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,fhyi@scnu.edu.cn
Received 30 December 2010; Accepted 11 February 2011
Academic Editor: Jin Liang
Copyrightq2011 H. Yan and F. Yi. This is an open access article distributed 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 inequalityPVI. 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 the free boundaries of PVI. Some properties of the free boundaries are studied in this paper. We show that the bondholder 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 debtbondsand equityshares 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.
In 2 the authors assume that a firm’s value is comprised of one equity and one convertible bond, the value of the issuing firm has constant volatility, the bond continuously pays coupons at a fixed rate, and the firm continuously pays dividends at a rate that is a fixed fraction of equity. Default occurs if the coupon payments cause the firm’s value to fall to zero, in which case the bond has zero value. In their model, both the bond price and the stock price are functions of the underlying of the firm value. Because the stock price is the difference between firm value and bond price and dividends are paid proportionally to the stock price, a nonlinear differential equation was established for describing the bond price as a function of the firm value and time.
As we know, it is difficult to obtain the value of the firm. However, it is easier to get its stock price. So we choose the bond priceVS, tas a function of the stock priceSof the firm and timetsee Chapter 36 in3or4–7.
InSection 2, we formulate the model and deduce that VS, t γ S in the domain {S ≥ K/γ}and VS, tis governed by the following variational inequality in the domain {0≤S≤K/γ}:
wherec,γ,K, andLare positive constants.cis the coupon rate,γis the conversion ratio for converting the bond into the stock of the firm,Kis the call price of the firm,Lis the face value of the bond with 0< L≤K, andL0is just B-S operatorsee8,
wherer,σ, andqare 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 thatc > rK andr ≥ q. From a financial point of view, the assumption provides a possibility of calling the bond back from the firmseeSection 2 or2. Furthermore, we suppose thatL ≤ K. Otherwise, the firm should call the bond back before maturity and the valueL makes no sensesee Section 2. It is clear that V K is the unique solution ifL K. So we only consider the problem in the case ofL < K.
Since1.1is a degenerate backward problem, we transform it into a familiar forward nondegenerate parabolic variational inequality problem; so letting
we have that are seldom results on the properties of the free boundaries—the optimal call and conversion strategies in the existing literature. The main aim of this paper is to analyze some properties of the free boundaries.
The pricing model of the convertible bond without call is considered in 9, where there exist two domains: the continuation domain CT and the conversion domain CV. The free boundary St between CT and CV means the optimal conversion strategy, which is dependent on the timetand more thanK/γ.
But in this model, their exist three domains: the continuation domain CT, the callable domain CL, and the conversion domain CV {x ≥ 0}. The boundary between CV and CT ∪ CL is x 0, which means the call strategy. The free boundaryhtis the curve between CT and CLseeFigure 1, which means the optimal call strategy. And there exist t0, T0such that
It means that the bondholder should convert the bond if and only if the stock price Sof the firm is no less thanK/γ, whereas, in the model without call, the bondholder may not convert the bond even ifS > K/γ. More precisely, the optimal conversion strategySt without call is more than that K/γ in this paper see9 orSection 2. When the time to the expiry date is more thanT0, the firm should call the bond back ifS < K/γ. Neither the
bondholder nor the firm should exercise their option if the time to the expiry date is less than t0 and S < Keht. Moreover, when the time to the maturity lies in t0, T0, the bondholder
should call the bond back ifKeht≤S < K/γ.
In Section 2, we formulate and simplify the model. In Section 3, we will prove the existence and uniqueness of the strong solution of the parabolic variational inequality1.4
and establish some estimations, which are important to analyze the property of the free boundary.
t
x t0
T0
u=K
CT Kex< u < K
CL
u=CVKex
h(t)
Figure 1:The free boundaryht.
in10if the difference betweenuand the upper obstacleK is decreasing with respect tot. But the proof is difficult if the condition is falsesee11–14. In this problem,∂tu−K≥0,
which does not match the condition. Moreover,∂xxulnL−lnK,0 ∞, and the starting
point0, t0of the free boundaryhtis not on the initial boundary, but the side boundary in
this problem. Those make the proof ofht∈C∞t0, T0more complicated. The key idea is to
construct cone locally containing the local free boundary and proveht ∈ C0,1t
0, T0; then
the proof ofC∞t0, T0is trivial. Moreover, we show that there is a lower boundh∗tofht andhtconverges to−∞astconverges toT0−inTheorem 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 ofc. In the life time of the bond, the bondholder has the right to convert it into the firm’s stock with the conversion factorγ and obtainsγ Sfrom the firm after converting, and the firm can call it back at a preset price ofK. The bondholder’s right is superior to the firm’s, which means that the bondholder has the right to convert the bond, but the firm has no right to call it if both sides hope to exercise their rights at the same time. If neither the bondholder nor the firm exercises their right before maturity, the bondholder must sell the bond to the firm at a preset valueLor convert it into the firm’s stock at expiry date. So, the bondholder receives max{L, γ S}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 spaceΩ,F, ; the stock price of the firmSsfollows
St,Ss S
s
t
r−qSt,Su du
s
t
σSt,Su dWu, S∈0,∞, s∈t, T, t∈0, T, 2.1
wherer,q, andσare positive constants, representing risk free interest rate, the dividend rate, and volatility of the stock, respectively.Wt is a standard Brown motion on the probability
Denote byFtthe natural filtration generated byWtand augmented by all the -null
sets inF. LetUt,Tbe the set of allFt-stopping times taking values int, T.
The model can be expressed as a zero-sum Dynkin game. The payoffof the bondholder is
whereτ, θ∈ Ut,T. The stopping timeτis the firm’s strategy, andθis the bondholder’s strategy.
The bondholder chooses his strategyθto maximizeRS, t;τ, θ; meanwhile, the firm chooses its strategyτ to minimizeRS, t;τ, θ.
Denote the upper valueVand the lower valueVas
VS, tΔess sup
then the value of the Dynkin game exists and
In the case of 0 < S < K/γ, applying the standard method in15, we see that the strong solution of the following variational inequality is the value of the Dynkin game:
−∂tV − L0V c, ifγ S < V < K, S, t∈DT,
IfL > K, then the firm is bound to call the bond back before the maturity because the firm paysKafter calling, but more thanLwithout calling. In this case, the valueLmakes no sense. So, we suppose thatL≤K.
Ifc ≤ rK, then the firm is bound to abandon its call right. From a financial point of view, the firm would payKto the bondholder at timetafter calling the bond, whereas, if the firm does not call in the time intervalt, tdt, then he would pay the coupon paymentcdt and at mostKof the face value of the convertible bond at timetdt. So, the discounted value of the bond without call is at mostKcdt−rKdt ≤K. Hence, the firm should not call the bond back at timet.
From a stochastic point of view, we can denote a stopping time
τ1inf
From a variational inequality point of view, since
−∂tK− L0KrK > c, 2.12
provided thatc < rK, which contradicts with the third inequality in2.8, so, ifc < rK, then V /Kin the domain{t < T,0< S < Kγ}.
To remain the call strategy, we suppose thatc > rK. We will consider the other case in another paper because the two problems are fully different.
Since we suppose thatc > rKandr ≥q, then
−∂t
γ S− L0
γ Sqγ S≤rK < c. 2.13
Hence,{V γ S}is empty in problem2.8. So, problem2.8is reduced into problem1.1. The model of pricing the bond without call is an optimal stopping problem
US, tΔess sup
θ∈Ut,T
QS, t;θ| Ft,
QS, t;θ
θ
t
cert−ruduert−rθγ St,S
θ I{θ<T}ert−rTmax
L, γ St,ST I{θT}.
2.14
It is clear that
US, t ess sup
θ∈Ut,T
RS, t;T, θ| Ft≥VS, t.
2.15
SinceU, V ≥γ S, then
Uγ S⊂V γ S, CV∗⊂CV, 2.16
where CV∗is the conversion domain in the model without call and CV is that in this paper.
3. The Existence and Uniqueness of
W
p,2,loc1Solution of Problem
1.4
Since problem1.4lies in the unbounded domainΩT, we need the following problem in thebounded domainΩn T
Δ −
n,0×0, Tto approximate to problem1.4:
∂tun− Lunc, ifun< K, x, t∈ΩnT,
∂tun− Lun≤c, ifunK, x, t∈ΩnT,
un−n, t L, un0, t K, 0≤t≤T,
unx,0 max{L, Kex}, −n≤x≤0,
3.1
Following the idea in10,16, we construct a penalty function βεs seeFigure 2,
which satisfies
ε >0 and small enough, βεs∈C∞−∞,∞,
βεs 0, ifs≤ −ε, βε0 C0Δc−rK >0,
βεs≥0, βεs≥0, βεs≥0,
lim
ε→0βεs
⎧ ⎨ ⎩
0, s <0,
∞, s >0.
3.2
Consider the following penalty problem of3.1:
∂tuε,n− Luε,nβεuε,n−K c, inΩnT,
uε,n−n, t L, uε,n0, t K, 0≤t≤T,
uε,nx,0 πεKex−L L, −n≤x≤0,
3.3
whereπεsis a smoothing function because the initial value max{L, Kex}is not smooth. It
satisfiesseeFigure 3
πεs
⎧ ⎨ ⎩
s, s≥ε,
0, s≤ −ε,
πεs∈C∞IR, πεs≥s, 0≤πεs≤1, πεs≥0, εlim→0πεs s.
3.4
Lemma 3.1. For any fixedε >0, problem3.3has a unique solutionuε,n∈Wp2,1ΩnT∩CΩnTfor
any1< p <∞and
max{L, Kex} ≤uε,n≤K inΩnT, 3.5
∂xuε,n≥0 in ΩnT. 3.6
Proof. We apply the Schauder fixed point theorem17to prove the existence of nonlinear problem3.3.
DenoteB CΩn
TandD {w ∈ B: w ≤ c/r}. ThenDis a closed convex set inB.
Defining a mappingFbyFw uε,nis the solution of the following linear problem:
∂tuε,n− Luε,nβεw−K c inΩnT,
uε,n−n, t L, uε,n0, t K, 0≤t≤T,
uε,nx,0 πεKex−L L, −n≤x≤0.
ε C0
s
Figure 2:The functionβε.
s
s
−ε ε
Figure 3:The functionπε.
Furthermore, we can compute
∂t
c r
− L
c r
βεw−K r
c
r βεw−K≥c, c
r > K≥uε,n on∂pΩ
n T,
3.8
where∂pΩnTis the parabolic boundary ofΩnT. Thusc/ris a supersolution of the problem3.7,
anduε,n≤c/r. HenceFD⊂D. On the other hand,
0≤βεw−K≤βε
c r −K
, 3.9
which is bounded for fixedε >0. So, it is not difficult to prove thatFDis compact inBand Fis continuous. Owing to the Schauder fixed point theorem, we know that problem3.3has a solutionuε,n∈Wp2,1ΩnT. The proof of the uniqueness follows by the comparison principle.
Here, we omit the details.
Now, we prove3.5. Since
∂tK− LKβεK−K rKβε0 c,
K≥uε,n on∂pΩnT.
Therefore,Kis a supersolution of problem3.3, anduε,n≤KinΩnT. Moreover,
∂tKex− LKex βεKex−K qKexβεKex−K≤qKexβε0 qKexc−rK≤c,
Kex|x−n Ke−n≤Luε,n−n, t, Kex|x0Kuε,n0, t,
Kex≤max{Kex, L} ≤πεKex−L Luε,nx,0.
3.11
Hence,Kexis a subsolution of problem3.3. On the other hand,
∂tL− LLβεL−K rLβεL−K≤rLβε0≤c,
Luε,n−n, t, L < Kuε,n0, t,
L≤max{Kex, L} ≤πεKex−L Luε,nx,0.
3.12
Thus,Lis a subsolution of problem3.3as well, and we deduceuε,n≥max{Kex, L}.
In the following, we prove3.6.
Indeed,uε,n ≤ Kanduε,n0, t K imply that∂xuε,n0, t ≥ 0. Furthermore,uε,n ≥ L
and uε,n−n, t Lthat imply ∂xuε,n−n, t ≥ 0. Differentiating3.3with respect toxand
denotingW∂xuε,n, we obtain
∂tW− LWβεuε,n−KW0 inΩnT,
W−n, t≥0, W0, t≥0, 0≤t≤T,
Wx,0 πεKex−LKex≥0, −n≤x≤0.
3.13
Then the comparison principle implies3.6.
Theorem 3.2. For any fixedn∈IN,n >lnK−lnL, problem3.1admits a unique solutionun∈
CΩn T∩W
2,1
p ΩnT \BδP0 for any1 < p < ∞, 0 < δ < n, whereP0 −lnKlnL,0,
BδP0 {x, t:xlnK−lnL2t2≤δ2}. Moreover, if n is large enough, one has that
max{L, Kex} ≤u
n≤K inΩnT, 3.14
∂xun≥0 inΩnT, 3.15
∂tun≥0 a.e. inΩnT. 3.16
Proof. From3.5and the properties ofβεs, we have that
0≤βεuε,n−K≤βε0 c−rK. 3.17
ByWp2,1andCα,α/20< α <1estimates of the parabolic problem18, we conclude that
uε,nW2,1
whereCis independent ofε. It implies that there exists aun∈Wp2,1ΩnT\BδP0∩CΩnTand
a subsequence of{uε,n}still denoted by{uε,n}, such that asε → 0,
uε,n un inWp2,1
Ωn
T\BδP0
weakly, uε,n−→un inC
Ωn T
. 3.19
Employing the method in16or19, it is not difficult to derive thatunis the solution
of problem3.1. And3.14,3.15are the consequence of3.5,3.6asε → 0.
In the following, we will prove3.16. For any smallδ > 0, wx, t Δ unx, tδ
satisfies, by3.1,
∂tw− Lwc, ifw < K, x, t∈−n,0×0, T −δ,
∂tw− Lw≤c, ifwK, x, t∈−n,0×0, T −δ,
w−n, t Lun−n, t, w0, δ Kun0, t, 0≤t≤T−δ,
wx,0 unx, δ≥max{L, Kex}unx,0, −n≤x≤0.
3.20
Applying the comparison principle with respect to the initial value of the variational inequalitysee16, we obtain
unx, tδ wx, t≥unx, t, x, t∈−n,0×0, T−δ. 3.21
Thus3.16follows.
At last, we prove the uniqueness of the solution. Suppose that u1n and u2n are two
Wp,2,loc1 Ωn
T∩CΩnTsolutions to problem3.1, and denote
NΔx, t∈ΩnT :un1x, t< u2nx, t. 3.22
Assume thatNis not empty, and then, in the domainN,
u1nx, t< u2nx, t≤K, ∂tu1n− Lu1nc, ∂t
u1n−u2n
− Lu1n−u2n
≥0. 3.23
DenotingWu1n−u2n, we have that
∂tW− LW ≥0 inN, W0 on ∂pN. 3.24
Theorem 3.3. Problem1.4has a unique solutionu∈CΩT∩Wp2,1ΩRT\BδP0 for any1< p <
∞,R >0, andδ >0. And∂xu∈CΩT\BδP0. Moreover,
max{L, Kex} ≤u≤K in ΩT, 3.25
∂xu≥0 a.e. inΩT, 3.26
∂tu≥0 a.e. inΩT. 3.27
Proof. Rewrite Problem3.1as follows:
∂tun− Lun fx, t, x, t∈ΩnT,
un−n, t L, un0, t K, 0≤t≤T,
unx,0 max{L, Kex}, −n≤x≤0,
3.28
whereun∈Wp2,1ΩnT\BδP0implies thatfx, t∈LplocΩnTand
fx, t cI{un<K}rKI{unK}, 3.29
whereIAdenotes the indicator function of the setA.
Hence, for any fixedR > δ >0, ifn > R, combining3.14, we have the followingWp2,1
andCα,α/2uniform estimates18:
unW2,1
p ΩRT\BδP0 ≤CR,δ, unCα,α/2ΩR
T ≤CR, 3.30
hereCR,δ depends on Rand δ, CR depends onR, but they are independent of n. Then, we
have that there is au∈Wp,2,1locΩT∩CΩTand a subsequence of{un}still denoted by{un},
such that for anyR > δ >0,p >1,
un u in Wp2,1
ΩR
T \BδP0
weakly asn−→∞. 3.31
Moreover,3.30and imbedding theorem imply that
un−→u inC
ΩRT
, ∂xun−→∂xu inC
ΩRT\BδP0
asn−→∞. 3.32
It is not difficult to deduce thatuis the solution of problem1.4. Furthermore,3.32implies that∂xu∈CΩT\BδP0. And3.25–3.27are the consequence of3.14–3.16. The proof
4. Behaviors of the Free Boundary
Denote
CT{x, t:ux, t< K} continuation region,
CL{x, t:ux, t K} callable region. 4.1
Thanks to3.26, we can define the free boundaryhtof problem1.4, at which it is optimal for the firm to call the bond, where
ht inf{x≤0 :ux, t K}, 0< t≤T 4.2
We claim thatwx, tpossess the following four properties.
iw∈Wp,2,loc1 ΩT∩CΩT,
iiw≤K, for allx, t∈ΩT,
iiiwx,0 L≤max{L, Kex}ux,0, for allx∈−∞,0, iv∂tw− Lw≤c, a.e. inΩT.
In fact, from the definition ofT0, we have that
wx, T0 c
then propertyiis obvious.
Combiningwx, T0 K, we have propertyii. It is easy to check propertyiiifrom
the definition ofw. Next, we manifest propertyivaccording to the following two cases. In the case of 0< t≤T0,
∂tw− Lwr
c r −L
e−rtr
c r −
c r −L
e−rt
c. 4.8
In the other case ofT0< t≤T,
∂tw− LwrK < c. 4.9
So, we testify propertiesi–iv. In the following, we utilize the properties to provew≤u. Otherwise,N{w > u}is nonempty; then we have that
ux, t< wx, t≤K, ∂tu− Luc, ∂tu−w− Lu−w≥0, inN. 4.10
Moreover,u−w ≥ 0 on the parabolic boundary ofN. According to the A-B-P maximum principlesee20, we have that
u−w≥0 inN, 4.11
which contradicts the definition ofN. So, we achieve thatw≤u. Combiningwx, t Kfor anyt≥T0, it is clear that
Kwx, t≤ux, t≤K, for anyt≥T0, 4.12
which means that CT ⊂ {0 < t < T0, x < 0}, CL ⊃ {t ≥ T0, x < 0}, andht −∞for any
t≥T0
Theorem 4.2. The free boundary ht is decreasing in the interval 0, T0. Moreover, h0 Δ
limt→0ht 0. Andht∈C0, T0.
Proof. 3.26and3.27imply that
∂xu−K≥0, ∂tu−K≥0 a.e.inΩT. 4.13
Hence, for any unit vectorn n1, n2satisfyingn1, n2 >0, the directional derivative
of functionu−Kalongnadmits
∂nu−K≥0 a.e.inΩT, 4.14
that is,u−Kis increasing along the directorn. Combining the conditionu−K≤0 inΩT, we
know thatxhtis monotonically decreasing. Hence, limt→0htexists, and we can define
h0 lim
Sinceu0, t K, soh0≤0. On the other hand, ifh0<0, then
ux, t K, ∀x, t∈h0,0×0, T, ux,0 max{L, Kex}< K, ∀x∈h0,0. 4.16
It is impossible becauseuis continuous onΩT.
In the following, we prove that ht is continuous in0, T0. If it is false, then there
existsx1< x2<0, 0< t1< T0such thatseeFigure 4
lim
t→t−1ht x1, tlim→t1ht x2. 4.17
Moreover,
∂tu− Luc inMΔ{x, t:x2 < x < ht,0< t≤t1}. 4.18
Differentiating4.18with respect tox, then
∂t∂xu− L∂xu 0 in M. 4.19
On the other hand,∂xux, t1 0 for anyx∈x1, x2in this case, and we know that∂xu≥0
by3.26. Applying the strong maximum principle to4.19, we obtain
∂xux, t 0, inM. 4.20
So, we can defineux, t gtin M. Consideringuht, t K and u ∈ CΩT, we see
thatux, t ≡ K inM, which contradicts thatux, t < K for anyx < ht. Thereforeht ∈ C0, T0.
Theorem 4.3. There exists somet0∈0, T0such thatht 0for anyt∈0, t0andht is strictly
decreasing ont0, T0.
Proof. Definet0sup{t:t≥0, ht 0}. In the first, we prove thatt0>0. Otherwise,h0 0
andht<0 fort >0.
Recalling the initial value, we see that
∂xux,0 Kex for anyx∈lnL−lnK,0, lim
x→0−∂xux,0 K. 4.21
Meanwhile,ux, t Kin the domain{x, t:ht< x <0,0< t < T0}implies that∂xu0, t
0 for anyt >0seeFigure 4; then∂xuis not continuous at the point0,0, which contradicts
∂xu∈CΩT\BδP0.
In the second, we prove thatt0 < T0. In fact, according toLemma 3.1,ht −∞for
anyt ≥ T0, hence,t0 ≤ T0. Ift0 T0, then the free boundary includes a horizontal linet
T0, x ∈ −∞,0. Repeating the method in the proof ofTheorem 4.2, then we can obtain a
t
x x1
x2
t1
T0
CT h(t)
CL
Figure 4:Discontinuous free boundaryht.
At last, we prove thathtis strictly decreasing ont0, T0. Otherwise,x hthas
a vertical part. Suppose that the vertical line isx x1, t∈ t1, t2, thenux, t Kfor any
x, t∈−∞, x1×t1, t2. Since∂xuis continuous across the free boundary, then∂xux1, t 0
for anyt∈t1, t2. In this case, we infer that
∂tux1, t 0, ∂t∂xux1, t 0 for anyt∈t1, t2. 4.22
On the other hand, in the domainN −∞, x1×t1, t2,uand∂tusatisfy, respectively,
∂tu− Luc inN, ux1, t K for anyt∈t1, t2,
∂t∂tu− L∂tu 0, ∂tu≥0, inN,
∂tux1, t 0 for anyt∈t1, t2.
4.23
Then the strong maximum principle implies that∂x∂tux1, t < 0, which contradicts the
second equality in4.22.
Theorem 4.4. ht> h∗tfor anyt∈0, T0withlimt→T0−ht −∞(seeFigure 1), where
h∗t ln L
K
1 αln
c−rLe−rt−c−rK
rK , 0≤t < T0, 4.24
whereαis the positive characteristic root ofLw0, that is, the positive root of the algebraic equation
σ2 2 α
2
r−q−σ
2
2 α−r 0. 4.25
Proof. Define
Wx, t c
r −
c r −L
e−rtK
α1
Lα e
αx, x, t∈Ω
We claim thatWx, t∈C2Ω
T0and possess the following three properties.
iWx,0≥ux,0for−∞< x <0 and W0, t≥K for 0< t≤T0,
In the first, we prove that there exists anM0>0 such that
M0∂xu−∂tu≥0 inN. 4.30
In fact,u, ∂tusatisfy the equations
t0
t1
t2
X CT
Γ1
Γ2
t
x h(t)
CL
Figure 5:The free boundaryht.
then the interior estimate of the parabolic equation implies that there exists a positive constant Csuch that
∂tux, t≤C onΓ1∪Γ2, 4.32
here
Γ1 Δ{xX, t1≤t≤t2}, Γ2Δ{X ≤x≤0, tt1}. 4.33
On the other hand, we see that∂tu≥0 inΩT from3.27, and∂tu0, t 0. Applying
the strong maximum principle to∂tux, t, we deduce that
∂txu0, t<0, t∈0, t0. 4.34
It means that∂xu0, tis strictly decreasing on0, t0. It follows that, by∂xu0, t0 0,
∂xu0, t1>0. 4.35
Moreover,
∂xu≥0, ∂t∂xu− L∂xu 0, x, t∈CT. 4.36
Employing the strong maximum principle, we see that there is aδ >0, such that
∂xux, t≥δ onΓ1∪Γ2, 4.37
Provided thatδis small enough. Combining4.32, there exists a positiveM0 C/δ 1
such that
2
1
−0.8 −0.6 −0.4 −0.2 0 0
t
x
h(t)
Figure 6:The free boundary.
Next, we concentrate on problem1.4in the domainN. It is clear thatusatisfies
∂tu− Luc, ifu < K, x, t∈ N,
∂tu− Lu≤c, ifuK, x, t∈ N,
uX, t uX, t, u0, t K, t1≤t≤t2,
ux, t1 ux, t1, X≤x≤0.
4.39
And we can use the following problem to approximate the above problem:
∂tuε− Luεβεuε−K c, inN,
uεX, t uX, t, uε0, t K, t1≤t≤t2,
uεx, t1 ux, t1, X≤x≤0.
4.40
Recalling4.38, we see that, ifεis small enough,M0∂xuε−∂tuε≥0 on the parabolic boundary
ofN. Moreover,wΔM0∂xuε−∂tuεsatisfies
Applying the comparison principle, we obtain
M0∂xuε−∂tuεw≥0 inN. 4.42
As the method in the proof ofTheorem 3.3, we can show thatuε weakly converges touin
Wp2,1Nand4.30is obvious.
On the other hand, we see thatM∂xu∂tu≥0 inNfor any positive numberMfrom
3.26and3.27. So,
M0∂xu±∂tu≥0 inN, 4.43
which means that there exists a uniform cone such that the free boundary should lies in the cone. As the method in9, it is easy to derive thatht ∈ C0,1t
1, t2. Moreoverht ∈
C∞t0, t2can be deduced by the bootstrap method. Sincet2is arbitrary and the free boundary
is a vertical line whilet∈0, t0, thenht∈C0,10, T0∩C∞t0, T0.
5. Numerical Results
Applying the binomial tree method to problem 1.4, we achieve the following numerical results—Figure 6:
Plot of the optimal exercise boundaryhtis a function oft. The parameter values used in the calculations arer 0.2,q0.1,σ 0.3,L 1,K 1.5,c 0.5,T 2, andn 3000. In this case, the free boundary is increasing withx0 0. The numerical result is coincided with that of our proofseeFigure 6.
Acknowledgments
The project is supported by NNSF of China nos. 10971073, 11071085, and 10901060and NNSF of Guang Dong provinceno. 9451063101002091.
References
1 M. Sˆırbu, 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. Sˆırbu 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.
9 Z. Yang and F. Yi, “A free boundary problem arising from pricing convertible bond,” Applicable Analysis, vol. 89, no. 3, pp. 307–323, 2010.
10 A. Friedman, “Parabolic variational inequalities in one space dimension and smoothness of the free boundary,”Journal of Functional Analysis, vol. 18, pp. 151–176, 1975.
11 A. Blanchet, “On the regularity of the free boundary in the parabolic obstacle problem. Application to American options,”Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1362–1378, 2006.
12 A. Blanchet, J. Dolbeault, and R. Monneau, “On the continuity of the time derivative of the solution to the parabolic obstacle problem with variable coefficients,”Journal de Math´ematiques Pures et Appliqu´ees. Neuvi`eme S´erie, vol. 85, no. 3, pp. 371–414, 2006.
13 L. Caffarelli, A. Petrosyan, and H. Shahgholian, “Regularity of a free boundary in parabolic potential theory,”Journal of the American Mathematical Society, vol. 17, no. 4, pp. 827–869, 2004.
14 A. Petrosyan and H. Shahgholian, “Parabolic obstacle problems applied to finance,” in Recent Developments in Nonlinear Partial Differential Equations, vol. 439 ofContemporary Mathematics, pp. 117– 133, American Mathematical Society, Providence, RI, USA, 2007.
15 A. Friedman, “Stochastic games and variational inequalities,” Archive for Rational Mechanics and Analysis, vol. 51, pp. 321–346, 1973.
16 A. Friedman,Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1982.
17 D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 2nd edition, 1983.
18 O. A. Ladyˇzenskaja, V. A. Solonnikov, and N. 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.