Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 309678, 21 pages
doi:10.1155/2011/309678
Research Article
A Variational Inequality from Pricing
Convertible Bond
Huiwen Yan and Fahuai Yi
School of Mathematics, South China Normal University, Guangzhou 510631, China
Correspondence should be addressed to Fahuai Yi, [email protected]
Received 30 December 2010; Accepted 11 February 2011
Academic Editor: Jin Liang
Copyright q 2011 H. Yan and F. Yi . This is an open a ccess article d istributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The model of pricing American-style convertible bond is formulated as a zero-sum Dynkin game,
which can be transformed into a parabolic variational inequality PVI. The fundamental variable
in this model is the stock price of the firm which issued the bond, and the differential operator in
PVI is linear. The optimal call and conversion strategies correspond to t he free boundaries of PVI.
Some properties of the free boundaries are studied in this paper. We show that the bo ndholder
should convert the bond if and only if the price of the stock is equal to a fixed value, and the firm
should call the bond back if and only if the price is equal to a strictly decreasing function of time.
Moreover, we prove that the free boundaries are smooth and bounded. Eventually we give some
numerical results.
1. Introduction
Firms raise capital by issuing debt bonds and equity shares of stock. The convertible bond
is intermediate between these two instruments, which entitles its owner to receive coupons
plus the return of principle at maturity. However, prior to maturity, the holder may convert
the bond into the stock of the firm, surrendering it for a preset number of shares of stock. On
the other hand, prior to maturity, the firm may call the bond forcing the bondholder to either
surrender it to the firm for a previously agreed price or convert it into stock as before.
∈ D
T
Δ
0,
K
γ
×
0,T
,
−∂
t
V −L
0
V ≤ c, if V K,
S, t
∈ D
T
,
V
K
γ
SS
V
r − q
S∂
S
V − rV,
1.2
where r, σ,andq are positive constants and represent the risk-free interest rate, the volatility,
and the dividend rate of the firm stock, respectively. In this paper, we suppose that c>rK
and r ≥ q. From a financial point of view, the assumption provides a possibility of calling
the bond back from the firm see Section 2 or 2. Furthermore, we suppose that L ≤ K.
Otherwise, the firm should call the bond back before maturity and the value L makes no
sense see Section 2. It is clear that V K is the unique solution if L K.Soweonly
consider the problem in the case of L<K.
Since 1.1 is a degenerate backward problem, we transform it into a familiar forward
nondegenerate parabolic variational inequality problem; so letting
u
x, t
V
S, T − t
,x ln S − ln K ln γ, 1.3
Advances in Difference Equations 3
we have that
∂
K, 0 ≤ t ≤ T,
u
x, 0
max
{
L, Ke
x
}
,x≤ 0,
1.4
where
Lu
σ
2
2
∂
xx
u
r − q −
σ
2
2
∂
x
u − ru. 1.5
There are many papers on the convertible bond, such as 1, 2, 9.Butasweknow,there
∈ C
t
0
,T
0
∩ C
∞
t
0
,T
0
, lim
t → T
−
0
h
t
−∞,
1.6
and ht is strictly decreasing in t
0
,T
0
0
,T
0
∩ C
∞
t
0
,T
0
. As we know, the proof of the smoothness is trivial by the method
4AdvancesinDifference Equations
t
x
t
0
T
0
u = K
CT
Ke
x
<u<K
CL
u = Ke
x
CV
h(t)
Figure 1: The free boundary ht.
in 10 if the difference between u and the upper obstacle K is decreasing with respect to t.
But the proof is difficult if the condition is false see 11–14.Inthisproblem,∂
t of ht
and ht converges to −∞ as t converges to T
−
0
in Theorem 4.4.
In the last section, we provide numerical result applying the binomial method.
2. Formulation of the Model
In this section, we derive the mathematical model of pricing the convertible bond.
The firm issues the convertible bond, and the bondholder buys the bond. The firm has
an obligation to continuously serve the coupon payment to the bondholder at the rate of c.In
the life time of the bond, the bondholder has the right to convert it into the firm’s stock with
the conversion factor γ and obtains γS from the firm after converting, and the firm can call
it back at a preset price of K. The bondholder’s right is superior to the firm’s, which means
that the bondholder has the right t o convert thebond,butthefirmhasnorighttocallitif
both sides hope to exercise their rights at the same time. If neither the bondholder nor the
firm exercises their r ight before maturity, the bondholder must sell the bond to the firm at a
preset value L or convert i t into the firm’s stock at expiry date. So, the bondholder r eceives
max{L, γS} from the firm at maturity. It is reasonable that both of them wish to maximize the
values of their respective holdings.
Suppose that under the risk neutral probability space Ω, F,
; the stock price of the
firm S
s
follows
S
t,S
s
S
s
and volatility of the stock, respectively. W
t
is a standard Brown motion on the probability
space Ω, F,
.Usually,thedividendrateq is smaller than the risk free interest rate r.So,we
suppose that q ≤ r.
Advances in Difference Equations 5
Denote by F
t
the natural filtration generated by W
t
and augmented by all the -null
sets in F.LetU
t,T
be the set of all F
t
-stopping times taking values in t, T.
The model can be expressed as a zero-sum Dynkin game. The payoff of the bondholder
is
R
S, t; τ,θ
τ∧θ
t
ce
rt−ru
du e
S, t
Δ
ess sup
θ∈U
t,T
ess inf
τ∈U
t,T
R
S, t; τ, θ
|F
t
,
V
S, t
Δ
ess inf
τ∈U
t,T
ess sup
θ∈U
t,T
≤
R
S, t; τ
∗
,θ
∗
|F
t
≤
R
S, t; τ, θ
∗
|F
t
, ∀τ, θ ∈U
t,T
, 2.4
then the value of the Dynkin game exists and
V
S, t
I
{tT}
R
S, t; t, t
|F
t
,
R
S, t; t, θ
|F
t
KI
{t<θ}
γSI
{θt}
|F
t
I
{tT}
, ∀S ≥
K
γ
.
2.7
6AdvancesinDifference Equations
Inthecaseof0<S<K/γ, applying the standard method in 15, we see that the
strong solution of the following variational inequality is the value of the Dynkin game:
−∂
t
V −L
0
V c, if γS<V <K,
S, t
∈ D
T
,
−∂
t
V −L
0
V ≥ c, if V γS,
S, t
∈ D
γ
.
2.8
If L>K, then the firm is bound to call the bond back before the maturity because the firm
pays K after calling, but more than L without calling. In this case, the value L makes no sense.
So, we suppose that L ≤ K.
If c ≤ rK, then the firm is bound to abandon its call right. From a financial point of
view, the firm would pay K to the bondholder at time t after calling the bond, whereas, if the
firm does not call in the time interval t, t dt, then he would pay the coupon payment cdt
and at most K of the face value of the convertible bond at time t dt. So, the discounted value
of the bond without call is at most K cdt − rKdt ≤ K.Hence,thefirmshouldnotcallthe
bond back at time t.
From a stochastic point of view, we can denote a stopping time
τ
1
inf
t ≤ u ≤ T : γS
t,S
u
≥ K
. 2.9
If t<T,0<S<K/γ,then
τ
1
>t1, and, for any θ ∈U
t,T
,wehave
R
−
c
r
I
{τ
1
∧θT }
e
rt−rT
max
L, γS
t,S
T
−
c
r
≤
c
r
e
rt−rτ
1
∧θ∧T
K −
τ
1
I
{τt}
,θ
|F
t
, 2.11
which means that τ is not the optimal call strategy, and the firm should not call in the domain
{t<T,0 <S<K/γ}.
Advances in Difference Equations 7
From a variational inequality point of view, since
−∂
t
K −L
0
K rK > c, 2.12
provided that c<rK, which contradicts with the third inequality in 2.8,so,ifc<rK,then
V
/
K in the domain {t<T,0 <S<Kγ}.
To remain the call strategy, we suppose that c>rK. We will consider the other case in
another paper because the two problems are fully different.
Since we suppose that c>rKand r ≥ q,then
−∂
t
γS
θ
t
ce
rt−ru
du e
rt−rθ
γS
t,S
θ
I
{θ<T}
e
rt−rT
max
L, γS
t,S
T
I
{θT}
.
2.14
It is clear that
U
S, t
is the conversion domain in the model without call and CV is that in this paper.
3. The Existence and Uniqueness of W
2,1
p,loc
Solution of Problem 1.4
Since problem 1.4 lies in the unbounded domain Ω
T
, we need the following problem in the
bounded domain Ω
n
T
Δ
−n, 0 × 0,T to approximate to problem 1.4:
∂
t
u
n
−Lu
n
c, if u
n
<K,
x, t
∈ Ω
n
T
,
∂
max
{
L, Ke
x
}
, −n ≤ x ≤ 0,
3.1
where n ∈ IN
and n>ln K − ln L.
8AdvancesinDifference Equations
Following the idea in 10, 16, we construct a penalty function β
ε
ssee Figure 2,
which satisfies
ε>0 and small enough,β
ε
s
∈ C
∞
−∞, ∞
,
β
ε
ε → 0
β
ε
s
⎧
⎨
⎩
0,s<0,
∞,s>0.
3.2
Consider the following penalty problem of 3.1:
∂
t
u
ε,n
−Lu
ε,n
β
ε
u
ε,n
− K
c, in Ω
n
T
} is not smooth. It
satisfies see Figure 3
π
ε
s
⎧
⎨
⎩
s, s ≥ ε,
0,s≤−ε,
π
ε
s
∈ C
∞
IR
,π
ε
s
≥ s, 0 ≤ π
∩ CΩ
n
T
for
any 1 <p<∞ and
max
{
L, Ke
x
}
≤ u
ε,n
≤ KinΩ
n
T
,
3.5
∂
x
u
ε,n
≥ 0 in Ω
n
T
.
3.6
Proof. We apply the Schauder fixed point theorem 17 to prove the existence of nonlinear
problem 3.3.
Denote B C
Ω
K, 0 ≤ t ≤ T,
u
ε,n
x, 0
π
ε
Ke
x
− L
L, −n ≤ x ≤ 0.
3.7
Advances in Difference Equations 9
ε
C
0
s
Figure 2: The function β
ε
.
s
s
−ε
ε
Figure 3: The function π
ε
ε,n
on ∂
p
Ω
n
T
,
3.8
where ∂
p
Ω
n
T
is the parabolic boundary of Ω
n
T
.Thusc/r is a supersolution of the problem 3.7,
and u
ε,n
≤ c/r.HenceFD ⊂ D. On the other hand,
0 ≤ β
ε
w − K
≤ β
ε
c
r
K ≥ u
ε,n
on ∂
p
Ω
n
T
.
3.10
10 Advances in Difference Equations
Therefore, K is a supersolution of problem 3.3,andu
ε,n
≤ K in Ω
n
T
.Moreover,
∂
t
Ke
x
−L
Ke
x
β
ε
ε,n
−n, t
,Ke
x
|
x0
K u
ε,n
0,t
,
Ke
x
≤ max
{
Ke
x
,L
}
≤ π
ε
Ke
x
− L
L u
−n, t
,L<K u
ε,n
0,t
,
L ≤ max
{
Ke
x
,L
}
≤ π
ε
Ke
x
− L
L u
ε,n
x, 0
.
3.12
Thus, L is a subsolution of problem 3.3 as well, and we deduce u
ε,n
ε
u
ε,n
− K
W 0inΩ
n
T
,
W
−n, t
≥ 0,W
0,t
≥ 0, 0 ≤ t ≤ T,
W
x, 0
π
ε
Ke
x
0
{x, t : x ln K − ln L
2
t
2
≤ δ
2
}. Moreover, if n is large enough, one has that
max
{
L, Ke
x
}
≤ u
n
≤ K in Ω
n
T
, 3.14
∂
x
u
n
≥ 0 in Ω
n
T
, 3.15
∂
t
u
W
2,1
p
Ω
n
T
\B
δ
P
0
u
ε,n
C
α,α/2
Ω
n
T
≤ C,
3.18
Advances in Difference Equations 11
where C is independent of ε. It implies that there exists a u
n
∈ W
2,1
\ B
δ
P
0
weakly,u
ε,n
−→ u
n
in C
Ω
n
T
. 3.19
Employing the method in 16 or 19,itisnotdifficult to derive that u
n
is the solution
of problem 3.1.And3.14, 3.15 are the consequence of 3.5, 3.6 as ε → 0
.
In the following, we will prove 3.16.Foranysmallδ>0, wx, t
Δ
u
n
x, t δ
satisfies, by 3.1,
w
−n, t
L u
n
−n, t
,w
0,δ
K u
n
0,t
, 0 ≤ t ≤ T − δ,
w
x, 0
u
n
x, δ
≥ max
{
∈
−n, 0
×
0,T − δ
. 3.21
Thus 3.16 follows.
At last, we pro ve the uniqueness of the solution. Suppose that u
1
n
and u
2
n
are two
W
2,1
p,loc
Ω
n
T
∩ CΩ
n
T
solutions to problem 3.1, and denote
N
Δ
2
n
x, t
≤ K, ∂
t
u
1
n
−Lu
1
n
c, ∂
t
u
1
n
− u
2
n
−L
u
1
n
− u
2
0
for any 1 <p<
∞, R>0,andδ>0.And∂
x
u ∈ CΩ
T
\ B
δ
P
0
.Moreover,
max
{
L, Ke
x
}
≤ u ≤ K in
Ω
T
,
3.25
∂
x
u ≥ 0 a.e. in Ω
T
, 3.26
∂
t
u ≥ 0 a.e. in Ω
T
K, 0 ≤ t ≤ T,
u
n
x, 0
max
{
L, Ke
x
}
, −n ≤ x ≤ 0,
3.28
where u
n
∈ W
2,1
p
Ω
n
T
\ B
δ
P
0
implies that fx, t ∈ L
p
loc
Ω
n
p
Ω
R
T
\B
δ
P
0
≤ C
R,δ
,
u
n
C
α,α/2
Ω
R
T
≤ C
R
,
3.30
here C
R,δ
depends on R and δ, C
R
weakly as n −→ ∞. 3.31
Moreover, 3.30 and imbedding theorem imply that
u
n
−→ u in C
Ω
R
T
,∂
x
u
n
−→ ∂
x
u in C
Ω
R
T
\ B
δ
P
0
as n −→ ∞. 3.32
It is not difficult to deduce that u is the solution of problem 1.4.Furthermore,3.32 implies
: u
x, t
K
}
callable region
.
4.1
Thanks to 3.26,wecandefinethefreeboundaryht of problem 1.4, at which it is
optimal for the firm to call the bond, where
h
t
inf
{
x ≤ 0:u
x, t
K
}
, 0 <t≤ T 4.2
see Figure 1. It is clear that
CT
{
,x<0
}
,h
t
−∞ for any t ≥ T
0
. 4.4
Proof. Define
w
x, t
⎧
⎪
⎨
⎪
⎩
c
r
−
c
r
− L
e
−rt
,wehavethat
w
x, T
0
c
r
−
c
r
− L
exp
−r
1
r
ln
c − rL
c − rK
K, 4.6
then property i is obvious.
Moreover, if 0 <t≤ T
0
,thenwededuce
∂
e
−rt
r
c
r
−
c
r
− L
e
−rt
c. 4.8
In the other case of T
0
<t≤ T,
∂
t
w −Lw rK < c. 4.9
So, we testify properties i–iv. In the following, we utilize the properties to prove w ≤ u.
Otherwise, N {w>u} is nonempty; then we have that
u
x, t
<w
0
, 4.12
which means that CT ⊂{0 <t<T
0
,x<0},CL⊃{t ≥ T
0
,x<0 },andht−∞ for a ny
t ≥ T
0
Theorem 4.2. Thefreeboundaryht is decreasing in the interval 0,T
0
.Moreover,h0
Δ
lim
t → 0
ht0.Andht ∈ C0,T
0
.
Proof. 3.26 and 3.27 imply that
∂
x
u − K
≥ 0,∂
t
u − K
0
lim
t → 0
h
t
.
4.15
Advances in Difference Equations 15
Since u0,tK,soh0 ≤ 0. On the other hand, if h0 < 0, then
u
x, t
K, ∀
x, t
∈
h
0
, 0
exists x
1
<x
2
< 0, 0 <t
1
<T
0
such that see Figure 4
lim
t → t
−
1
h
t
x
1
, lim
t → t
1
h
t
x
2
.
∂
x
u
0inM. 4.19
On the other hand, ∂
x
ux, t
1
0foranyx ∈ x
1
,x
2
in this case, and we know that ∂
x
u ≥ 0
by 3.26. Applying the strong maximum principle to 4.19,weobtain
∂
x
u
x, t
0, in M. 4.20
So, we can define ux, tgt in M.Consideringuht,tK and u ∈ C
Ω
T
,wesee
that ux, t ≡ K in M, which contradicts that ux, t <Kfor any x<ht. Therefore ht ∈
for any x ∈
ln L − ln K, 0
, lim
x → 0
−
∂
x
u
x, 0
K.
4.21
Meanwhile, ux, tK in the domain {x, t : ht <x<0, 0 <t<T
0
} implies that ∂
x
u0,t
0foranyt>0 see Figure 4;then∂
x
u is not continuous at the point 0, 0, which contradicts
∂
x
u ∈ CΩ
T
\ B
δ
P
1
x
2
t
1
T
0
CT
h(t)
CL
Figure 4: Discontinuous free boundary ht.
At last, we prove that ht is strictly decreasing on t
0
,T
0
.Otherwise,x ht has
a vertical part. Suppose that the vertical line is x x
1
,t∈ t
1
,t
2
,thenux, tK for any
x, t ∈ −∞,x
1
× t
1
,t
2
.Since∂
0foranyt ∈
t
1
,t
2
. 4.22
On the other hand, in the domain N −∞,x
1
× t
1
,t
2
, u and ∂
t
u satisfy, respectively,
∂
t
u −Lu c in N,u
x
1
,t
K for any t ∈
t
1
,t
1
,t
2
.
4.23
Then the strong maximum principle implies that ∂
x
∂
t
ux
1
,t < 0, which contradicts the
second equality in 4.22.
Theorem 4.4. ht >h
∗
t for any t ∈ 0,T
0
with lim
t → T
−
0
ht−∞ (see Figure 1), where
h
∗
t
ln
L
2
α − r 0. 4.25
Proof. Define
W
x, t
c
r
−
c
r
− L
e
−rt
K
α1
L
α
e
αx
,
x, t
L
K
α1
L
α
e
αx
≥
⎧
⎪
⎪
⎨
⎪
⎪
⎩
L max
{
L, Ke
x
}
u
x, 0
if x ≤ ln L − ln K,
L
K
α1
L
c
r
− L
e
−rt
c. 4.28
Hence, we have property ii.
It is not difficult to check that, for any t ∈ 0,T
0
,
W
h
∗
t
,t
K, ∂
x
W
αK
α1
L
α
e
αx
1
. Repeating the method in the
proof of Theorem 4.2, then we can obtain a contradiction. So, lim
t → T
−
0
ht−∞.
Theorem 4.5. The free boundary ht ∈ C
0,1
0,T
0
∩ C
∞
t
0
,T
0
.
Proof. Fix t
1
∈ 0,t
0
and t
2
∈ t
0
,T
0
, and denote X h
∗
u −Lu c, ∂
t
∂
t
u
−L
∂
t
u
0,
x, t
∈ CT, 4.31
18 Advances in Difference Equations
t
0
t
1
t
2
X
CT
Γ
1
Γ
, Γ
2
Δ
{
X ≤ x ≤ 0,t t
1
}
.
4.33
On the other hand, we see that ∂
t
u ≥ 0inΩ
T
from 3.27,and∂
t
u0,t0. Applying
the strong maximum principle to ∂
t
ux, t,wededucethat
∂
tx
u
0,t
< 0,t∈
0,t
0
∂
x
u
0,
x, t
∈ CT. 4.36
Employing the strong maximum principle, we see that there is a δ>0, such that
∂
x
u
x, t
≥ δ on Γ
1
∪ Γ
2
, 4.37
Provided that δ is small enough. Combining 4.32, there exists a positive M
0
C/δ1
such that
M
0
∂
x
∈N,
u
X, t
u
X, t
,u
0,t
K, t
1
≤ t ≤ t
2
,
u
x, t
1
u
x, t
1
,X≤ x ≤ 0.
K, t
1
≤ t ≤ t
2
,
u
ε
x, t
1
u
x, t
1
,X≤ x ≤ 0.
4.40
Recalling 4.38,weseethat,ifε is small enough, M
0
∂
x
u
ε
−∂
t
u
ε
≥ 0 on the parabolic boundary
of N.Moreover,w
ε
− ∂
t
u
ε
w ≥ 0inN. 4.42
As the method in the proof of Theorem 3.3, we can show that u
ε
weakly converges to u in
W
2,1
p
N and 4.30 is obvious.
On the other hand, we see that M∂
x
u ∂
t
u ≥ 0inN for any positive number M from
3.26 and 3.27.So,
M
0
∂
x
u ± ∂
t
u ≥ 0inN, 4.43
which means that there exists a uniform cone such that the free boundary should lies in
the cone. As the method in 9,itiseasytoderivethatht ∈ C
0,1
t
in the calculations are r 0.2, q 0.1, σ 0.3, L 1, K 1.5, c 0.5, T 2, and n 3000.
In this case, the free boundary is increasing with x00. The numerical result is coincided
with that of our proof see Figure 6.
Acknowledgments
The project is supported by NNSF of China nos. 10971073, 11071085, and 10901060 and
NNSF of Guang Dong province no. 9451063101002091.
References
1 M. S
ˆ
ırbu, I. Pikovsky, and S. E. Shreve, “Perpetual convertible bonds,” SIAM Journal on Control and
Optimization, vol. 43, no. 1, pp. 58–85, 2004.
2 M. S
ˆ
ırbu and S. E. Shreve, “A two-person game for pricing convertible bonds,” SIAM Journal on
Control and Optimization, vol. 45, no. 4, pp. 1508–1539, 2006.
3 P. Wilmott, Derivatives, the Theory and Practice of Financial Engineering, John Wiley & Sons, New York,
NY, USA, 1998.
4 Y. Kifer, “Game options,” Finance and Stochastics, vol. 4, no. 4, pp. 443–463, 2000.
5 Y. Kifer, “Error estimates for binomial approximations of game options,” The Annals of Applied
Probability, vol. 16, no. 2, pp. 984–1033, 2006.
6 C. K
¨
uhn and A. E. Kyprianou, “Callable puts as composite exotic options,” Mathematical Finance,vol.
17, no. 4, pp. 487–502, 2007.
7 A. E. Kyprianou, “Some calculations for Israeli options,” Finance and Stochastics, vol. 8, no. 1, pp.
73–86, 2004.
8 F. Black and M. Scholes, “The pricing of options and coperate liabilities,” Journal of Political Economy,
vol. 81, pp. 637–659, 1973.
Advances in Difference Equations 21
9 Z. Yang and F. Yi, “A free boundary p roblem arising from pricing convertible bond,” Applicable
zenskaja,V.A.Solonnikov,andN.N.Uralceva,Linear and Quasi-linear Equations of Parabolic
Type, American Mathematical Society, Providence, RI, USA, 1968.
19 F. Yi, Z. Yang, and X. Wang, “A variational inequality arising from European installment call options
pricing,”
SIAM Journal on Mathematical Analysis, vol. 40, no. 1, pp. 306–326, 2008.
20 K. Tso, “On Aleksandrov, Bakel’man type maximum principle for second order parabolic equations,”
Communications in Partial Differential Equations, vol. 10, no. 5, pp. 543–553, 1985.
Copyright: Tài liệu đại học ©