NANZAN-TR-2004-03

The Valuation and Optimal Strategies of Callable
Convertible Bonds
Kyoko Yagi and Katsushige Sawaki

September 2004

Technical Report of the Nanzan Academic Society
Mathematical Sciences and Information Engineering

The Valuation and Optimal Strategies of Callable
Convertible Bonds
Kyoko Yagi∗ and Katsushige Sawaki †Nanzan University(27 Seirei, Seto, Aichi, 489-0863, Japan)

Abstract
This paper presents a simple pricing model for valuing callable convertible bonds under
the setting of a coupled stopping game between the issuer (firm) and the investor (holder).
We use the model to provide the valuation formula for such bonds and to explore someanalytical properties of optimal conversion and call strategies as well as several numericalresults by using the finite element method. Furthermore, optimal critical prices for calland conversion are examined numerically.
Keywords : Asset pricing, callable discount, convertible bonds, optimal stopping time,
optimal boundaries
AMS subject Classification : 91B28

1

Introduction

Consider a zero-coupon callable convertible bond (abbreviated by CB) which permits aninvestor to exchange the bond for a certain number of the underlying stock at any time of hischoosing and gives the issuer the right to purchase back the bond at any time for a specificamount. Since the convertible bond with such a call feature should clearly be evaluated lessthan the convertible bond without that feature, it is an attractive and complex security.
In this paper we present a pricing model for callable CBs where the underlying stock
pays dividends. In section 2 we formulate the valuation of the callable CB under the settingof a coupled stopping game based upon the Kifer’s game option model [8]. In section 3 wediscuss optimal call and convertible policies for the issuer and the investor, respectively. It
will be shown that the CB with the call feature is worth less than the bond without andthat the conversion feature increases the value of the CB to the investor. In section 4 weshow that the value of the callable CB can be decomposed into four terms of the bond price,
the value of the European call option with the face value as the exercise price, the early
conversion premium and the callable discounted value. Section 5 provides some numericalresults to recognize some analytical properties of the value of the callable and non-callable∗†

d04mm003@nanzan-u.ac.jp
sawaki@nanzan-u.ac.jp1

CBs by using the finite element approach. Finally, in section 6 we summarize the paper withsome concluding remarks.
Brennan and Schwartz [1] provides the groundwork for the pricing model of convertible
bond together with Ingersoll [5]. On the other hand, Kifer[8] paves a way of the pricing
model of American options with the call feature. McConell and Schwartz [10] provides andanalyzes a typical example, LYON, of the callable American option. Seko and Sawaki [12]considers the valuation of the callable American put and call options following the Kifer’s
approach. Myneni [11] and Bunch and Johnson [2] study the pricing model for American
options. Longstaff and Schwartz [9] presents a new computational approach based upon
the least-squares simulation for callable option. Kariya and Tsuda [7] proposes a statisticalmodel for valuing the CB. Yagi and Sawaki [13] studies a simple binomial model for valuing
the convertible bond but analyzes neither the optimal boundaries nor the decomposition of
the CB price. The inspiration behind our work was the work of [1], [8] and [12]. However,these ideas cannot be transfered directly but must be revised to the callable CB because thecall feature does not rise the penalty lost in CBs. So far, there are not so many analytical
papers which analyze the value and optimal boundaries of callable CBs although there aremany empirical studies on CBs, [3] and [4].

2

The Pricing Model for Callable Convertible Bonds

Consider a firm financing stocks and convertible bonds in the capital market. The value ofthe firm at time t Vt is given by
Vt ≡ CB(t, Vt ) + mStb

(1)

where CB(t, Vt ) is the total value of the CB with the face value F and the maturity T , andStb the stock value with the number of shares m before conversion. Let Sta be the stock value
after conversion and n the number of stocks converted from the CB. Then, equation (1) canbe rewritten as
Vt = (n + m)Sta .
(2)
Letting z = n/(n + m) as the dilution ratio, the conversion value at time t CV (t, Vt ) is givenbyCV (t, Vt ) = nSta = zVt .
(3)
Denote CP (t) = F the call price when the firm calls the CB at time t. Then the firm pays
the investor max(zVt , F ).
A riskless asset price Bt with the interest rate r is given by
dBt = rBt dt, B0 > 0, r > 0.

(4)

Let Zt be a standard Brownian motion defined on a probability space (Ω, F, {Ft }, P ). Define
the risk neutral probability measure P̃ which is equivalent to P as()µ¶
1 µ−r 2dP̃ ¯¯µ−rt−Z(t) .
(5)¯ = exp −
dP Ft
2
κ
κ
2

Then, the stochastic differential equation of the firm value can be written as the well known
geometric Brownian motiondVt = (r − δ)Vt dt + κVt dZ̃t(6)where δ is the constant rate of dividend payments for the firm value, κ the volatility and Z̃tthe standard Brownian motion defined on the probability space (Ω, F, {Ft }, P̃ ).
The CB holder as the investor should follow an optimal conversion policy so as to maximize
the value of the CB. In other words, if the value of the CB exceeds its conversion value givenby (3), it is optimal for the investor not to convert. On the other hand, since the issuer seeks
to minimize the value of the CB, it is optimal for the issuer not to call the CB when thegreater of call price or the conversion value is greater than the value of the CB. It follows
from these arguments that the value of the callable CB must satisfy the inequalities
zVt ≤ CB(t, Vt ) ≤ max(zVt , F )(7)which provides the lower and upper bounds of the CB value and allows the firm to declare
the bankruptcy in the maximum of zVt and F .
Let σ be a call time (a stopping time) by the issuer and τ a conversion time (a stoppingtime) by the investor. Denote Tt,T the set of stopping times with respect to the filtration{Ft ; 0 ≤ t ≤ T }. The payoff function at the stopping time σ ∧ τ is given byR(σ, τ ) = max(zVσ , F )1{σ<τ } + zVτ 1{τ ≤σ}(8)where 1{σ<τ } = 1 if the issuer stops first and 1{τ ≤σ} = 1 if the investor stops first, and theissuer pledges to pays the investor at any stopping time. At the time of σ = τ the investor’sright dominates the issuer’s one. At the maturity T the value of the callable CB is the smallerof the value of the firm or the maximum between the conversion value and the face value of
the CB, that is,
CB(T, VT ) = min(VT , max(zVT , F )).
(9)
After a minor revision , we can use the following theorem given by Kifer [8] which paved
a way of the pricing model for a game option. We use the model to value a callable CB as a
special case.
Theorem 1. Definehi¡¢
Jtv (σ, τ ) = Ẽ e−r(σ∧τ ∧T −t) R(σ, τ ) + min(VT , max(zVT , F ))1{σ=T,τ =T } |Vt = v .

(10)

Suppose that for all v > 0
"
E

#
¯
¯
sup e−rτ zVτ ¯ V0 = v < ∞.

0≤τ ≤T

There is a unique no-arbitrage price process CB(t, v) of the game option in the sense ofBlack-Scholes model such that CB(t, v) can be represented by the right continuous process
CB(t, v) ==inf

sup Jtv (σ, τ )supinf Jtv (σ, τ ).

σ∈Tt,T τ ∈Tt,Tτ ∈Tt,T σ∈Tt,T

3

(11)

Moreover, the optimal stopping times σ̂t and τ̂t for the issuer and the investor, respectively,
are given byσ̂t = inf {σ ∈ [t, T ) | CB(σ, Vσ ) = max(zVσ , F ) } ,τ̂t = inf {τ ∈ [t, T ) | CB(τ, Vτ ) = zVτ } .

(12)Furthermore, there exists a self-financing portfolio π ∗ such that (σ̂, π ∗ ) is a hedge against
the callable contingent claim with initial capital J0v (σ̂, τ ) = CB(0, v) for all τ and with P̃ probability such portfolio π ∗ is unique.
Proof should be referred to Kifer [8] after putting Yt = zVt as the payoff function of theinvestor.

3

Optimal Call and Conversion Policies and Their Optimal
Boundaries to Value the Callable CB

To value the callable CB we must identify the call and conversion policies for the firm andthe investor to follow. In according to the spirit of Brennan Schwartz [1] we assume that it is
optimal for the firm to minimize the value of CB and for the investor to maximize the valueof the CB at each stopping time.
Optimal call policy The firm will never call the CB when its value is less than the greaterof the call price or its conversion value because the firm wishes to minimize the valueof the CB. Under such an optimal call policy the following condition holds;
CB(t, Vt ) ≤ max(zVt , F ), at all t, Vt
Optimal conversion policy The investor will never convert the CB when its value isgreater than the conversion value because the investor seeks to maximize the valueof the CB. Under an optimal conversion policy we have
CB(t, Vt ) ≥ zVt .
Define the stopping region S f for the firm, S i for the investor and the continuation region C
for the both of them as follows;Sf= {(t, v) | CB(t, v) = max(zv, F )}S i = {(t, v) | CB(t, v) = zv}(13)

C = {(t, v) | zv < CB(t, v) < max(zv, F )}On the other hand, defining the value of non-callable CB by CB(t, v), the stopping region S
and the continuation region C for the investor isSi= {(t, v) | CB(t, v) = zv}

C = {(t, v) | CB(t, v) > zv}
4ibecause the non-callable CB allows only the investor to exercise the right for the conversion.
For each t define
Stf = {v| (t, v) ∈ S f }, Sti = {v| (t, v) ∈ S i }, Ct = {v| (t, v) ∈ C}
and similarlyii

S t = {v| (t, v) ∈ S }, C t = {v| (t, v) ∈ C}.
Proposition 2. The value of callable CB is less than or equal to the value of non-callable,that is,
CB(t, v) ≤ CB(t, v) f or all t and v(14)
and moreover,iSti ⊇ S t f or all t.

(15)

The proof immediately follows from Theorem 1 because only the investor seeks to maximize the value of CB in equation (11). Equation (15) follows from equation (14).
The optimal call boundary for the firm can be defined as the graph of vtf ≡ inf{v| v ∈ Stf }.
Similarly, the optimal conversion boundary for the investor is the graph of vti ≡ inf{v| v ∈ Sti },t ∈ [0, T ].
Set vt∗ = min(vtf , vti ) and similarly v it the optimal conversion boundary for the non-callable
CB.
Theorem 3. The following relationships hold between the callable and non-callable CB,(i) Stf = [vtf , ∞), Sti = [vti , ∞), Ct = [0, vt∗ )i(ii) S t = [v it , ∞), C t = [0, v it )(iii) vti ≤ v it and vtf ≤ F/zi

Proof of (i) and (ii) follows from the definitions of Stf , Sti and Ct , S t and C t . Property (iii)
can be proved from Proposition 2. Note that the optimal call and conversion policies depend
upon the conversion ratio, the value of the CB and the time remained before the maturity.

4

A Decomposition of the Value of the Convertible Bond

Ingersoll [5] has shown that it is optimal for the investor not to convert the non-callable CB
before the maturity when the stock pays no dividend, and hence the value of the non-callable
CB can be presented in term of the sum of the bond price and the price of the European
call option when the CB pays no coupon. This means that when the stock pays a constantdividend, the value of the non-callable CB can be rewritten as the sum of the bond price andthe price of American call option.

5

Theorem 4. Let B(t, v) be the bond price when the value of firm is v at time t and C theprice of the American call option. Then, the value of the non-callable CB can be decomposed
as follows;
CB(t, v) = B(t, v) + C(t, zv; B(t, v))= B(t, v) + Ce (t, zv; F ) + p(t, v)(16)

where Ce (t, zv; F ) is the European call option when the exercise price is the face value withthe underwriting asset value zv and p(t, v) is the premium of the early conversion given by
·Z T
¸
¯
¯
−r(s−t)p(t, v) = Ẽ(17)eδzVs 1{vis ≤Vs } ds¯Vt = v .
t

Proof. It is well known from Ingersoll [5] that we have
CB(t, v) = B(t, v) + C(t, zv).
Putting f (t, v) = e−rt C(t, zv) and applying the Ito’s lemma we obtain
¶
Z T
Z Tµ
∂f
1 2 2 ∂2f∂f
f (T, VT ) = f (t, v) +
dVs +
κ Vs
ds.
+
2
∂v 2
∂s
t ∂v
t

(18)

Therefore,
e−rT C(T, zVT )
=e

−rt

Z T
e

C(t, zv) +t−rs ∂C

∂v

µ

Z T
dVs +

e

−rs

t

1 2 2 ∂2C∂C
κ Vs
− rC +
2
∂v 2
∂s

¶
ds

Z T
∂C
= e−rt C(t, zv) +e−rs κVs
dZ̃s
∂v
t
µ
¶
Z T
2
∂C
∂C
−rs 1 2 2 ∂ C+eκ Vs+ (r − δ)Vs
− rC +
ds.
2
∂v 2
∂v
∂s
t
Note that1 2 2 ∂2C∂C∂Cκ v
+ (r − δ)v− rC +
2
2
∂v
∂v
∂t

(t, v) ∈ C
 0,
µ
¶
=
1 2 2 ∂2B
∂B
∂Bi
 −δzv −κ v+ (r − δ)v− rB +, (t, v) ∈ S
2
2
∂v
∂v
∂t
and

1 2 2 ∂2B
∂B∂B
κ v
+ (r − δ)v− rB +
= 0.
2
2
∂v
∂v
∂t
Equation (19) can be rearranged as
Z T
Z T
∂C
−rT
−rt
−rs
e
C(T, zVT ) = e C(t, zv) +e κVs
dZ̃s −
e−rs δzVs 1{vis ≤Vs } ds.
∂vtt
After multiplying by ert and taking the expectation conditioning on Vt = v, we have
¸
·Z T
¯
¯
−r(T −t)−r(s−t)Ẽ[eC(T, zVT ) | Vt = v] = C(t, zv) − ẼeδzVs 1{vis ≤Vs } ds ¯ Vt = v
t

6

(19)

Next, we present the decomposition of the value of the callable CB. Let C(t, v) be thevalue of conversion option which is the difference between the value of the callable CB andthe bond price.
Theorem 5. The value of the callable CB can be represented by the following decomposition;
CB(t, v) = B(t, v) + C(t, v)= B(t, v) + Ce (t, zv; F ) + p(t, v) − d(t, v),
where

·Z T
p(t, v) = Ẽe−r(s−t)

t

and

·Z T
d(t, v) = Ẽµ−r(s−t)

e
t

(20)

¸
¯
¯
δzVs 1{vs∗ ≤Vs } ds¯Vt = v

(21)

¶
¸
¯
∂C
∂C
∗+
∗−
v ∗ ¯
(s, vs ) −(s, vs ) dLs (vs )¯Vt = v .
∂v
∂v

(22)

where we can call p(t, v) the early conversion premium and d(t, v) the callable discounted
value, respectively, and Lvt (vt∗ ) the local time of Vt at the level vt∗ in the time interval [0, t].
Proof. Put g(t, v) = e−rt C(t, v) and applying the generalized Ito’s lemma for convex
functions [6]. Then we obtain
Z T

Z Tµ

1 2 2 ∂ 2 g ∂gκ Vs+
g(T, VT ) = g(t, v) +
2
∂v 2
∂s
t
t
¶
Z Tµ
∂g
∂g
(s, vs∗+ ) −(s, vs∗− ) dLvs (vs∗ ).
+
∂v
∂v
t
∂g
dVs +
∂v

¶
ds
(23)

Note from the free boundary conditions that1 2 2 ∂2C
∂C∂C
κ v
− rC +=+ (r − δ)v
2
2
∂v
∂v
∂t

(

0,
(t, v) ∈ C−δzv, (t, v) ∈ S i ∪ S f .

(24)

Therefore, we have
Z T
Z T
∂C
e−rT C(T, VT ) = e−rt C(t, v) +e−rs κVs
dZ̃s −
e−rs δzVs 1{vs∗ ≤Vs } ds
∂v
t
t
µ
¶
Z T
∂C
∂C
−rs
∗+
∗−
+
e
(s, vs ) −(s, vs ) dLvs (vs∗ ).
∂v∂vtMultiplying by ert and taking the expectation conditioning on Vt = v, we obtain
·Z T
¸
¯
¯
−r(T −t)−r(s−t)Ẽ[eC(T, VT ) | Vt = v] = C(t, v) − ẼeδzVs 1{vs∗ ≤Vs } ds ¯ Vt = v
t
¶
¸
·Z T
µ
∂C
∂C
(s, vs∗+ ) −(s, vs∗− ) dLvs (vs∗ )+ Ẽ
e−r(s−t)
∂v
∂v
t
(25)
which can be rewritten as in equation (20).
7

Corollary 6. The early conversion premium for the callable CB is less than or equal tothe one for the non-callable CB, that is, p(t, v) ≥ p(t, v) for all t, v and hence d(t, v) ≥ 0 for
all t, v.
Proof. From definition of vt∗ and Theorem 3 (iii) we have vt∗ ≤ v it for all t. Then,from equation (17) and (21) we obtain p(t, v) ≥ p(t, v) for all t, v because the conditional
expectations should be taken over the time interval [t, T ].

5

Numerical Examples

In this section we present the numerical valuation of the callable and non-callable CBs, andthe optimal call and conversion boundaries through a simple finite element method. Table 1
shows the data we use to evaluate the values of the CB.
Table 1: Data
Face value F

100

Maturity T

3Conversion ratio z

0.4

Interest rate r

0.05

Dividend rate δ

0.02

Volatility κ

0.3

Figure 1 illustrates the values of the callable and non-callable CB and the bond prices
as functions of v. We may identify the relationship between the callable and non-callable
CB given by equation (14). Figure 2 demonstrates the differences between the callable andnon-callable CB prices which present the callable discounted values. Table 2 shows the valuesof the callable and non-callable bonds and the bond for the various v, T , z, r, δ, κ. Figure
3 shows the behavior of the optimal conversion and call boundaries for the non-callable andthe callable CBs, respectively, for the different levels of t.

8

CB
200

firm value

conversion value

150

face value
100

non-callable convertible bondcallable convertible bond
bond

50

0
50

100

150

200

v* =F/z

300

350

400

450

500

v

400

450

500

v

Figure 1: The value of CB

C100

non-callable convertible bondcallable convertible bond

80

60

40

20

0
50

100

150

B(0,v)/z

v*

300

350

Figure 2: The value of the conversion option

9

Table 2: CB: callable convertible bond, CB: non-callable convertible bind, B: bond
v
50
100
150
200
250
300
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100

T
3
3
3
3
3
3
1
3
5
7
10
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3

z
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.2
0.4
0.6
0.8
1.0
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4

r
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.01
0.05
0.1
0.2
0.3
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05

δ
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.01
0.02
0.03
0.04
0.05
0.02
0.02
0.02
0.02
0.02

κ
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.1
0.2
0.3
0.4
0.5

CB
45.1250
71.6472
82.981690.8296
100.0000
120.0000
85.001371.6472
62.731056.2548
49.392071.3818
71.6472
73.7240
81.8913100.0000
76.0225
71.6472
65.8969
54.8890
46.5636
72.5999
71.6472
70.6660
69.6573
68.622583.0857
77.3445
71.6472
66.2945
61.4190CB
45.1250
72.078485.2760
96.3348109.1865
124.3606
85.008172.0784
63.770957.6278
50.7265
71.3818
72.0784
75.6884
83.5720100.0000
76.4018
72.0784
66.369455.3233
46.828573.106172.0784
71.0328
69.9691
68.8870
83.0857
77.376172.0784
67.6892
64.010210

B
45.122271.379080.9672
84.2080
85.345485.7698
84.999971.379061.529453.580543.9595
71.3790
71.3790
71.3790
71.3790
71.3790
75.9804
71.379065.126452.055439.954772.3026
71.3790
70.4244
69.4401
68.4276
83.0857
77.3195
71.379065.495159.7349

CB − B
0.00280.26832.01446.621614.6546
34.23020.00150.26831.20162.67435.43250.00280.26832.345010.5123
28.62100.04210.26830.77052.83366.60890.29730.26830.24160.21720.19490.00000.02510.26830.79941.6842CB − B
0.00280.69944.308812.1267
23.841138.5908
0.00820.69942.24154.04736.76700.00280.69944.309412.1930
28.62100.42140.69941.24303.26796.87380.80350.69940.60850.52900.45950.00000.05670.69942.19414.2753CB − CB
0.00000.43112.29445.50529.18654.36060.00670.43111.03991.37301.33450.00000.4311
1.96441.68070.00000.37930.43110.47260.43430.26480.50620.43110.36690.31180.26450.00000.0316
0.43111.39472.5912

v
500
450
400
350
300
250
200
150

optimal conversion boundary (non-callable convertible bond) -v i
vf
optimal call boundary (callable convertible bond)

100
50
0
0

0.5

1

1.5

2

2.5

3 t

Figure 3: Optimal conversion and call boundaries of the CB

6

Summary and Concluding Remarks

In this paper we have studied the pricing of the callable convertible bonds and the optimal calland conversion policies of the callable CB by means of a coupled stopping game between theissuer and the investor. We have shown that the value of the callable CB can be decomposed
into the sum of the bond price, the value of the European call and the difference between theearly conversion premium and the callable discounted value. Furthermore, we have embodied
the optimal conversion and call boundaries numerically computed by the finite differential
method.
We assume that the bond has no coupon payment beside the stock pays a constantdividend. Further investigation is left for future research when the bond pays coupons to thebond holder because the bond usually pays coupons. It is interesting to investigate how the
coupon payments affect the value and optimal boundaries of the callable CB. What happens
to the value of the CB if the call time is restricted ? For example, the firm is allowed tocall only in some certain time interval. Considering these questions leads more practicalevaluation of existing financial commodities traded in the actual market.

References[1] M. J. Brennan and E. S. Schwartz, Convertible bonds: Valuation and optimal strategies
for call and conversion, The Journal of Finance. 32 (1977) 1699-1715.
[2] D. S. Bunch and H. E. Johnson, The American put option and its critical stock price,
Journal of Finance. 55 (2000) 2333-2356.

11

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