, note that

(

∫ Tn

0

e−rucdu+ e−rTn(minvλf (XxTn

),K1n<NK/γ + γXxTn1n=NK/γ)

)

1≤n≤NK/γ

,

is a G-submartingale, and a G-martingale with n replaced by n∧Nx⋆ca,λ

. The proof is

similar to the one in Lemma 4.2 (see also Theorem 1 in [9]), and is therefore omitted.In turn, for any N ∈ N (λ) such that N ≤ NK/γ , since 1 < NK/γ = XT1

< K/γ,by the submartingale property we have

minvλf (XxT1),K1XT1

<K/γ + γXxT11XT1

≥K/γ

≤ EXx

T1

[

∫ TN

T1

e−r(u−T1)cdu+ e−r(TN−T1)(minvλf (XxTN

),K1N<NK/γ + γXxTN1N=NK/γ)

]

≤ EXx

T1

[

∫ TN

T1

e−r(u−T1)cdu+ e−r(TN−T1)(K1N<NK/γ + γXxTNK/γ

1N=NK/γ)

]

.

Substituting the above inequality into the recursive equation (4.17) yields that

vλf (x) ≤ Ex

[

∫ TN

0

e−ruc du+ e−rTN (K1N<NK/γ + γXxTNK/γ

1N=NK/γ)

]

.

The above inequality turns out to be an equality with N = Nx⋆ca,λ

, which shows theoptimality of TNx⋆

ca,λ.

Lemma 4.5 implies that the whole region (0,∞) can be divided into the con-tinuation region (0, x⋆

ca,λ) and the stopping region [x⋆ca,λ,∞). The next proposition

provides a relationship between the optimal thresholds x⋆co,λ in Lemma 4.2 and x⋆

ca,λ

in Lemma 4.5, which also completes the proof of Lemma 4.5.Proposition 4.6. Suppose that r > 0, q > 0 and K ∈ ( cr , γx

⋆co,λ). Then,

x⋆co,λ > K

γ if and only if x⋆ca,λ < K

γ .

Proof. Recall the definition θ = Kγx⋆

ca,λ. Hence, x⋆

ca,λ < Kγ is equivalent to θ > 1.

In turn, (4.15) implies

θ > 1 ⇔1

A(αλ − βλ)

λK

λ+ q

(

1−(r − q)βλ

λ+ r

)

> 1

⇔

(

α−rβλ

λ+ r

)

(

K −c

r

)

<λK

λ+ q

(

1−(r − q)βλ

λ+ r

)

⇔ [α(λ+ r) − rβλ](λ+ q)(

K −c

r

)

< λK[(λ+ r) − (r − q)βλ]

⇔ (λ+ r)K[α(λ + q)− λ− qβλ] <c

r(λ+ q)[α(λ + r)− rβλ].

On the other hand, it follows from (4.3) that this is equivalent to x⋆co,λ > K

γ .

Proposition 4.7. Suppose that r > 0, q > 0 and K ∈ ( cr , γx⋆co,λ). Then, the

value function satisfies vλf (x) ≥ γx for x ≤ x⋆ca,λ.

Proof. Note that vλf (x) is increasing in x, vλf (0) =cr > 0 and vλf (x

⋆ca,λ) = K >

γx⋆ca,λ, so it is sufficient to show that vλf does not cross γx on (0, x⋆

ca,λ]. In turn, this


will follow if (vλf )′(y) < γ at any crossing point y ∈ (0, x⋆

ca,λ] — then vλf may crossdown below γx but cannot cross back above.

Let y ∈ (0, x⋆ca,λ] be such that vλf (y) = γy. Then c

r +(

K − cr

)

(

yx⋆ca,λ

)α

= γy and

(vλf )′(y) < γ ⇔

α

y

(

γy −c

r

)

< γ ⇔ y <1

γ

α

α− 1

c

r.

The last inequality in the above follows from Lemma 4.1 since we have y < x⋆ca,λ <

Kγ < x⋆

co,λ < 1γ

αα−1 . This proves the claim.

To conclude this section, we provide the dynamic programming equation for theoptimal stopping problem (4.10). The proof follows along similar arguments in Propo-sition 4.4 and is thus omitted.

Proposition 4.8. The value function satisfies the dynamic programming equa-tion(4.22)

vλf (x) = Ex[

∫ η

0

e−ruc du+ e−rσ∗

maxγXxσ∗ ,K1σ∗=η + e−rηvλf (X

xη )1η<σ∗].

for any η ∈ R(λ) with η ≤ σ∗.

5. Pricing the callable convertible bond. We divide our analysis into threecases according to the value of the surrender price K. Observe that

∫∞

0 e−rucdu ≡ cr

is the value of the corresponding perpetuity if the bond is never converted nor called.Define Jx and Jx by

(5.1) Jx(σ, τ) = Ex

[∫ σ∧τ

0

e−rucdu+ e−rτγXxτ 1τ≤σ + e−rσK1σ<τ

]

.

and(5.2)

Jx(σ, τ) = Ex

[∫ σ∧τ

0

e−rucdu+ e−rτγXxτ 1τ≤σ + e−rσ maxγXx

σ ,K1σ<τ

]

.

5.1. Case K ≤ cr . In this case, since the surrender price K is smaller than the

corresponding perpetuity value cr , the firm will choose to call at the first opportu-

nity. On the other hand, the bondholder will preempt the firm’s stopping action byconverting the bond if γXx

T1 > K.Theorem 5.1. Suppose that r > 0, q > 0 and K ≤ c

r . Then, (σ∗, τ∗) =(T1, TNK/γ

) is a saddle point for Jx in (3.3)-(3.4), and the game value is given by

(5.3) vλca(x) =

c+λKλ+r +A

(

γxK

)αλ , x < K/γ;c

λ+r + λγxλ+q +B

(

γxK

)βλ , x ≥ K/γ,

with A > 0 and B > 0 given by

A =(λ+ r)− βλ(r − q)

(αλ − βλ)

λK

(λ+ q)(λ+ r);

B =(λ+ r)− αλ(r − q)

(αλ − βλ)

λK

(λ+ q)(λ+ r).


Proof. The signs of A and B follow from Lemma 3.2. For the main result it issufficient to verify the conditions (i)-(iii) in Corollary 2.4 to show (σ∗, τ∗) is a saddlepoint for Jx in (3.3)-(3.4) where Jx and Jx are as in (5.1) and (5.2) respectively.

First, we apply integration by parts to Jx to obtain

Jx(σ∗, τ∗) =c

r+ E

x[

e−rT1

h (γXxτ∗)1τ∗=T1 + h

(

maxγXxT1,K

)

1τ∗>T1

]

where to save space we write h(z) = z− cr . By the definitions of TNK/γ

, maxγXxT1,K =

K on the event τ∗ > T1. Hence,

Jx(σ∗, τ∗) =c

r+ E

x[

e−rT1

h (γXxτ∗)1τ∗=T1 + h (K)1τ∗>T1

]

= Jx(σ∗, τ∗),

which verifies (i). To show (ii), for any τ ∈ R(λ), we have

Jx(σ∗, τ)

=c

r+ E

x[

e−rT1

h (γXxτ )1τ=T1 + h

(

maxγXxT1,K

)

1τ>T1

]

≤c

r+ E

x[

e−rT1

h (maxγXxτ ,K)1τ=T1 + h

(

maxγXxT1,K

)

1τ>T1

]

=c

r+ E

x[

e−rT1

(

maxγXxT1,K −

c

r

)]

.

Then, (ii) follows by observing that

(5.4) Jx(σ∗, τ∗) = Jx(σ∗, τ∗) =c

r+ E

x[

e−rT1

(

maxγXxT1,K −

c

r

)]

.

For (iii), we have, for any σ ∈ R(λ),

Jx(σ, τ∗) =c

r+ E

x[

e−r(σ∧τ∗)

h (γXxτ∗)1τ∗≤σ + h (K)1τ∗>σ

]

=c

r+ E

x[

1τ∗=T1e−rT1h

(

γXxT1

)

+1τ∗>T1e−rT1

e−r(τ∗−T1)h (γXxτ∗)1τ∗≤σ + e−r(σ−T1)h(K)1τ∗>σ

]

.

Since γXxτ∗ ≥ K and K − c

r ≤ 0, we further obtain

Jx(σ, τ∗) ≥c

r+ E

x[

1τ∗=T1e−rT1

(

γXxT1

−c

r

)

+ 1τ∗>T1e−rT1

(

K −c

r

)]

= Jx(σ∗, τ∗),

which verifies (iii).To calculate the game value vλca(x) = Jx(σ∗, τ∗) in (5.4), we may use the expo-

nential distribution of T1 to rewrite (5.4) as

vλca(x) = Ex

[∫ ∞

0

e−(r+λ)u(c+ λmaxγXxu ,K)du

]

.

which, in turn, implies that vλca(x) is the solution of the following HJB equation

LλV + c+ λmaxγx,K = 0


such that V is bounded at zero, and of linear growth at infinity. It is immediate that,on [0,K/γ)

V (x) =c+ λK

λ+ r+A

(γx

K

)αλ

,

and on [K/γ,∞)

V (x) =c

λ+ r+

λγx

λ+ q+B

(γx

K

)βλ

.

where A and B are constants to be determined. Value matching and first order smoothfit at K/γ give that A and B solve

λK

λ+ r+A =

λK

λ+ q+B;

Aαλγ

K=

λγ

λ+ q+Bβλ

γ

K,

which yield the expressions of A and B in the theorem, and we conclude.

Initial Stock Price x

Bon

d P

rice

K

x

Vca

Fig. 5.1. The convertible bond value vλca for Case K ≤ cr.

5.2. Case cr < K < γx⋆

co,λ. In this case, since the surrender price is larger thanthe value of the perpetuity, the firm will not seek to call the bond when the stockprice is low; nor will the bondholder seek to convert in this case. Conversely, whenthe stock price is large both the firm will seek to call, and the bondholder will seekto convert, with the bondholder’s action taking precedence. Finally, when the stockprice is moderate the firm will seek to call. The bondholder would prefer that thebond is not called and would not choose to convert of their own volition, but, giventhat the bond is being called, may elect to convert.

In particular, there are no circumstances where the bondholder wants to convertand the firm does not want to call. Hence the game option reduces to an optimalstopping problem for the firm (subject only to the fact that the bondholder may chooseto pre-empt the call to receive γXx

t rather than K.) The resulting optimal stoppingproblem is precisely the problem we studied in Section 4.2. Then σ∗ = TNx⋆

ca,λis

the candidate optimal stopping time of the firm. Moreover, Proposition 4.7 impliesthat in the continuation region (for the firm), the value function vλf (x) dominates


the conversion payoff γx. This confirms that indeed, there is no incentive for thebondholder to convert even if they were given such an opportunity. Instead, if wedefine another stopping time τ∗ = TNK/γ

, then since Kγ ≥ x⋆

ca,λ we have that

(5.5) τ∗ = TNK/γ≥ TNx⋆

ca,λ= σ∗.

The idea is to show in that (σ∗, τ∗) is a saddle point for J in (3.3)-(3.4).Theorem 5.2. Suppose that r > 0, q > 0 and c

r < K < γx⋆co,λ. Then, (σ∗, τ∗) =

(TNx⋆ca,λ

, TNK/γ) is a saddle point for Jx in (3.3)-(3.4), and the game value is given

by vλca(x) = vλf (x), with vλf (x) given in Lemma 4.5.

Proof. Recall the definitions of Jx and Jx from (5.1) and (5.2). We want to showthat (i)-(iii) of Corollary 2.4 are satisfied.

By Proposition 4.6, we know that x⋆ca,λ < K

γ . This means σ∗ ≤ τ∗. In turn, thedefinition of TNK/γ

implies that

Jx(σ∗, τ∗) =c

r+ E

x[

e−rσ∗

h (γXxτ∗)1τ∗=σ∗ + h (maxγXx

σ∗ ,K)1τ∗>σ∗

]

=c

r+ E

x[

e−rσ∗

h (γXxτ∗)1τ∗=σ∗ + h (K)1τ∗>σ∗

]

= Jx(σ∗, τ∗).

Moreover, integration by parts yields

Jx(σ∗, τ∗) =c

r+ E

x[

e−rσ∗

(

maxγXxσ∗ ,K −

c

r

)]

= Ex

[

∫ σ∗

0

e−rucdu+ e−rσ∗

maxγXxσ∗ ,K

]

= vλf (x),(5.6)

where we used Lemma 4.5 in the last equality. This shows condition (i) and thatvλf (x) is the game value.

To verify condition (ii), note that, for any τ ∈ R(λ),

Jx(σ∗, τ)

=c

r+ E

x[

e−r(σ∗∧τ)(

γXxτ 1τ≤σ∗ +maxK, γXx

σ∗1σ∗<τ −c

r

)]

=c

r+ E

x[

e−r(σ∗∧τ)(

γXxτ 1τ<σ∗ + γXx

τ 1τ=σ∗ +maxK, γXxσ∗1σ∗<τ −

c

r

)]

.

Using Xxτ < x⋆

ca,λ on the event τ < σ∗, whence γXxτ ≤ vλf (X

xτ ) on τ < σ∗ by

Proposition 4.7, we obtain

Jx(σ∗, τ)

≤c

r+ E

x[

e−rτh(

vλf (Xxτ ))

1τ<σ∗ + e−rσ∗ (

h(γXxσ∗)1τ=σ∗ + h(maxK, γXx

σ∗)1σ∗<τ

)

]

≤c

r+ E

x[

e−rτh(

vλf (Xxτ ))

1τ<σ∗ + e−rσ∗

h (maxγXxσ∗ ,K)1σ∗≤τ

]

= Ex

[

∫ τ∧σ∗

0

e−ruc du+ e−rτvλf (Xxτ )1τ<σ∗ + e−rσ∗

maxγXxσ∗ ,K1σ∗=τ,

]

= vλf (x),

where the last equality follows from Proposition 4.8 with η = σ∗ ∧ τ . Then

Jx(σ∗, τ) ≤ vλf (x) = J(σ∗, τ∗) = J(σ∗, τ∗),


where we used (5.6) in the second last equality.Next we prove (iii). Let σ ∈ R(λ). With σ := σ ∧ τ∗, and using γXx

τ∗ ≥ K sothat γXx

σ1σ=τ∗ +K1σ<τ∗ = maxγXxσ ,K, we have

Jx(σ, τ∗) =c

r+Ex

[

e−r(σ∧τ∗)(

γXxτ∗1σ≥τ∗ +K1σ<τ∗ −

c

r

)]

=c

r+Ex

[

e−rσ(

γXxσ1σ=τ∗ +K1σ<τ∗ −

c

r

)]

=c

r+Ex

[

e−rσ(

maxγXxσ ,K −

c

r

)]

.

= Ex

[

∫ σ

0

e−rucdu+ e−rσ maxγXxσ ,K

]

.

By (4.10) and Lemma 4.5, we further have

Jx(σ, τ∗) ≥ infη∈R(λ),η≤TNK/γ

Ex

[∫ η

0

e−rucdu + e−rη maxγXxη ,K

]

= vλf (x) = Jx(σ∗, τ∗),

where we used (5.6) in the last equality. This completes the proof.


Bon

d P

rice

K

x

Vca

Fig. 5.2. The convertible bond value vλca for Case cr< K < γx⋆

co,λ.

5.3. Case K ≥ γx⋆co,λ. In this case, since the surrender price is very high, the

firm will postpone their call time to avoid payingK unless the stock price is very high.On the other hand, in the region where the firm does want to call, the bondholderalready wants to convert, and therefore, the callable feature has no impact. Thismeans that the game problem for a callable convertible bond reduces to an optimalstopping problem without the callable feature, i.e. the first auxiliary problem westudied in Section 4.

Recall that in Section 4.1 we defined the (optimal) stopping time τ∗ = TNx⋆co,λ

.

Now define another stopping time σ∗ = TNz where the level z is such that vλco(z) = K.Since K

γ ≥ x⋆co,λ,

Kγ is in the stopping region, meaning that vλco(

Kγ ) ≤ γK

γ = K. In

turn, z ≥ Kγ ≥ x⋆

co,λ, and

(5.7) τ∗ = TNx⋆co,λ

≤ TNz = σ∗.


Theorem 5.3. Suppose that r > 0, q > 0 and K ≥ γx⋆co,λ. Let z be such that

vλco(z) = K. Then, (σ∗, τ∗) = (TNz , TNx⋆co,λ

) is a saddle point for J in (3.3)-(3.4),

and the game value is given by vλca(x) = vλco(x), with vλco(x) given in Lemma 4.2.

Proof. By (5.7), we know that τ∗ ≤ σ∗. In turn, Lemma 4.2 implies that

Jx(σ∗, τ∗) = Ex

[

∫ τ∗

0

e−rucdu+ e−rτ∗

γXxτ∗

]

= Jx(σ∗, τ∗) = vλco(x).(5.8)

This shows condition (i) and that vλco(x) is the game value.

Next, we verify condition (ii). Since Xσ∗ is in the stopping region of the optimalstopping problem (4.1), it follows that K = vλco(z) ≤ vλco(Xσ∗) ≤ γXσ∗ . Hence, forany τ ∈ R(λ),

Jx(σ∗, τ) =c

r+ E

x[

e−rτ(

γXxτ −

c

r

)

1τ≤σ∗ + e−rσ∗

(

maxK, γXxσ∗ −

c

r

)

1τ>σ∗

]

≤c

r+ E

x[

e−rτ(

γXxτ −

c

r

)

1τ≤σ∗ + e−rσ∗

(

γXxσ∗ −

c

r

)

1τ>σ∗

]

=c

r+ E

x[

e−r(τ∧σ∗)(

γXxτ∧σ∗ −

c

r

)]

= Ex

[

∫ τ∧σ∗

0

e−rucdu+ e−r(τ∧σ∗)γXxτ∧σ∗

]

.

It follows that

Jx(σ∗, τ) ≤ supη≤R(λ)

Ex

[∫ η

0

e−rucdu+ e−rηγXxη

]

= vλco(x) = Jx(σ∗, τ∗) = Jx(σ∗, τ∗),

where we used (5.8) in the last two equalities.

Finally, using Xxσ < x⋆

co,λ on the event σ < τ∗, whence K ≥ vλco(Xxσ) on

σ < τ∗ by Corollary 4.3, we have

Jx(σ, τ∗) =c

r+ E

x[

e−r(τ∗∧σ)(

γXxτ∗1τ∗≤σ +K1σ<τ∗ −

c

r

)]

≥c

r+ E

x[

e−r(τ∗∧σ)(

γXxτ∗1τ∗≤σ + vλco(X

xσ)1σ<τ∗ −

c

r

)]

= Ex

[

∫ τ∗∧σ

0

e−rucdu+ e−r(τ∗∧σ)(

γXxτ∗1τ∗≤σ + vλco(X

xσ )1σ<τ∗

)

]

for any σ ∈ R(λ). Then, using the dynamic programming equation (4.9) in Proposi-tion 4.4 with η = σ ∧ τ∗, we further have

Jx(σ, τ∗) ≥ vλco(x) = Jx(σ∗, τ∗)

where we used (5.8) in the last equality, which verifies condition (iii), and we conclude.



Bon

d P

rice

K

x

Vca

Fig. 5.3. The convertible bond value vλca for Case K ≥ γx⋆co,λ.

6. Comparison with convertible bonds with forced conversion/no liq-uidity constraints. In this section, we compare our results with two situations. Thefirst one is with a forced conversion condition, and the second one is without liquidityconstraints.

6.1. Comparison with convertible bonds with forced conversion. In [22],a forced conversion condition is introduced, i.e. the firm will force a conversion of theconvertible bond to γ shares of stock at the earliest Poisson arrival time after thestock price exceeds a predetermined threshold K

γ . By introducing the triggering and

conversion times HK/γ := infu ≥ 0 : Xxu ≥ K

γ and TM := infTN ≥ HK/γ : N ≥ 1,the expected discounted payoff of the convertible bond at HK/γ can be calculated as

Lλ(XxHK/γ

) = EXx

HK/γ

[

∫ TM

HK/γ

e−r(u−HK/γ)cdu+ e−r(TM−HK/γ)γXxTM

]

=c

r + λ+

λ

q + λγXx

HK/γ.

By introducing such a forced conversion condition, it is sufficient to solve the problemfor t ∈ [0, HK/γ ], so the state space reduces from x ∈ (0,∞) to x ∈ (0, Kγ ] for which

the upper payoff U := K always dominates the lower payoff L := γx (so the existencetheorem of constrained Dynkin games in [18] applies). See Chapter 2.5 in [22] forfurther details.

Although the forced conversion guarantees the order condition L ≤ U , its draw-back is significant: there is a possibility that the stock price Xx

TMdrops below any

small number ε > 0 when the forced conversion is taking place, i.e. Px(XxTM

< ε) > 0.Neither the firm nor the bondholder have incentive to stop in such a situation. Inparticular, the problem with forced conversion studied in [18] and [22] is not the truecallable convertible bond problem as studied in this paper.

In the following table, we make a comparison between our results and the con-vertible bond with forced conversion in [22]. Note that ‘small’, ‘moderate’ and ‘large’values of K refer to different reference points in the two situations. They are c

r and

γx⋆co,λ without forced conversion, and c

r and c(λ+q)q(λ+r) with forced conversion.


Table 6.1

Comparison of convertible bonds with/without forced conversion

Without forced conversion With forced conversion

State space (0,∞) (0, Kγ]

Small K LλV + c+ λmaxγx,K = 0 LλV + c+ λK = 0Call precedes conversion Call precedes conversion

Moderate K LλV + c+ λ(minV,K1x<K/γ + γx1x≥K/γ) = 0 LλV + c+ λminV,K = 0Call precedes conversion Call precedes conversion

Large K LλV + c+ λmaxV, γx = 0 LλV + c+ λmaxV, γx = 0Conversion precedes call Conversion precedes call

6.2. Comparison with convertible bonds without liquidity constraints.Finally, we investigate the asymptotic behavior of the optimal stopping strategiesand the convertible bond values when λ → ∞. Intuitively, they will converge to theircounterparts without liquidity constraints. Let us first recall a result from [24] (seealso Proposition 7.1 in [18]) about the convertible bond without liquidity constrains.

Recall that α > 1 is the positive root of Q(z) = 0 with Q(z) = 12σ

2z2 + (r − q −12σ

2)z − r and Hy := inft ≥ 0 : Xxt ≥ y.

Proposition 6.1. Suppose that r > 0, q > 0 and that the admissible stoppingtimes τ (of the firm) and σ (of the bondholder) are chosen as FW -stopping times. Let

(6.1) x⋆co :=

(

α

α− 1

)

c

γr.

Then,(i) Case K ≤ c

r : (σ∗, τ∗) = (0, HK/γ) and the convertible bond value is given asvca(x) = maxK, γx.

(ii) Case cr < K < γx⋆

co: (σ∗, τ∗) = (HK/γ , HK/γ) and the convertible bond value

is given as

vca(x) =

cr +

(

γxK

)α (K − c

r

)

, x < Kγ ;

γx, x ≥ Kγ .

(iii) Case K ≥ γx⋆co: (σ∗, τ∗) = (HK/γ , Hx⋆

co) and the convertible bond value is

given as

vca(x) =

cr +

(

xx⋆co

)α(

γx⋆co −

cr

)

, x < x⋆co;

γx, x ≥ x⋆co.

We conclude the paper with an asymptotic analysis for the case where the Poissonintensity λ increases to ∞.

Proposition 6.2. When λ → ∞, x⋆co,λ → x⋆

co, x⋆ca,λ → K

γ , and vλca(x) → vca(x).Hence, the convertible bond with liquidity constraints will converge to its counterpartwithout liquidity constraints when λ → ∞.

Proof. We prove the claims using Theorems 5.1-5.3 and Proportion 6.1. Fromthe definitions of αλ and βλ as roots of quadratics we have that as λ ↑ ∞, αλ → ∞,


αλ

λ → 0, βλ → −∞ and βλ

λ → 0. Moreover, from the expression for x⋆co,λ in (4.3) and

(6.1) we have x⋆co,λ → x⋆

co, and from Lemma 4.1, x⋆co,λ < x⋆

co.For Case K ≤ c

r , from the properties of αλ and βλ above we have A,B → 0.Hence, vλca(x) → K for x ≤ K/γ and vλca(x) → γx for x > K/γ, which means

vλca(x) → maxK, γx = vca(x).

For Case cr < K < x⋆

co we have that K < x⋆co,λ for large enough λ. Then by

the expressions for x⋆co,λ and A,B,C in (4.13) — (4.16) we have x⋆

ca,λ → Kγ and

A,B,C → 0. Hence, from the expression for vλf (x) in (4.12),

vλca(x) = vλf (x) → vca(x).

For case K ≥ x⋆co,λ, the expression for vλco(x) in (4.4) implies that

vλca(x) = vλco(x) → vca(x),

and we conclude.

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abstract. arxiv:2111.02554v1 [q-fin.mf] 3 nov 2021

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